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Cite this: DOI:10.56748/ejse.234003
Received Date: 17 December 2022
Accepted Date: 05 June 2023
1443-9255
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Nonlinear finite element Analysis of laterally
loaded piles in Layered Soils
Haitham H. Saeed a* and Huda Saad Abed a
a Building and Construction Technology Engineering Department, Northern Technical University, Mosul, Iraq
Corresponding author: HaithamSaeed@ntu.edu.iq
Abstract
This work is based on Winkler’s theory to analyze laterally loaded single piles using the finite element method.
Soil nonlinearity is considered via nonlinear springs “p-y” curves. Exact element displacement shape functions
and stiffness matrix are used for the element in the case of linear Winkler’s modulus of subgrade reaction. In the
nonlinear stage, an averaging technique for the element secant Winkler modulus is used to calculate the shape
functions and stiffness matrix. An iterative technique is used to consider the nonlinearity of the soil. Few elements
are required to simulate the pile efficiently. Unlike other analytical methods, the current method can be used to
analyze piles with any load-transfer curves with arbitrary variation with depth.
Keywords
Laterally loaded piles, ‘‘p-y’’ Curves, Elastic foundation, Finite Element, Layered soils
1. Introduction
Piles are increasingly used to support axial and lateral loads in many
structures such as tall buildings, earth retaining structures, and bridge
substructures. The important aspects in the design of laterally loaded
piles are the amount of deflection and the moment in the pile [Poulos and
Davis 1980]. Laterally loaded piles can be analyzed by different methods
[Liao and Lin, 2003; Zhang et al., 2013; Gupta and Basu, 2019; Franklin
and Scott, 1979; Basu et al., 2009; Budhu and Davies, 1987; Sun, 2011].
Some methods treat the soil as an elastic continuum [Basu and Prezzi,
2009; Budhu and Davies, 1987; Sun, 2011]; which requires significant
modeling and computation effort. These methods are used primarily for
research and not for design purposes [Sun, 2011]. One of the most
suitable and popular methods is known as the Winkler foundation
method in which the supporting soil is represented by closely spaced
springs. For the nonlinear analysis, these springs are assumed to have a
nonlinear force-displacement relationship [Reese and Van, 2011]. These
relationships are sometimes called p-y curves. The p-y method is very
versatile and can be used to analyze and design a variety of cases
encountered in practice. The main advantage of the p-y analysis method
of laterally loaded piles is that it does not require discretizing the
supporting soil into elements. Therefore, the API standard and many
commercial software packages such as SAP, pileLAT, GEO5, LAP and
many others use this method to analyze and design laterally loaded piles.
Therefore, many researchers [Amirmojahedi et al., 2023; Wei et al.,
2023; Franke, and Rollins, 2013; Lianyang, 1992; Xiaoling et al, 2020]
focused on developing p-y curves for different types of soils. Psaroudakis
et al. in 2021 developed a semi-analytical method based on Winkler’s
theory to analyze laterally loaded single piles. Shape functions that
describe the elastic displacements of the pile under loading were used in
combination with the principle of virtual work in the analysis. Soil
nonlinearity was simulated via nonlinear ‘‘p-y’’ springs located along the
pile axis. Using the finite difference method, Yin et al. in 2018 proposed
a simplified iterative method based on p-y curves to analyze laterally
loaded single piles in or near a sloping ground. Basu et al. in 2011
developed an analytical method to analyze piles subjected to horizontal
loads and moments at the pile head. The soil supporting the pile was
considered a multi-layered and elastic continuum. The energy principles
were used in deriving the governing differential equations. An iterative
method based on the finite difference method was used to solve the
differential equations and determine the displacements and forces along
the pile. This study makes use of the p-y curves with arbitrary shapes to
efficiently analyze piles using the finite element method with very few
elements.
2. Problem Description and Modeling of
the Pile-soil System
The pile model under consideration has length L with flexural
rigidity EI. The pile material is considered linear elastic. The pile is
considered embedded in multi-layered soil of random mechanical
properties. Each soil layer may have a nonlinear reaction modulus (k)
and varying ultimate lateral resistance along with its depth. The reaction
modulus (k) is defined as the soil reaction per unit length of the pile due
to unit lateral displacement of the pile at that location. A force P0 and
moment M0 are applied at the pile head.
3. Element shape functions and stiffness
matrix
3.1 Homogeneous elastic soil
For the case of homogeneous elastic soil, the reaction modulus k,
sometimes referred to as the modulus of subgrade reaction is constant.
