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Effect of Pile Spacing on the Static Lateral Behavior of Vertical and Battered
Pile Groups

Ahmad Souri, Ph.D., M.ASCE1; Murad Y. Abu-Farsakh, Ph.D., P.E., M.ASCE2;

and George Z. Voyiadjis, Dist.M.ASCE3
1
Louisiana Transportation and Research Center, Louisiana State Univ., Baton Rouge, LA 70808.
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E-mail: asouri2@lsu.edu
2
Louisiana Transportation and Research Center, Louisiana State Univ., Baton Rouge, LA 70808.
E-mail: cefars@lsu.edu
3
Dept. of Civil and Environmental Engineering, Louisiana State Univ., Baton Rouge, LA 70808.
E-mail: cegzv1@lsu.edu

Abstract
The lateral resistance of pile groups is greatly affected by the pile spacing. The presence of
neighboring piles at close spacings results in a reduction in the lateral capacity of each pile in the
group due to the greater interaction between the piles. In this study, the effect of pile spacing on
the static lateral behavior of two pile group configurations (vertical and battered piles) is
investigated using finite element (FE) modeling. A total of six three-dimensional FE models
were developed in Abaqus version 6.12 for 4 × 4 vertical and battered pile groups at three pile
spacings (3D, 5D, 7D), where D is the pile width. The nonlinear pile group behavior was
accounted for using elastoplastic constitutive models and pile-soil interface model. The influence
of spacing on pile group efficiency, internal axial load in the piles, and p-multipliers is discussed.
The results showed that the battered pile group had larger group efficiency (70–90%) than the
vertical pile group (40–80%) at all pile spacings. The vertical pile group had a significant boost
in pile group efficiency (28%) when the spacing increased from 3D to 5D, while it was 14% in
the battered pile group. The internal axial load decreased only in two pile rows in both pile
groups with increased pile spacing. The p-multipliers increased with increased pile spacing in
both pile groups, in which the rate was highest in the middle and trailing rows.

Keywords: pile group, lateral behavior, battered piles, group efficiency, p-multipliers

INTRODUCTION
Pile group foundations are commonly used for supporting structures subjected to lateral loads
such as bridge piers and offshore platforms. The lateral capacity of single piles can be predicted
by several methods such as elastic solutions (Poulos and Davis 1980), the p-y curve method
(Matlock 1970; Reese et al. 1974), and the finite element (FE) method (e.g., Comodromos and
Pitilakis 2005; Karthigeyan et al. 2006, Mroueh and Shahrour 2009). The commonly used p-y
curve method is based on the beam on elastic foundation theory, in which the nonlinear soil
reaction is represented by the empirical p-y curves. The lateral capacity of pile groups is typically
less than the sum of individual piles capacities due to the pile-soil-pile interaction, which is
referred to as the group effect (Figure 1). The explanation for it is that the stress zones from
surrounding piles overlap and results in apparent weakening in the soil in front of the piles. The
group effect intensity is affected by pile spacing and soil type (e.g., McVay et al. 1995, Rollins et

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al. 1998, Chandrasekaran et al. 2010). The p-y curve method is used for predicting the lateral
capacity of pile groups and the group effect is accounted for using scalar factors called p-
multipliers.
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Figure 1. Sketch explaining the mechanism of pile-soil-pile interaction or the group effect
In this paper, the effect of pile spacing on the lateral behavior of two pile group configurations
(vertical and battered) is investigated. The FE method is adopted in this study for its cost
efficiency and reliability. The three-dimensional nature of the problem and multiple sources of
nonlinearity can be accounted for using the FE method. A number of three-dimensional FE
models for the pile groups was developed and analyzed using Abaqus v6.12 software package.
The influence of pile spacing on the pile group efficiency, axial reaction in the piles, and p-
multipliers is presented and discussed.
FINITE ELEMENT MODEL DESCRIPTION
Geometry
The effect of pile spacing in clay soil is studied for vertical and battered pile groups. Each pile
group (PG) comprised of eight square concrete piles in a 4x4 arrangement, in which each pile
measured 0.9 m in width and 33.5 m in length and the angle of batter in the battered PG was
1H:6V. The geometry of the PGs was selected after the dimensions of the M19 pier foundation
of the I-10 Twin Span Bridge over Lake Pontchartrain in Louisiana (Abu-Farsakh et al. 2011,
Souri et al. 2015). Due to the symmetry of the PGs about the axis of loading, the developed FE
models resembled half of the PGs geometry, which greatly reduced the solution time. Three
center-to-center pile spacings (S) (equal in both directions) were considered for the parametric
study: 3D, 5D, and 7D measured at the pile cap level and normalized by the pile width (D)
(Figure 2). The PG and soil body models were made of two separate FE meshes, in which the
piles were placed in pre-bored holes in the soil mesh. The interaction between the PG and soil
models was considered using the contact interaction feature at the interface. Both the PG and soil
meshes were built of the eight nodes solid continuum brick element (C3D8R) with the total
number of elements used was ~72000. The soil body was created from single clay soil material
(medium stiff clay) and was sized large enough to eliminate the influence of boundaries.

