Simplified Approach to Estimate Lateral Load on Drilled
Shafts Resulting from a Heavily Loaded Adjacent Shallow
Foundation Using Horizontal Stress Isobars
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Pravin Jha, S.M.ASCE1; and Sanjeev Kumar, P.E., F.ASCE2
Abstract: Growing urbanization is leading to more structures being constructed close to each other, resulting in substantial interaction
between their foundations. Although the interaction between two shallow foundations and between piles in a pile group has been studied by
several researchers, the effects of interaction between shallow and deep foundations are not very well understood or studied. Not much
published literature is available for practicing engineers to analyze and design shallow and deep foundations that are constructed adjacent to
each other. The second author of this paper was recently involved in a project in which a structure designed to be supported on drilled shafts
was supposed to be constructed adjacent to another structure designed to be supported on a large mat foundation with high floor load.
Although the authors have performed a detailed parametric study using finite-element techniques to better understand stress distributions
and develop a simplified method to analyze this type of problem, this paper is only focused on estimating lateral stresses on the shafts and
their lateral load analysis. A stress bulb for lateral stresses under a uniformly loaded square foundation is proposed, which is a significant
tool for practicing engineers to understand lateral stress distribution below a uniformly loaded square area and estimate lateral stresses on
nearby deep foundations. DOI: 10.1061/(ASCE)GM.1943-5622.0000521. © 2015 American Society of Civil Engineers.
Author keywords: Drilled shafts; Mat foundation; Lateral load analysis; Foundation; Horizontal stress isobars; Stress bulbs.
Introduction
Construction of two or more foundations close to each other has
become a common practice, especially in urban areas. When two
foundations are constructed close to each other such that their
stress influence zones overlap, the design of each foundation must
consider the impact of stresses from one foundation on the other.
Interference effects between two or more shallow foundations
and pile–soil–pile interactions have been the subject of many
investigations in the last several decades. Stuart (1962) studied
the interference between two strip foundations resting on the
surface of sand and presented a theoretical solution for the ultimate
bearing capacity. Das and Larbi-Cherif (1983) conducted several
laboratory model tests on two closely spaced rough strip foundations on sand to determine the bearing capacity and results were
compared with the theoretical solution given by Stuart (1962).
A reasonable agreement was observed in the results and their
study concluded that the bearing capacity of closely spaced
foundations increases with the decrease in center-to-center spacing
of the foundations, whereas the settlement increases with the
decrease in the spacing. Graham et al. (1984) studied the bearing
capacity of three closely spaced footings on sand and concluded
that the close spacing permits higher load than that of a similar
1
Graduate Student, Dept. of Civil and Environmental Engineering,
Mail Code 6603, Southern Illinois Univ., Carbondale, 1230 Lincoln
Drive, Carbondale, IL 62901. E-mail: pravinjha@siu.edu
2
Chair, Professor and Distinguished Teacher, Dept. of Civil and
Environmental Engineering, Southern Illinois Univ. Carbondale, 1230
Lincoln Drive, Carbondale, IL 62901 (corresponding author). E-mail:
kumars@ce.siu.edu
Note. This manuscript was submitted on June 9, 2014; approved on
March 17, 2015; published online on June 17, 2015. Discussion period
open until November 17, 2015; separate discussions must be submitted for
individual papers. This paper is part of the International Journal of
Geomechanics, © ASCE, ISSN 1532-3641/04015032(8)/$25.00.
© ASCE
isolated footing. Their study showed that the bearing capacity may
increase by 150% when the friction angle of the sand is 35° and
the centerline spacing between the footings is twice the width of
the footing. Unlike the study done by Stuart (1962) where failure
mechanism of the two adjacent structures were taken as
independent of each other, Harrop-Williams and Grivas (1985)
presented a technical note that summarized a simple procedure to
determine the degree of dependence between the adjacent geotechnical structures for failure mechanism. Khing et al. (1992)
studied the bearing capacity efficiency of two closely spaced strip
foundations on geogrid-reinforced sand by conducting a series of
laboratory model tests. Al-Ashou et al. (1994) studied the effect
of the number of reinforcement layers in sand to see the effect
on the bearing capacity of closely spaced square and strip footings.
