Zhang Thesis
Zhang Thesis
Zhang Thesis
2012
Recommended Citation
Zhang, Yida, "Numerical Study of Laterally Loaded Batter Pile Groups with the Application of Anisotropic
Modified Cam-Clay Model" (2012). LSU Master's Theses. 3496.
https://digitalcommons.lsu.edu/gradschool_theses/3496
This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has
been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital
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NUMERICAL STUDY OF LATERALLY LOADED BATTER PILE
GROUPS WITH THE APPLICATION OF ANISOTROPIC MODIFIED
CAM-CLAY MODEL
A Thesis
in
by
Yida Zhang
B.S., Zhejiang University, 2010
December 2012
To my wife Yuxi, for her unwavering love and trust.
ii
ACKNOWLEDGEMENTS
My deepest appreciation goes to my advisor Dr. Murad Abu-Farsakh for his invaluable
guidance, inspiration and suggestions through this entire study. Dr. Murad’s broad knowledge on
foundations, soil mechanics and numerical methods has influenced me very much, which directly
research. I am especially grateful for his endless tolerance and kindness in every situation
occurred during this study. I will not achieve so many things without his trust on me.
I would like to express my sincere gratitude to my co-advisor Dr. George Voyiadjis for
guiding me exploring the kingdom of material mechanics. His profound insight on the subject
and rigorous attitude towards academics will have lasting influence in my future voyage in the
ocean of knowledge. I also appreciate Dr. Guoping Zhang for agreeing to serve on my thesis
committee and for his advices. I would like to thank Dr. Xinbao Yu for his help in performing
FB-MultiPier analysis of this research. I also thank Mr. Danial Faghihi for his discussions of the
I would like to extend my gratitude to many friends and colleagues in LTRC and LSU. I
appreciate Dr. Qiming Chen, Dr. Xiaochao Tang, Hao Ying and Xiaoling Tan for their help on
my living and studying. I would also like to thank Sanjay Dhakal, Imran Akond and Md. Nafiul
Haque for sharing the office in the past two years and helping me with the thesis writing.
Last but not least, I would like to thank my wife and my parents for their unstinting support
ABSTRACT ................................................................................................................................... X
v
LIST OF FIGURES
Figure 1. 1: Lateral response of soil................................................................................................ 3
Figure 1. 2: Shadow and edge effects on a laterally loaded pile group (Walsh, 2005). ................. 4
Figure 1. 3: The concept of p-multiplier (Rollins et al. 1998). ....................................................... 4
Figure 1. 4: Type of batter piles (after Prakash and Subramanyam, 1965) .................................... 6
Figure 1. 5: Sketch illustrating probable distribution of soil reactions during a pull-over test of
3-pile model Dolphin (Tschebotarioff, 1953) ......................................................................... 7
Figure 2. 1: M19 east and west bound piers ................................................................................. 13
Figure 2. 2: Summary of in-situ exploration and testing of site M19 pier.................................... 14
Figure 2. 3: Layout of instrumented piles ..................................................................................... 15
Figure 2. 4: a) Eastbound pier with steel strands anchored at the dead end side; b) Jacking system at
the live-end of westbound pier. ............................................................................................. 16
Figure 2. 5: Schematic diagram of lateral load test setup ............................................................. 17
Figure 2. 6: Model of laterally loaded pile (Reese, 1997) ............................................................ 19
Figure 2. 7: Element for beam-column (Hetenyi, 1946) ............................................................... 19
Figure 2. 8: Characteristics shape of p-y curves for (a) soft clay (Matlock, 1970); (b) stiff clay
(Reese, 1975); (c) sand (Reese et al., 1974) ......................................................................... 22
Figure 2. 9: Effect of pile-soil separation and soil plasticity on lateral head load-deflection
behavior of single pile (Trochanis et al., 1991) .................................................................... 27
Figure 2. 10: Effects of pile-soil separation and soil plasticity on horizontal soil surface
displacements away from pile loaded laterally (Trochanis et al., 1991) .............................. 28
Figure 2. 11: P-y curves obtained from different methods (Brown and Shie, 1990) .................... 29
Figure 2. 12: The plastic zones by different soils (Yang and Jeremic, 2002) ............................... 30
Figure 3. 1: Illustration of the principle of effective stress (Lambe & Whitman, 1979) .............. 34
Figure 4. 7: as function of sleeve friction and overconsolidation ratio (Masood and Mitchell,
1993) ..................................................................................................................................... 73
Figure 4. 8: FB-MultiPier model for M19 pier ............................................................................. 79
Figure 5. 1: Comparison between partially and the fully drained analysis ................................... 83
Figure 5. 2: Excessive pore water pressure and total geostatic pressure along the path............... 84
Figure 5. 3: Excessive pore water pressure dissipation curves of soil at various depths .............. 84
Figure 5. 4: At a lateral load of 1870kips, the a) contours of lateral displacement of the whole
model and b) displacement vectors of the pile nodes on the deformed pile group. .............. 86
Figure 5. 5: Contour of the void ratio at first clay later after 1870kips lateral load ..................... 87
Figure 5. 6: Lateral deflection under different lateral load ........................................................... 88
Figure 5. 7: Lateral deflection profiles from field test, FEM and FB-MultiPier .......................... 90
Figure 5. 8: Lateral deflection profiles of piles in different location ............................................ 91
Figure 5. 9: lateral deflection profiles obtained from different FE models .................................. 92
Figure 5. 10: Contour of the mean effective stress for clays a) before and b) after lateral loading
............................................................................................................................................... 94
Figure 5. 11: Contour of volumetric strain after lateral loading ................................................... 95
Figure 5. 12: Failure surface in positive and negative batter piles (Prakash and Subramanyam,
1965) ..................................................................................................................................... 95
vii
Figure 5. 13: Deviatoric stress of piles in different columns ........................................................ 96
Figure 5. 14: Contour of a) deviatoric stress and b) deviatoric strain of the first three soil layers97
Figure 5. 15: Deviatoric stress distribution at the second layer .................................................... 98
Figure 5. 16: Deviatoric stress distribution of various FE models ............................................... 99
Figure 5. 17: Bending moment profiles under different lateral loads from FE analysis ............ 101
Figure 5. 18: Bending moment profiles from the strain gauges, FEM and FB-MultiPier analyses
............................................................................................................................................. 102
Figure 5. 19: Bending moment profiles of piles in different location ........................................ 104
Figure 5. 20: Bending moment profiles from various FE models .............................................. 104
Figure 5. 21: lateral load distribution pattern evaluated at pile head for various FE models ..... 106
Figure 5. 22: Increment of axial load from strain gauges and FEM ........................................... 109
Figure 5. 23: Axial load distribution pattern for piles in different rows from various FE models
............................................................................................................................................. 111
Figure 5. 24: The trapezoidal zone under batter pile group foundation...................................... 111
Figure 5. 25: Soil resistance profiles at different lateral loads from FEM ................................. 113
Figure 5. 26: Soil resistance profiles for different piles from FEM and FB-MultiPier analysis 114
Figure 5. 27: Soil resistance profiles from various FE models ................................................... 116
Figure 5. 28: Soil resistance profiles under different lateral loads from single pile and vertical pile
group model ........................................................................................................................ 117
Figure 5. 29: P-y curves from FEM and FB-MultiPier analysis ................................................. 119
Figure 5. 30: P-y curves from different FE models .................................................................... 121
Figure 5. 31: Load-deflection curves for single pile and normal spacing GB ............................ 123
viii
LIST OF TABLES
Table 1. 1: Some reported p-multipliers ......................................................................................... 5
Table 2. 1: Designed loading time table ....................................................................................... 17
Table 2. 2: The FE models on laterally loaded piles in literatures ............................................... 32
Table 3. 1: Initial values and material parameters for numerical tests ......................................... 52
Table 4. 1: Summary of all the FE models ................................................................................... 64
Table 4. 2: Soil properties determined from in-situ and UU tests ................................................ 67
Table 4. 3: Typical void ratio for some soils (Das, 1990) ............................................................ 71
Table 4. 4: Parameters for AMCCM and DP model ..................................................................... 74
Table 4. 5: Typical permeability k for various soils (Das, 1990) ................................................. 77
Table 4. 6: Summary of input parameter of p-y curves used in FB-Multipier ............................. 78
Table 4. 7: Input parameters for FB-MultiPier analysis ............................................................... 80
Table 5. 1: Total axial load developed at middle column in different FE models at 1870kips lateral
load...................................................................................................................................... 112
Table 5. 2: P-multipliers obtained from FE analysis .................................................................. 124
Table 5. 3: Variation of p-multipliers at different depth for normal spacing GB model ............ 125
ix
ABSTRACT
This study presents a series of numerical studies of laterally loaded batter pile groups based
on data of the full-scale lateral load test on M19 eastbound pier foundation of the new I-10 Twin
Span Bridge, Louisiana. The numerical studies include several continuum-based 3D finite
element analyses on batter/vertical pile/pile groups and a FB-MultiPier analysis of the pile
foundation. The Anisotropic Modified Cam Clay Model, has been implemented into UMAT and
applied for describing clay behavior in all FE models. The explicit substepping scheme with
modified Euler algorithm is selected to implement the model in ABAQUS software. The
resultant UMAT shows good accuracy compared to the ABAQUS in-built Modified Cam Clay
model. Also it exhibits wonderful computational stability and efficiency in the pile group
analyses, which greatly accelerated the whole research processes. The results of FE analyses
were compared with the measured field data from lateral load test and those predicted by
FB-MultiPier. All of them showing good agreement on lateral deformation profiles and bending
moment profiles. The comparison of the lateral deflection, bending moment, soil resistance and
lateral/ vertical load distributions between different spacing batter/ vertical pile groups and single
isolated pile illustrate that small spacing and the vertical piles will produce intensified group
effect. The concept of “trapezoidal zone” is firstly proposed to explain the axial load distribution
pattern of batter pile group foundation. An additional coupled pore fluid diffusion and stress
analysis on a single pile model demonstrated that the resultant excessive pore pressure caused by
INTRODUCTION
Piles are slender structural members that are often made of steel, concrete, timber, polymers
or composite materials. They can be used in single or group to construct pile foundations. Several
conditions that require the use of pile foundations include: 1) the upper soil layers are highly
compressible and the use of shallow foundations cannot support the upper structure; 2) the
structure are subjected to horizontal forces and 3) the presence of large amount of expansive and
collapsible soils. Typically pile foundations can provide much higher vertical and lateral bearing
capacities than shallow foundations, since they are able to transmit the vertical load to deeper
stronger soil layers (or bedrock) as well as providing bending and lateral resistance to a horizontal
loads.
Pile foundations are primarily designed for vertical support of structures, while specific
consideration should be taken on their lateral response for supporting high buildings, bridges and
offshore structures due to wind loads, frequent wave loads, possible huge lateral impact and
earthquake loadings. According to Rao et al. (1998), lateral loads are usually in the order of 10-15%
of the vertical loads in case of onshore structures and 25-30% in case of costal and offshore
structures. The lateral load capacity and maximum lateral deflection of the pile foundation are the
1
two most concerned aspects for engineers. It is believed that the later one is the major criterion on
the design.
The resistance of a pile to a horizontal displacement is provided by its surrounding soils via
two components: 1) the side resistance from friction and adhesion due to relative movement
between soil and pile and 2) the normal stresses due to the difference between the soil passive
pressure in front of the pile and the soil active pressure behind the soil (Figure 1.1a). The soil
resistance to a certain portion of a pile is related with the lateral displacement of that portion, and
their relations are always presented in the so-called p-y curves (Figure 1.1b).
The p-y curves are influenced by many factors such as soil types, soil properties, soil
saturation and drained conditions, pile geometries and pile group interactions. Various p-y curves
The p-y curves for group pile foundation can be different from those obtained in single pile
tests due to the group interaction effect. In pile groups, each pile pushes against the soil behind it
creating a shear zone which can be enlarged and overlapped with each other as the lateral load
increases (Figure 1.2). Such overlapping will be amplified with the decreased pile spacing. The
overlapping between two piles in the same row is called “edge effects” and the one between piles
in different rows is known as “shadowing effects.” These group interaction effects result in
reduced lateral resistance and thus produce larger bending moment and lateral displacement than
2
the single isolated pile for the same given average load per pile. To quantify the reduction of later
resistance in group pile foundation, Brown et al. (1987) introduced a reduction factor or
“P-multiplier ( )” concept to deduct the value of p in the single pile p-y curves (Figure 1.3).
The lateral resistance of the piles within the group is a function of row location. It has been
proved in many literatures that the lead row piles carried more load than the trailing row piles.
Many researchers found that the last row carries the least load. However, some other researchers
3
(Rollins et al., 1998; Zhang et al., 1999; Rollins et al., 2003b) reported that the last row carries a
Figure 1. 2: Shadow and edge effects on a laterally loaded pile group (Walsh, 2005).
Spacing also affects the lateral resistance of the pile groups. Rollins et al. (2003b) reported
that the group spaced at 5.6D (D is the diameter of the pile) showed very little reduction on lateral
4
resistance. However, lateral resistance consistently decreases with closer spacing. The
p-multipliers from past experiments by several researchers are summarized in Table 1.1.
Batter piles or group batter pile foundations are often used in foundation design in order to
achieve a higher lateral capacity of the foundation. According to their direction of inclination,
batter piles are classified into positive batter piles which are inclined against the loading direction
and negative batter piles which are battered toward the loading direction (Figure 1.4). It is reported
(Prakash and Subramanyam, 1965; Ranjan et al., 1980) that negative batter piles show a definite
5
increase in lateral resistance over that of vertical piles while positive batter piles exhibit less lateral
resistance than that of vertical piles. Lu (1981) supported these observations by explaining that the
soil reaction at ground level is zero for a positive batter pile and maximum for a negative batter pile,
indicating that the upper layer soil offer much higher lateral resistance to a negative batter pile than
a positive batter pile. However, for the batter pile groups, such soil resistance distribution will also
be influenced by the pile alignment. Tschebotarioff (1953) reported that the positive batter pile in
the leading row of the group batter pile foundation took the most stress during their 1:10 scaled
lateral load tests. Possible soil reaction distribution for a batter pile group is shown in Figure 1.5.
6
Figure 1. 5: Sketch illustrating probable distribution of soil reactions during a pull-over test of
3-pile model Dolphin (Tschebotarioff, 1953)
Recently, a full-scale lateral load test was performed on the batter pile group foundation of
the M19 eastbound pier of the new I-10 Twin Span Bridge over Lake Pontchartrain, Louisiana.
The test provides complete subsurface information and large amount of measured data on pile
deflection and moment distribution during the lateral loading, which allows one to evaluate
current design methodology for laterally loaded piles, or to carry out comprehensive analyses of
Currently, the most popular method in designing laterally loaded piles is the p-y method,
which will be discussed in Chapter 2. FB-MultiPier, a nonlinear finite element method software
7
based on the concept of p-y curves, has been used during the design of the new I-10 Twin Span
Bridge by LADOTD. Hence, the study of the full-scale lateral load test will provide a direct
evaluation of the FB-MultiPier program and, of course, the p-y design method itself.
most advanced method in analyzing pile foundations, of the whole lateral load test mentioned
earlier. Such FE analysis will provide much more details of the problem that p-y methods cannot
provide, which helps the reader to better understand the response of the soil-pile system under
lateral loads.
In order to achieve the objective, considering one FE model of the original batter pile group
geometry is not enough. The author will study the same problem with a single vertical pile model,
a vertical pile group model with the same spacing as the batter pile model, and two batter pile
group model with larger and smaller spacings. These analyses will provide complete information
of how the group effect and pile inclination will influence the lateral response of pile foundations.
Among all the factors that affect the quality and reliability of FEA, the selection of
constitutive model for soil is critical. An advanced soil model, Anisotropic Modified Cam Clay
(AMCC) Model, has been selected to describe the clay behavior in this study.
In addition, there is a section discussing the coupled pore water FE analysis. Since the test is
performed on the foundation in the lake with clay-dominating subsurface, a correct assumption of
Chapter 2 includes a brief review of past lateral load tests on pile foundations and the
description of the recent full-scale lateral load test on M19 pier of the new I-10 Twin Span Bridge.
A detailed literature review covering current approaches with emphasis on finite element method
Modified Cam Clay model. Its numerical implementation will also be discussed. Some numerical
Chapter 4 presents the details of finite element models including FE mesh and the
determination of various parameters for soil constitutive models. The coupled pore fluid diffusion
and stress analysis in ABAQUS will also be introduced. In addition, the FB-MultiPier model is
briefly described.
Chapter 5 presents the results of finite element analysis, including lateral deflection, bending
moment, soil resistance, lateral and axial load distribution and back calculated p-y curves.
9
CHAPTER 2
LITERATURE REVIEW
Full-scale load tests are always desirable by researchers for providing direct field
measurements that can be used to verify their models or to develop empirical methods. However,
only a few full-scale lateral load tests on piles were reported in the literature due to their high
The first full-scale lateral load test shown in the literatures is conducted by Feagin (1937)
on timber and concrete piles. Matlock (1970) carried out a lateral load test on steel pipe piles of
12.75 diameters for both static and cyclic loading. Kim et al. (1976) conducted a full-scale test
on batter pile group foundation and found out that battered piles provide more lateral resistance
with less bending moment. Brown et al. (1987, 1988) conducted full-scale test on vertical pile
groups on steel piles and generated the corresponding p-y curves. Later, Brown et al. (1988)
introduced the p-multiplier concept to modify p-y curves of single piles to consider the reduced
resistance of the group pile. Ruesta and Townsend (1997) carried out full-scale tests on
reinforced concrete pile and noted that the outer piles took more loads than the inner pile of same
row due to the smaller influence of shadowing effect they encountered than inner piles. Rollins et
al. (1998) suggested various p-multipliers after conducting a full-scale test on vertical pile group
in clay. They observed that the displacement of pile group is 2-2.5 times higher than the single
10
pile for the same average pile load. They also reported that the back row (trailing) carried
somewhat higher loads than middle row, which completely conflicted with the conclusion of
Brown et al. (1988) that the back rows resist lowest loads.