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Material constitutive models

Concrete
The nonlinear material behavior is incorporated using elastoplastic constitutive models for pile
concrete and soil materials. The nonlinear concrete material behavior was accounted for by using
the concrete damaged plasticity (CDP) model. The CDP model provides a distinct stress-strain
behavior in the tension and compression regions (Figure 3) and introduces tension stiffening and
stiffness degradation in the post-cracking region (Abaqus 2011).
- D
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D D D

D D

Figure 2. Geometry and dimensions of the FE models

In the CDP model, stiffness degradation is applied to the elastic modulus using the scalar damage
parameter ′d′. The value of ′d′ is defined as a function of plastic strain and increases nonlinearly
from zero to unity with plastic strains. For the current work, „d‟ was back-calculated using
existing analytical concrete models and data from the literature (Mander et al. 1988, Gupta and
Maestrini 1990, Cicekli et al. 2007). The evolution of concrete yield/cracking strength is
controlled by the yield surface function and a non-associated flow rule with Mohr-Coulomb
plastic potential function for the plastic strain rate. More details about the model can be found in
Abaqus documentation (Abaqus 2011). The elastic properties used for the concrete model were
Young‟s modulus GPa and Poisson‟s ratio .

Figure 3. Concrete stress-strain response in the CDP model, (a) compression, (b) tension

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Soil
The nonlinear clay soil behavior was simulated using the Anisotropic Modified Cam-clay
(AMCC) model (Dafalias et al. 2006). This model is based on the original of Modified Cam-clay
model (Roscoe and Burland 1968) with additional rotational hardening rule to account for the
anisotropic soil behavior. The elastic material stiffness in the model is dependent on the mean-
stress ( ) and defined by the bulk modulus ( ), where e is the void ratio and is
the slope of the reload line in the consolidation test. The plastic behavior is controlled by the
yield surface and the associative flow rule for the evolution of plastic strains. The anisotropic
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soil behavior is controlled by the variable „ ‟ which defines the rotation of the yield surface.
More details for the AMCC model formulation and features can be found in Dafalias et al.
(2006). The AMCC model was incorporated in Abaqus using the UMAT subroutine (Abaqus
2011). The model parameters used in the study are summarized, which represented the properties
of medium-stiff clay soil.

Table 1. Clay soil constitutive model parameters

Unit Shear
Poisson‟s
Soil Type weight strength eo M x C
ratio
(kN/m3) su (kPa)
Medium-stiff
19 36 0.25 0.8 0.01 0.11 0.8 0.03 1.33 4.0
clay
* eo is void ratio; is slope of the reload line in e-log p’ plot; is slope of the virgin consolidation line in e-log p’ plot; M is
slope of the critical state line; is the initial anisotropy ( ) and is the coefficient of lateral earth pressure; x
and C are model constants

Pile-soil interface model

The interface model provided the capabilities for the gap formation at the pile-soil interface and
the transfer of interface friction shear stress. The model allows the transfer of the normal bearing
stress when the pile push into the soil and prevents the transfer of tensile stress to the soil when
the pile moves away. Once the normal stress at the interface vanishes and the pile continue to
move away from the soil, the pile and soil nodes separate and the gap forms behind the pile. The
frictional stress is fully transferred at the interface and its magnitude is limited by the Coulomb
friction criteria ( ) and the user-defined maximum shear stress limit ( ), whichever
smaller. The reason for using is to constrain the interface shear stress limit for the cases
with very large normal stress ( ). The value of was assumed to be 0.5 and was chosen
equal to be , where is the undrained shear strength for the soil.