A similar study was done by Kumar and Saran (2003) on closely
spaced square and strip footings on sand. They all observed
a similar trend and concluded that the effect on the bearing
capacity of closely spaced square footings on reinforced sand is
insignificant but it has a pronounced effect in closely spaced strip
footings. However, Ghazavi and Lavasan (2008) showed that for the
interfering square footings, the bearing capacity increases with the
use of geogrid layers depending on the distance between the footings.
Lateral load analysis of deep foundations and pile–soil–pile
interaction have been a subject of wide interest and many studies
have been completed on this topic. Davison (1970) proposed a
technique for analyzing piles subjected to lateral loads and
moments to determine the deflections and stresses in a selected
pile–soil system. Reese et al. (1974) proposed the p–y method to
analyze piles subjected to lateral loads, where p = soil resistance
and y = pile deflection. Trochanis et al. (1991) examined the
nonlinear soil behavior on the axial and lateral response of piles
subjected to monotonic and cyclic loading. The study was based
on the FEM technique and aimed at developing a simplified
model for representing pile–soil–pile interaction effects. Duncan
et al. (1994) proposed a characteristic load method for analyzing
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laterally loaded single piles and drilled shafts, which they claim
to be simpler than the p–y method but closely approximates the
p–y analysis results. Prakash and Kumar (1996) used Khmax approach to predict the lateral-load deflection response of single piles
in sand.
Rollins et al. (1998) studied the lateral load behavior of a fullscale pile group in clay. Their study concluded that the pile group
deflected two times more than the single pile under the same
average lateral load, and group effects significantly reduced the
load-carrying capacity of the pile group compared with the single
pile with the maximum moment in the group being 50–100%
higher compared with a single pile. Ashour et al. (1998), Ashour
and Norris (2000), and Ashour et al. (2004) presented the results
of pile–soil–pile interaction for layered soil using a strain wedge
model. Rollins et al. (2005) studied the lateral load behavior of a
full-scale pile group in sand. Their study concluded that group
effects led to less resistance offered by piles in the group compared
with an individual pile subjected to the same lateral load. Basu and
Salgado (2008) developed a method to determine the lateral load
response of rectangular piles in a layered elastic medium. The
differential equation governing the displacement of the pile–soil
system was derived and the closed-form solution was presented to
determine pile deflection, slope of the deflected curve, the bending
moment, and the shear force profiles for the entire length of the
pile. Hussein et al. (2010) studied the effect of pile–soil separation
on the behavior of laterally loaded single piles and piles in a group
for static and dynamic loading using a field test and finite-element
(FE) analysis. The study concluded that for a single-pile analysis,
ignoring soil–pile separation underestimates both the deflection
and the maximum bending moment. Similarly, the effect of soil–
pile separation significantly affects the load-carrying capacity,
bending moment and the load distribution among the piles in a pile
group. Kim and Jeong (2013) studied the resistance of laterally
loaded piles using three-dimensional (3D) soil–pile interaction.
A 3D FE analysis was used to generate a framework for
determining p–y curves based on the surrounding soil stresses
that represent soil–pile interaction for laterally loaded piles in clay.
Zhang et al. (2013) presented semianalytical solutions for the
behavior of laterally loaded vertical piles embedded in multilayer
soil. Ashour and Helal (2014) studied the contribution of vertical
skin friction to the lateral resistance of large diameter shafts. The
study concluded that the vertical side resistance enhances the
lateral load performance of large diameter shafts, and it varies with
the shaft deflection. Basu et al. (2014) presented the results of onedimensional, quasi-axisymmetric FE analysis that models the
installation and loading of drilled-displacement (DD) piles in sand
and proposed a set of equations to estimate the unit shaft resistance
of DD piles in sand.
Recently, the second author of this article was involved in
a unique project in which a structure, designed to be supported on
drilled shafts, was supposed to be constructed adjacent to another
structure, designed to be supported on a heavily loaded mat
foundation. Fig. 1 shows a slightly simplified version of foundation plan for the project.