In the nearest decades, Huang et al. (2001) conducted a full-scale test on bored and driven
precast pile groups to investigate the influences of installation procedure of pile in lateral soil
resistance. He reported that the driven pile installation increased the group interaction by causing
the soil to move laterally and hence becomes denser; while bored pile installation loosens the soil
and hence decreases the group interaction. Rollins et al. (2003a, 2003b and 2005) carried out
several full-scale lateral load tests of piles in clayey and sandy soils at various pile spacings. The
major findings out of their study are: 1) the middle pile of the same row carries the smallest load
in that row while the side piles carry 20-40% higher loads in the sandy soil, which agrees with
other studies conducted in sands (Ruesta and Townsend. 1997; McVay et al., 1998) and conflict
with some tests results in clays (e.g. Brown et al., 1987; Rollings et al., 1998); 2) The group
effect becomes negligible when the spacing between rows increased to more than 6D, where D is
the pile diameters; 3) Group pile has significantly higher bending moments than those in isolated
2.1.2 The Full-Scale Lateral Load Test at the New I-10 Twin Span Bridge
A new (5.4 mile long) I-10 Twin Span Bridge has been recently constructed over Lake
Pontchartrain between New Orleans and Slidell, Louisiana, to replace the old bridge that was
seriously damaged from the storm surge and water waves associated with Hurricane Katrina that
11
hit the southern part of Louisiana in August of 2005. The new bridge is located 300 ft east of the
old bridge. It was built with an elevation of 30 ft, which is 21 ft higher than the old bridge, and
an 80 ft high-rise section near Slidell to allow for marine traffic, making it less susceptible to
high storm surge. The bridge consists of two parallel structures with three 12 ft travel lanes and
The M19 pier is the second pier south of the marine traffic underpass. It supports 200 ft
long steel girders in the north side and 135 ft long concrete girders in the south side. The
foundations of M19 piers consist of 24 precast prestressed concrete (PPC) battered piles of 110 ft
long and a 3 ft ×3 ft square section with an outer width of 36 inch and a circular void of 22.5
inch. All piles are inclined with a slope of 1:6; half of them are negative or reverse batter and the
rest are positive or forward batter. The piles were spaced at 4.3 pile width in the direction of
lateral loading and 2.5 pile width in the other direction. The average embedded length of the
piles was 87ft. The size of M19 pile cap (or footing) is 44 ft long × 42.5 ft wide × 7 ft deep. The
water depth is 12 ft. Figure 2.1 presents a photo of the M19 eastbound and westbound piers site.
Subsurface exploration and testing programs were performed to characterize the subsurface
soil conditions along the entire I-10 Twin Span Bridge, including the M19 eastbound pier
location. This includes soil sampling, laboratory testing and in-situ standard penetration tests
(SPT) and cone penetration tests (CPT). One soil boring was performed close to M19 pier to a
depth of 200 ft and 48 Shelby tube samples were extracted from cohesive layers for laboratory
testing, such as unconsolidated undrained (UU) triaxial tests. SPT were also conducted in sandy
12
layers. In addition, five CPT tests were conducted at M19 pier site down to a depth of 160 ft each,
one CPT at the center of M19 pier foundation, and four CPTs at distances 5-10 ft out from the
four corners of the pile cap. The purpose of the CPT was to define the subsurface soil profile and
identify any variations across the site, and to locate the depth of the bearing sand layer to support
the piles. The depth of sand layer at M19 pier was found to be at depths ranging from 100 ft to
110 ft below the water surface. However, the piles at M19 eastbound pier were tipped in stiff
consists of a medium to stiff silty clay to clay soil with silt and sand pockets and seams down to
110 ft depth (the undrained shear strength, Su, ranging from 0.28 tsf to 2.02 tsf) with a layer of
medium dense sand located between 35 ft to 47 ft depth (the SPT-N values ranging from 16 to
22). Medium dense to very dense sand with interlayers of silty sand, clayey sand, and silty clay
13
soil was found between 110 ft and 160 ft depth (the SPT-N values ranging from 3 for loose
clayey sand to 86 for very dense sand). The subsurface soil description from soil boring, profiles
of Su and SPT-N values, profiles of cone tip resistance (qc) and sleeve friction (fs), and the CPT
Figure 2. 2: Summary of in-situ exploration and testing of site M19 pier(Abu-Farsakh et al.,
2011)
Eight selected piles were instrumented with Micro-Electro-Mechanical Sensor (MEMS)
In-Place Inclinometers (IPIs) pile deflection profile and twelve selected piles were instrumented
14
with resistance type sister bar strain gauges. The locations of instrumented piles are presented in
Figure 2.3. The IPI consists of a string of tilt sensors connected together and placed permanently
in the pile through a PVC casing. Six IPI-MEMS sensors were installed in each of the eight piles
at depths of 5, 15, 25, 35, 45, and 65 ft from the bottom level of the pile cap with the lowest one
tied to an anchor point at the bottom of PVC casing at 85 ft from the bottom of the pile cap. Two
pairs of strain gauges were installed in each of the 12 piles at locations of -16 ft and -21 ft from
the pile top before pile cut off to measure the strain distribution. The profile of pile deformation
due to applied lateral load can be calculated from the IPI data; and the bending moment and axial
other using high strength steel tendons run through two 4 in PVC pipes installed in both pile caps.
For setup of lateral load test, the M19 eastbound pier was designed as dead end and the M19
15
west bound pier was designed as live end. The steel strands were first anchored at the dead-end
side, and then were threaded one-by-one through the two 4 in PVC pipes from the dead-end at
M19 eastbound pier toward hydraulic jack of the live-end at M19 westbound pier. Each steel
tendon includes 19-0.62 in diameter strands of low relaxation, high yield strength steel
( ). The lateral load was applied using 600 ton jacks with piston-end facings to
pull M19 eastbound and west bound pier toward each other using the steel tendons. Figure 2.4
presents the photos of M19 eastbound dead end and westbound piers live end design.
(a) (b)
Figure 2. 4: a) Eastbound pier with steel strands anchored at the dead end side; b) Jacking
system at the live-end of westbound pier (Abu-Farsakh et al., 2011)
The designed sequence of lateral load test includes preloading each tendon to 300 kips, then
loading, unloading and reloading. The design maximum applied load was 2000 kips. However,
the test was unloaded earlier at a maximum applied load of 1870 kips when the stroke in one of
the 600-ton jacks reached its maximum limit. The schematic diagram of lateral load setup at M19
pier is depicted in Figure 2.5 and the designed loading sequence is presented in Table 2.1.
16
Figure 2. 5: Schematic diagram of lateral load test setup
Due to different objectives of research studies, laterally loaded piles were investigated in
various methods, which can be generally classified into two categories: simplified methods and
continuum-based methods. The former emphasize on capturing the problem in the most effective
way with maximum engineering applicability, while the later discusse the problem in a more
17
physical-sounded way which allows one to see how a particular factor affect the pile lateral
behavior.
The simplified methods basically model the soil through a set of independent linear or
nonlinear springs attached to the piles, which provide resistance to the lateral movement or
vertical movement of piles. In this type of approach, which also known as the “p-y” or “t-z”
methods (Matlock and Reese, 1960; Reese et al., 1974), the load displacement curves of springs
are assumed to be known a priori. These curves are obtained from observations of past
experiments results. Continuum-based methods treat the soils surrounding piles as elastic or
elasto-plastic continuum media. It allows one to investigate how the lateral performance of the
pile is influenced by many particular issues such as soil-pile interaction, soil inelastic behavior or
In the following sections, both the simplified methods and continuum-based methods will
be discussed and the recent works on continuum-based finite element analysis of laterally loaded
The well-known p-y method is proposed by McClelland and Focht (1956) and developed by
Reese and his coworkers (Matlock and Reese 1960; Matlock 1970; Reese et al. 1974). This
method can conveniently incorporate the effect of soil nonlinearity by indicating that the soil
resistance p is nonlinear function of local lateral deflection y. This method has become widely
used for design with the development of computer techniques for solving nonlinear fourth-order
18
differential equations and the remote-reading strain gauges for obtaining p-y curves from field
experiments.
Figure 2.6 illustrated the analytical framework of the laterally loaded pile problem for p-y
analysis. Based on this model, the governing differential equations can be obtained by analyzing
the equilibrium of momentum of an infinite small element of the pile (Figure 2.7):
19
dM dy
Px Vv 0 (2.1)
dx dx
or
d 2M d 2 y dVv
Px 0 (2.2)
dx 2 dx 2 dx
Substituting
d 2M d4y
E I
p p
dx 2 dx 4
dVv
p
dx
p E py y
into Eq. (2.2), the governing differential equation take the form:
d4y d2y
E p I p 4 Px 2 E py y 0 (2.3)
dx dx
here = axial load on pile; y = lateral deflection of pile at point x ; p = soil resistance per unit
Some typical p-y curves for sand and clay are presented in Figure 2.8. Figure 2.8a
represents the p-y curve for soft clay, proposed by Matlock (1970). The initial portion can be
established by using . The middle portion of the curve is described by the equation
( ) , where is the lateral soil resistance per unit length; is the ultimate lateral
soil resistance per unite length; y is the lateral displacement; represents the lateral
displacement corresponding to one half of the ultimate lateral soil resistance and can be
computed according to .
20
Reese et al. (1975) conducted a series of lateral load tests on overconsolidated stiff clay and
proposed the typical p-y curve for stiff clay (Figure 2.8b). Mathematical description of the curve
is presented below:
Figure 2.8c illustrated typical p-y curves for sand, which described as Reese model (Reese
et al. 1974). Similarly, the initial straight line is ; The slope of linear line between u
and .
Based on the p-y approaches, several computer programs such as COM624P (Wang and
Reese, 1993), LPILE (Reese et al., 1997) and FB-MultiPier (University of Florida, 2000) are
developed to facilitate the analysis and design of laterally load piles. COM624P and LPILE
program are constructed by finite difference method. Nonlinear behavior of soils based on prior
established p-y curves are implemented into the softwares. Piles are simulated as beams with
lateral stiffness and typically concrete pile stiffness is assumed nonlinear whereas steel pile
stiffness is assumed linear. One significant deficiency of LPILE and COM624P program is that
21
(a) (b)
(c)
Figure 2. 8: Characteristics shape of p-y curves for (a) soft clay (Matlock, 1970); (b) stiff clay
(Reese, 1975); (c) sand (Reese et al., 1974)
The FB-MultiPier, developed by University of Florida, is programed based on finite
element method for carrying out more flexible analysis of piles in different configurations and
alignments. The FB-MultiPier program couples the nonlinear structural finite element analysis
and the nonlinear static soil models for axial, lateral and torsional soil behavior, thus being
characteristics of the FB-MultiPier, it will be adopted in this study to analyze the lateral load test
22
on M19 foundation and the results will be compared to those produced by ABAQUS model and
The P-y methods successfully take the nonlinearity of soil property into account and are
Continuum-based analyses are attractive for their direct physical basis. Continuum-based
methods treat the soils surrounding piles as elastic or elasto-plastic continuums. Solutions of this
kind of analysis are attempted by many scholars both analytically and numerically (e.g. Poulos,
The analytical solutions of continuum-based method are of highly complexity, thus only
some analytical or semi-analytical elastic solutions were achieved so far. The first systematic
analysis based on the theory of elasticity was reported by Poulos (1971 a, b). In this approach,
the soil is idealized to be an isotropic homogeneous elastic material with constant elastic
parameters and , which are unaffected by the presence of the pile, while the pile is
accuracy and computational cost, the pile is divided into 21 portions, and each portion is
assumed to have a uniform horizontal stress p distributed along width and length. The solution of
the problem starts from equating soil displacements and pile displacements. The former is
23
evaluated from the Mindlin equation for horizontal displacement due to a horizontal load within
a semi-infinite mass, and the latter is obtained from the equation of flexure of a thin strip. Two
necessary boundary conditions for solving the equations are considered: a free-head pile and a
fixed-head pile. The major obstacle of application of this theory is the determination of the
in-situ soil modulus . Poulos suggests that such modulus should be determined from a
Sun (1994) developed an elastic solution for cylindrical pile based on the modified Vlasov
model for the static analysis of beams on elastic foundations. The soil and the pile are idealized
similarly to those by Poulos (1971). Slippage and separation at the pile/soil interface are also not
u F ( z ) (r ) cos( )
v F ( z ) (r ) sin( )
w0
where ( ) is the displacement along the pile axis; and ( ) is a dimensionless function
representing the variation of the soil displacement in the r direction. Then governing equations
d 4F d 2F
2t kF 0 (2.4)
dz 4 dz 2
or
d 2 d 2
2
r 2
r r 0 (2.5)
dr 2
dr R
24
here is a nondimensional parameter that is considered to be a principal parameter in this
research. The determination of parameter requires iterative technique that was discussed by
Vallabhan and Das (1988). Solving the differential equations with different boundary conditions
using the above procedures, the closed-form solutions of dimensionless displacement of vertical
pile in homogenous medium, ̅ ( ̅), was expressed in equations. In this research, the effect of
various factors to principal parameter was studied and discussed, and a detailed computing
Later on, Basu and Salagado (2007) extend Sun’s analysis to multi-layered elastic continua,
which can take into account an arbitrary number of soil layers. The governing differential
equations were obtained using the principle of minimum potential energy, and were solved
analytically through the method of initial parameters. Similarly, the principal parameter in Eq.
The advantage of above analytical or semi-analytical approaches is that the soils are treated
in a more physically founded and mathematically rigid way. The obvious disadvantages are that
they over simplified the soil property and can hardly take into account the nonlinear soil
behavior as well as the gapping and slippage effect of pile-soil interface. Therefore, many
scholars yield to numerical methods including finite element method, finite difference method
and boundary element method to extend their interpretation of the problem by investigating the
influence of soil nonlinearity and interface effect to the lateral response of pile. Among them, the
25
finite element method is considered to be the most powerful method in analyzing the laterally
Several representative works on recent approaches by finite element method are presented
herein. Muqtadir and Desai (1986) developed a 3D FEM model to simulate axially and laterally
loaded steel batter pile groups. Soil was governed by nonlinear (hyperbolic) elastic model, and a
special “thin layer element” was adopted to simulate the debonding and slippage of pile soil
interface. The constitutive law at normal direction for such interface elements is similar to those
of surrounding soil, whereas tangential behavior is calibrated by direct shear test of steel-soil
interface. The research reveals that the inclusion of soil non-linearity and interface effects can
Trochanis et al. (1991) focused on investigating the behavior of single pile subjected to
combinations of both lateral and axial loads with incorporation of nonlinear soil behavior and the
soil-pile interaction effects by means of 3D finite element elastoplastic model which is built on
ABAQUS 4.6. It is stressed that the vast amount of information obtained from the 3D FEM
model can provide the reader much insight into the behavior of pile foundations, whereas such
information could not be easily gained in field tests. The soil materials is idealized as
elements, and 2D quadratic 18-node interface elements governed by modified Coulomb’s friction
theory were adopted to account for pile-soil separation and slippage. The model is verified by the
26
comparison of the vertical and horizontal soil surface displacements to those obtained by Poulos
and Davis (1980) and the field data from a test conducted in Mexico City. Both of them show
good agreements. This study showed that the pile-soil slippage could produce dominating
influence of the vertical response of the pile, while pile-soil separation leads to significant
change on the lateral behavior of the pile. Both cases were more or less affected by soil plasticity.
Figure 2.9 and Figure 2.10 illustrate the effect of pile-soil separation and soil yielding on the
lateral response of a single pile and the soil displacements in the vicinity of the pile.
Brown and Shie (1990, 1991), which put more emphasis on comparison with empirical design
procedures. The FEM model was constructed on ABAQUS 4.7. Two types of soil, clay and sand,
were considered in this model. The Von Mises yield surface with associate flow rule was used
for clay and the extended Drucker-Prager model with nonassociated flow was adopted for sand.
Gapping and slippage of pile-soil interface were also simulated by 18-node interface elements.
27
The development of this FEM model stressed on reveal some limitations of currently analytical
techniques and design procedures rather than being applied to practical uses directly. The results
of von Mises model were compared to those given by subgrade reaction approaches by
comparing the p-y curves given by the FE model and those obtained from the COM624 output
using soft clay criteria (as shown in Figure 2.11). The comparison of the two set of p-y curves
shows that the finite element results gave greater soil stiffness near the ground surface than the
1) The loading path that the soil near ground surface experienced is more close to triaxial
extension than triaxial compression that has been used to develop empirical p-y curves.
2) The Von Mises constitutive model is basically unable to simulate saturated clay subjected to
undrained loading.
