In general, pile foundation is used when heavy engineering buildings
structures have to be transfer superstructure loads to the deep strong
stratum. Most piles are generally subjected to lateral load as well as
vertical load. The design of pile under lateral loading will be dependent
on the satisfying a limiting lateral-deflection requirement that may result
in the specification of allowable lateral loads which much less than ultimate
lateral capacity of the pile. Accurate ground water level information
is needed for the estimation of soil densities, determination of effective
soil pressures and preparation of effective soil pressure diagrams. This
information is vital for performing foundation design.
Current method of analysis available to study single pile and pile groups depends
on the direction of loading, i.e., pure vertical or pure horizontal. For pure
lateral loading on piles, there are three methods of analyses which can be categorized
into (Poulos and Davis, 1980): (1) beam on elastic foundation
method-subgrade reaction; (2) elastic continuum method and (3) the finite element
method. Trochanis et al. (1991), Yang
and Jeremiae (2005), Johnson (2006) and Tahghighi
and Kongai (2007) used the finite element method to numerically study piles
response to pure lateral loads.
In most practical cases, the piles are subjected to both vertical and horizontal
loads. The influence of the vertical load intensity on lateral response of the
piles was studied by Karthigeyan et al. (2006,
2007) using a general three-dimensional finite element analysis for cohesion
and cohesionless soil. According this study, the response of pile in both soil
type under lateral loads in influenced by the response of vertical loads.
This study reports the result of preliminary three-dimensional finite
element analysis on laterally loaded pile embedded in cohesionless soil,
considering it considers load intensity and water table elevation variation
of the foundation soils in consideration of the lateral pile deformation,
lateral soil pressure, lateral shear stresses and induced stresses on
FINITE ELEMENT MODELS AND MESH GENERATION
Finite element analysis were performed using the software PLAXIS 3D FOUNDATION
version 1.1. In the finite element method a continuum is divided into
a number of (volume) elements. Each element consists of a number of nodes.
Each node has a number of degrees of freedom that correspond to discrete
values of the unknowns in the boundary value problem to be solved.
In order to perform the finite element calculations, the geometry has
to be divided into elements. A composition of finite elements is called
finite element mesh. The basic soil elements of a 3D finite element mesh
are represented by the 15-node wedge elements as shown in Fig.
1a. These elements are generated from the 6-node triangular elements.
||3D finite element mesh for soil mass and location of
pile, (a) 15-node wedge element and (b) 3DFE mesh
The 15-node wedge element is composed of 6-node triangles in horizontal
direction and 8-node quadrilaterals in vertical direction. According to
Karthigeyan (2006, 2007), the soil mass dimension depends on the pile
diameter and length. The width of soil mass is taken as 40B, in which,
B is the pile diameter or pile width. The soil mass affection the pile
response diminishes when the width is greater than 40B. The height of
soil mass is L+20B, in which, L is the length of pile as shown in Fig.
Linear-elastic model: Hookes law is the main constitutive law in the
theory of elasticity as shown in Fig. 2. It provides the relationship
between the stresses and strains at all points within the body and it is used
here for modeling the stress-strain relationship of the pile material. The model
involves two elastic stiffness parameters, namely Youngs modulus, E and Poissons
ratio, v. It is primarily used for modeling of stiff structural member for example
piles in the soil (Johnson et al., 2006).
|| Stress-strain curve
|| Mohr coulombs failure surface
Mohr-coulomb model: The Mohr-coulomb model is used in this study to
compute realistic bearing capacities and collapse loads of footings or foundation
soils. As well as other applications in which the failure behavior of the soil
plays a dominant role. This elasto-plastic model is based mainly on two soil
parameters that are known in most practical situations. These parameters are
the effective cohesion intercept, c and the friction angle, φ. In addition
three parameters namely Youngs modulus, E, Poissons ratio, v and the dilatancy
angle, ψ are needed to calculate the complete σ-ε behavior. Mohr
coulombs failure surface criteria are shown in Fig. 3. According
to Johnson (2006), the failure envelope only depend on
the principal stresses (σ1, σ3) and is independent
of the intermediate principle stress (σ2). When mapped in three-dimensional
stress space, Mohr-coulomb criteria resolved into an irregular hexagonal pyramid.
This pyramid forms the failure/yield envelope, which in turn governs how soil
will behave. The material behaves elastically if the stress point lies within
the failure envelope. However, if the stress reaches the yield surface the material
will undergo plastic deformation. In the Mohr-coulomb model used herein, it
is assumed that the soil has a linear elastic relation until failure.
