Lateral Response of 2 × 2 Pile Group under Combined Axial and Lateral Loading
IGC 2009, Guntur, INDIA
LATERAL RESPONSE OF 2 × 2 PILE GROUP UNDER COMBINED AXIAL
AND LATERAL LOADING
S. Karthigeyan
Scientist, Central Building Research Institute, Roorkee–247 667, India.
E-mail: [email protected]
K. Rajagopal
Professor, Civil Engineering, Indian Institute of Technology Madras, Chennai–600 036, India.
E-mail: [email protected]
ABSTRACT: This paper presents some numerical results for investigating the effect of vertical load on the lateral capacity of
2 × 2 pile group in homogeneous sand. The lateral response of pile groups under combined loading has been analyzed by
using 3-D finite element based GEOFEM-3D program. In the analysis, the pile is treated as a linear elastic material and the
elasto-plastic stress-strain behaviour of soil has been idealized by using the Drucker-Prager constitutive model with nonassociated flow rule. The effects of some of the key parameters like method of loading and c/c spacing between the piles are
studied. Numerical results obtained from analyses have shown that the influence of combined loading is found to increase the
lateral capacity of pile group. The influence of combined loading is also found to increase the efficiency of pile group. The
critical c/c spacing between the piles to achieve the optimal performance of the group under combined loading has also been
brought out.
1. INTRODUCTION
Pile foundations are extensively used to support various
structures built on loose/soft soils, where shallow
foundations would undergo excessive settlements or shear
failure. These piles are used to support vertical loads, lateral
loads and combination of vertical and lateral loads. However,
in view of the complexity involved in analyzing the piles
under combined loading, the current practice is to analyze the
piles independently for vertical loads to determine their
bearing capacity and settlement and for the lateral load to
determine their flexural behavior. Procedures are available
(IS 2911 – Part I/ Sec. 1 to 4–1979) for analyzing the piles
separately under pure vertical loads and pure lateral loads.
This approach holds good as long as the magnitudes of the
lateral loads are nominal, say less than 5% of the vertical
loads. However, in situations where the lateral loads are
significantly high; the interaction effect due to the combined
vertical (axial) and lateral loads can become critical. As such,
no reliable or rational method is available for analyzing the
lateral response of piles under combined vertical and lateral
loading.
Piles are generally grouped together with two or more piles
in different configurations. The lateral load capacity of pile
group is dependent on the method of loading, c/c spacing
between the piles and the type of soil. Piles are grouped with
different configurations in such a manner that optimal
performance of the group can be achieved. The c/c spacing
between the piles to achieve the optimal performance of the
group is defined as critical spacing. Several investigators
(e.g. Trochanis et al. 1991, Narasimha Rao et al. 1998, Patra
& Pise, 2001) have studied the critical spacing to achieve
optimal performance of the pile group under pure lateral
loads. However, there are very limited investigations on the
pile groups subjected to combined vertical and lateral
loading. The performance of pile groups under combined
loading is mainly dependent on c/c spacing between the
piles.
In view of the above, the present paper focuses on the effect
of method of loading and c/c spacing on the lateral capacity
of 2  2 pile group under combined vertical and lateral
loading in sandy soils. The details of the numerical model,
the parameters studied and their influences on the lateral
capacity of piles are discussed.
2. FINITE ELEMENT BASED NUMERICAL MODEL
All finite element analyses in this investigation were
performed by using the 3-dimensional finite element
program GEOFEM3D. Figure 1 shows a schematic threedimensional finite element mesh discretization of 2  2 pile
group and the soil continuum. Based on symmetry, only half
of the pile group in the direction of lateral load is analyzed
(Figure 1, lateral load is applied along the x-axis). 20-node
isoparametric brick elements are used to discretise the pile
group and soil continuum. The interface between the pile and
soil has been modelled using 16-node joint elements of zero
thickness to represent possible slip and separation between
the pile and the soil.