28
(a) FEM, VM model (b) COM624, Clay soil
Figure 2. 11: P-y curves obtained from different methods (Brown and Shie, 1990)
Yang and Jeremic (2002) developed a FE model based on the OpenSees finite element
framework to study the soil layering effect on laterally loaded piles. The FE model was used to
generate the p-y curves and were compared to those obtained from the centrifuge test and design
method (LPILE program). The constitutive models adopted in this research were simple von
Mises model for clay and Drucker-Prager with nonassociated flow rule for sand. Interface effects
are taken into account by implementing interface elements. The influence of layered soil on pile
lateral response was investigated in four cases: 1) uniform clay; 2) clay-sand-clay; 3) uniform
sand; 4) sand-clay-sand. They investigated the distribution of plastic zone by visualizing the
plastified Gauss points. It was illustrated that for unified clay soil the plastic zone propagates
very deep while does not extend far horizontally (Figure 2.12a), whereas for uniform sand case
the plastic zone was more likely to propagate toward the surface (Figure 2.12b). The main
findings of their study were 1) When a sand layer is present within a clay deposit, the increase in
29
lateral pressure in clay near the interface is confined to a narrow zone, up to two times of pile
width, therefore the layering effect in this case is not prominent; 2) When a clay layer is present
within a sand deposit, the reduction in pressures spread well into the sand layer (up to four times
of pile width). The layering effects are of more importance in this case since the disturbance
zone is large and the pressure reduction is significant. Reduction factors were given in terms of
finite element model to further investigate the two proposed reasons that account for the
disagreement of FEM and p-y methods on soil stiffness near ground surface. They analyzed the
different stress paths behind and in front of the pile, and pointed out that the soil anisotropy
effect was responsible for the discrepancy of shear strength values subjected to different stress
paths. Thus, an isotropic von Mises constitutive law incorporation different back-calculated
30
shear strength values was used to account for anisotropy soil behavior. The results showed that
the displacements decrease more rapidly in the horizontal direction than in the vertical direction,
which was also obtained by Yang and Jeremic (2004). The predicted displacements showed good
agreement with field data. Some discrepancy between the predicted and field measured bending
moment was attributed to inaccuracies in simulation of the initial stress, constitutive law, and
Abbas et al. (2009) studied the consolidation effect of laterally loaded pile problems. A 2D
finite element program was employed in their study and produced the development of pile lateral
displacement, lateral soil pressure and soil friction stress with respect to time. Biot (1941)
consolidation equations were implemented in incremental form at the element level, which can
be expressed as:
[ ] [ ] { }
[ ]{ } { } (2.6)
[ ] [ ] [ ]{ }
where [ ] and [ ] are the solid stiffness and fluid conductivity matrices; [ ] is the
represents the external load vector which may itself be time dependent; & are the
displacement and access pore water pressure, respectively. Their results showed that the pile in
cohesionless soil has more resistance in the rapid loading and less resistance in the long-term
loading, while the pile in cohesive soil had the opposite property.
Detailed information of soil constitutive models and simulation scopes of above mentioned
31
Table 2. 2: The FE models on laterally loaded piles in literatures
Pile/soil Pore-water
Drained/Undrained Constitutive
References interface pressure Soil type
condition modeling
model analysis
Trochanis et al., Coulomb
Undrained N Clay/ sand Drucker-Prager
1991 Friction
Muqtadir and Nonlinear
Not specified N Sand Nonlinear Elastic
Desai, 1986 Elastic
Brown and Shie, Coulomb Von Mises/
Not specified N Not Specified
1990,1991 Friction Drucker-Prager
Yang and Coulomb Von Mises/
Not specified N Sand & Clay
Jeremic, 2002 Friction Drucker-Prager
No interface
Abbas et al., 2009 Biot Consolidation N Not Specified Mohr-Coulomb
elements
Ahmadi and Coulomb
Undrained N Clay Von Mises
Ahmari, 2009 Friction
In summary, using three dimensional FEA for laterally loaded pile problems has the
1) It can simulate the soil surrounding piles as 3D continuums rather than simplify it as a series
2) It is able to capture the different elasto-plastic behavior of clay and sand soils rather than
3) It is able to realistically capture some essential features of the lateral loading problems. For
example, the pile-soil interface behavior can be simulated using interface elements, which
allow separation and slippage of the interface while pile is subjected to large lateral loads.
However, it is noticed that most of the past and recent numerical approaches suffer the
problem of over simplifying the subsurface soil conditions of the pile foundations (e.g.,
32
homogenous soil or maximum three layers of soil was considered). More importantly, nearly all
of previous works adopted some simple constitutive laws (such as Mohr-Coulomb or Von Mises)
to simulating natural soils, which has already been criticized to be inadequate (Brown and Shie,
1990) to represent the actual behavior of clay surrounding the laterally loaded piles.
In our case, the subsurface investigations of the M19 pile of 1-10 Twin Span Bridge
indicate that the dominant of the soil layers consist of soft or stiff clays. Thus the selection of a
proper constitutive model for clay will lead to a big difference of the reliability of the FEA
results. As the great improvement of computer speed in the past decade, a FEA with improved
constitutive models for soils are desirable to achieve more reliable finite element results.
33
CHAPTER 3
The starting point of classical soil mechanics should be the principle of effective stress
(Terzaghi, 1936), which can be illustrated by establishing equilibrium across a wavy plane
joining interparticle contacts (Figure 3.1). The principle of effective stress can be expressed as
( )
where is the effective stress, is the total stress and is the pore water pressure
Figure 3. 1: Illustration of the principle of effective stress (Lambe & Whitman, 1979)
Such simple illustration highlights several basic principles that underlying classical soil
mechanics (Gens, 2010): the material is multiphase; the microstructure is implicitly considered; a
new variable, pore water pressure, is incorporated; and there is coupling between mechanical
One of the most critical issue in classical soil mechanics is to describe the consolidation
behavior of soils. The pioneer work on one-dimensional consolidation theory was accomplished
34
by Terzaghi and Frolich (1936). The general three-dimensional consolidation theory was
established by Biot (1941). Assuming the soil to be isotropic and linear elastic, the consolidation
𝜕 𝜕 𝜕 𝜕 𝜕 𝑡
∇ ( + + ) ( )
𝜕𝑥 𝜕 𝜕 𝜕 𝜕
where is the permeability of the soil; is the unit weight of water; is the inverse of the
(solid mass balance, water mass balance and equilibrium) and constitutive equations that
governing water flow and soil deformation. In fact the constitutive equations adopted in this
( + ) ( )
( )
Obviously, the isotropic linear elasticity is inadequate in representing complex soil behavior.
The blank of soil elasto-plastic model has been well filled by the later development of critical
state soil mechanics (CSSM) (Roscoe et al., 1958; Roscoe and Burland, 1968; Schofield and
Wroth, 1968). CSSM provide a complete and unified framework that brought together many key
features of saturated soil behavior, such as shear and volumetric behavior, strength, dilatancy and
yielding, in a consistent manner. Based on the framework of CSSM, the Cam Clay model was
35
developed for mathematically describing the complex elastoplastic behavior of soils. Such model
has been modified later (Roscoe and Schofield, 1963) into Modified Cam Clay Model (MCCM)
by redefining the shape of the yield surface. The model itself is very attractive due to its
simplicity; hence it has been widely used both in engineering practice and academic study. The
MCCM also provide a foremost framework that can be extended to describe other features such
as soil anisotropy (e.g. Ohta and Hata, 1971; Banerjee and Yousif, 1986; Dafalias, 1987), thermo
behavior (e.g. Graham et al., 2001; Cui et al., 2000) and cyclic behavior (e.g. Dafalias and
Hemnann, 1980; Yu et al., 2006). Before extending our discussion to higher soil constitutive
One should firstly consider the volumetric behavior of clay under normal compression and
unloading-reloading conditions, as shown in Figure 3.2a. It is often found that the linearity of
normal compression lines and unloading-reloading lines in the compression plane is improved if
data are plotted with a logarithmic scale for the mean stress axis. Mathematically, the normal
( )
( )
where and are slope of the two lines and and are the intercepts on the lines at p’=1;
36
It is very important to notice that the unloading-reloading line actually represents a
nonlinear elastic behavior (often referred as porous elastic), which differs from the linear
elasticity of metal. Thus it is convenient to discuss clay’s elastic response in incremental form:
[ ] [ ][ ] ( )
where p’ and q are effective mean stress and deviatoric stress; and are volumetric strain
and deviatoric strain; K and G are tangent bulk modulus and shear modulus, respectively.
Obviously, the bulk modulus and the shear modulus of clay are dependent on current mean
effective stress and specific volume, which can be obtained by taking derivative of both side of
Eq. (3.6):
( )
A complete elastoplastic model needs a yield surface to separate the elastic and plastic
(a) (b)
Figure 3. 2: a) Normal compression line and unloading-reloading line in plane; b)
Elliptical yield surface for Modified Cam Clay model in p’-q plane.
37
The corresponding yield function is expressed by
[ ( )] ( )
where M is the slope of critical state line on p-q plane; is the pre-consolidation pressure.
The shape and size of the yield surface is determined by both M and . However, only
pre-consolidation pressure controls the expansion (or shrinkage) of the yield surface, which
means the yield surface can change size but keeps constant shape. In this way, the expansion of
the yield surface and the hardening of the soil are linked with the normal compression of the soil.
The relation between and specific volume v completely follows the normal consolidation
line which can be expressed the same as Eq. (3.5). Rewriting Eq. (3.5) and Eq. (3.6) in
incremental form and replace specific volume by volumetric strain, one obtains the hardening
𝜕
𝜕
( )
𝜕
{𝜕
The formulation of MCCM has been completed now. The model can successively predict
the response of soil samples in conventional triaxial drained and undrained compression tests.
Besides, the model itself is simple, and the corresponding parameters , , M and have direct
physical meanings and can be easily obtained from triaxial or oedometer tests.
38
3.2 Anisotropic Modified Cam-Clay Model
Experimental results have shown that the yield surface for naturally deposited clay tends to
align along the consolidation line (e.g., Graham et al., 1983). Banerjee et al. (1981) distinct
such soil anisotropy into two types: the “inherent anisotropy” comes from the formation of the
Stems from this finding, several researchers proposed anisotropic models based on
extension of Modified Cam-Clay Model (Ohta and Hata, 1971; Banerjee and Yousif, 1986;
Dafalias, 1987). Ohta and Hata (1971) presented an anisotropic model for normally consolidated
clay. The yield surface can be inclined at the origin of the stress space at the before shearing but
it does not include rotational hardening when subjected to consecutive shearing. Banerjee and
Yousif (1986) developed an incremental plasticity theory to describe the mechanical behavior of
anisotropically consolidated clays. The yield ellipse is allowed to initially align along the
line and rotate, when it is subjected to shearing, following isotropic and anisotropic hardening
law, Moreover, the concept of bounding surface is introduced in order to simulate the inelastic
material behavior under monotonic and cyclic loading. Started with a different energy dissipation
assumption, Dafalias (1987) developed an Anisotropic Modified Cam Clay Model (AMCCM)
which extended the Modified Cam-Clay Model by introducing an anisotropic variable α and two
anisotropic parameters c and x. This model is attractive due to its simplicity and the effectiveness
in capturing both “inherent” and “induced” anisotropic of clay. Hence, the AMCCM by Dafalias
39
(1987) will be implemented into UMAT in this study and then be assigned as the clay
In triaxial space, the formulation of the yield surface of traditional Modified Cam-Clay
Model started from the plastic work dissipation assumption Eq. (3.11) proposed by Burland
(1965):
+ √( ) +( ) ( )
Note that all the stresses we discussed in this chapter are effective stresses, so the prime
used to denote “effective” is omitted in this section. Dafalias (1987) added a non-dimensional
anisotropic variable into the plastic work dissipation assumption to account for the effect of
internal residual stresses and the coupling of ̇ and ̇ . Eq. (3.11) becomes:
+ √( ) +( ) + ( )
( )
( )
Noticing , we have + . Substituting in (3.13) and rearranging the terms:
( )
+
( )
( )
+
Replacing with and rearranging the terms, one obtains the yield function:
+ ( + ) ( )
40
which is the identical to the yield function of AMCCM. Dafalias (1987) proposed an evolution
anisotropy and c controls the pace at which such anisotropy develops. When setting c=0 and
An illustration of the yield surface is presented in Figure 3.3. The yield ellipse is rotated and
distorted comparing to the yield locus of MCCM. The degree of such rotation and distortion is
controlled by . Several characteristic points are shown in this figure. The normal of the ellipse
at points C and C’ are along q-axis and also are the intersection point between yield locus and
critical state line. The normal of the ellipse at points O and A are along p-axis with point A
intersection of yield locus and p-axis but is the projection of point A on p-axis.
Equation (3.16) and (3.17) are discussed under the triaxial loading space. Generalization of
the above works into multiaxial stress space can be achieved by replacing the scalar by a
√ √
41
Figure 3. 3: Schematic illustration of the anisotropic yield surface in the triaxial p-q space
(Dafalias, 1987)
Then the generalized AMCCM yield function and hardening law for is presented as
+ (( )( )+( ) ) ( )
+ 𝜕
〈 〉 | | ( 𝑥 ) ( )
𝜕
global equilibrium equations. The solution scheme is schematically presented in Figure 3.4.
42
Figure 3. 4: Materially nonlinear finite element method solution strategy (Hashash and Whittle,
1992)
As summarized by Hashash and Whittle (1992), the implicit nonlinear finite element
+ 𝛥𝑈 𝑅 + + ( )
𝑈 + 𝑈 + 𝛥𝑈 + ( )
+ + 𝑈 + 𝑈
where is the increment step number; 𝑖 is the iteration number of current step; K is the global
tangent stiffness matrix; U is the nodal displacement vector, R is the vector of the applied nodal
forces and F is the vector of nodal force due to stresses in the element. The term 𝑅 + + at
43
the right hand side of Eq. (3.20) is often recognized as residual load vector. Every global Newton
iteration start with a predicted incremental nodal displacement Δ𝑈. Such Δ𝑈 is passed into each
element and processed with different interpolation methods (which depend on the type of the
element) to obtain the incremental strains Δ at each Gauss points. Then the material
constitutive equations will be integrated via some algorithms with respected to this strain
increment Δ at each Gauss point. As a requirement of the global Newton iteration, the output of
the material level calculation must contain the updated stress and the Jacobian matrix 𝐽
𝜕 𝜕 . Finally the updated stress and Jacobian will be assembled into the global nodal force
vector and global tangent stiffness matrix for the next iteration.
The user-subroutine UMAT is a user interface provided by ABAQUS that allows customers
to use self-defined constitutive relations at material calculation level. In this research, the
AMCCM will be implemented into the UMAT and then the subroutine will be numerically tested
by a series of triaxial compression procedure under constant volume (CV) or constant pressure
(CP) conditions.
Numerically implementing the Cam Clay plasticity has been attempted by many researchers.
Only several selected works are presented herein. Borja and Lee (1990) firstly implemented the
Modified Cam-Clay Model under the framework of implicit return mapping algorithm with
closest point projection for associative flow rule and central return mapping for non-associative
flow rule. In the following work, Borja (1991) point out that an ‘average’ bulk modulus along
44
with an iterative scheme are required to calculate the nonlinear elastic behavior described in
modified cam clay model. Hashash and Whittle (1992) also developed an integration algorithm
for the Modified Cam-Clay Model using return mapping algorithm with general closest point
projection algorithm. Both Borja and Lee (1990) and Hashash and Whittle (1992) emphasized
the use of consistent tangent stiffness matrix (Simo and Taylor, 1985), which has been proved
that it can guarantee a quadratic convergence for the global Newton iteration, in their works.
Devi and Singh (2008) implemented the MCCM in an Object-Oriented FE system. Yamakawa et
al. (2010) developed a return mapping algorithm of the subloading surface Cam-clay model
The use of implicit return mapping algorithm is conceptually appealing for its accuracy and
unconditional stability. However, such method requires second order derivatives of the yield
function, which will result in tedious Jacobian for a complex constitutive relation. One can see
that just for Modified Cam-Clay model it becomes fairly complicated of the resultant consistent
tangent matrix and the Jacobian for local Newton iteration. Noticing that the AMCCM has one
straightforward explicit integration algorithm to develop a sufficient robust and efficient code
which can work with a 68,000-element FEA model. Sloan (1987) proposed an explicit
substepping scheme for implementing general elastoplastic relations. Modified Euler algorithm
is used in order to achieve a controlled error during the integration of the constitutive equations.
This method has been concluded to have superior robustness and efficiency than implicit
45
achievements when applied to Modified Cam-Clay Model (Potts and Ganerdra, 1992; 1994).
Sloan et al. (2001) refined this method with adding elastoplastic unloading algorithm and
extending the applicability to Cam-Clay plasticity. In this work an explicit integration algorithm
The complete pseudo algorithm for implementing AMCCM can be found in Appendix I. In
the following sections we will focus on discussing some key issues in realizing the AMCCM via
As introduced in section 3.1, clay exhibits nonlinear behavior even in elastic regime. Recall
the tangent bulk modulus that relates infinite small stress and strain increments
( )
As indicated by Borja (1991) and Sloan et al. (2001), such tangent modulus cannot be directly
utilized in numerically implementing the elastic response of MCCM since the calculation is over
a finite time step 𝛥 during which the stress and strain increments are so large that the
( )
and integrating its both side over time span ( , + ), one obtains:
+
Δ ( )
46
Rearrange (3.24) equation and substitute + where e is the void ratio:
+ +
+
+( + )
+
+
+[ ( + )] Δ ( )
Δ
+
̅ ( ) ( )
Δ
then Eq. (3.26) becomes the familiar form as traditional linear elasticity:
+
+̅ ( )
This secant bulk modulus ̅ actually represents the averaged changing of over ( ,
+ ). Assuming Poisson’s ratio to be a constant throughout the elastic regime, the secant
̅( )
̅ ( )
( + )
Keep in mind that both ̅ and ̅ are functions of the void ratio and the stress state at
the beginning of the current step and the strain increment of this step. Hence the assembled
̅+ ̅ ̅ ̅ ̅ ̅
̅+ ̅ ̅ ̅
̅ ̅( ) ( )
̅+ ̅
̅
̅
[ ̅]
47
3.3.3.2 Yield Surface Intersection
Stresses can be updated using the secant stiffness matrix ̅ defined in Eq. (3.29) for a
given strain increment if the stress path is completely in the elastic realm. When the strain
increments causes current stress state exceeds the yield surface, it is necessary to find the
intersection point of the stress path with the yield surface. Introducing a multiplier that
controls the strain increments, such problem is equivalent to find a 𝑡 that satisfying:
(( + 𝑡
̅ ) ) ( )
For 𝑡 the stress state is inside the yield surface while for 𝑡 the stress state
is outside the yield surface where integration algorithm should be applied. Since no yielding
occurs within elastic regime, the anisotropic variable and preconsolidation pressure will
not develop and hence can be treated as constant. An algorithm for finding the intersection point
based on the work of Sloan et al. (2001) is developed and presented in Appendix I. The
difference here is that the method of false position is used for determining 𝑡 instead of the
Pegasus algorithm.