Using Winkler’s hypotheses, the governing differential equation for
bending of laterally loaded piles in homogeneous elastic soil is the same
as the equation of beam on elastic foundation [Reese and Van, 2011]:
(1)
Where = Lateral deflection of the pile
= Flexural rigidity of pile
= Soil reaction modulus
Element nodal displacements and their corresponding forces and nodal
degrees of freedom are shown in Fig. 1.
Fig.1 (a) Element nodal displacements; (b) Nodal forces; (c)
Nodal D.O.F
Putting λ =
(2)
Equation 1 can be written as
(3)
The general solution of this equation is [Reese 1977]:
(x) =
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Electronic Journal of Structural Engineering
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Electronic Journal of Structural Engineering, 2023, Vol 23, No. 3
(4)
Where A, B, C and D are arbitrary constants and can be determined
from the boundary conditions of the pile element.
The slope at any point within the pile element is the derivative of the
general solution of the differential equation (Eq.4). Therefore,
=
(5)
To derive the element shape functions, the following four boundary
conditions are set for the element:
1: , (6 a)
2: , (6 b)
Where and represent displacement and rotation respectively
at node 1 of the element. Similarly, and represent displacement
and rotation respectively at node 2 of the element.
Substituting these four boundary conditions into equations 4 and 5,
the arbitrary constants can be determined by solving the
resulting four simultaneous equations. And Eq. 4 can be written in terms
of nodal displacements as:
= + + + (7)
Where N1, N2, N3 and N4 are the element displacement shape
functions and are given by:
=
- +
+
+
(8a)
=
+
+
(8b)
=
- +
+
+
(8c)
=
+
+
(8d)
Where and
These shape functions depend on the value of λ and reduce to the
Lagrange interpolation polynomials when λ approaches zero. Figure 2
shows the shape functions for different λ values.
Fig. 2 Shape functions for beam on elastic foundation.
The element stiffness matrix is derived by applying the principle of
minimum potential energy and may be written as follows.
(9)
The terms in the stiffness matrix [K] corresponding to the degrees of
freedom shown in Fig. 1 are given by:
,
,
, =
,
=
, = ,
=
, = , = - ,
=
Where = .
When the value of λ approaches zero, the stiffness matrix reduces to
the ordinary beam element stiffness matrix.
3.2 Nonlinear layered soil
Assuming the supporting soil as a linear elastic material is an
oversimplification of reality. Soil behaves non-linearly in the range of
loading of practical interest [Reese, 1977; Davies and Budhu 1986].
Moreover, the hardness of soil profiles changes with depth due to soil
stratification. Therefore, the value of soil reaction () is not constant
along the element length. Therefore, the differential equation of the
laterally loaded pile element will be as follows:
(10)
Where represents the variation of the reaction modulus
with depth and displacements. The exact solution for Eq.10 is available
for special cases in which the reaction modulus is elastic and varies
linearly with depth. Psaroudakis et al. in 2021 proposed a solution for an
elastic and parabolic profile of reaction modulus. However, for nonlinear
analysis and randomly varying soil profiles, some approximation must
be done to solve the governing differential equation.
In this study, an average reaction modulus is calculated for each
element as follows:
kav=
(11)
In the nonlinear case, the soil side resistance R(x) can be written as
R(x)= (12)
-0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N1
ξ
λ =0
λ =3
λ =6
λ = 9
-0.1
0
0.1
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N2
ξ
λ = 0
λ = 3
λ = 6
λ =9
-0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N3
ξ
λ = 0
λ = 3
λ = 6
λ = 9
-0.2
-0.1
0
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N4
ξ
λ = 0
λ = 3
λ = 6
λ = 9
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Where is the secant soil reaction modulus which is presented
graphically in Figure 3.
Fig. 3 Secant reaction modulus
Therefore, the average secant reaction modulus can be written as:
(13)
In which represents the total lateral soil reaction within the
element divided by the total element lateral displacements. The absolute
value notation is to take account of displacements in the negative
direction. The value of ks depends on the p-y curve of the soil layers, and
explicit integration of the numerator may not be possible for many p-y
models. Therefore, numerical integration is used to evaluate integrals in
equation 13. Using Gaussian quadrature rule, Eq. 13 is evaluated as:
(14)
where are lateral displacement, secant soil reaction
modulus, and Gaussian weight at Gauss point on the element
respectively. The number of Gaussian points (n) used in this study is five.