As shown in Fig. 1, a 15:24 × 15:24 m (50 × 50 ft) mat foundation, designed to support a pressure of 124.48 kPa (2,600 psf),
was planned to be constructed approximately 0.9 m (3 ft) away
from another structure that is designed to be supported on 1.22 m
(4 ft) diameter and approximately 30.48 m (100 ft) deep drilled
shafts bearing in soil. It is obvious that the drilled shafts within the
stress influence zone of the mat foundation will experience additional lateral stresses and down drag. Similarly, stress conditions
below the mat foundation are likely to be influenced by the additional stiffness provided by the adjacent drilled shafts.
© ASCE
Fig. 1. Foundation construction plan
A review of readily available literature by the authors did
not reveal much information on how to estimate additional
stresses resulting from interaction between these foundations. The
accurate determination of stresses on each shaft is complicated
because of several factors, including the interaction between
overlapping zone of stresses and the shadowing effects. Therefore, in 2012 the authors (Jha et al. 2012) used basic geotechnical
principles and readily available information in the published
literature to estimate the additional lateral stresses and down
drag on the drilled shafts, referred to as preliminary analysis in
this paper. However, after this preliminary analysis, authors performed a more detailed parametric study using FE techniques to
better understand stress distributions and develop a simplified
method to analyze this type of problem (see Jha 2015). This paper
is focused on estimating and analyzing drilled shafts for lateral
stresses only. The preliminary analysis procedure used to estimate
lateral stresses is also summarized herein. More detailed information about the preliminary analysis procedure is presented by
Jha et al. (2012).
Preliminary Analysis for Estimating Lateral Stresses
Boussinesq (1885) proposed a solution for calculating vertical
stresses at any point in the semi-infinite soil mass due to a point
load at the surface. The solution assumes that the soil is linear
elastic, homogeneous, isotropic, and semi-infinite. On the basis of
further analysis of the Boussinesq solution for the point load,
isobars for square loaded area, similar to the one shown in Fig. 2,
are readily available in the geotechnical engineering literature,
e.g., Bowles (1996). This pressure bulb is widely used in geotechnical engineering practice to assess vertical stresses at various
depths in the soil mass because of uniformly distributed load on a
square-loaded area at the surface. Fig. 2 also shows the location of
the drilled shafts relative to that of the mat foundation.
Using the isobars shown in Fig. 2, vertical stresses were estimated at various depths on the leading face (face toward the mat
foundation) of each drilled shaft. Fig. 3 shows vertical pressures at
various depths on Shafts S1, S2, and S3 obtained using the isobars.
The lateral pressure at the leading face of each shaft at various
depths was estimated by assuming the coefficient of lateral earth
pressure at rest (i.e., lateral-to-vertical pressure ratio) of 0.5. Lateral force per unit length of the shaft was then calculated
by multiplying the lateral pressure with the shaft diameter and
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lateral load-deflection analysis was performed using commercially
available software LPILE (Reese et al. 2004). Additional information about the lateral load analysis and results are available in
Jha et al. (2012).
The authors realized that the preliminary analysis procedure has
a lot of limitations and inherit assumptions. Therefore, more detailed analyses were carried out. Owing to the complexity of the
problem, it is not practical to obtain meaningful results using
closed-form solutions. To have a better understanding of stress
distributions on the drilled shafts because of uniformly distributed
load on the mat foundation, FE analyses were performed using
the research version of the commercial software ANSYS. Results
from ANSYS were compared with those obtained from readily
available closed-form equations and commercially available
software PLAXIS. Further analyses were performed to propose a
simplified method that practicing engineers can use to estimate the
additional lateral stresses on drilled shafts located within the
influence zone of a heavily loaded mat foundation.