The constitutive equations of AMCCM can be summarized in tensor form (where a tensor is
- Yield function:
+ (( ) ( )+( ) ) ( )
- Hardening laws:
48
+ 𝜕
〈 〉 | | ( 𝑥 ) ( )
𝜕
+ + 𝜕
〈 〉 | | ( )
𝜕
- Flow rule:
𝜕
〈 〉 ( )
𝜕
A key point in explicitly implementing AMCCM is the derivation of the explicit expression
𝜕 𝜕
+ + ( )
𝜕 𝜕
̅ ( ) ̅ ̅ ( )
Combining Eqs. (3.33) and (3.34), the loading index can be derived as follows:
̅ 𝜕
+
𝜕 ( )
̅ 𝜕
𝜕
One can notice the presence of the term in the right hand side of Eq. (3.35). On the
other hand, also partially depend on the value of according to Eq. (3.31b). Substituting
Eq. (3.31b) into Eq. (3.35), the completely explicit expression of the loading index is
expressed as follows:
49
̅
( )
̅ 𝜕 + 𝜕
| | ( 𝑥 )
𝜕 𝜕
The terms , and are derived by Voyiadjis and Song (2000) and are rewritten in
For explicit integration algorithms, the stresses may drift away from the yield surface at the
end of each substep. This kind of drifting may be very small compare to the stress increment in
that step but can accumulate to a large scale of error after thousands of steps. Sloan et al. (2001)
proposed a combined consistent correction and normal correction scheme which provides an
enhanced stability of the whole correction procedure. Such algorithm is adopted in the present
work.
Firstly the uncorrected stresses and hardening parameters will be processed through a
consistent correction scheme. The 1st Taylor polynomial of the yield function about
( ) is:
𝜕 𝜕 𝜕
+ + + ( )
𝜕 𝜕 𝜕
Here , and will be viewed as a small correction to the current , and .
Such corrections make the change of stress and hardening parameters together while remaining
50
the total strain increment unchanged, which is consistent with the philosophy of the
displacement finite element procedure (Potts and Gens, 1985). Assume a correction index
̅ ( )
( )
̅ 𝜕 + 𝜕
| | ( 𝑥 )
𝜕 𝜕
The correction of the stresses and hardening parameters can be proceeded using ,
If the previous consistent correction scheme cannot achieve a convergence, such method
will be abandoned for one step and use the backup normal correction scheme (Sloan et al., 2001).
The hardening parameters and are assumed to be constant and only stresses are corrected
( )
51
3.4 Numerical Testing
Firstly, the correctness of the implemented AMCCM will be tested by comparing the stress
path, stress-strain curve and volumetric behavior with the corresponding curves produced by the
inbuilt MCC model provided by ABAQUS. Then its capability of capturing soil anisotropy will
be checked by comparing the yield locus of AMCCM with different anisotropic parameters.
Two types of triaxial tests are carried out under an axial strain of 10%. The CV test is aimed
to simulate the undrained condition, while CP test is for drained condition. Table 3.1
summarized the parameters adopted in the numerical tests which are arbitrarily chosen from the
AMCCM is reduced to classical MCCM (denote as UMAT-MCC). All tests will be run on the
samples with two different initial mean stresses and in order to observe
the yielding behavior from both wet side and dry side. The same thing will be done using
ABAQUS inbuilt MCC model (denote as ABA-MCC) with identical material parameters. The
stress paths of all the eight numerical tests are presented in Figures 3.5. The corresponding
curves and the curves of the CP test are illustrated in Figure 3.6 and 3.7
respectively.
52
Figure 3. 5: Stress paths predicted by the UMAT-MCC and ABA-MCCM
Figure 3.5 demonstrates a perfect match of the stress paths calculated by the UMAT and the
inbuilt MCCM in all four cases. This figure is presented via MATLAB software and the yield
ellipse is mathematically drawn according to the MCCM yield function Eq. (3.9) with input
parameters M=0.9 and . It can be seen that the sample entered the plastic zone exactly
at the moment the stress paths touch the ellipse. The successive yield surfaces are expanding
when the sample yields from the wet side, while they are shrinking when the specimen yields
The stress strain curves given by UMAT-MCC and inbuilt ABA-MCC shown in Figure 3.6
exhibit some discrepancies in elastic regime. The UMAT-MCC indicates a stiffer elastic
response than the ABA-MCC and thus these curves achieve their peaks faster.
53
(a) Constant Volume Tests
54
One possible explanation is that these differences are due to the use of secant modulus in
the elastic part calculations. According to ABAQUS Theory Manual 4.4.1 Porous elasticity, the
shear modulus for porous material with zero tensile strength is:
( )( + )
( )
( + )
Comparing to Eq. (19), one can find the expression of bulk modulus adopted by ABAQUS
is:
+
( )
which has been called “instantaneous” modulus in the Manual. This modulus clearly differs from
the secant modulus adopted in this research. As we explained earlier the tangent bulk modulus
modulus. Hence, the use of a secant modulus (averaged over the step) instead of the tangent
modulus (evaluated at the beginning of that step) will produce a stiffer stress-strain response.
After the sample starting plastic deformation, good agreements are observed at both peak stresses
The volumetric behaviors predicted by both UMAT-MCC and ABA-MCC under the
constant pressure tests show excellent agreement (Figure 3.7). It is observed that for lightly and
heavily overconsolidated clay, the volumetric behavior in elastic regime is almost the same.
However, as soon as the plastic strain develops, the former will continue shrinking in volume
55
Figure 3. 7: Volumetric behavior predicted by UMAT-MCC and ABA-MCCM
Both triaxial compression and extension tests under CV condition are performed to observe
the anisotropic behavior. The use of CV condition is for the easiness of distinguishing the elastic
- No anisotropy: , CV compression;
[ ]
56
which is arbitrarily chosen for convenience. All the tests will be performed on the samples with
two different initial mean stresses and in order to observe the yielding
Figure 3.8 illustrates the stress paths of the samples yield from dry side and wet side
compression/extension procedures. The figure is presented via MATLAB and the two yield
surfaces are mathematically drawn according to the AMCCM yield function Eq. (3.18) with
1) The samples under triaxial compression with initial anisotropy ( ) goes through a
much longer elastic path before entering the plastic zone than the one with initial anisotropy
( ); while for those under triaxial extension the former will begin yielding much
quicker than the latter. This effect is consistent with the concept of “inherent anisotropy”.
2) For the samples without inherent anisotropy effect ( ), the yield locus of the sample
with c=0 only experience expanding/shrinking; while for those with c=1 this
3) As soon as the stress paths touch the analytically drawn ellipse, yielding occurs and they start
to develop towards the CSL. This exact match between numerically obtained stress paths and
57
Figure 3.9 further displays such rotational hardening and the effect of inherent anisotropy in
plain. In conclusion, the validity of the AMCCM UMAT has been well proved through
the series of numerical tests above. Setting the anisotropic factor c=0, the implemented AMCCM
produces identical results as ABAQUS inbuilt MCC does except exhibiting a slight stiffer
stress-strain response in elastic regime. CV tests running under different anisotropic parameter
combinations illustrate that the implemented AMCCM can successively capture both inherent
anisotropy and induced anisotropy of the soil. During the numerical verification, the authors
noticed the efficiency and stability of the code is amazingly good, which greatly benefit the
59
CHAPTER 4
4.1 Introduction
As the rapid development of computers, the application of numerical methods for solving
geotechnical problems is becoming more recognized by both geotechnical researchers and other
engineers. The numerical methods that are frequently used to study pile foundations includes finite
difference method (FDM) (e.g. Fakharian et al., 2008; Huh et al., 2008), finite element method
(FEM) (e.g. Muqtadir and Desai, 1986; Trochanis et al., 1991), boundary element method (BEM)
(e.g. Filho, 2005) and discrete element methods (DEM) (e.g. Uchida and Kawabata, 2004;
Lobo-Guerrero and Vallejo, 2007). Among those, the FEM have been regarded as the most
versatile and widely used approach for analyzing boundary value geotechnical engineering
(such as a set of partial differential equations and boundary/initial conditions) of the problem that
they are dealing with, and then develop their FEM codes to numerically solve these equations.
Alternatively, there exist many commercial FEM packages (e.g. ANSYS, ABAQUS and PLAXIS)
that can be directly used by geotechnical researchers. These commercial FEM softwares greatly
promoted the application of FEM in geotechnical engineering and significantly facilitated the
studies based on FE analysis. However, such advancement does not guarantee any enhanced
reliability of the numerical results. In fact, the users of ANSYS or ABQUS, especially
geotechnical engineers, should handle their FEA with great cautions since these softwares were
developed for general-purpose analysis and design, which need a strong background on both FEM
and soil mechanics to ensure the reliability of the outputs. As indicated by Potts and Zdravkovic
60
(1999), proficient experience with the finite element codes (i.e. understanding their capabilities
and limitations) as well as understanding the relevant theories behind soil mechanics and soil
constitutive models are the key issues in solving numerical problems accurately.
In this chapter, we will discuss in details the FE model used for the batter pile group
foundation including its mesh, interface modeling, constitutive models for materials and the
determination of the model parameters. The coupled pore fluid diffusion and stress analysis
provided by ABAQUS will be discussed. A brief introduction of the FB-MultiPier model will also
be presented since we intended to compare the result of FEM with the p-y approach.
A three-dimensional finite element model with exact geometry of the M19 pier foundation
was developed on ABAQUS. The lateral load was applied on the left side of pile cap and directed
horizontally to the right. Only half of the foundation including 12 piles is simulated due to the
symmetric nature of this problem. The size of the mesh was selected such that the length of soil
media is 220 ft, which is 5 times the size of pile cap (44 ft), and the depth of the soil media is 165
ft, which is 1.7 times the embedded depth of the piles (97 ft). To assess the effect of boundary
distance on the results of FE analysis, another model was developed using a mesh with 440 ft
length (10 times of pile cap size) and the resulted lateral deformation of pile cap was about 2%
different than the original model. This small discrepancy demonstrates that the boundary effect on
the model is negligible. Thus, the FE model with 220 ft wide is adopted in this study for its
economic computation cost. The mesh includes 12 piles of 110 ft long and 3 ft ×3 ft square section
that are inclined with a slope of 1:6 and spaced exactly the same as the M19 pier foundation. The
cap is located 12 ft above the mudline. The lateral load was applied on the left side of pile cap and
61
directed horizontally to the right (along +y direction). A uniformly distributed pressure load is
The final FE mesh (Figure 4.1a) consists of a total of 68,229 three-dimensional linear
interpolation reduced integration solid elements (C3D8R). This type of elements is superior to the
fully-integrated linear elements in representing more variations in bending and also avoided the
“shear locking” problem that occurs during shearing. However, the linear reduced integration
elements are allowing spurious singular modes (“hourglassing”) which can only be avoided by
setting proper hourglassing stiffness by the users. A hourglassing stiffness of 1.0 is proved to be
The boundary conditions for the FE model include restraining the horizontal movement along
the side boundaries ( or ), restraining the vertical movement along the bottom boundary
distributed hydrostatic pressure (10ft water=624psf) on the mud surface to simulate the effect of
The initial geostatic balance condition of layered soil is installed in the soil body using
command “*initial conditions, type=geostatic” for each layer according to its total unit weight.
Meanwhile, the gravity is applied to the whole model body with each soil layers assigned
corresponding density. In this way the inner geostatic stress and the external gravity are able to
reach a balance at the beginning of the analysis without any settlements (or with a neglectable
amount of settlements due to the weight of pile group). The information of soil unit weight or
density is directly obtained from the UU tests and the CPTs performed near M19 piers. It should be
noticed that this procedure actually simulated the cast-in place types of pile since it assumes
62
initially undisturbed soil stress with the presence of piles. This is not the case for M19 foundation
Trochanis et al. (1991) discussed the significance of incorporating slippage and separation in
pile-soil interface in the FE simulation of laterally loaded piles. ABAQUS provides several
approach, and assignment of “master” and “slave” roles to the contact surfaces. The tracking
approaches are to account for the relative motion of two interacting surfaces in mechanical contact
simulations. Two options of tracking approaches are offered: finite sliding, which is the most
general and allows any arbitrary motion of the surfaces, and small sliding assuming relatively little
sliding of one surface along the other. ABAQUS applies conditional constraints at various
locations on interacting surfaces to simulate contact conditions. The locations and conditions of
these constraints depend on the contact discretization used in the overall contact formulation. Two
discretization methods enforce the slave nodes not to penetrate into the master surface, however
the node-to-surface contact discretization allows the master surface to penetrate into the slave
surface while large undetected penetrations of master nodes into the slave surface do not occur
contact conditions in an average sense over regions nearby slave nodes rather than only at
individual slave nodes. In this study, the surface-to-surface contact discretization with
The mechanical contact property consists of two components: one normal to the surfaces and
one tangential to the surfaces. The interface in the normal direction is assumed to be “hard” contact
which minimizes the penetration of the slave surface into the master surface at the constraint
63
locations and does not allow the transfer of tensile stress across the interface, while the tangential
interaction behavior is governed by Coulomb friction model which relates the normal force to its
shear behavior. A friction coefficient of 0.424 has been assigned to the tangential behavior, which
corresponds to an angle of interface friction =23o between the soil and the piles. Separation is
allowed after contact and slippage can happen when the tangential stress reached certain limit.
One of the objectives of this research is to investigate the effect of group interaction and pile
inclination. Such objective can be achieved by comparing the lateral deflection, bending moment
and soil resistance profiles produced by FE models with different spacing and pile inclinations.
Repeating the same techniques, four additional FE models with varied geometry features from the
original group pile model are constructed. They are summarized in Table 4.1. The meshes of all the
five FE models developed in this research are presented in Figure 4.1. The various colors in Figure
4.1b distinguish the soil stratification which will be discussed in next section. Figure 4.2 presents
the pile plane view of the pile layout and the numbering of the piles.
Subsurface condition of M19 pier foundation site is of high heterogeneity (as shown in Figure
2.2), so it is convenient to classify the soil into certain layers and then discuss their constitutive
models. Combining the soils with similar properties, the subsurface soil can be divided into eight
layers including two sand layers and six clay layers (Figure 4.2).
The unit weights and corresponding overburden pressure, the clays’ undrained shear
strength , and plasticity index , and the sands’ friction angle are determined based on the
overall evaluation of the five CPTs, the SPT and the UU tests and listed in Table 4.2.
4.3.1 Sands
The sand soil layers are simulated using the Drucker-Prager (DP) model with non-associated
flow rule. DP model had been used in many studies to describe the behavior of sand soils (e.g.,
Brown and Shie, 1990; Trochanis et al., 1991; Yang and Jeremic, 2002; Karthigeyan et al., 2006).
The Drucker-Prager yield function incorporated the effect of hydrostatic stress, which is expressed
as follows:
( )
where
( + ( )( ) ) ( )
which describes the shape of the yield stress on the -plane; describes the slope of the yield
surface in the p–t stress plane and is referred as the angle of friction; d describe the cohesion of the
material, p is the mean stress, q is the Mises equivalent stress which has the expression
67
√ ; is the is the ratio of the yield stress in triaxial tension to the yield stress in triaxial
compression, and r is the third invariant of deviatoric stress. The yield surface of Drucker-Prager
It is worth mentioning that the angle of friction and cohesion d in the Drucker-Prager
criterion are different from the friction angle and cohesion c in the Mohr-Coulomb criterion.
𝑖
( )
√ ( 𝑖 ) √ ( 𝑖 )
The angle of friction of sand layers was calculated from the estimated friction angle
from the corrected SPT-N values, and the cohesion is artificially set to a small value to avoid
convergence difficulties. The Young’s modulus of cohesionless soil was estimated from the
corrected SPT-N value using the following formula proposed by Kulhawy and Mayne (1990):
(4.4)
where is the atmospheric pressure, is 10 for clean normally consolidated sand and is the
68
4.3.2 Clays
All the clay layers were simulated using the AMCCM that was implemented into ABAQUS
through UMAT subroutine. The FE analysis is based on the assumption that all the clay layers are
fully drained so no excessive pore water pressure exists (the validity of such assumption will be
discussed in chapter 5). In other words, if we consider the whole soil body as the superposition of
water and soil skeleton, the stresses contributed by water will be exactly the hydrostatic water
pressure while the stresses offered by soil skeleton are under the governing of AMCCM or DP
constitutive law. Hence the stresses given by ABAQUS must be processed with:
( )
before passing the stresses into AMCCM. After successful updating of the stresses and state
variables, remember to recover the effective stress to total stress before passing back to ABAQUS:
+ ( )
The Above treatment can guarantee the response of the whole system is driven by total stress
while remain the implemented AMCCM only manipulate the effective stress. Keep in mind that
The AMCCM has six material constants: Poisson’s ratio; M slope of critical state line;
slope of unloading-reloading line (or logarithmic bulk modulus); slope of normal compression
line (or logarithmic hardening constant); and the anisotropic parameters c and x. Also, it has three
state variables (e void ratio, preconsolidation pressure and back stress), thus requires the
users to provide three initial values of these state variables ( , and ). These
parameters will be either calculated from empirical correlations from CPT data or estimated based
on past experiences of the normal range of the parameter for a certain type of clay.
69
The preconsolidation pressure can be calculated from the OCR of soil which is defined
as
𝑅 ( )
where is the effective overburden pressure of each layer, which can be simply calculated by
averaging the overburden pressure at the top and bottom of that layer. The OCRs are obtained by
and plastic index via the diagram (Figure 4.5) proposed by Andresen et al. (1979)
The initial void ratio of clay can be estimated from the typical range of soil void ratio
constant should be systematically calibrated via hydrostatic compression test and triaxial
compression tests. Unfortunately, undisturbed soil samples were unavailable at the time of this
research and also these parameters belong to the school of critical state soil mechanics that can
hardly be correlated from CPT or SPT data. However, the in-situ tests provided some direct soil
70
strength information, which can help us make the proper judgments within the common range of
these parameters. It will be helpful to make such judgments if one firstly obtains the Clay’s
1110-1-1904, the Young’s modulus is related with undrained shear strength by the following
relation:
( )
where is a function of OCR and plasticity index and can be determined according to Figure
4.6. Although the Young’s modulus cannot directly be used in AMCCM, they are good indicator
of the relative elastic stiffness of all the six clay clays. According to Yu (2006), the slope of
swelling line for clay ranges from 0.01 to 0.06 and the Poisson’s ratio is among 0.15 and 0.35
for both clay and sand. The typical value of for clay is in the range of 0.1-0.2. Combining all
71
Figure 4. 6: Chart for estimating (EM 1110-1-1904)
The slope of critical state line M is determined based on the back-calculation of undrained
shear strength interpreted from CPT data. According to critical state soil mechanics, the
relationship between clay undrained shear strength and Cam-clay parameters is expressed as
𝑅
( ) ( )
where ; is the initial effective mean stress; is the ratio of tip pressure to critical state
pressure and for modified cam clay model . Using this equation, one can back-calculate the
M value with , 𝑅, and that were determined earlier for each layer.