The usual definition of the equation of Mohr-coulomb surface is (Potts
and Zdravkovic, 1999):
Which, when rewritten in terms of invariants and Lode angle θ becomes:
Interface element: The response of a soil-structure system subjected
to static and dynamic loadings can be influenced significantly by the
characteristics of the contacts between the structure and the soil. Often,
in the soil-soil and soil-structure interaction complete bonding in the
contact plane was assumed to perform the design analysis. Although that
assumption simplifies the analytical procedure, it underestimates the
actual response as no relative motions are included. Thus, for a realistic
simulation of these problems it may be appropriate to incorporate such
motions using special finite elements, like the interface elements.
Interfaces are modeled as 16-node interface elements. Interface elements
consist of eight pairs of nodes, compatible with the 8-noded quadrilateral
side of a soil element (Fig. 4). Along degenerated soil
elements, interface elements are composed of 6-node pairs, compatible
with the triangular side of the degenerated soil element. Each interface
has a virtual thickness assigned to it which is an imaginary dimension
used to obtain the stiffness properties of the interface. The virtual
thickness is defined as the virtual thickness factor times the average
element size. The average element size is determined by the global coarseness
setting for the 2D mesh generation. The default value of the virtual thickness
factor that is used in this study is 0.1. The stiffness matrix for quadrilateral
interface elements is obtained by means of Gaussian integration using
3x3 integration points. The position of these integration points (or stress
points) is chosen such that the numerical integration is exact for linear
stress distributions. The 8-node quadrilateral elements provide a second-order
interpolation of displacements. Quadrilateral elements have two local
coordinates (ξ and η).
Verification problem: Verification example is worked out to compare
results obtained by finite element analysis to those obtained from experimental
study. The case study deals with lateral load in which the deflection response
of bored piles in cemented sand were examined by field test on single pile under
lateral load (Ismael, 1998). All piles were 0.3 m in
diameter and had a length of 3 or 5 m. The site of this load test was in Kuwait.
The surface soil to depth of 3.5 m was characterized as having both components
of shear strength, both effective parameters. The soil profile consists of a
medium dense cemented silty sand layer to a depth 3 m. This is underlain by
medium dense to very dense silty sand with cemented lumps to the bottom of the
borehole. The geotechnical properties of the soil layers are shown in Table
1. Ground water was not encountered within the depth of the borehole.
The comparison between the finite element results and field test data
is shown in Fig. 5. Comparable data were obtained between
the experimental results.
||Local numbering and positioning of nodes (●) and
integration points (x) of an 8-node quadrilateral element
|| Geotechnical properties of the soil layers
The numerical simulation is reasonably accurate
for the problem of laterally loaded piles and pile-soil interaction over
a wide range of deformation for 3 and 5 m piles long. It can also be observed
that the 5 m pile length has a higher lateral resistance compared to the
3 m pile.
RESULTS AND DISCUSSION
The lateral soil resistance is one of the most important factors that
directly effect on the pile response under such loads. This performance
depends on the interaction between pile material and the surrounding soil.
This study includes: (i) effect of changing the water table elevation
possibly due to the weather and other environmental conditions on the
behavior of pure laterally loaded pile and (ii) effect of lateral load
intensity on the behavior of pile under pure lateral load. Two load intensities
were studied, i.e., 50 and 250 kN.
Parametric studies: The analysis consists of modeling of single
short pile using linear-elastic model with 15-node wedge elements. The
cross-section of the pile is circular with a diameter of 1.2 m. The pile
was loaded under pure lateral load in two stages. Circular pile was used
to simulate the behavior of piles under lateral loads. These piles were
embedded into a sandy soil layer. The tested pile, dimensions and soil
properties are summarized in Table 2.
||Failure modes of vertical piles (a) short pile and (b) long
pile under lateral loads (Poulos and Davis, 1980)
Four soil conditions were studied with respect to the ground water table.
First of all, dry soil condition was analyzed. Then the water table was
set at the base of the pile and the middle length of the pile. Finally
the whole soil was considered fully saturated.
Assessment of lateral deflection and lateral soil pressure: In the design
of pile subjected to lateral load, the ultimate lateral resistance of pile is
required to satisfied two criteria (Patra and Pise, 2001;
Poulos and Davis, 1980), (i) normal deflection at working
loads should be within the permissible limit ; and (ii) pile should be safe
against ultimate failure.
The upper part of pile is the most critical part in the case of laterally loaded
pile (Poulos and Davis, 1980) because of its greater deflection
and its ability to carry higher lateral loads than the lower part as shown in
Fig. 6. The short pile theory assumed that the point of rotation
is near the base of pile which means that soil failure will take place but no
fracture occurs as in the case of long pile as shown in Fig. 6a
and Fig. 6b, respectively. According to Zhang
et al. (2005) the overall soil resistance occurred in the soil beneath
the piled foundation is the summation of front soil pressure and side shearing
stress. The lateral soil resistance distributed uniformly between the two sides
of pile in opposite direction to the lateral load with greatest value occurred
at the middle on pile, while the maximum shear stress occurred in the pile side
as shown in Fig. 7.