Finite-Element Analysis using ANSYS
A 3D model of the foundation system was created and analyzed
using a research version of the commercially available software
ANSYS. ANSYS is an engineering simulation software based on the
FE technique that is capable of solving a wide variety of problems
including static/dynamic structural analysis (linear/nonlinear), heat
transfer and fluid problems, acoustic, and electromagnetic problems. In static structural analysis, ANSYS solves for the boundary
value problem using the three equilibrium equations involving
three displacement components. The physics behind the static
structural analysis is that the stress components in the equilibrium
equations are replaced by strain components, which in turn are
replaced by displacement components (Lee 2011). For this study, a
linear elastic model was chosen to perform the analyses. A detailed
parametric study and numerous analyses were performed as a part
of the entire study. However, analyses relevant to this paper are
discussed herein.
Soil and Concrete Properties
Fig. 2. Stress influence zone of the mat foundation along with the
location of drilled shafts
The soil and drilled shafts were both modeled as linear elastic
material. However, the shaft–soil interface was modeled as a
nonlinear frictional contact with a coefficient of friction of 0.35.
The modulus of elasticity and Poisson’s ratio of the soil were taken
as 4:788 × 104 kPa (1 × 106 psf) and 0.4, respectively. The modulus of elasticity and Poisson’s ratio of the concrete shaft were
taken as 2:485 × 107 kPa (5:191 × 108 psf) and 0.15, respectively.
Fig. 3. Vertical pressures on Shafts S1, S2, and S3
© ASCE
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0.00
To avoid the interference of boundaries, a boundary check was
performed by applying 124.48 kPa (2,600 psf) uniformly distributed load on a 15.24 m (50 ft) diameter foundation and pushing the
boundaries away from the model in steps so that the stresses under
the foundation and at various vertical cross sections away from the
foundation match those obtained from Boussinesq solution, which is
commonly available in geotechnical engineering literature. A convergence study was performed to select an optimal mesh size. Using
these preliminary analyses, a semi-infinite soil body of 183 × 183 m
(600 × 600 ft) in lateral direction and 91.5 m (300 ft) deep was
selected for the final analyses. Convergence study led to a mesh size
of 0.6 m (2 ft) for the soil and shaft body and 0.15 m (0.5 ft) at the
interface. The advantage of the symmetry was taken into the model,
and thus only half of the model was analyzed in ANSYS.
0.10
Stress Intensity
0.20
0.30
0.40
0.50
0.60
0.80
0.90
1.00
0.0
0.5
1.0
1.5
2.0
Distance from the center of the mat foundation in terms of B
Fig. 4. Vertical stresses at a depth of 0:25B below a circular loaded
area in terms of diameter of the loaded area, B
Cases Analyzed
The following analyses relevant to this paper were performed.
1. No shaft: mat foundation only with uniformly distributed load
of 124.48 kPa (2,600 psf) resting at the ground surface without
any shaft;
2. One shaft: one shaft (S1) adjacent to the mat foundation along
the centerline of the mat foundation;
3. Two shafts: two shafts (S1 and S2) in a row adjacent to the
mat foundation along the centerline of the mat foundation;
4. Three shafts: three shafts (S1, S2, and S3) in a row adjacent to
the mat foundation along the centerline of the mat foundation; and
5. Fifteen shafts: fifteen shafts in a grid arrangement as shown in
Fig. 1.
Analyses were also performed by introducing Shafts S1, S2,
and S3 individually at their respective locations.
0.00
0.05
0.10
0.15
0.20
Results from ANSYS and Verification of ANSYS
Results
ANSYS
PLAXIS
Alhvin & Ulery
0.25
0.30
0.35
0.0
0.5
1.0
1.5
2.0
Distance from the center of the mat foundation in terms of B
Fig. 5. Lateral stresses at a depth of 0:25B below a circular loaded
area in terms of diameter of the loaded area, B
Boussinesq (1885) presented a solution for calculating horizontal
and vertical stresses inside a semi-infinite mass due to a point load
acting on the surface/boundary of the semi-infinite mass. To make
sure that the results from ANSYS are appropriate, first a pressure of
124.48 kPa (2,600 psf) was applied on a 0.3048 m (1 ft) square area
(to simulate a point load). Stresses obtained from ANSYS at various
depths and distances from the center of the loaded area were
compared with those obtained from the closed-form solution given
by Boussinesq, which is readily available in the published literature
(e.g., Timoshenko and Goodier 1951; Das 2008). The stresses
obtained from ANSYS were observed to be very close to those
obtained from the closed-form solution. It was concluded that the
slight difference in the results was due to the fact that in ANSYS the
load was applied on 0.3048 m (1 ft) square area instead of a point.