Some suggestions of anisotropic constants c and x from literatures can be used in this study.
Wheeler (1997) suggested a value of 4/3 for x. In the same work by Wheeler (1997), a new
parameter is introduced to describe the rate at which approaching to its target value 𝑥 :
𝑥( + )
( )
( )
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Wheeler (1997) suggested that in the absence of suitable experimental data, a value of 30 for
can be considered for a typical value. Having each layer’s , , 𝑥, , and , one is can back
As indicated by Graham et al. (1983), the yield surface for naturally deposited clay tends to
align along the consolidation line, which means the initial inclination angle can be
determined if one knows the of the in-situ soil. For each layer, is obtained according to the
relationship between , OCR and proposed by Masood and Mitchell (1993) (Figure 4.7).
Then is set to be identical to the angle of the consolidation line on the plane,
( ) ( )
( )
( + ) ( + ) +
Figure 4. 7: as function of sleeve friction and overconsolidation ratio (Masood and Mitchell,
1993)
Finally, the basic soil properties directly correlated from the in-situ tests and the material
parameters used for AMCCM and Drucker-Prager model are summarized in Table 4.4.
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Table 4. 4: Parameters for AMCCM and DP model
Soft Clay 0-15 71.7 0.30 3.2 0.9 1459.3 1.5 6.1 0.9 0.03 0.1 1.33 1.10
Stiff Clay 15-25 1307.9 0.15 2.7 1.0 3226.3 1.2 0.0 1.1 0.01 0.14 1.33 0.87
Medium Clay 25-38 780.0 0.25 2.2 0.9 3902.9 1.3 6.1 1.0 0.01 0.12 1.33 0.77
Stiff Clay 49-70 1307.9 0.20 1.3 0.8 4206.1 0.9 13.2 0.9 0.01 0.12 1.33 0.57
Stiff Clay 70-81 2400.2 0.20 1.0 0.9 4094.8 0.9 6.1 1.1 0.01 0.12 1.33 0.44
Stiff Clay 81-99 1620.0 0.20 1.0 0.8 4685.2 0.9 13.2 1.1 0.01 0.12 1.33 0.44
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4.3.3 Concretes
The piles and pile cap are made of concrete which is simulated using a linear elastic model in
this study. The elastic Young’s modulus was estimated based on the average results of 28-day
concrete compressive strength on cylindrical specimens ( = 9064 psi). Since the lateral
load test was conducted six months after pile construction, the average compressive strength was
increased by 20% to account for concrete curing and the prestressed confinement effect. The
Young’s modulus of the concrete was then estimated using the following equation:
√ ( )
The value of pile Young’s modulus was estimated to be 8.56 × 108 psf. A common value of
As mentions earlier, we assume the soils to be under fully drained conditions when handling
the pore water pressure problem. However, the adoption of such assumption will not be validated
until one performs a coupled pore fluid diffusion analysis of this problem and compares its results
with the non-coupled one. In this section, the coupled pore fluid diffusion and stress analysis in
ABAQUS will be discussed, and then it will be used to carry out a series of drained,
In ABAQUS, a coupled pore fluid diffusion/stress analysis is used to model single phase,
partially or fully saturated fluid flow through porous media and can be performed in terms of either
total pore pressure or excess pore pressure by including or excluding the pore fluid weight. In this
75
study we are focusing on the saturated fluid flow since all the soils are submerged under the lake’s
ABAQUS can provide the solutions either in terms of total or “excess” pore fluid pressure.
The excess pore fluid pressure at a point is the pore fluid pressure in excess of the hydrostatic
pressure required to support the weight of pore fluid above the elevation of the material point. In
ABAQUS the total pore pressure solutions are provided when the gravity distributed load is used
to define the gravity load on the model, while excess pore pressure solutions are provided in all
other cases; for example, when gravity loading is defined with body force distributed loads.
Recalling the use of “gravity” type of load in our FE models, the pore fluid obtained from the FEA
The pore fluid flows are governed by Forchheimer's law, which can be expressed as follows:
𝜕
( + √ ) ( )
𝜕
where is the volumetric flow rate of wetting liquid per unit area of the porous medium;
is the fluid saturation (s=1 for a fully saturated medium); n is the porosity of the porous
medium; is the fluid velocity; ( ) is a “velocity coefficient” which may be depend on the
void ratio of the material; is the dependence of permeability on saturation of the wetting liquid
such that at ; is the density of and specific unit weight the fluid respectively;
( ) is the permeability of the fully saturated medium, which can be a function of void ratio
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The Forchheimer's law tells us that high flow velocities have the effect of reducing the
effective permeability and, therefore, “choking” pore fluid flow. As the fluid flow velocity reduces,
In this research, a coupled pore fluid diffusion and stress analysis is carried out on single pile
model. All the soil constitutive models and their parameters are the same as the uncoupled models.
Clays and sands are assigned different permeability according to the typical permeability k for
various soils (Table 4.5. Das, 1999). Times allowed for consolidation after each load increments is
exactly following the time table of the full scale load test on M19 pier eastbound foundation,
which is listed in Table 2.1. In the finite element model, the C3D8R elements are replaced by
C3D8RP elements to perform the coupled analysis, since the latter one has an additional degree of
freedom on pore water pressure. The upper surface of the soil bulk is assigned the boundary
from other part of the soil bodies including the pile-soil interface are not allowed.
Table 4. 5: Typical permeability k for various soils (Das, 1990)
Type of soil k (cm/sec)
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4.5 FB-MultiPier Analysis
The FB-MultiPier program is based on finite element approach and can be used to analyze the
entire components of the bridge from bridge slab to soil layer. It uses the iterative solution
technique to predict the lateral displacements. During iteration, the stiffness of soils and piles are
calculated and eventually generated the stiffness matrix to predict the lateral displacement of piles
as output. Then, this displacement is used to predict the internal forces of structure members. The
advantages of FB-Multipier for simulating laterally loaded piles is that it has an in-built library of
soil p-y curves, and the group effect can be easily taken into account by user defined p–multipliers
for pile group analysis. Some in-built p-y curves included in FB-Multipier and their input
* is the internal friction angle; is subgrade modulus; is the unit weight; is the
undrained shear strength; is the major principal strain at 50% soil strength.
78
The whole structure of M19 eastbound pier was modeled including 24 batter piles, 2-pier
columns, shear wall, and a cantilever bent (Figure 4.8). Piles are modeled as three dimensional
discrete elements. The nonlinear behavior of concrete material are modeled by using input or
default stress-strain curves that are a function of compressive stress of concrete ( ) and modulus
of elasticity of concrete ( ). The pile model is generated by inputting the same geometry
properties as the original foundation. The fixed headed pile cap is modeled using nine nodded shell
elements which is based on Mindlin’s theory that can consider the bending and shear deformations.
The soils are classified into eight layers to be consistent with the FE models introduced earlier in
this chapter. Each soil layer surrounding piles are modeled as an attached nonlinear spring that
characterized by a proper selected of p-y curve. The input parameters ( for sand and
for clay) are determined using the UU tests or in-situ CPT and SPT tests results. The input
Soft Clay 0-15 Clay (Soft < Water) 123 240 0.02
Stiff Clay 15-25 Clay (Stiff < Water) 119 1560 120 0.005
Medium Clay 25-38 Clay (Stiff < Water) 108 1104 60 0.007
Stiff Clay 49-70 Clay (Stiff < Water) 113 1533.6 100 0.005
Stiff Clay 70-81 Clay (Stiff < Water) 122 3162 150 0.004
Stiff Clay 81-99 Clay (Stiff < Water) 128 1796.4 150 0.005
80
CHAPTER 5
5.1 Nomenclatures
Before systematically presenting the FEA results, it is desirable to reintroduce the pile
numbering and their position in the pile group (recall Figure 4.2) for the convenience of
The nomenclature for the rows is the same as the one adopted by Zhang et al. (1999), where
the 1st row is named as “lead row” and the 4th row is called “trail row”. The most outside column
is called “side column”, followed by “middle column” and then the most inside one is named
“inner column”. In the following sections many aspects will be compared between different rows
(e.g., piles 2, 4, 5, 6) and different columns (e.g., piles 1, 2, 3). Among all, pile 2 is selected here
to be the reference pile that connects the comparisons between rows and columns.
5.2 Coupled Pore Fluid Diffusion and Stress Analysis on Single Pile Model
First of all, it is necessary to examine the soundness of our fully drained assumption the
fully drained assumption adopted in the construction of all the FE models, as introduced in
section 4.3. A coupled pore fluid diffusion analysis is performed on a single pile model. In the
coupled analysis, the time duration of each loading increment becomes very important since the
consolidation process is involved. Such time durations are set to be the exactly the same as the
time schedule followed in the full-scale lateral load test that carried on M19 east bound pier
foundation. Unloading-reloading processes are not included in this analysis for the sake of
81
comparing with the previous uncoupled FE analysis. In this analysis, we neither assume its fully
drained nor fully undrained condition, but set the upper surface of the soil body to be the drained
boundary and let the whole soil body consolidate according to the testing time. Hence, this
coupled pore fluid diffusion analysis can be regarded as “partially drained” analysis.
The lateral deflection and bending moment profiles obtained from the partially drained
model and the previous fully drained model at 1870 kips lateral load are presented in Figure 5.1.
It can be seen that the results from the partially drained analysis and the fully drained analysis
are very close to each other with a maximum difference of about 3% in lateral deflection and 4%
in bending moment. To investigate possible causes of such small discrepancy between the two
draining conditions, we extracted the excessive pore water pressure developed along depth in
front of the pile at the end of loading (when the steel strands are cut) and plot it with the total
geostatic pressure in Figure 5.2. It can be seen that the major excessive pore water pressure is
developed within the top 15ft of the soil, which is the soft clay soil layer. Such excessive pore
pressure is really small compared to the total stress due to soil self-weight.
The excessive pore water pressure dissipation curve of the point at 3.75 ft, 11.25 ft and
21.65 ft below ground surface are plotted in Figure 5.3. It is observed that there is a bouncing up
of the pore water pressure after each load increment applied, and then dissipation occurs during
the resting period. In this four-hour loading procedure the excessive pore pressure is kept
accumulating until the steel strands are cut. The overall excessive pore water pressure developed
at 3.75 ft is lower than that at 11.25 ft since this point is very close to the ground surface which is
82
the drained boundary. The excessive pore pressure of soil at 21.65 ft depths is neglectable
Figure 5. 3: Excessive pore water pressure dissipation curves of soil at various depths
84
Based on Figures 5.2 and 5.3, the small discrepancy of the lateral deflection and bending
moment profile observed between partially drained and fully drained analysis in Figure 5.1 can
be explained. The excessive pore water pressure are mainly concentrated near the ground surface
which is the drainage boundary, hence the accumulated excessive pore water pressure after the
four-hour loading procedure is still small compared to the initial geostatic pressure of the site.
In conclusion, although some variations (maximum 3% - 4%) are observed between the
partially drained and fully drained analyses, the latter can still provide quite reliable results
representing the lateral response of the piles in the full-scale lateral load test on M19 eastbound
pier foundation while having a significant saving in the computational cost of the analysis.
The lateral displacement contour of the whole soil body after applying a lateral load of 1870
kips is illustrated in Figure 5.4a. It can be seen that the pile cap and the surface soil close to piles
have the largest displacement. The soils within 28 ft width and 29 ft depth from the center of
pile location are mobilized from 0.21 in to 0.63 in due to the maximum applied lateral load of
1870 kips. Such influence is neglectable for the soil beyond this region. Figure 5.4b presents the
displacement vectors for all nodes of the pile group at 1870 kips lateral load. A scale factor of
250 is adopted to visualize the lateral deflections of the piles. It is noticed that the toes of left
side piles and right side piles are not in the same level. The negative battered piles (left side piles)
are subjected to uplifting while the positive battered piles (right side piles) are subjected to down
85
(a)
(b)
Figure 5. 4: At a lateral load of 1870kips, the a) contours of lateral displacement of the whole
model and b) displacement vectors of the pile nodes on the deformed pile group.
86
As a result, the whole pile foundation exhibits a tendency of rising of the elevation due to
the lateral load. This phenomenon is consistent with our experience: piles are easier to pull out
than drive in, thus when rotation occurs the whole foundation are tend to be lift up. Figure 5.5
shows the distribution of void ratio of the first layer. As expected, the soil exhibit densification in
front of the lead row and expansion behind the trail row.
Figure 5. 5: Contour of the void ratio at first clay later after 1870kips lateral load
Figure 5.6 presents the profiles of lateral deflection generated from the FE model on batter
pile group foundation under different lateral loads. The FE predicted maximum pile cap lateral
deformation is 0.79 in. Due to the “fixed” condition of pile head (enforced by pile cap), the
deflection curves always tend to keep perpendicular to bottom of the pile cap.
87
Figure 5. 6: Lateral deflection under different lateral load
Figure 5.7 compares the FE predicted lateral deflection profiles obtained from FEM using
AMCCM with those measured in the field using the IP inclinometers and those predicted by the
FB-MultiPier analysis under lateral load 1870kips. Here the group factors of 0.9, 0.8, 0.8 and 0.7
for lead row to trail row were used in the FB-MultiPier analyses. The deflection profiles from the
FEM show slightly larger deflection near the pile cap than the measured values. However, the
overall deflection curves along the pile have very good match with the field measured data. The
FB-MultiPier predicted pile head displacement is close to the measured value; however the
deflection curve drifts away as the depth increasing along the pile. The “Stationary point” (the
point with no lateral deformation) obtained from FB-MultiPier analysis is much shallower than
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those given by field measurements and FEM. Such discrepancies were also reported by McVay
et al. (2005) in there barge impact tests on a bridge pier at St. George Island Causeway. A
possible explanation of such discrepancy could be due to the simple way that FB-MultiPier
account for group interaction effects. In FB-MultiPier, the group effect is empirically considered
using the p-multiplier, which is determined by comparing the single pile p-y curves with group
pile p-y curves from field tests. The p-multiplier of the pile in the same row assumed to be
constant along the pile depth regardless of the soil type. First of all, the unified p-multiplier for
the pile representing average value of the different group interaction effects of various soil types.
Secondly, such assumption neglected the possible variation of group interaction effect for the
same soil at different depth. Thus, for the soil, the shallower layers may suffer more group
interaction effect than the deeper layers due to larger mobilization of shallower soil and thus
more shadowing effect. For the batter pile group cases, it becomes even more questionable to use
a unified p-multiplier to reduce all the p-y curves along the entire pile depth, since the spacing of
rows spacing vary dramatically from pile top to the pile toe. Take the M19 pier foundation as an
example, the spacing between 2nd and 3rd rows is 4.5D (13.6ft) at ground level and becomes
15.4D (46.1ft) at pile toes. Obviously the pile-soil system near the ground surface will
experience more pile-soil-pile interaction effects than at deeper depths. This statement will be
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Figure 5. 7: Lateral deflection profiles from field test, FEM and FB-MultiPier
The lateral deflection profiles of the piles aligned in the same row but different columns
(pile 1, 2 and 3) and those in the same column but varied rows (pile 2, 4, 5 and 6) under 1870
kips lateral load are presented in Figures 5.8a and 5.8b, respectively. The lateral deflections of
the piles near the pile cap are almost the same due to the confinement of the pile cap. However,
some variation occurs at deeper portion of the pile and such difference reaches its maximum at
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30 – 35 ft below pile cap, which is corresponding to the second soil layer (stiff clay). The piles
located in the side column, lead row and trail row show larger bent than the other piles,
indicating that these piles are encountered more soil resistance than the rest. This observation
lateral load of 1870 kips is presented in Figure 5.9. In the figure GV represents group vertical
pile foundation and GB stands for group batter pile foundation. All the comparison is based on
the reference pile (pile 2). Many conclusions can be drawn from Figure 5.9:
1) All the pile groups regardless of their pile inclination or spacing show much higher lateral
displacement than that of a single pile, which is consistent with most field or model tests.
2) For the batter pile groups, the small spacing model produces largest lateral deformation
(0.94in) while large spacing model shows relatively small lateral deformation (0.71in).
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3) The vertical pile group model exhibits significant large lateral deformation (1.22in), which is
54% greater than the batter pile group with the same spacing (0.79in) and 130 % larger than
chapter regarding the oversimplification of group effect using a unified p-multiplier for an entire
pile. The “fixed point” for single vertical pile model is much shallower than the group piles, and
is very close to the FB-Multipier predicted depth. It suggests that when pile group is subjected to
lateral loading, the soils near the ground surface are mobilized as a block, extending the soil’s
influence zone due to lateral load to be deeper than the single pile case. This comparison
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confirms the authors’ believe that the group effect has influence on both pile head movement and
the deflection pattern along depth. The p-y curves of the soils located near ground level are
subjected to greater reduction due to group interaction effects which can be incorporated with
increased p-multipliers.