665
Lateral Response of 2 × 2 Pile Group under Combined Axial and Lateral Loading
22 piles
S
Plan view
H
The response of the pile group under pure vertical and lateral
load was analysed initially. The ultimate vertical load (Vult)
capacity of a single pile was evaluated a priori by separate
numerical analyses Then the response of pile group under
combined loading was analyzed separately with the vertical
load in terms of 0 (pure lateral load case), 0.2Vult, 0.4Vult,
0.6Vult and 0.8Vult. The analysis in the lateral direction was
performed using displacement control (rather than load
control) so as to be able to evaluate the lateral loads developed
at various lateral displacement levels as a percentage of the
size of the pile.
Lateral loading direction
Pile
cap
Pile
Z
Z
X
X
Y
Y
3. RESULTS AND DISCUSSIONS
3.1 Influence of Method of Loading
Fig. 1: Typical 3-d Finite Element Mesh for Pile Group
2700
2400
Table 1: The Properties of Pile and Soil
Pile
Soil
square concrete pile B = 1.2 m
 : 36
M25 grade concrete
Es: 50 MPa
Ep: 25000 MPa
µs: 0.30
µp: 0.15
s: 20 kN/m3
3
Ko: 0.42
p: 24 kN/m
H
S
S
2100
Lateral load (kN)
All the nodes on the lateral boundaries are restrained from
moving in the normal direction to the surface representing
rigid, smooth lateral boundaries and all the nodes on the
bottom surface are restrained in all the three directions
representing rough, rigid bottom surface. The constitutive
behaviour of piles in the group is assumed to be linear
elastic. The soil is assumed to be elastic-plastic based on
Drucker-Prager constitutive model with non-associated flow
rule. The set of pile—soil properties considered in the
analyses are shown in Table. 1.
A series of finite element analyses have been carried out on
2  2 pile group under both Simultaneously Applied Vertical
and Lateral load (SAVL) and Vertical load Prior to Lateral
load (VPL). Figures 2a, b. shows typical lateral loaddeflection responses of pile group (response of only the
symmetric half-group is shown here) in sand at c/c spacing of
3B for both SAVL and VPL cases.
1800
1500
Vertical load (V)
1200
V = 0.8 Vult
900
V = 0.6 Vult
600
V = 0.4 Vult
300
V = 0.2 Vult
V=0
0
Note:
B : Pile width in ‘m’
Ep : Young’s modulus of pile in ‘MPa’
µp : Poisson’s ratio of pile
p : Unit weight of concrete in ‘kN/m3’
 : Angle of internal friction in degree
Es : Young’s modulus of soil in ‘MPa’
µs : Poisson’s ratio of soil
s : Unit weight of soil in ‘kN/m3’
Ko : Co-efficient of earth pressure at rest condition
0
50
100 150 200 250
Lateral deflection (mm)
300
350
(a) SAVL case
2700
2400
S
H
S
The analysis was performed in two stages. In the first stage,
vertical loads on the pile group were applied. Then in the
second stage, the vertical loads were kept constant and equal
lateral deformations were applied to the nodes on the pile top
surface. During this phase, the vertical deformations of the
pile head nodes were kept constant to prevent the rotation of
the pile cap. This process of analysis corresponds to a rigid
fixed pile cap that does not rotate during lateral translation.
The elaborate details of this numerical model, analysis
scheme using incremental finite element equations adopted
etc. have been in general as brought out in Karthigeyan et al.
(2006 & 2007).
666
Lateral load (kN)
2100
1800
1500
Vertical load (V)
1200
V = 0.8 Vult
900
V = 0.6 Vult
600
V = 0.4 Vult
300
V = 0.2 Vult
V=0
0
0
50
100 150 200 250
Lateral deflection (mm)
300
350
(b) VPL
Fig. 2: Lateral Loads–Deflection Curves of 2 × 2 Pile Group
in Sand with Respect to Method of Loading
Lateral Response of 2 × 2 Pile Group under Combined Axial and Lateral Loading
stresses along the length in the case of front piles as
compared to the back piles in the group.