Alhvin and Ulery (1962) integrated the Boussinesq equation
and presented a solution to calculate stresses, strains, and deflections at any point in the homogeneous half-space due to uniformly
distributed load on a circular area. The details of the solution are
readily available in the published literature (e.g., Das 2008). To
verify the results, a model with a 15.24 m (50 ft) diameter circular
foundation was created in ANSYS. Soil and concrete properties
were kept the same as stated earlier. Stresses under the circular
foundation were calculated using ANSYS and compared with those
obtained from the solution presented by Alhvin and Ulery (1962).
To further verify the ANSYS model, a similar circular footing with
© ASCE
ANSYS
PLAXIS
Alhvin & Ulery
0.70
Stress Intensity
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Boundary Check and Convergence Study
the same applied load and soil and concrete properties was modeled
in another commercially available software, PLAXIS 2D. PLAXIS
2D is also based on a FE technique, but is specifically designed to
model and analyze geotechnical engineering problems. Fig. 4
shows the vertical stresses from ANSYS, Alhvin and Ulery (1962),
and PLAXIS 2D on a horizontal cross section at a depth of 3.81 m
(12.5 ft), i.e., at a depth of 0.25B, where B is the diameter of the
circular loaded area. Fig. 5 shows lateral (horizontal) stresses from
ANSYS, Alhvin and Ulery (1962), and PLAXIS 2D on the same
cross section at the same depth. The figures are plotted in terms of
stress intensity, i.e., ratio of stress at a particular point and the
pressure on the circular loaded area.
Figs. 4 and 5 show that the stresses obtained from ANSYS are in
agreement with those obtained from Alhvin and Ulery (1962) and
PLAXIS 2D. Therefore, various models for the problem shown in
Fig. 1 were then created and analyzed using ANSYS.
FEM Analysis of the Problem Using ANSYS
To understand stresses around the drilled shafts, cross sections
were taken at several faces of the drilled shafts. However, results
presented in the paper refer to Faces A and B shown in Fig. 6.
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B
A
Fig. 6. Faces of shafts for identification of cross sections
60
50
Stress, kPa
30
20
10
40
were plotted. Fig. 9 shows the lateral stress at a depth of 3.81 m
(12.5 ft), i.e., 0:25B from the ground surface without shaft and
with Shaft S1. A similar response at a depth of 7.62 m (25 ft), i.e.,
0:5B from the ground surface, is shown in Fig. 10.
Figs. 9 and 10 show that the presence of the shaft changes
lateral stresses in the vicinity of Faces A and B of the shafts.
However, analysis of these figures and other similar figures for
multiple shafts did not reveal any simple trend that a practicing
engineer can use for analyzing such a problem. Therefore, motivated by the readily available and used stress bulbs (isobars) for
vertical stresses below a square foundation as shown in Fig. 2, the
authors decided to develop similar stress bulbs for lateral (horizontal) stresses. The current version of ANSYS does not have
capability of automatically producing such stress bulbs that could
be used for geotechnical engineering analysis. Therefore, from the
ANSYS analyses, numerous cross sections were taken at various
depths and distances to draw stress bulbs for lateral (horizontal)
stresses. Fig. 11 shows lateral stress bulbs below a square mat
foundation for a Poisson’s ratio (υ) of 0.4 and various stress intensities. Stress intensity is defined as the lateral stress normalized
by the applied vertical pressure on the square mat foundation. For
example, an intensity of 0.85 means that the lateral stress value is
85% of the applied vertical pressure on the square mat foundation.
As evident from this figure, the shape of the stress bulb is very
different from that of the vertical stress bulb shown in Fig. 2.