The Contour of the mean effective stress (SDV10 in the UMAT) distribution for soil layer
clays before and after lateral load is presented in Figure 5.10. The mean effective stress after
geostatic process (Figure 5.10a) shows generally uniform distribution at the same elevation with
only a little disturbance near the pile due to gravity of the pile group. Significant change of the
mean effective stress field for the top three soil layers is observed after application of a lateral
load of 1870 kips (Figure 5.10b). The soils behind the piles (on the left side of the pile in the
figure) show a decrease of mean effective stress while the soils in front of the piles (on the right
side of the pile in the figure) are subjected to increased mean effective stress. This finding
indicates that the stress path of the soil near ground surface behind the pile is similar to that of
triaxial extension test while the stress path of the soil in front of the pile is close to the one in
triaxial compression test, which further supported the statement made earlier that the soil
anisotropy could be a key issue in simulating the laterally loaded pile behaviors. The influence of
lateral load to the mean effective stress distribution becomes less significant for the soil layers
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(a)
(b)
Figure 5. 10: Contour of the mean effective stress for clays a) before and b) after lateral loading
The change in mean effective stress should be discussed together with the volumetric strain
(SDV12 in the UMAT) distribution after loading (shown in Figure 5.11). Obviously the lateral
load transmitted from the pile pushes the soil in front of the piles to a denser state and the gaps
94
left behind the piles are instantaneously filled by soils and thus result in a relaxation of the soil
behind the piles. Such densification effect is most significant for the soil in front of the lead row
and the maximum relaxation occurs at the soils behind the trail row. It is worth to notice that the
volumetric strain contour is very similar to the possible failure surfaces for batter piles described
Figure 5. 12: Failure surface in positive and negative batter piles (Prakash and Subramanyam,
1965)
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5.4.2 Deviatoric Stress &Deviatoric Strain
The deviatoric stress q (also called “Mises stress” in ABAQUS) of piles at different
columns is shown in Figure 5.13. Only the top halves of the piles are presented since the
deviatoric stress variation of the bottom half are neglectable compares to the top portion. The
yellow-red area indicates the largest concentration of the deviatoric stress. It is notice that such
high deviatoric stress area occurs at the cap-pile connection and 0-10ft and 24-35ft below pile
cap. The piles located in the lead row show larger deviatoric stress at the cap-pile connection
than those in the rest rows. Also, it is observed that the piles in the side column share more
observed that a large fraction of the lateral load is absorbed by the second stiff clay layer, while
the rest are mainly taken by the third medium clay layer. The first clay layer is so soft that only a
small fraction of deviatoric stress is allocated in this layer. Such distribution pattern will be
observed again when we discuss the soil resistance profiles along the piles in section 5.8. The
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situation is reversed for the contours of deviatoric strain (Figure 5.14b) of the first three soil
layers. For example, a large region of the first soft clay layer undergoes deviatoric strains while
only a small area near the pile of the second stiff clay layer developed some deviatoric strains.
Extracting the second layer and readjust the limit ranges of the contour (Figure 5.15) allows
one to overlook the deviatoric stress distribution at the cross-section that offer major lateral
resistance. The deviatoric stress distribution pattern indicates that the soils in front of the lead
row contributed much more soil resistance that those in between the piles or behind the trail row.
(a)
(b)
Figure 5. 14: Contour of a) deviatoric stress and b) deviatoric strain of the first three soil layers
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Figure 5. 15: Deviatoric stress distribution at the second layer
The shadowing\edge effects caused by pile group interaction have been discussed in many
literatures (e.g. Rollins et al., 1998; Zhang et al., 1999). However, no literature has illustrated
how such effect influence the stress distribution of the soil under lateral loads. In this research,
the shadow effect can be well visualized by comparing the plane view of deviatoric stress
contours of the second stiff clay layer (which offer the major lateral resistance) of all the three
varied-spacing group batter pile modes and the vertical group pile model, as shown in Figure
5.16. Notice that all the contours are plotted under the same color limit (from 300 psf to 2430 psf)
Firstly, there is a significant increase of the deviatoric stress of the soil in front of the lead
row for the vertical pile group (Figure 5.16b) comparing to the batter pile group (Figure 5.16a) at
same spacing. The main reason is that major portion of the lateral load on pile cap is directly
transferred to the soils for vertical piles, while for inclined piles part of the lateral load is
transmitted into axial component of the piles and then digested by skin friction and toe resistance.
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Hence, the deviatoric stress developed in the soil surrounding the vertical piles will be much
with various spacing. The negative batter piles of large spacing model (Figure 5.16c) produce
relatively independent stress concentration zones under a lateral load of 1870 kips, while these
isolated zones turn out to be a united region when comes to normal spacing model. Such region
even extends into a larger area when the pile spacing further narrowed (Figure 5.16d). Obviously,
this phenomenon is caused by increased overlapping of the piles’ influenced zones when pile
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spacing getting closer. It is also interesting to notice that the maximum deviatoric stress in front
of the lead row of large spaced piles is 1485 psf with a relatively spread out pattern, while for the
normal spacing model it becomes 1924 psf and distributed more compactly. The small spacing
model has the most concentrated stress zone with an average deviatoric stress of 2324psf located
in front of the lead row. Still, such zone is formed due to the highly overlapped shadowing and
The piles bending moment profiles along depth were extracted from the FE model at
different load increments and presented in Figure 5.17. Evidently, the maximum positive
bending moments occur at pile head for all piles due to rigid cap-pile connection. The maximum
negative bending moments occurs at approximately 28 ft below pile cap for all lateral loads.
As introduced in Chapter 2, two pairs of strain gauges were installed on each of the 12
selected piles at two different locations. The strain gauge data for each pair was used to calculate
the bending moment and axial load. The moments can be calculated using the following
equation:
( 𝑡 )
( )
where 𝑡 is the tensile strain; is the compressive strain; h is the horizontal distance between
Figures 5.18 present the comparison of bending moments generated from the FEM, strain
gauges and the FB-MultiPier analyses at two different load increments (970 kips and 1745 kips).
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Figure 5. 17: Bending moment profiles under different lateral loads from FE analysis
101
There is a missing of the second pair strain gauge data at pile 4 since one gauge was damaged
during pile installation. The position of pile 8 is located in the trail row of the other half of the
pile group foundation which is not simulated in the FE model. The counterpart to pile 8 is pile 1
in the FE simulated half, thus the FE and FB-MultiPier results shown in the bottom two figures
of Figure 5.18 are actually the same and only the SG data changed.
Figure 5. 18: Bending moment profiles from the strain gauges, FEM and FB-MultiPier analyses
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Generally speaking, both FEM and the FB-MultiPier predicted bending moments agree well
with those deduced from the strain gauges data. The point of zero bending moment (which is
corresponding to the inflection point of the deflection curves) occurs at about 15-18ft below the
bottom of the pile cap. For both case (970 kips and 1745 kips), the maximum positive moment
predicted by the FEM are very close to those by FB-MultiPier; while the maximum negative
bending moment obtained from FEM is 21%-24% lower than those predicted by the
FB-MultiPier analysis. Besides, the location of maximum negative moment from the FEM
occurs at slightly shallower depth (27.5 ft below pile cap) than the location from the
The bending moment profiles of the piles aligned in the same row but different columns
(pile 1, 2 and 3) and those in the same column but different rows (pile 2, 4, 5 and 6) under 1870
kips lateral load are plotted in Figure 5.19a and 5.19b, respectively. A very important finding
from this figure is that the pile located at side column (pile 1) shows 11.7% higher maximum
positive bending moment (at pile head) and 18.0% higher maximum negative bending moment
(at 27.5ft below pile cap) than those piles in inner column. Such significant variation of bending
moment for the piles in the same row is normally neglected in practice. This phenomenon will be
further discussed in section 5.8 when we comparing the soil resistance profiles of the piles in
different columns. For the piles aligned in the same column, it is found that the piles in lead row
and trail row take an averagely 7% more bending moment than those in the 2nd and 3rd rows, with
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(a)Different columns (b)Different rows
Figure 5. 19: Bending moment profiles of piles in different location
Figure 5.20 presents the different bending moment distribution of the reference pile (pile 2)
obtained from all the five FE models described in Figure 4.1 at lateral load of 1870 kips. The
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1) The batter pile groups show very similar bending moment profile with the small spacing
model produces a slightly larger bending moment compared to the normal spacing and large
spacing models.
2) The batter pile groups have smaller (30%) bending moment than the vertical pile groups
regardless of the spacing, but these are still significantly higher (60%) than the bending
Researchers are very interested to know how the lateral load is distributed at each pile
among the group pile foundation. Figure 5.21 illustrated the lateral load distribution pattern of all
the piles in different rows and columns evaluated at pile head for various FE models. The lateral
load distributions among piles in different columns (Figure 5.21a) show that the side column
piles take the most lateral load followed by the middle column and then the inner column.
Similar finding was also reported by Ruesta and Townsend (1997) in their lateral load test at
Roosevelt Bridge. This variation exists in all pile group models and more significantly in small
spacing group and less obvious in large spacing group, which indicates that the edge effect is
For the piles in same column (Figure 5.21b), there is no surprise that the one located in lead
row is subjected the highest lateral load, followed by the 2nd and 3rd rows. For the large spacing
and normal spacing batter group models, the trail row takes the smallest portion of lateral load.
This lateral load distribution pattern is well known by researchers and also obeys our knowledge
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Lateral load at pile head
120
100
Lateral load (Kips)
80
60
40
20
0
Side column Middle column Inner column
Large spacing-GB 78.81 67.32 63.99
Small spacing-GB 94.70 72.17 66.23
Normal spacing-GB 84.81 68.56 64.15
Normal spacing-GV 93.29 81.56 78.51
100
Lateral load (Kips)
80
60
40
20
0
Trial row 3rd row 2nd row Lead row
Large spacing-GB 67.32 81.61 85.99 88.04
Small spacing-GB 72.17 71.18 75.58 96.30
Normal spacing-GB 68.56 77.79 82.19 90.26
Normal spacing-GV 81.56 69.88 72.95 101.82
spacing batter group and the vertical group model have slightly higher lateral load distributed on
the trail row than that of the 3rd row or even the 2nd row.
This phenomenon is very interesting and there are some contradict observations to what
reported in the literatures. Many researchers concluded that the lateral load distribution is rigidly
decreasing from lead row to trail row (Brown et al., 1987; Ruesta and Townsend, 1997).
However, according to the lateral load tests reported by other researchers (Rollins et al., 1998;
Rollins et al., 2003b), trail row sometimes takes higher lateral load than 3rd or 2nd rows. In this
study, we noticed that such effect occurred at small spacing batter group and vertical group
models, which are believed to have the highest group effects among all the four models (as
illustrated in Figure 5.16). It can be inferred that when piles comes closed to each other, the most
influenced piles are those located in between lead and trail rows because they are suffering the
shadowing effect of the rows behind them and meanwhile their supporting zone (the shadow they
created) are disturbed by rows in front of them. On the contrary, the lead row and trail row are
less influenced because the shadow created by the lead row remains integrity and the trial row is
free from the shadow effect from other piles. Therefore when group interaction effect increases
(due to closer spacing or changing of batter to vertical), the rows in between will suffer higher
reduction of lateral strength while the lead and trail row are less affected. Based on this
observation, a possible explanation of the phenomenon that the trail row takes higher lateral load
than the 3rd or even the 2nd rows in high-group-effect pile foundations could be due to the
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decreases of the lateral stiffness of the 2nd and 3rd row been faster than the lead and trail rows
during development of group effect, which finally result in a higher load distribution on the trail
row.
Batter pile group foundation is able to transfer part of the lateral load on pile cap to the axial
load of the piles, which will increase the lateral capacity of the foundation. In the full-scale test,
the changes in pile axial load can be derived from measurements of strain gauge data. Figures
5.22 compares the measured axial load increments for piles 1 and 7 with those obtained from the
FE model. Good agreement is observed for the measured data and FE predicted curves. As
expected, the negative batter pile (pile 1) is subjected to axial extension and the positive batter
pile (pile 7) is subjected to axial compression. Both FEM and full-scale test results show that
such increment or reduction in axial load is proportional to the increase of applied lateral load.
Figure 5.23 shows the axial load distribution pattern of all the piles in different rows for
various FE models. Notice that the axial load shown in this figure are purely caused by applied
lateral load at cap, which is a result of subtraction of the total axial load by the axial load after
geostatic balance. The axial load developed on vertical pile groups are in accordance with our
empirical judgments. The lead row is subjected to highest compressive axial load while the trail
row takes the maximum axial traction load. The 2nd row and 3rd row are subjected to a small
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(a) Pile 1
(b) Pile 7
Figure 5. 22: Increment of axial load from strain gauges and FEM
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A dramatic difference of axial load distribution patter among vertical pile group and batter
pile groups is observed. The major axial load is no longer taken by the lead and trail rows but
shifted onto the 2nd and 3rd row piles. One possible explanation is that the trapezoidal soil block
formed between of 2nd and 3rd rows strengthened its resistance to rotation and then more axial
load is shifted to the piles beside this block, as illustrated in Figure 5.24. One can find that for
both vertical and batter pile group, there is a zone between 2nd and 3rd row where soil is subjected
to rotational stress states. The resistance to such rotation comes from the soil-pile interface shear
force developed along sides of the block and the normal force by the soil bed underneath the
block. Obviously, the trapezoidal zone formed by the batter pile foundation is able to create
higher rotation resistance by providing higher value of both components than the narrow
rectangular zone of vertical pile foundation do, since the former has a broader side area and
wider base zone. It is also noticed that the axial load developed in lead and trail rows are
increased as the pile-pile spacing decreased, indicating that a stronger group effect will reduce
the “trapezoidal zone effect” and thus shift more axial load to lead and trail row piles.
Summary of total axial load developed on both models at middle column is presented in
Table 5.1. It shows that regardless of pile spacing, the 1:6 batter piles are able to transmit part of
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Axial load distribution
200
150
100
Axial load (Kips)
50
-50
-100
-150
-200
Trial row 3rd row 2nd row Lead row
Large spacing-GB 28.60 156.26 -156.46 -27.96
Small spacing-GB 103.53 146.57 -147.23 -106.13
Normal spacing-GB 55.31 159.60 -160.20 -55.08
Normal spacing-GV 137.35 19.86 -20.61 -137.45
Figure 5. 23: Axial load distribution pattern for piles in different rows from various FE models
Figure 5. 24: The trapezoidal zone under batter pile group foundation
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Table 5. 1: Total axial load developed at middle column in different FE models at 1870kips
lateral load
Normal Normal Small Large
spacing-GV spacing-GB spacing-GB spacing-GB
Total Axial Load
at middle column 315.27 430.19 503.46 369.28
(Kips)
Increased
Percentage in Reference 36.5% 59.7% 17.1%
Axial Component
Figure 5.25 presents the soil resistance profiles obtained from normal spaced batter pile
group model at different loads for all the piles. It can be seen that the first soft clay layer (0-15 ft
below mudline or 12-27 ft below pile cap) provides a relatively small and uniform lateral
resistance regardless of the depth at all levels of lateral load. Significant rise in the lateral
resistance profile occurs when entered the second stiff clay layer (15-25 ft below mudline or
27-37 ft below pile cap). A peak of positive lateral resistance is achieved at 27.5 ft and a
maximum negative lateral resistance occurs at 35.5 ft below pile cap for all the six piles under
any lateral loads. Both peaks are within the second stiff clay layer. It is interesting to notice that
the point 27.5 ft below the pile cap takes the maximum positive lateral resistance and the
maximum negative bending moment at the same time. Soil resistance reduced to zero at 33 ft
below cap. The third medium clay layer (25-38 ft below mudline or 37-50 ft below pile cap) are
basically offering some negative lateral resistance to the piles. Under this layer, the soils
resistance is negligible.
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Figure 5. 25: Soil resistance profiles at different lateral loads from FEM
Figures 5.26 present the soil resistance profiles for different piles under 1870 kips lateral
load, which are also compared with the FB-MultiPier profiles. The Figure 26a shows that the
side pile 1 encountered higher lateral resistance in the first soft clay layer than the piles located
in the middle and inner columns do. It has also a slightly higher maximum positive soil
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resistance at 27.5 ft depth than the middle and inner piles. The negative lateral resistance
provided by soils below 33 ft shows little variance for the three piles.
resistance in the first soft clay layer than the 2nd and 3rd row piles. The lead row has the highest
maximum positive soil resistance (21.1 kips) at 27.5 ft depth, followed by the 2nd row pile (19.5
kips) and the 3rd row pile (18.4 kips). The trail row takes a slightly higher maximum resistance
(18.8 kips) than the 3rd row but still less than the 2nd row and lead row. One major difference
between the Figure 26a and 26b is that the negative lateral resistance below 33 ft has significant
variation among different rows. The lead row takes the highest maximum negative resistance
(-7.3 kips) followed by the trail row (-6.0 kips) and then the 2nd and 3rd rows (-4.0 and -3.9 kips
respectively).
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According to both figures, the soil resistances near the ground surface and the maximum
positive lateral resistance predicted by FB-MultiPier are generally close to those generated by the
FEM model. However, it predicts a deeper zero soil resistance point (43 ft below cap) which
makes the fourth medium sand layer offer the major part of the negative lateral resistance.
The soil resistance profiles of the reference pile (pile 2) obtained from the different FE
models are presented in Figure 5.27. Some very interesting facts can be observed:
1) All the three different spaced batter pile groups has similar soil resistance distribution pattern.
Among them, the large spacing model encountered the highest soil resistance at all depth
regardless of soft or stiff clay soil layer, followed by the normal spacing model and then the
small spacing model. This phenomenon indicates: a) the group interaction effect will reduce
the lateral resistance offered by soils; b) such reduction of resistance nearly covers the soil at
all depth except those not influenced by the lateral load on pile cap; c) closely spaced pile
group suffer more reduction in lateral soil resistance due to group interaction effect than the
2) The comparison of the vertical pile group and the batter pile groups shows that the soil
resistance offered by the former is higher than that of the latter. The maximum positive
lateral resistance from the vertical group (26.4 kips) is 28.7% higher than that of the normal
3) It is also noticed that the vertical pile group exhibits extremely sharp fluctuation in soil
resistance, while the single isolated vertical pile model produces a relatively smooth
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distribution on the resistance. This phenomenon is very interesting since it shows that the
group interaction not only affects the soil resistance but also influences its distribution
pattern. For the easiness of discussion, we plot their soil resistance profile under different
resistance to the single isolated pile than to the vertical group piles. Such stiffer response by soft
layer directly reduced the responsibility of the second stiff clay layer and hence less lateral
resistance is occurred in this layer compared to vertical group piles. An explanation of this could
be that when isolated piles assembled into a group, their interaction (shadowing\edge effect)
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between each other significantly reduced the strength/stiffness of the first layer, hence a large
portion of lateral load have to be transmitted in to the second stiff clay layer, due to which a
sharp increase in lateral resistance at the soft-stiff joint surface (27 ft below pile cap) is occurred.