Due to soil nonlinearity, an iterative procedure is used to determine
and the related lateral displacements. A flow chart of the solution
procedure is shown in Figure 4.
Fig. 4 Flow chart of the proposed solution algorithm
The iterative solution algorithm is shown graphically in Fig. 5.
Fig. 5 Schematic diagram of iterative secant stiffness method.
The proposed method can analyze piles in multilayered soils with
different p-y curves. A computer program was developed to implement
the proposed analysis method. Only a few elements are required for
good accuracy.
Validation of the of the proposed method
To check the feasibility of the proposed analysis method, two
examples were solved and compared with others well documented in the
literature.
Example 1
A case history analyzed by Reese [Reese 1977] is reanalyzed and the
results are compared with those of Reese. The geometrical properties
and loading of the pile are shown in Fig. 6. The load transfer “p-y “curves
of the soil are shown in Fig. 7 in which the soil properties vary with
depth. Linear variation is assumed between any two curves. The soil
properties are constant for a depth of more than 3.91 m. This example
was reanalyzed using three elements only with five Gauss points along
each element. The lateral deflection of the pile is calculated using
displacement shape functions of each element and is shown in Fig. 8. On
the same figure is shown the results of [Reese, 1977] using the finite
difference method obtained by dividing the pile into 72 segments. The
bending moment diagram of the pile is shown in Fig. 9. The results are
very close to those of Reese by using only three elements.
Fig.6 Properties and loading of pile.
Fig. 7 p-y curves of supporting soil
R (Lateral reaction)
1
(Lateral displacement)
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Electronic Journal of Structural Engineering, 2023, Vol 23, No. 3
Fig. 8 Load deflection curve of laterally loaded pile.
Fig. 9 Bending moment along the pile.
Example 2
Reese and William in 2011 analyzed a laterally loaded steel pipe pile
driven into stiff clay. The p-y curves for the clay layers are shown in Fig.
10. The experimental p-y curves in Fig. 10 were derived from data from
the tests at Manor- Texas [Reese, and Robert 1975]. The soil properties
were assumed constant below a depth of 3.05m below the ground
surface. Linear variation is assumed between any two curves. The pile
consists of two pieces with a total length of 15.24 m. The length of the
upper piece is 7.01 mm, and its outside diameter is 641 mm. The bottom
piece has a length of 8.23 m and an outer diameter of 610 mm. The
flexural rigidity (EI) of the two pieces is 493,700 and 168,400 kN.m2
respectively. The load was applied 0.305m above the ground surface.
Fig. 10 p-y curves of supporting soil of example 2 [Reese and
William 2011]
This example was reanalyzed using four elements with only five Gauss
points along each element. The load-deflection relationship of the pile
head is shown in Fig. 11. In the same Figure, the experimental results and
analytical solution of [Reese and William, 2011] are shown for
comparison.
Fig. 11 Load- Deflection curve of the pile head
The pile was instrumented with strain gauges along its shaft for
measurement of the bending moments. The maximum bending moment
of the pile using the present study in addition to the experimental and
analytical results of [Reese and William, 2011] are also shown in Figure
12. The agreement is excellent among the experimental, analytical
[Reese and William, 2011] and the present study for both the load-
deflection and the load-maximum moment on the pile.
Fig. 12 Relationship of maximum bending moment and lateral
load
4. Conclusion
A simple and efficient technique is presented for the nonlinear
analysis of laterally loaded piles in multilayered soils using p-y curves.
Averaging technique is used to calculate the soil reaction modulus for
each element. The corresponding exact shape functions are used to
calculate displacements, soil reaction and stiffness matrix for the
elements. An iterative method using the secant stiffness method is used
to track the soil nonlinearity. The present method does not necessitate
discretizing the pile into many elements. Only two to four elements are
sufficient for good accuracy.
Conflict of Interest:
The authors declare that they have no conflicts of interest.
References
Amirmojahedi M., Abu-Farsakh M., Voyiadjis G., and Souri A. (2023)
Development of p-y Curve Model for Sand Using Finite Element Analysis
of Laterally Loaded Piles. Canadian Geotechnical Journal.