0
40
Stress, kPa
1 Shaft (S1)
2 Shafts (S1 and S2)
3 Shafts (S1, S2, and S3)
15 Shafts
20
30
20
10
5
30
35
Fig. 7. Lateral stresses on Face A of Shaft S1 for different cases
analyzed
20
20
10
0
0
10
5
Shaft S1-Face A
Shaft S2-Face A
Shaft S3-Face A
15
20
Depth, m
10
5
10
15
20
Distance from the center of the mat foundation, m
Face A
Face B
18
Stress, kPa
30
0
Fig. 9. Lateral stresses on horizontal cross section at a depth of 3.81 m
(0:25B)
Stress, kPa
40
No Shaft
Shaft S1
Edge of Mat
Foundation
25
15
25
50
Shaft (S1)
35
10
15
Face B
10
5
60
Face A
45
0
Depth, m
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Fig. 7 shows the lateral stresses on Shaft S1, Face A for various
cases analyzed, i.e., one shaft, two shafts in a row, three shafts in
a row, and fifteen shafts. In other words, this figure shows the
influence of the existence of other shafts on Face A of Shaft S1. The
results presented show that the introduction of second shaft behind
Shaft S1 increases the lateral stress at Face A of Shaft S1. However,
no further increase in the lateral stress was observed for the case of
three shafts in a row or fifteen shafts in a grid arrangement.
Fig. 8 shows the lateral stresses on Face A of Shafts S1, S2, and
S3. As expected, results presented in Fig. 8 show that the magnitude of peak lateral stress decreased and the depth of peak lateral
stress increased as the location of the shaft moved away from the
mat foundation. To understand how the lateral stresses are changing with distance from the mat foundation at various depths, the
lateral stresses at various depths, without shafts and with shafts,
16
No Shaft
Shaft S1
14
Edge of Mat
Foundation
12
Shaft (S1)
10
25
8
30
35
Fig. 8. Lateral stresses on Face A of Shafts S1, S2 and S3
© ASCE
0
5
10
15
20
Distance from the center of the mat foundation, m
Fig. 10. Lateral stresses on horizontal cross section at a depth of
7.62 m (0:5B)
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Distance from the center of the mat foundation in terms of B
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0
0.85
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.15
0.1
0.05
0.5
1
1.5
0
2
4
0.25B Depth
0.5B Depth
0.75B Depth
1B Depth
6
Edge of Mat
Foundation
8
10
Fig. 11. Lateral stress bulb for square mat foundation for υ = 0:4 in
terms of width of the mat foundation, B
12
Fig. 13. Ratio of lateral-to-vertical stress on horizontal cross sections
at various depths below the mat foundation in terms of width of the
mat foundation, B
Distance from the center of the mat foundation in terms of B
0
0.5
1
1.5
2
Distance from the center of the mat foundation in terms of B
0
0.5
1
1.5
2
0
0.7- v= 0.4
0.7- v=0.3
0.5
0.1- v= 0.4
0.1- v=0.3
0.05- v= 0.4
0.05- v=0.3
1
Depth in terms of B
Depth in terms of B
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Edge of Mat
Foundation
Ratio of lateral to vertical stress
Depth in terms of B
Distance from the center of the mat foundation in terms of B
0
0.5
1
1.5
2
0
Edge of mat
foundation
1
Face A
Edge of Mat
Foundation
1.5
Note: Dotted lines show the zone of
Face B
Shaft (S1) influence of shaft on the adjacent lateral
stresses
1.5
Fig. 12. Change in lateral stress bulb for square mat foundation for
υ = 0:4 and 0.3 in terms of width of the mat foundation, B
However, this stress bulb can be used similar to the vertical stress
bulb to estimate lateral stress anywhere below the foundation
owing to a uniformly distributed load on a square foundation.
Because the lateral stress generated inside the semi-infinite soil
body beccause of the uniformly distributed vertical load on the
boundary depends on the Poisson’s ratio of the soil, the isobars for
stress intensities of 0.7, 0.1, and 0.05 and the Poisson’s ratios of
0.4 and 0.3 were drawn on the same graph as shown in Fig. 12. It
can be seen that the stress bulbs shifts slightly inward and upward
with the decrease in the Poisson’s ratio.