Figure 5. 28: Soil resistance profiles under different lateral loads from single pile and vertical
pile group model
As mentioned in chapter 2, the advantage of 3D finite element model is that it can consider
the effect of pile geometry, group pile spacing, soil-pile interface and inelastic behavior of soil in
a more physical sounded way. Such 3D FEM analysis can produce the p-y curves rather than
assuming them as input parameters as the case for the FB-MultiPier. The p-y curves at selected
depths can be deduced from the soil reaction p profiles obtained at different load levels and the
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Observation of the soil resistance profiles of various models shows that the first soft clay
layer and the second stiff clay layer provide the major part of positive lateral resistance. Hence, it
is desirable to extract the p-y curves for both of them. The points locate at 16 ft and 28 ft below
pile cap (or 4 ft and 16 ft below ground level) are selected to produce the p-y curves representing
the first soft clay layer and the second stiff clay layer.
Figures 5.29 present the p-y curves obtained from the normal spacing group batter pile
model for piles 2, 4, 5, and 6 at two different depths, 16ft and 28ft below pile cap, respectively.
For comparison, the p-y curves generated by single pile model and those extracted from the
FB-MultiPier for these two layers are also included in the figure.
For the p-y curves of piles in different rows of the batter group model, the one that belong
to the lead row shows the highest stiffness at both depth, and that of 2nd and 3rd rows indicates a
significant soften due to group effect. For both depths, the trail row pile produces higher stiffness
p-y curves than the 2nd and 3rd rows but less stiff compare to the lead row. It is noticed that the
p-y curves from the lead row and the trail row are very close for the point at 16 ft below pile cap
but separated for deeper depth (28ft below pile cap). Recall that for batter pile group, the pile
spacing goes larger as the position going deeper, indicating a larger group effect near ground
surface. Relating such fact to this case, it seems that larger group interaction effect will shift
more responsibility of lateral resistance to the boundary piles such as lead row and trail row piles.
Surprisingly, such conclusion was also obtained at section 5.6 when we comparing the lateral
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(a) 16 ft below pile cap
the absence of group effect. Comparing the discrepancy between the single pile and the group
pile p-y curves at 16 ft depth and 28 ft depth, it is again confirmed that the soils near ground
level will experience higher pile group interaction effect than those in deeper location.
The FB-MultiPier p-y curves generally give larger soil resistance than the FEM generated
p-y curves. At 16 ft, the resulted FB-MultiPier p-y curve suggests that the soil has entered a
plastic range, which is not the case for the FEM predicted p-y curves. Interestingly, the tangent
stiffness of single pile p-y curves are very close to the FB-MultiPier p-y curves at elastic stage.
The p-y curves of the reference pile (pile 2) that obtained from different FE models at both
depths are presented in Figure 5.30. It further supported some conclusions that we made earlier
in previous sections:
1) Large spacing batter group pile model has the least group interaction effect, while small
spacing one has the most group effect. Normal spacing vertical group model also has high
2) Soils near ground surface suffer higher group effect than those in deeper locations.
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(a) 16 ft below pile cap
The p-multiplier for reducing the single pile p-y curve to the group pile p-y curve is a
function of the location of the pile but is a constant over the entire depth of pile. According to the
definition of P-multiplier, it should be determined by directly comparing the p-y curve obtained
from single pile model and group pile model. However, such directly determined p-multiplier
can be varied at different depth (e.g. 16 ft and 28 ft below pile cap). In practice, the p-multipliers
are often determined by comparing the load-deflection curves of pile head of the group piles and
that of the single pile, so that they can represent an average reduction of the piles over their
entire depths.
The load-deflection curves for single pile and the piles 1 to 6 in different pile group model
is presented in Figure 5.31. It can be seen that the single pile has the highest stiffness of the
lateral responses, while due to group effect the piles in different locations in the batter pile group
are more or less subjected to a reduction in lateral stiffness. By applying proper p-multipliers, the
single pile load-deflection curve can be reduced to match those from batter pile group foundation.
The back-calculated p-multipliers for different pile locations are summarized in Table 5.2. In the
similar way, the calculated p-multipliers for the small-spacing GB model, large-spacing GB
model and the group vertical pile model are also obtained and summarized in Table 5.2.
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Figure 5. 31: Load-deflection curves for single pile and different pile group models
The comparison of the p-multipliers for the vertical pile group model and the batter pile
model shows a significant reduction, indicating a better performance of batter pile group
foundation than vertical pile group foundation when subjected to lateral loads. In all cases the
lead row has the least reduction in soil reaction and such reduction will be intensified for the
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trailing rows; while for those models suffer higher group effects, the trail row sometimes will
take more load than the 3rd rows. Side columns always have higher p-multipliers than the middle
Normal-spacing
4.3D 2.5D 0.78 0.74 0.67 0.59 0.73 0.55
GB
Small-Spacing
3.0D 1.5D 0.70 0.57 0.51 0.52 0.68 0.48
GB
Large-Spacing
6.0D 3.5D 0.84 0.82 0.78 0.64 0.75 0.61
GB
Normal-spacing
4.3D 2.5D 0.57 0.41 0.39 0.45 0.52 0.45
GV
the soil resistance of the pile over its depth. However, directly comparing the p-y curves at
different depths for the normal spacing batter pile group model (Figure 5.29) allows one to
evaluate the variation of p-multiplier for different soil layers. These directly obtained
p-multipliers are summarized in Table 5.3. As the author expected, the shallower soils for batter
pile group foundation will suffer much higher group interaction effect than deeper soils. Also
one can observe that under high group interaction effect, the 2nd and 3rd row piles will have
highest reduction in lateral reaction and shift more responsibilities to the boundary piles (the lead
124
Table 5. 3: Variation of p-multipliers at different depth for normal spacing GB model
Lead Row 2nd Row 3rd Row Trail Row Side Column Inner Column
Average value
Load-deflection curves)
125
CHAPTER 6
6.1 Summary
In this study, a series of finite element analyses are carried out based on the data obtained
from the full-scale lateral load test on the batter pile group foundation of M19 eastbound pier of
the new I-10 Twin Span Bridge over Lake Ponchartrain, Louisiana.
A finite element model with exactly the same pile geometry, inclination and pile to pile
spacing as of the M19 eastbound foundation is constructed using the finite element analysis
software ABAQUS. The subsurface exploration data (including five CPTs, a SPT and a soil
boring test) from the same site of full-scale lateral load test project is directly used to interpret
the subsurface condition of the M19 eastbound pier site. Soils are classified into eight layers
including two sandy layers and six clayey layers. Different soil constitutive models are assigned
to these layers, and the parameters associated with these models are correlated or estimated from
For the sake of investigating the effect of group interaction and pile inclination, four
additional FE models (including a large spacing / small spacing batter pile group model, a
normal spacing vertical pile group model and a single vertical pile model) are developed to
perform the same lateral load tests. A coupled pore fluid diffusion and stress analysis on a single
pile model is performed to estimate the influence of excessive pore water pressure on laterally
loaded piles. To compare the continuum-based FEM results with the widely used p-y curve
126
methods, the FB-MultiPier software is used to perform a p-y curve based analysis of the
All the results of FE and FB-MultiPier analysis are compared with the field measured data
including lateral deflection profiles, bending moment profiles and axial load evolution curves.
Besides, the lateral/axial load distribution, soil resistance profiles and the p-y curves obtained
from these analyses are compared with each other and some interesting conclusions were
obtained.
Another major contribution of this research is the use of a more advanced soil constitutive
model for clay in the FE analysis of laterally loaded piles. Many researchers (e.g. Brown and
Shie, 1990; Ahmadi and Ahmari, 2009) realized the importance of incorporating the anisotropic
strength behavior of soils into numerical analysis of laterally loaded pile problems. In this
research, the Anisotropic Modified Cam-clay Model (AMCCM) proposed by Dafalias (1987) is
implemented in UMAT which is a user interface that provided by ABAQUS to allow the use of
self-defined constitutive models in the FE analysis. Being aware that the UMAT is going to be
used in a large 3D FE model, the algorithm that adopted to implement AMCCM needs to have
scheme with modified Euler algorithm is selected to carry out such task since the combination of
substepping and modified Euler algorithm provides a mean to control the error in explicit
integrations. In the later FE analysis of the full-scale tests, the implementation of AMCCM
127
shows a wonderful computational stability and efficiency, which greatly accelerated the whole
research process.
6.2 Conclusions
Cam-clay model and the FE analysis of the laterally loaded batter pile groups are summarized
below:
1. The Anisotropic Modified Cam-clay model (Dafalias, 1987) is capable to capture both
“inherent” and “induced” anisotropic of the clay behavior in a simple but effective
formulation.
2. The explicit substepping scheme proposed by Sloan (1987) shows excellent stability,
models.
3. The profiles of lateral deflection and bending moment from partially drained and fully
drained analysis are very close to each other with a maximum difference of about 3% in
128
4. The effect of excessive pore pressure caused by lateral load on pile cap is really small
compared to the total geostatic pressure due to soil self-weight. Therefore, the presence of
Lateral deformations
5. The soils within 28 ft width and 29 ft depth were mobilized ranges from 0.20 in to 0.65 in
due to the applied lateral load. Such influence is neglectable for the soil beyond this region.
6. The lateral deflection profiles from the FEA show very good match with the field measured
data. The one by FB-MultiPier has good prediction in pile cap movement, however, the
“Stationary point” (the point with no lateral deformation) is much shallower than those given
by the field measurements and the FEM. Such discrepancies were attributed to the simple
way that the FB-MultiPier accounts for the group interaction effects.
7. The comparison between the five FE models that have various pile spacing and inclination
shows that: 1) Pile groups have much higher lateral displacement than that of a single pile; 2)
For the batter pile groups, small spacing model produces largest lateral deformation (0.94in)
while large spacing model shows relative small lateral deformation (0.71 in); 3) Vertical pile
group model exhibits significant large lateral deformation (1.22 in), which is 54% greater
than the batter pile group with the same spacing (0.79 in) and 130 % larger than the single
129
Contours of stress\strain distribution
8. The lateral load transmitted from the pile pushes the soil in front of the piles to a denser state
and the gaps left behind the piles are instantaneously filled by soils and thus result in a
9. The second stiff clay layer takes major portion of the deviatoric stresses caused by lateral
movement of piles, while the first soft clay layer accumulated highest deviatoric strains.
10. The comparison between the five FE models shows that: 1) There is a significant increase of
the deviatoric stress of the soil in front of the lead row for the vertical pile group comparing
to the batter pile group at same spacing. 2) Deviatoric stresses become more concentrated
and intensified when pile spacing reduces due to stronger shadowing and edge effect between
the piles.
11. Both the FEM and the FB-MultiPier predicted bending moments agree well with those
deduced from the strain gauges data. The maximum positive moment predicted by the FEM
are very close to those by FB-MultiPier; while the maximum negative bending moment from
FEM is 21%-24% lower than those predicted by the FB-MultiPier. Besides, the location of
maximum negative moment from the FEM occurs at slightly shallower depth (27.5 ft below
pile cap) than the location obtained from the FB-MultiPier analysis (31.5 ft below pile cap).
12. The bending moment profiles of the piles aligned in different columns (pile 1, 2 and 3) show
that the pile located at side column has 11.7% higher maximum positive bending moment (at
130
pile head) and 18.0% higher maximum negative bending moment (at 27.5 ft below pile cap)
than those of the pile in inner column. For the piles aligned in different rows, it is found that
piles in lead row and trail row have an averagely 7% more bending moment than those in 2nd
13. The comparison between the five FE models shows that: 1) Small spacing batter group model
produces a slightly larger bending moment compared to the normal spacing and large spacing
models; 2) All batter pile groups have smaller (30%) bending moments than the vertical pile
group regardless of the spacing; however, they are still significantly higher (60%) than the
14. For the piles in different columns, the side column piles take the most lateral load followed
by the middle column and then inner column. This variation exists in all pile group models
and more significantly in small spacing group and less obvious in large spacing group, which
indicates that the edge effect is intensified when spacing between columns reduced.
15. For the piles in different rows, the one located in the lead row has the highest lateral load,
followed by the 2nd and the 3rd rows. For the large spacing and normal spacing batter group
models, the trail row takes the smallest portion of lateral load; while for the small spacing
batter group and vertical group model the trail row takes slightly higher lateral load than 3rd
row or even 2nd row. The proposed explanation is that the increased group interaction effect
131
have more influence on softening the soil in between the rows and thus shift more lateral load
16. The results of axial load from FEA show good agreement with strain gauge data, showing
that the negative batter pile is subjected to axial extension and the positive batter pile is
17. For the vertical pile group model, more axial load is taken by the lead and trail rows; while
more axial load are taken by the 2nd and 3rd rows piles in the batter pile group model. This
phenomenon is attributed to the trapezoidal soil block formed in between of 2nd and 3rd rows,
18. For the piles in different columns, the side piles encountered higher lateral resistances than
the piles located in the middle and inner columns. For the piles in different rows, the piles
located in the lead and trail rows have higher soil resistance than the 2nd and 3rd row piles.
19. The soil resistances by FB-MultiPier are generally close to those generated by the FEM
model. However it predicts a deeper zero soil resistance point (43ft below cap).
20. The comparison between the five FE models shows that: 1) All the three different spaced
batter pile groups has similar soil resistance distribution pattern; 2) The soil resistance by the
vertical pile group is higher than that of the batter pile groups. The maximum positive lateral
resistance from the vertical group (26.4 kips) is 28.7% higher than that of the normal spacing
132
batter group (18.8 kips); 3) The vertical pile group exhibits extremely sharp fluctuation in
soil resistance, while the single isolated vertical pile model produces a relatively smooth
From present work, it was found that piles in side columns take innegligible larger bending
moments and lateral loads than those of the inner columns, which is also observed by Ruesta and
Townsend (1997) in their full-scale lateral load test. However, in most literatures on laterally
loaded pile groups (e.g. Brown et al., 1988; Rollins et al., 1998), such variations between
columns are normally ignored and the only piles in different rows are treated with different
p-multipliers. Therefore, it is suggested that researchers should pay more attention on lateral load
distribution of piles in different columns in the future numerical studies or field tests on laterally
In this study, it was also observed that for some group pile models that the lateral load taken
by lead row and trailing rows are descending, while in some other models show that the trail row
takes larger lateral load than the 3rd or even the 2nd rows. Interestingly, the lateral load
distribution patterns observed from many lateral load tests are also varied from each other. This
phenomenon is attributed to the group interaction effect, which should be examined by future
The concept of “Trapezoidal zone” for batter pile groups is firstly introduced in this study.
The author believes this zone is responsible for the increased allocated axial load on the 2nd and
133
the 3rd row piles as compared to the lead and trail row piles. However, such axial load
distribution pattern is fully obtained from the FE analysis without any support from field. It is
suggested that the axial load distribution pattern in batter pile group foundation to be further
investigated by conducting more field lateral load tests and/or numerical studies such as finite
element analyses on batter pile group foundations to study the effect of trapezoidal zone to the
134
REFERENCES
Abbas, J. M., Chik, Z. H., Taha, M. R. and Shafiqu, Q. S. M. (2010). “Time-dependent lateral
response of pile embedded in elasto-plastic soil”, Journal of Central South University of
Technology, Vol 17, Issue 2, pp 372-380.
Abu-Farsakh, M., Yu X., Pathak B., and Zhang, Z., 2011 “Field Testing and Analyses of a Batter
Pile Group Foundation under Lateral Loading,” in press, Journal of Transportation Board.
Ahmadi M., and Ahmari S., 2009, “Finite-element modeling of laterally loaded piles in clay,”
Proceedings of the Institution of Civil Engineers, Geotechnical Engineering, Issue 162, pp.
151-163.
Andresen, A., Berre, T., Kleven, A. and Lunne, T. (1979), “Procedures used to obtain soil
parameters for foundation engineering in the North sea”. Marine Geotechnology, Vol 3, pp
201-66.
Banerjee, P.K., and Yousif, N.B., 1986, "A plasticity model for the mechanical behavior of
anisotropically consolidated clay," International Journal for Numerical and Analytical
Methods in Geomechanics, Vol. 10, pp.521-541.
Banerjee, P.K., Stipho, A.S., and Yousif, N.B. (1981), "A simple analytical model of the bi-axial
stress strain behavior of anisotropically consolidated clays," In Implementation of Computer
Procedure and Stress Strain Laws in Geotechnical Engineering (Eds. Desai and Saxena),
Acorn Press, N.C., pp. 535-545.
Basu, D., and Salgado, R., 2007, “Elastic analysis of laterally loaded pile in multi-layered soil,”
Geomechanics and Geoengineering, Vol. 2, No. 3, 2007, pp. 183-196.
Borja, R.I. (1991), “Cam-clay plasticity. Part II: implicit integration of constitutive equations
based on a non-linear elastic stress predictor’’, Computer Methods in Applied Mechanics and
Engineering, Vol. 88, pp. 225-40.
Borja, R.I. and Lee, S.R. (1990), “Cam-clay plasticity. Part I: implicit integration of elasto-plastic
constitutive relations’’, Computer Methods in Applied Mechanics and Engineering, Vol. 78, pp.
49-72.
Bowles, J., 1996. Foundation analysis and design. 5th ed. McGraw Hill, New York.
135
Brown, D. A., Reese, L. C., and O’Neill, W. M. (1987) “Cyclic lateral loading of a large-scale pile
group.” Journal of Geotechnical and Geoenvironmental Engineering. Vol. 114(11), pp: 1261–
1276.