3000
2500
Lateral load (kN)
In this figure, the dotted line represent the case of pure lateral
load and solid lines for the influence of combined vertical
and lateral loading. It can be seen that the influence of
combined loading under SAVL case is noticeable only at
higher lateral deflection levels. However, the influence of
combined loading is felt at all deflection levels in the VPL
case. It can be noted that the influence of combined loading is
higher at larger vertical load levels. Similar trends have also
been observed for pile groups with other c/c spacing between the
piles.
3.2 Influence of Spacing between the Piles (S)
2000
1500
Vertical load (V)
V = 0.8 Vult
1000
V = 0.6 Vult
V = 0.4 Vult
500
V = 0.2 Vult
V=0
0
0
50
100 150 200 250
Lateral deflection (mm)
300
350
Fig. 3: Lateral Load-Deflection Curves of 2  2 Pile Group at
a c/c Spacing between the Piles of 4B
3000
Lateral load (kN)
2500
2000
1500
Vertical load (V)
V = 0.8 Vult
1000
V = 0.6 Vult
V = 0.4 Vult
500
V = 0.2 Vult
V=0
0
0
50
100 150 200 250
Lateral deflection (mm)
300
350
Fig. 4: Lateral Load-Deflection Curves of 2  2 Pile Group at
a c/c Spacing between the Piles of 5B
667
15
12
PVC (%)
The influence of combined loading on the lateral behaviour
of 2  2 configuration of piles in sand have been studied at
various c/c spacing of 2B to 6B (where ‘B’ is the pile width).
To study this influence, all finite element analyses were
carried out under VPL case (Vertical load Prior to Lateral
load case). This type of loading is similar to that in field
situations like the pile jetties, transmission line towers,
overhead water tanks, etc. Here, the piles are first subjected
to vertical loading from the weight of the deck or super
structure, etc. The lateral loading may be due to wind, wave
loading, ship impact, etc. while the piles are subjected to
vertical loads. Figures 3 and 4 shows a typical lateral load
deflection curves of pile group at a c/c spacing between the
piles of 4B and 5B respectively. In the figure, the dotted line
represent the case of piles under pure lateral load and solid
line represents the case of the pile with the presence of
different vertical load levels. It can be seen that the lateral
load corresponding to specific lateral deflection level of pile
group increases with the increase in vertical load levels. Also
it is noted that the influence of vertical load on the lateral
capacity of pile group decreases with increase in c/c spacing
between the piles. In order to quantify this phenomenon at
individual pile level, the Percentage Variation in lateral
Capacity (PVC) on front and back piles within the group has
been assessed separately. The PVC values are estimated for
various c/c spacing between the piles at different deflection
levels of 0.01B, 0.025B, 0.05B and 0.1B. The PVC is
defined in terms of the Lateral Capacity of pile group With
Vertical load (LCWV) and the Lateral Capacity of pile group
under Pure Lateral loading (LCPL).
LCWV  LCPL
PVC 
 100 %
LCPL
Figures 5, 6 shows the typical PVC values of a group of piles
at a delfection level of 0.025B for front and back piles
respectively with respect to various c/c spacing between the
piles. It can be seen from these figures that the PVC values
increases with increase in vertical load levels for all c/c
spacing of piles in the group. It is further noted from the
figures that the influence of combined loading is almost
constant beyond a c/c spacing of 5B. It shows that the
influence combined loading on the performance of pile group
is significant up to a critical c/c spacing of 5B. The PVC
values are found to be higher for the front piles as compared
to back piles. Similar trends have also been observed for
piles in all other deflection levels. This is mainly attributed
due to the development of relatively higher lateral soil
9
c/c pile spacing (S)
6
S = 2B
S = 3B
S = 4B
3
S = 5B
S = 6B
0
0.2
0.4
0.6
0.8
1.0
Vertical load in terms of Vult
Fig. 5: Percentage Variation in Lateral Capacity (PVC) of
Front Piles in the Group
Lateral Response of 2 × 2 Pile Group under Combined Axial and Lateral Loading
10
4. CONCLUSIONS
In this paper an attempt has been made to examine the lateral
behaviour of 2  2 pile groups under combined vertical and
lateral loading. Based on the study, the following conclusions
can be drawn:
 The influence of combined loading is found to increase
the lateral capacity of pile group. However, the influence
depends on the sequence of loading and c/c spacing
between the piles.