Figure 8 shows the lateral deformation along the length
of pile with different loadings and water table elevation. It was found
that for both magnitudes of loadings (50 and 250 kN) the lateral deformation
of pile in the case of dry soil and W.T. in the middle of pile were almost
||Lateral pile deformation, (a) lateral load = 250 kN,
(b) lateral load = 50 kN
||Distribution of front soil pressure along pile length
(depth), (a) lateral load = 250 kN, (b) lateral load = 50 kN
Also when the water table elevation was at the base of the pile,
largest lateral deformation occurred. From Fig. 8 it
can be predicted that the lateral deformation shape is not straight line
especially in the upper part between e = 1 L and e = 0.6 L. This means
that the pile has point of fracture at this critical part of pile. In
the lower part of pile one can see the point of rotation at 0.2 L, which
means the negative lateral deformation occurred.
It is important to study the lateral soil pressure along pile length in order
to understand which part of the pile carry larger soil pressure that may cause
pile collapse. Figure 9 shows the distribution of the lateral
soil pressure with depth of pile under different water table elevation. In the
case of small lateral loading (50 kN) it was predicted that the pressures increased
with depth in the four soil conditions with a maximum value of 68.5 kN m-2
at the base under dry soil condition. In the case when the water table is at
the base of pile it was noticed that at depth 0.6 L the lateral soil pressure
started to decrease and reached the minimum value at about 0.4 L. However, when
the lateral load was increased to 250 kN, the soil performance changed. The
figure show that maximum soil pressure is concentrated at 0.8 L and thus this
point can be considered as the critical point in the design as well as the fracture
point. This is in line with those recommended by Poulos and
||Distribution of front soil pressure along pile length
(depth), (a) lateral load = 250 kN, (b) lateral load = 50 kN
The lateral soil shear resistance is shown in Fig. 10. The
value of lateral shear soil resistance is small compared with lateral soil pressure.
Therefore the lateral response of pile depends mainly on the front soil pressure
and less for side shear. The maximum value occurred at e = 0.8L and reduced
to very low near the base of pile for both loading intensity (50 and 250 kN).
Also the reduction in the side shear can be seen at e = 0.6L and could change
to negative at this depth when the lateral load of 250 kN was applied. This
means that at this depth the side shear resistance is neglected for calculation
of the total soil resistance.
Assessment of internal pile stresses: The pile can be improved
as a structural column under lateral load or at least as cantilever beams.
By considering the effect of surrounding soil on the behavior of this
structural member. Due to the load, the pile showed two zones of stresses;
the first is compression and it occurred always in the opposite side of
applied load and the second zone represent the tension part of the member
and is always on the same side of the applied load as shown in Fig.
11. The tension zone is very critical as the cracks normal occur in
Figure 12 shows the compression and tension stresses
that developed in the pile under pure lateral load. In the case of a low
loading value (50 kN) no different found in the values of compression
and tension stresses. For a high loading (250 kN) one can observe the
difference in values especially for fully saturated soil condition. This
means that the pile in this soil condition (saturated soil) is less resistant
than that in other soil conditions. The greatest value for both compression
and tension stresses occurred in the zone between e = 0.6 L and 0.8 L
in the case of low loading. However the maximum values in the case of
high loading occurred at e = 0.6 L. From the figure it can be seen that
the tension stresses change in sign between e = 0 and 0.2 L for low loading
as the pile rotate at this point. For higher pile loading the change in
sign appears very clearly at the pile base.
||Compression and tension zones of a laterally loaded
||Pile structural stresses, (a, b) Effect of low loading
value (H = 50 kN) on compression and tension and (c, d) Effect of
high loading value (H = 250 kN) on compression and tension
A three-dimensional finite element analysis have been used for the assessment
of the behavior of laterally loaded pile. From this investigation, the
following conclusions can be drawn:
||The water table elevation affected the behavior of laterally loaded
||The response of the pile in cohesionless soil under lateral load
is influenced by the magnitude of horizontal load
||The greatest magnitude of lateral soil pressure occurred at e =
0.8 (near to surface) and this section is more critical than other
parts of pile
||Lateral shear soil resistance is small comparing with lateral soil
pressure (between 10-13%) which means that the soil resistance depends
mainly on the front soil pressure
||Pile structural stresses is influenced by the magnitude of load
and water table elevation