A side note: Geotechnical practicing engineers often use a
concept that lateral stress at any location below a loaded foundation can be taken as half of the vertical stress, whereas half (0.5) is
the coefficient of lateral earth pressure at rest. To clarify this, ratios
of the lateral to vertical stresses obtained from ANSYS at various
depths and distances from the center of the mat foundation were
plotted as shown in Fig. 13. It is very clear from Fig. 13 that lateral
stresses below a loaded foundation cannot be taken as the product
of vertical stress and the coefficient of lateral earth pressure at rest.
The ratio of lateral to vertical stress increases along the horizontal
cross section beneath a loaded area and could be significantly more
than 1 at shallow depths.
In an effort to understand how the lateral stress bulb changes
owing to the presence of a shaft, the stress bulbs for various intensities were redrawn (as shown in Fig. 14) using the results obtained
© ASCE
0.5
0.85
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.15
0.1
0.05
Fig. 14. Lateral stress bulb for square mat foundation with Shaft S1 in
terms of width of the mat foundation, B
with Shaft S1. The dotted lines shown adjacent to the shaft indicate
the influence zone. Results shown in Fig. 14 do provide some information on how the presence of a shaft is influencing the lateral
stresses and lateral stress bulbs around Faces A and B of the shaft,
but has limited value for practicing engineers. Therefore, further
analyses were performed to understand how the lateral stresses
caused by the vertical load on the mat foundation and estimated
from the lateral stress bulb shown in Fig. 11 influence the lateral
load-deflection behavior of the shafts. A commercially available and
frequently used software LPILE was used for these analyses.
Lateral Load-Deflection Response of the Shafts
Lateral load-deflection responses of the shafts were developed
using LPILE Plus 5.0 computer software. LPILE is a special
purpose program that is commonly used for lateral load analysis of
a single pile or a shaft. The program is capable of computing the
deflection, bending moment, shear and soil reaction with respect to
the depth of pile or shaft considering the nonlinear behavior of soil
(Reese et al. 2004). The soil is modeled using p–y curves that can
be internally generated by the program using published literature
and recommendations for various types of soils, or it can be input
as user-defined site specific p–y curves. The program is capable
of analyzing a pile or a shaft for various pile-head conditions.
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400
400
350
350
300
300
Load, kN
450
250
200
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Only Design Load
Design Load + Load from Stress Bulb
Design Load + Load from ANSYS
Design Load + Lateral Load using k = 0.5
100
Design Load + Load from Stress Bulb
Design Load + Load from ANSYS
50
50
Design Load + Lateral Load using k=0.5
0
0
2
4
6
Deflection, mm
8
10
Fig. 15. Lateral load versus deflection response at ground level for
Shaft S1
The properties of soil and pile or shaft can be varied as a function
of the depth. For this study, each shaft was assumed to have a
design lateral load of 444.82 kN (100 kip) at the shaft head,
applied in increments of 44.482 kN (10 kip) to develop lateral
load-deflection responses. Lateral load analysis of the shafts was
performed with and without the additional lateral load caused by
the load on the adjacent mat foundation.
At each shaft location S1, S2, and S3, LPILE analysis was
performed for four different cases of lateral loading on the shafts.
It should be noted that the shafts were introduced at its respective
location one at a time in the third case of LPILE analysis as
explained as follows.
1. Design lateral load of 444.82 kN only, applied at the free-head
of the shaft (referred to as Only Design Load).
2. Design lateral load of 444.82 kN (100 kip), applied at the freehead of the shaft plus the lateral load along the vertical length of
the shaft generated from the lateral stress bulbs shown in
Fig. 11. Stresses obtained from the stress bulb were multiplied
by the projected width of the shaft to convert stress into lateral
force (referred to as Design Load + Load from Stress Bulb).