PVC (%)
8
6
c/c spacing of piles
4
S = 2B
S = 3B
2
S = 4B
S = 5B
S = 6B
0
0.2
0.4
0.6
0.8
1.0
Vertical load in terms of Vult
Fig. 6: Percentage Variation in Lateral Capacity (PVC) of
Back Piles in the Group
3.3 Influence of Combined Loading on the Efficiency of
Pile Group
The influence of combined loading on the pile group
efficiency (g) was studied by calculating the group
efficiency factor as defined in equation 1.
Qg
(1)
g 
100 %
nQs
In which ‘Qg’ is the lateral capacity of pile group in kN, ‘Qs’
is the lateral capacity of single pile in kN and ‘n’ is the
number of piles in the group. The influence of combined
loading on the performance of pile has been assessed by
estimating the group efficiency for different c/c spacing of
piles at a deflection level of 0.1B. The computed group
efficiency under pure lateral load and combined vertical and
lateral loading at a deflection level of 0.1B against c/c
spacing between the piles in terms of pile width (B) is shown
in Figure 7. It can be seen that the influence of combined
loading is found to improve the efficiency of pile group as
compared to that of a pile group under pure lateral loading.
Also, it is found that the influence of combined loading on
the efficiency of pile group remains constant beyond a c/c
spacing of 5B.
110
Group efficiency (%)
100
90
c/c pile spacing (S)
80
S = 6B
S = 5B
S = 4B
70
 The influence of combined loading on the lateral capacity
of pile group decreases with increase in c/c spacing
between the piles and the influence is almost constant
beyond a c/c spacing of 5B.
 The efficiency of pile group increases with the increase in
the level of vertical loads. In general, the combined
loading influence is found to be more significant on the
front piles than on the back piles in the group.
 The results presented here are for 2  2 group of piles in
square configuration. This phenomenon needs to be
further verified by performing a series of similar analyses
for other pile group configurations.
REFERENCES
IS: 2911: Part I: sec. 1–4 (1979). “Code of Practice for
Design of Piles”, Bureau of Indian Standards.
Karthigeyan S, Ramakrishna VVGST and Rajagopal K.
(2006). “Influence of Vertical Load on the Lateral
Response of Piles in Sand”, Computers and Geotechnics,
l.33 (.2): 121–131.
Karthigeyan S, Ramakrishna VVGST and Rajagopal K.
(2007). “Numerical Investigation of the Effect of Vertical
Load on the Lateral Response of Piles”, Jl. of Geotech.
and Geoenvi. Engg., Proc. ASCE, 133(5): 512–521.
Narasimha Rao S, Ramakrishna VVGST and Babu Rao M.
(1998). “Influence of Rigidity on Laterally Loaded Pile
Groups in Marine Clay”, Jl. of Geotech. and Geoenvir.
Engg., ASCE, 124(6): 542–549.
Patra N.R. and Pise P.J. (2001). “Ultimate Lateral Resistance
of Pile Groups in Sand”, Jl. of Geotech. and Geoenvi.
Eng, Proc. ASCE, 127(6): 481–487.
Trochanis A.M., Bielak J. and Christiano P. (1991). “ThreeDimensional Nonlinear Study of Piles”, Jl. of Geotech.
Eng., Proc. ASCE, 117(3): 429–447.
S = 3B
S = 2B
60
0.0
0.2
0.4
0.6
0.8
Vertical load in terms of
1.0
1.2
Vult
Fig. 7: Pile Group Efficiency Vs. Vertical Load at a Deflection
Level of 0.1B.
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Lateral Response of 2 × 2 Pile Group under Combined Axial and Lateral Loading
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