Loads and boundary conditions

Three types of loads were used in each FE model: gravity, weight of superstructure, and lateral
load. The gravity load provided geostatic stress equilibrium in the soil body. The weight of
superstructure was applied as uniform distributed load on top of the pile cap in the global vertical
direction. The lateral load was incrementally applied as uniformly distributed load on the side of
the pile cap in the global horizontal direction, and its magnitude was determined following the
level of pile cap displacement. For boundary conditions, the displacement was constrained on the
far sides and bottom boundaries of the soil mesh (i.e. pin), while on the symmetry plane the
displacement constraint was similar to a „roller‟ (Figure 2).

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RESULTS
The effect of pile spacing on the pile group efficiency, internal axial load, and p-multipliers is
discussed in the following sections. For the purpose of discussion, each pile group row is labeled
as in Fig The first row is referred to as the “leading row” (L) the second row as the “middle
leading row” (ML) the third row as the “middle trailing row” (MT) and the fourth row as the
“trailing row” (T) The results from the FE models were obtained for each pile in the group
separately, and then they were averaged and reported for each pile row in the following sections.
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Figure 4. Nomenclature used for the pile group rows

Effect of pile spacing on group efficiency
Pile group efficiency is a simple measure of pile group lateral capacity in reference to the sum of
individual piles capacities. The lateral capacity is defined as the magnitude of lateral load
causing a certain displacement at the pile cap (for pile groups) or pile top (for single piles). For
the individual pile capacity, an equivalent single pile FE model to the pile group was used, in
which the geometry (i.e. total and embedment lengths, pile width, angle of batter), material, head
fixity condition, and soil profile were similar to the pile group FE model. The group efficiency is
defined as

Where HPG is the lateral capacity of the pile group, Hsingle is the lateral capacity of individual
pile, and n is the number of piles in the group.
The influence of lateral displacement on the group efficiency was investigated. The lateral
capacity of the pile groups at different pile cap displacements (0.5, 1.0, 1.5, and 2.0 inches) is
presented in Figure 5Fig. The figure also shows the sum of individual pile capacity ( ) at
similar displacements for n = 16 piles (for 4x4 PG configuration). The pile group efficiency was
estimated using the previous equation and presented at different displacements in the bottom
plots in Fig. It can be seen that the pile group efficiency remained constant at different pile cap
displacements in both pile groups.
The effect of pile spacing on the group efficiency is presented in Figure 6. Increasing the pile
spacing resulted in higher efficiency for both PGs. This follows the fact that the group effect
become weaker at larger spacing, and the lateral capacity of each pile in the group is closer to the
individual pile capacity. The largest improvement in PG efficiency occurred when the pile
spacing increased from 3D to 5D, in which the vertical and battered PGs had 28% and 16% boost
in efficiency, respectively. When the spacing was increased from 5D to 7D, the improvement
was less than 10% in both PGs. The latter indicates that the influence of the group effect
becomes minimal at pile spacings greater than 5D (McVay et al. 1995). The battered PG had
notably higher group efficiency than the vertical PG at all pile spacings. The difference was 30%

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at 3D spacing and 14% at 5D and 7D spacings. The efficiency results suggest that switching the
piles from vertical to battered configuration at 3D spacing is a viable design alternative to
improve the lateral capacity.
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Figure 5. Effect of lateral displacement on the group efficiency

Figure 6. Effect of pile spacing on the group efficiency

Effect of pile spacing on the internal axial load

The internal axial load here refers to the reaction generated in the piles from the lateral load
along their major axis (Figure 7). The axial load was obtained at the pile cap elevation for four
pile cap displacements (0.5, 1.0, 1.5, and 2.0 inches) to investigate the influence of displacement
on the axial load. The average axial load per row vs displacement results are shown in Figure 8
at pile spacing 3D, in which the axial load was normalized by the group lateral load (HPG).

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Normalizing the axial load provides a better measure that is independent of the group lateral load
magnitude (HPG). It is noticeable in Figure 8 that the normalized axial load was relatively
constant with lateral displacement for each row in the PGs and therefore considered independent
of pile cap displacement. This observation also holds for other pile group spacings (5D and 7D),
but their results are not shown here for brevity.
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Figure 7. Definition of internal axial load in the pile groups

Figure 8. Effect of lateral displacement on the normalized axial load at pile spacing = 3D
The effect of pile spacing on the normalized axial load is depicted in Fig. From statics, the axial
load resists the rotation of the pile cap and is expected to decrease at larger pile spacing due to
larger moment arm. In the vertical PG, the significant percentage of axial load was found only in
the leading and trailing rows (L, T) with an average of 29% at 3D spacing. When the spacing
increased to 7D, the percentage in rows L and T dropped 17%. In the middle rows (ML, MT), the
axial load percentage was notably smaller at 5% at 3D spacing and dropped to 3% at 7D spacing.
In the battered PG, the axial load percentage was significant in all piles with an average of 20%
in rows L and T and 25% in rows ML and MT at 3D spacing. After the spacing was increased,
the axial load percentage decreased 12% in rows L and T only, while the average percentage
remained relatively constant in rows ML and MT.