https://doi.org/10.1139/cgj-2021-0014
Basu D., Salgado R., and Prezzi M. (2009). A continuum-based model
for analysis of laterally loaded piles in layered soils. Ge´otechnique 59,
No. 2, 127–140. https://doi.org/10.1680/geot.2007.00011
Basu, D., Salgado, R., and Prezzi, M. (2011). A new model for analysis
of laterally loaded piles”, Geo-Frontiers: Advances in Geotechnical
Engineering. doi.org/10.1061/41165(397)14
Basu D., Salgado R., and Prezzi M. (2021) A new model for analysis
of laterally loaded piles. Geo-Frontiers: Advances in Geotechnical
Engineering. doi.org/10.1061/41165(397)14
0
100
200
300
400
500
600
700
0 5 10 15 20 25
Lateral load, P (kN)
Ground line deflection (mm)
Experimental
Analysis (Reese and William, 2011)
Analysis ( present study)
0
100
200
300
400
500
600
700
0500 1000 1500
Lateral load (kN.m)
Maximum bending moment (kN.m)
Experimental
Analysis (Reese and William,
2011)
Analysis ( present study)
5
Electronic Journal of Structural Engineering, 2023, Vol 23, No. 3
Budhu M., and Davies T. G. (1987) Nonlinear analysis of laterally
loaded piles in cohesionless soils. Can. Geotech. J. 24, 289–296.
https://doi.org/10.1139/t87-034
Davies T. G. and Budhu M. (1986) Non-linear analysis of laterally
loaded piles in heavily overconsolidated clays. Geotechnique 36(4):527–
538. https://doi.org/10.1680/geot.1986.36.4.527
Franke K. W. and Rollins K. M. (2013) Simplified Hybrid p-y Spring
Model for Liquefied Soils. Journal of Geotechnical and Geoenvironmental
Engineering, ASCE, Vol. 139, No. 4, pp. 564-576.
https://doi.org/10.1061/(ASCE)GT.1943-5606.0000750
Franklin J. F., and Scott R. F. (1979) Beam equation with variable
foundation coefficient. J. Eng. Mech. ASCE 105(5):811–
827.https://doi.org/10.1061/JMCEA3.0002525
Gupta B. K. and Basu D. (2019) Nonlinear solutions for laterally
loaded piles. Can. Geotech. J., 57, 1566–1580.
https://doi.org/10.1139/cgj-2019-0341
Lianyang Z. and Zhuchang C. (1992) Method for calculating p-y curve
of cohesive soil. Ocean. Eng., 4, 50–58.
Liao J. C. and Lin S. S. (2003) An analytical model for deflection of
laterally loaded piles. J. Mar. Sci. Technol. 2003, 11, 149–154.
https://doi.org/10.51400/2709-6998.2274
Psaroudakis E. G., Mylonakis G. E. and Klimis N. S. (2021) Non-linear
Analysis of Laterally Loaded Piles Using ‘‘p-y’’ Curves. Geotech Geol Eng.
https://doi.org/10.1007/s10706-020-01575-0
Poulos H. G. and Davis E. H. (1980) Pile foundation analysis and
design. Willey, Hoboken.
Reese L. C. and Robert C. (1975) Lateral loading of deep foundations
in stiff clay. Journal of the Geotechnical Engineering Division, ASCE
101(GT7): 633–649. https://doi.org/10.1061/AJGEB6.0000177
Reese L. C. (1977) Laterally loaded piles: program documentation. J.
Geotech. Engng Div., ASCE 103, 287-305.
https://doi.org/10.1061/AJGEB6.0000401
Reese L. C. and William V. (2011) Single Piles and Pile Groups Under
Lateral Loading. 2nd Edition, Taylor & Francis Group, LLC.
Sun K. (1994) Laterally loaded piles in elastic media. J. Geotech.
Engng ASCE 120, No. 8, 1324–1344.
Wei L., Wang J., Zhai S. and He Q. (2023) Analysis of Internal Forces
and Deformation for a Single Pile in Layered Soil Based on the p-y Curve
Method. Appl. Sci., 13, 3520. https://doi.org/10.3390/app13063520
Xiaoling Z., Dongzhi Z., Chengshun X., and Xiuli D., (2020) Research
on p-y curve of pile-soil interaction in saturated sandy soil under the
condition of weakened strength. Rock Soil Mech., 41, 2252–2260.
Yin P., He W. and Yang Z. J. (2018) A Simplified Nonlinear Method for
a Laterally Loaded Pile in Sloping Ground. Advances in Civil Engineering
Volume 2018, Article ID 5438618, 9 pages.
https://doi.org/10.1155/2018/5438618
Zhang L., Zhao M. H. and Zou X. J. (2013) Elastic-plastic solutions for
laterally loaded piles in layered soils. J. Eng. Mech. 139, 1653–1657.
https://doi.org/10.1061/(ASCE)EM.1943-7889.0000580