Brown, D., and Shie, C. (1990), “Three dimensional finite element model of laterally loaded piles,”
Computers and Geotechnics, Vol. 10, pp. 59-79.
Brown, D., Reese, L., O’Neill, W., 1987, “Cyclic lateral loading of a large-scale pile group,”
Journal of Geotechnical Engineering, Vol. 113, No. 11, pp. 1326-1343.
Brown, D.A., Morrison, C., and Reese, L. C (1998). “Lateral load behavior of pile group in sand.”
Journal of Geotechnical Engineering. Vol 114, No 11, pp: 1261-1276.
Cui, Y. J., Sultan, N., and Delage, P. (2000), “A thermomechanical model for saturated clays”,
Canadian Geotechnical Journal, Vol 37, pp 607–620.
Dafalias, Y.F., 1987, "An anisotropic critical state clay plasticity model," Constitutive Laws for
Engineering Materials: Theory and Applications, Elsevier Science Publishing Co. Inc., pp.
513-521.
Dafalias, Y.F., Hemnann, L.R. (1980). “Bounding surface formulation of soil plasticity”. In: Soil
Mechanics-Transient and Cyclic Loads. Wiley, New York, pp. 253–282.
Das, B. M. (1990), Principles of Foundation Engineering, 2nd ed. PWS-Kent, Boston, MA.
Devi, D., Singh, A.K. (2008), “On Finite Element Implementation for Cam Clay Model”, The 12th
International Conference of International Association for Computer Methods and Advances in
Geomechanics (IACMAG), Goa, India.
Ensoft, Inc., LPILE Plus 4 for Windows-A Program for the Analysis of Piles and Drilled Shafts
under Lateral Loads, May; 1999.
Fakharian, K., Ahmari, S., Amiri, A. (2008), “3-D Numerical Investigation of Piles under
Monotonic and Cyclic Lateral Loads in Clay”, Proceedings of the Eighteenth International
Offshore and Polar Engineering Conference, Vancouver, BC, Canada, July 6-11, 2008.
Feagin, L.B. (1937). “Lateral pile loading test,” Transactions, ASCE, vol 102, paper no 1959,
pp:236-254.
Florida Bridge Software Institute, 2005. FB-MultiPier user’s manual, University of Florida, 2005.
136
Gens, A. (2010) “Soil–environment interactions in geotechnical engineering”, Géotechnique, Vol
60, No. 1, pp 3–74.
Graham, J., Noonan, M.L., and Lew, K.V. (1983), "Yield states and stress-strain relationships in a
natural plastic clay," Canadian Geotechnical Journal, Vol. 20, pp. 502-516.
Graham, J., Tanaka, N., Crilly, T. and Alfaro M. (2001), “Modified Cam-Clay modeling of
temperature effects in clays”, Canadian Geotechnical Journal, Vol 38, pp 608–621.
Hashash Y.M.A., Whittle A.J., (1992), “Integration of the modified Cam-clay model in nonlinear
finite element analysis”. Computers and Geotechnics, Vol 14, pp 59–83.
Hibbitt, Karlson and Sorensen, 2002. ABAQUS Standard User’s Manuals, Version 6.3-1,
Pawtucket, RI, USA.
Huang, A., Hsueh, C., O’neill, M.W., Chern, S., and Chen, C (2001). “Effects of construction on
laterally loaded pile groups.” Journal of Geotechnical and Geo-environmental Engineering, vol
127, No 5, pp 385-397
Huh, J., Haldar, A., Kwak, K., Park, J. (2008), “Probabilistic Reliability Estimation of an Axially
Loaded Pile”, The 12th International Conference of International Association for Computer
Methods and Advances in Geomechanics (IACMAG), Goa, India, 1-6 October, 2008.
Karthigeyan, S., Ramakrishna, V., and Rajagopal K., 2006, “Influence of vertical load on the
lateral response of piles in sand,” Computer and Geotechnics, Vol. 33, pp. 121-131.
Kim, J. B., and Brungraber, R. J. (1976), “Full scale lateral load tests of pile groups”. Journal of
the Geotechnical Engineering Division, ASCE, Vol. 102, No. GT1, pp: 87-105.
Kulhawy, F., Mayne, P., 1990. Manual on estimating soil properties for foundation design. Report
EL-6800. Electric Power Research Institute, Palo Alto, 306 p.
Lobo-Guerrero, S., Vallejo, L.E. (2007), “Influence of pile shape and pile interaction on the
crushable behavior of granular materials around driven piles: DEM analyses”, Granular Matter,
Vol 9, pp 241–250.
Lu, S.S (1981). “Design load of bored pile laterally loaded.” Proceeding, 10th International
conference on Soil mechanics and Foundation Engineering., Balkema, Rotterdam. The
Netherlands, vol. 2, 767-770.
137
Masood, T., Mitchell, J. K. (1993), “Estimation of insitu lateral stresses in soils by cone
penetration test”, J. Geotech. Eng., Vol 119(10), pp 1624–1639.
Matlock, H. (1970), “Correlations for Design of Laterally Loaded Piles in Soft Clay,” Proceedings,
Offshore Technology Conference, Houston, Texas, Paper No. 1204, pp. 307-594.
Matlock, H., and Reese, L., 1960, “Generalized solutions for laterally loaded piles,” Journal of the
Soil Mechanics and Foundations Division, Vol. 86, No. 5, pp. 63–91.
Matlock, H., Ingram, W., Kelley, A., and Bogard, D., 1980, “Field tests of the lateral load behavior
of pile groups in soft clay,” Proceedings, 12th Annual Offshore Technology Conference.
Houston, TX, Vol. 4, p. 163-174.
Matos Filho R, Mendonca A.V., Paiva J.B. (2005), “Static boundary element analysis of piles
submitted to horizontal and vertical loads”, Eng Anal Boundary Elem, Vol 29, pp 195–203.
Mcclleland, B., and Focht, J. A. (1958). “Soil modulus for laterally loaded piles.” Trans. ASCE.,
vol 123, pp:1049-1063.
McVay, M., Wasman, S., Bullock, P., 2005, Barge impact testing of St George island causeway
bridge geotechnical investigation. Report Number 4910 4554 016-12, Florida Department of
Transportation, Tallahassee, FL., 150 p.
Muqtadir, A., and Desai, C., 1986, “Three dimensional analysis of a pile-group foundation,”
International Journal for Numerical and Analysis Methods in Geomechanics, Vol. 10, 1986, pp.
41-58.
Ohta, H., Hata, S. (1971), “On the state surface of anisotropically consolidated clay,” Proc., Jpn.
Soc. Civil Eng., Vol. 196, pp. 117-124.
Potts, D.M. and Ganendra, D. (1992), “A comparison of solution strategies for non-linear finite
element analysis of geotechnical problems’’, Proceedings of the 3rd International Conference
on Computational Plasticity, Barcelona, pp. 803-14.
Potts, D.M. and Ganendra, D. (1994), “An evaluation of substepping and implicit stress point
algorithms’’, Computer Methods in Applied Mechanics and Engineering, Vol. 119, pp. 341-54.
Potts, D.M. and Zdravkovic, L. (1999). Finite element analysis in geotechnical engineering:
theory, Thomas, Telford, London.
Poulos, H. G, Davis, E. H, 1980, Pile foundation analysis and design. John Wiley, New York, 410
p.
138
Poulos, H.G. 1971, “Behaviour of laterally loaded piles: II-pile groups,” Journal of Soil
Mechanics and Foundation Engineering Division 97, 5, pp. 733-751.
Prakash, S., and Subramanyam, G. (1965). “Behavior of battered piles under lateral loads.”
Journal of Indian National Society of Soil Mechanics and Foundation Engineering, New Delhi,
vol 4, Pp 177-196.
Ranjan, G., Ramasamy, G., and Tyagi, R. P (1980). “Lateral response of battered piles and pile
bents in clay.” Indian Geotechnical Journal, New Delhi, vol 10,No 2, pp: 135-1742.
Rao, S. Ramakrishna, V., and Rao, M., 1998, “Influence of rigidity on laterally loaded pile groups
in marine clay” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 124, pp.
542-549.
Reese, L., Cox, W., and Koop, F. (1974), “Analyses of laterally loaded piles in sand,” Proceeding
of the 6th Annual Offshore Technology Conference, Dallas, TX, pp. 473–483.
Reese, L.C. and Welch, R.C. (1975). “Lateral Loadings of Deep Foundations in Stiff Clay.”
Journal of Geotechnical Engineering, Div, ASCE, vol 101, No 7, pp: 633-649.
Reese, L.C., Wang, S.T., Arrellaga, J.A., and Hendrix, J. (1997). LPILE Plus 3.0 for windows,
Ensoft, Inc., Austin, Texas.
Rollins, K.M., Johnson, S.R., Petersen, K.T., and Weaver T.J. (2003a), “Static and dynamic lateral
load behavior of pile groups based on full-scale testing,” 13th International Conference on
Offshore and Polar Drilling, International Society for Offshore and Polar Engineering, paper
2003-SAK-02, pp. 8.
Rollins, K.M., Olsen, R.J., Egbert, J.J., Olsen, K.G., Jensen, D.H., and Garrett, B.H. (2003b),
“Response, analysis and design of pile groups subjected to static and dynamic lateral loads,”
Utah Department of Transportation Research and Development Division, Report No.
UT-03.03.
Rollins, M. K., Lane, D. J., and Gerber, M. T. (2005). “Measured and computed lateral response of
a pile group in sand. Journal of Geotechnical and Geoenvironmental Engineering., Vol 121, pp:
103-114.
Rollins, M. K., Peterson, T. K., and Weaver, J. T. (1998). “Lateral load behavior of full scale pile
group in clay.” Journal of Geotechnical and Geoenvironmental Engineering. Vol. 124 (6), pp:
468-478.
139
Rollins, M., Peterson, T., and Weaver, J. (1998), “Lateral load behavior of full scale pile group in
clay,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 124, No. 6, pp.
468-478.
Roscoe, K. H. & Burland, J. B. (1968). “On the generalized stress strain behavior of ‘wet’ clay”. In
Engineering plasticity (Eds J. Heyman and F. A. Leckie), pp. 535–609. Cambridge: Cambridge
University Press.
Roscoe, K. H., Schofield, A. N. & Worth, C. P. (1958). “On the yielding of soils”, Géotechnique
Vol 8, No. 1, pp 22–52.
Roscoe, K.H., Schofield, A.N. (1963), “Mechanical behaviour of an idealized wet clay”.
Proceedings of the European Conference on Soil Mechanics and Foundation Engineering 1,
47–54.
Ruesta, P.F., and Townsend, F.C. (1997), “Evaluation of laterally loaded pile group at Roosevelt
bridge,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 123, No. 12,
pp. 1153-1161.
Schofield, A. N. & Wroth, C. P. (1968). Critical state soil mechanics. London: McGraw-Hill.
Simo, J.C. and Taylor, R.L. (1985), “Consistent tangent operators for rate-independent
elastoplasticity’’, Computer Methods in Applied Mechanics and Engineering, Vol. 48, pp.
101-18.
Sloan S.W., Abbo A.J., Sheng D.C. (2001), “Refined explicit integration of elastoplastic models
with automatic error control”. Engineering Computations, Vol. 18(1/2), pp. 121–154.
Sloan, S.W. (1987), ``Substepping schemes for the numerical integration of elastoplastic stress
strain relations’’, International Journal for Numerical Methods in Engineering, Vol. 24, pp.
893-911.
Sun, K. (1994), “Laterally loaded piles in elastic media,” Journal of Geotechnical Engineering,
Vol. 120, No. 8, pp. 1324-1344.
Terzaghi, K. & Frolich, O. K. (1936). Theorie der Setzung von Tonschichten. Leipzig/Vienna:
Deuticke.
Terzaghi, K. (1936). “The shearing resistance of saturated soils and the angle between the planes
of shear”. Proc. 1st Int. Conf. on Soil Mechanics, Cambridge, MA 1, 54–56.
Trochanis, A., Bielak, J., and Christiano, P. (1991), “Three-dimensional nonlinear study of piles.”
Journal of Geotechnical Engineering, Vol. 117, No. 3, pp. 429-447.
140
Tschebotarioff, G.P. (1952). “ The resistance to lateral loading of single piles and pile groups”.
ASCE special publication, No 154, ASTM, West Conshohocken, pp: 38-48.
Uchida, K., Kawabata, T., Nakase, H., Imai, M., Syoda, D., Ooishi, J. (2004), “Evaluation of Load
Bearing Mechanism for Pile with Multiple Stepped Two Diameters ” Proceedings of The
Fourteenth International Offshore and Polar Engineering Conference, Toulon, France, May
23-28.
Voyiadjis, G.Z., Abu-Farsakh, Y.M. (1997), "Coupled theory of mixtures for clayey soils,"
Computer and Geotechnics, Vol. 20, pp. 195-222.
Voyiadjis, G.Z., and Song, C.R. (2000), "Finite strain, anisotropic modified Cam clay model with
plastic spin. I: Theory," Journal of Engineering Mechanics, Vol. 126, pp. 1012-1019.
Voyiadjis, G.Z., and Song, C.R. (2000), "Finite strain, anisotropic modified Cam clay model with
plastic spin. II: Application to piezocone test," Journal of Engineering Mechanics, Vol. 126, pp.
1020-1026.
Walsh, J. M., (2005), Full-scale lateral load test of a 3 x 5 pile group in sand, Thesis, Master of
Science in Civil Engineering, Brigham Young University.
Wheeler, S.J. (1997), “A rotational hardening elasto-plastic model for clays”. In Proceedings of
the 14th International Conference on Soil Mechanics and Foundation Engineering, Hamburg,
A.A. Balkema, Rotterdam. Vol. 1, pp. 431–434.
Wood, D.M.(1990), Soil behavior and Critical State Soil Mechanics, Cambridge Univ. Press, New
York.
Yamakawa, Y., Hashiguchi, K., Ikeda, K. (2010), “Implicit stress-update algorithm for isotropic
Cam-clay model based on the subloading surface concept at finite strains”, International
Journal of Plasticity, Vol 26, pp 634–658.
Yang, Z., Jeremic, B. (2002), “Numerical analysis of pile behavior under lateral loads in layered
elastic plastic soils,” International Journal for Numerical and Analytical Methods in
Geomechanics, Vol. 26, No. 14, pp. 1385–1406.
Yu, H.S, Khong, C., Wang, J. (2007), “A unified plasticity model for cyclic behaviour of clay and
sand”, Mechanics Research Communications, Vol 34, pp 97–114.
Zhang, L., McVay, M.C., Han, J. H., Lai, P. W. (1999a) Centrifuge modeling of laterally loaded
single battered piles in sand. Canadian Geotechnical Journal., vol 36(6), pp 561-575.
141
Zhang, Z. and Tumay M, (1999), “Statistical to Fuzzy Approach toward CPT Soil Classification,”
Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 125, No. 3, 1999, pp.
179-186.
Zhang., L., McVAy, M.C., Han, J.H., Lai, P.W. (1999b), “Numerical Analysis of laterally loaded
3X3 to 7X3 pile groups in sands”. Journal of Geotechnical and Geoenvironmental Engineering,
ASCE, Vol 125(11), pp 936-946.
142
APPENDIX: PSEUDO ALGORITHM FOR IMPLEMENTING AMCCM
2) Set
3) Calculate
̅ ̅( )
̅ ̅( )
( + )
( + )
4) If , the stress path is within elastic regime. Update stress and void ratio:
𝑡
( + )
Else if & ≥ , the stress path is crossing yield locus. Perform steps 5 to 9
MAXITS times to determine 𝑡 and update stress and void ratio for the 𝑡 part.
Else, perform substepping algorithm for elastoplastic loading, goto step 11.
5) Calculate
( )
143
̅ ̅( )
̅ Δ
( + )
7) If then set , ;
Else, set , .
← +
←
( + )
10) Set Δ ← ( )Δ , pass the current Δ , and e into step 11 to perform substepping
integration.
̅ Δ
Δ ̅
+
Δ
+
Δ | | ( 𝑥 )
Where
144
̅ ̅( Δ )
( )
𝛥 𝑥{ 𝛥 ( 𝛥 )}
And
, ,
+ , + , +
(26).
13) Update the stresses and hardening parameters, the bar denotes that these quantities are not the
̃ + ( + )
̃ + ( + )
̃ u dated
+ ( + )
‖ ‖ ‖ ‖ | |
𝑅 𝑥{ }
‖̃ ‖ ‖̃ ‖ ̃ u dated
15) If 𝑅 then this substep has failed. Reduce the size of current time step using the
factor q
𝑥{ √ 𝑅 }
← m x{ }
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16) This substep has succeeded. Permanently update the stresses and hardening parameters.
𝑡
̃ 𝑡
𝑡 ( )
( + )
Yield Surface Correction Scheme for AMCCM
17) If ( 𝑡
) then perform step 18-21 MAXIT times. In
the following equations the subscribe u and c denotes uncorrected and corrected quantities
respectively.
18) Set
( 𝑡
),
+
+ | | ( 𝑥 )
+
+ ( )
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20) If | ( )| | ( 𝑡
)| then abandon previous correction
and calculate:
22) Convergence cannot be achieved after MAXIT iterations, exit with error message.
24) Estimate the size of the next step by using the following formula:
𝑖 { √ 𝑅 }
Update pseudo time, ensure the next step size larger than the minimum step size and less than
← 𝑖 { 𝑥{ } }
← +
25) Integration of stresses at this whole step is successful. Pass the updated stresses, hardening
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parameters and state variables back to ABAQUS. Set the Jacobian the same as the secant
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VITA
Yida Zhang was born in Taiyuan, Shanxi, China, on March 19th, 1987. He received his
Bachelor of Science in Civil Engineering from Zhejiang University, China, on July of 2010. He
went to the United States in pursuing higher level of education. He expects to receive the degree
of Master of Science in Civil Engineering in the area of geotechnical engineering from Louisiana
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