3. Design lateral load of 444.82 kN (100 kip), applied at the freehead of the shaft plus the lateral load along the vertical length
of the shafts obtained directly from ANSYS when Shafts S1,
S2, and S3 are introduced one at a time at their respective
locations (referred to as Design Load + Load from ANSYS).
4. Design lateral load of 444.82 kN (100 kip), applied at the
free-head of the shaft plus the lateral load along the vertical
length of the shafts calculated assuming that the lateral
load is equal to half of the vertical load, i.e., using a coefficient of lateral pressure of 0.5 (referred to as Design Load +
Lateral Load using k = 0:5).
Fig. 15 shows the lateral load-deflection responses of Shaft S1
for all of the preceding cases. Results presented in the figure
clearly show the effect of additional lateral load caused by the
adjacent mat foundation. It is interesting to note that the deflection
at the ground surface calculated using the additional lateral stresses
from the stress bulb shown in Fig. 11 (i.e., lateral stress bulb
for the mat foundation only) is higher than that calculated using
the lateral loads obtained directly from ANSYS when Shaft S1 is
introduced. Although the peak lateral load caused by the mat
foundation obtained from ANSYS when Shaft S1 is present is
higher than the lateral load obtained using lateral stress bulb when
no shaft is present, the depth at which the peak lateral load occurs
(from ANSYS) is lower than that obtained from lateral stress bulb.
© ASCE
200
Only Design Load
100
0
250
150
150
0
1
2
3
4
Deflection, mm
5
6
7
Fig. 16. Lateral load versus deflection response at ground level for
Shaft S2
450
400
350
300
Load, kN
Load, kN
450
250
200
Only Design Load
150
Design Load + Load from Stress Bulb
Design Load + Load from ANSYS
100
Design Load + Lateral Load using k = 0.5
50
0
0
1
2
3
4
Deflection, mm
5
6
7
Fig. 17. Lateral load versus deflection response at ground level for
Shaft S3
This is the reason that the lateral defection at the shaft head is
smaller, even though the peak lateral load is higher. Given the
results presented in Fig. 15, it is concluded that for all practical
purposes, lateral analysis can be performed using the lateral
stresses obtained from the lateral stress bulb shown in Fig. 11.
Figs. 16 and 17 show similar lateral load deflection results for
Shafts S2 and S3, respectively. All these figures show the influence of the larger shaft stiffness compared with that of soil. It is
clear from these figures that as the shafts move away from the mat
foundations, the influence of load on the mat foundation on the
shafts decreases. These figures also support the conclusion that the
lateral load-deflection response can be performed using the lateral
stresses obtained from the lateral stress bulb shown in Fig. 11 for
all practical purposes. Therefore, availability of the lateral stress
bulb is a significant tool for practicing engineers to estimate lateral
forces on deep foundations within the influence of shallow foundations and significantly simplifies the analysis.
Conclusions
A detailed study focusing on estimating lateral stresses on deep
foundations within the influence zone of a heavily loaded shallow
04015032-7
Int. J. Geomech., 2016, 16(1): 04015032
Int. J. Geomech.
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foundation is presented in this paper. A stress bulb for lateral
stresses under a uniformly loaded square foundation is proposed,
which is a significant tool for the practicing engineers to understand lateral stress distribution below a uniformly loaded square
area. The shape of the lateral stress bulb is very different from that
of the vertical stress bulb commonly available in the published
literature. However, the proposed lateral stress bulb can be used
similarly to the vertical stress bulb to estimate lateral stresses.
Results of the lateral load-deflection analysis presented show that
for practical purposes, the analysis can be performed using the
lateral stresses estimated from the lateral stress bulb, which eliminates the need to perform sophisticated FE analysis on these types
of problems.
Acknowledgments
The authors would like to thank ANSYS for providing them access
to the ANSYS Academic Research Mechanical Release 14.5 license. Thanks are also due to Dr. Om Agrawal and Dr. Vijay K.
Puri, professors at Southern Illinois University Carbondale, and
Dr. Shamsher Prakash, Professor Emeritus at Missouri S&T, for
providing valuable help during analysis.
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