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Figure 9. Effect of pile spacing on the normalized axial load

Effect of pile spacing on p-multipliers
The p-multiplier is a scalar factor used to account for the group effect in the beam-spring FE
modeling tools (e.g., FL-Pier, FB-Multipier, Ensoft GROUP), and assumes values less than
unity. This factor is applied to the p-y curves for single pile to obtain the p-y curves for the pile-
in-group, see Figure 10. The process of calculating p-multipliers from the FE generated p-y
curves can be laborious and time consuming especially when analyzing multiple PG models.
This is because it requires obtaining the p-y curves at several points over depth for the single pile
and each pile in the group. Then, the p-multiplier at each depth is determined, and then the
average value over depth is reported for each pile.

Figure 10. Illustration for the p-multiplier concept

Instead, a faster procedure is adopted in this study to obtain the p-multipliers. The new
procedure, illustrated in Figure 11, starts with obtaining the soil resistance profiles for the single
pile and the piles in the group at a presumed displacement ( ) for the pile top and pile cap.
Then, the soil resistance ratio (pg/ps) is estimated over depth.

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Figure 11. Procedure for obtaining the p-multipliers

The p-multiplier for the pile-in-group is estimated from the average of soil resistance ratios over
the influence depth (~15D). The pile top and pile cap displacement ( ) was assumed 0.1D,
which was chosen to ensure a fully mobilized soil resistance. The procedure has singularity
issue, as it can be seen in Figure 11, at depths 13-15D arising from the mathematical operation.
The authors used judgment to exclude those values from calculations.
The effect of pile spacing on the p-multipliers is shown in Figure 12. In literature, the value of
the p-multiplier depends on the pile‟s row location within the group and is assumed the equal for
all piles within a specific row (e.g., Rollins et al. 1998). Therefore, the results in Figure 12
represents the averaged p-multipliers per row. Similar to the group efficiency, the results show
that the p-multiplier increased at a greater rate when the spacing increased from 3D to 5D. In
both PGs, the increase in p-multipliers was 0.3 for rows ML, MT, and T, while in row L it was
smaller at 0.2. When the spacing increased from 5D to 7D, the increase in p-multipliers was
relatively less at 0.1-point. It can be noticed that row L had the largest p-multiplier at any
spacing and the lowest rate of increase. This due to the fact that row L is the leading row, in
which the group effect has less influence compared to the remaining rows ML, MT, and T.

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Figure 12. Effect of pile spacing on the p-multipliers

CONCLUSIONS
The effect of pile spacing on the lateral behavior of pile groups was studied using three-
dimensional finite elements modeling. The study investigated the vertical and battered pile group
configurations at three pile spacings (3D, 5D, 7D). The influence of pile spacing on pile group
efficiency, internal axial load, and p-multipliers was presented. The study findings are:

 The pile group efficiency increased with increased pile spacings in both pile groups. the
largest boost in group efficiency was observed in the vertical pile group (+28%) when the
pile spacing increased from 3D to 5D. The battered pile group had higher group
efficiency (70-90%) than the vertical pile group (40-80%) at all pile spacings. The latter
indicates that vertical pile groups lose a significant part of its lateral capacity due to the
influence of the group effect. Also, the results indicate that the pile spacing 5D can be
considered as the borderline for diminishing the influence of the group effect.

 The normalized axial load decreased with increased pile spacing only in two rows in each
pile group. In the vertical pile group, the normalized axial load percentage was significant
only in the leading and trailing rows (L and T), and the percentage dropped from 30% to
12% when the spacing increased from 3D to 7D. While in the battered pile group, the
significant axial load percentage was in all rows (20-25%), and it dropped only in rows L
and T.

 The back-calculated p-multipliers in the vertical pile group was lower than the battered
pile group. Therefore, it is recommended to adjust the p-multipliers according to the
group configuration when using the design software to evaluate the lateral capacity of
pile groups.

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