# Numerical Analysis of Pile–Soil Interaction under Axial and Lateral Loads

- Yasser Khodair
^{1}and - Ahmed Abdel-Mohti
^{2}Email author

**8**:75

https://doi.org/10.1007/s40069-014-0075-2

© The Author(s) 2014

**Received: **29 July 2013

**Accepted: **23 April 2014

**Published: **12 July 2014

## Abstract

In this paper, the analysis of a numerical study of pile–soil interaction subjected to axial and lateral loads is presented. An analysis of the composite pile–soil system was performed using the finite difference (FD) software LPILE. Two three dimensional, finite element (FE) models of pile–soil interaction have been developed using Abaqus/Cae and SAP2000 to study the effect of lateral loading on pile embedded in clay. A lateral displacement of 2 cm was applied to the top of the pile, which is embedded into the concrete pile cap, while maintaining a zero slope in a guided fixation. A comparison between the bending moments and lateral displacements along the depth of the pile obtained from the FD solutions and FE was performed. A parametric study was conducted to study the effect of crucial design parameters such as the soil’s modulus of elasticity, radius of the soil surrounding the pile in Abaqus/Cae, and the number of springs in SAP2000. A close correlation is found between the results obtained by the FE models and the FD solution. The results indicated that increasing the amount of clay surrounding the piles reduces the induced bending moments and lateral displacements in the piles and hence increases its capacity to resist lateral loading.

## Keywords

## 1 Introduction

*p*is the soil resistance per unit length of the pile,

*K*

_{ h }is the modulus of subgrade reaction, and

*y*is the lateral deflection of the pile.

*x*is length along pile, and

*E*

_{ p }

*I*

_{ p }is the flexural stiffness of pile. The solution for the differential equation are readily available and can be found in Hetenyi (1946). The subgrade reaction has been widely accepted in the analysis of soil-structure interaction problems (Reese and Matlock 1956; Broms 1964). However, a drawback of the method is its inability to account for the continuity of soil. Additionally, the linear representation of the subgrade reaction for the soil elements along the depth of the pile fails to account for the non-linear nature of the soil. The p-y approach is another method for handling pile–soil interaction. The only difference between the p-y method and the subgrade reaction method is that the former is based on defining a nonlinear relationship between the soil reaction and the lateral deflection at each point along the depth of the pile. Therefore, a p-y relationship is defined at each distinctive point along the depth of the pile. The solution to Eq. (2) can be obtained using the finite difference method and computers. Appropriate boundary conditions must be imposed at the pile head to insure that the equations of equilibrium and compatibility are satisfied at the interface between the pile and the superstructure. The concept of a p-y curve was first introduced by McCelland and Focht (1958). The development of a set of p-y curves can introduce a solution to the differential equation in Eq. (2), and provide a solution for the pile deflection, pile rotation, bending moment, shear, and soil reaction for any load capable of being sustained by the pile. Several methods to obtain p-y curves have been presented in the literature (Georgiadis and Butterfield 1982; O’Neill and Gazioglu 1984; Dunnavant and O’Neill 1989). These methods rely on the results of several empirical measurements. Some researchers such as Ruesta and Townsend (1997) and Gabr et al. (1994) have attempted to enhance p-y curve evaluation based on in situ tests such as cone penetration, pressuremeter and dilatometer. However, such attempts have focused on the soil part of soil pile interaction behaviors. Robertson et al. (1985) developed a method that used the results of a pushed in pressuremeter to evaluate p-y curves of a driven displacement pile. Attempts towards deriving p-y curves using three dimensional finite element model has been provided by Brown Dan and Shie (1990, 1991). A simple elastic–plastic material model is used for the soil to model undrained static loading in clay soils. p-y curves are developed from the bending stresses in the pile, where nodal stresses along the pile are used to obtain bending. The finite element method (FEM) is considered the most powerful tool in modeling soil-structure interaction. The FEM has several advantages over the other methods, some of which are the: (1) versatility of the method allows for modeling different pile and soil geometries, (2) capability of using different boundary and combined loading conditions, (3) discretization of the model into small entities allows for finding solutions at each element and node in the mesh, (4) feasibility for modeling different types of soil models and various material behavior for piles, and (5) ability to account for the continuity of the soil behavior. Several researchers have used the FEM to model pile–soil interaction. Desai and Appel (1976) presented a finite element procedure that can allow for nonlinear behavior of soils, nonlinear interaction effects, and simultaneous application of axial and lateral loads. The pile was modeled as a one-dimensional beam element and the interaction between the pile and the soil was simulated by a series of independent springs. The variations of the generalized displacements and internal forces were described by means of energy functionals incorporating the adjoint structure concept. Thompson (1977) developed a two-dimensional finite element model to produce p-y curves for laterally loaded piles. The soil was modeled as an elastic-hyperbolic material. Desai and Kuppusamy (1980) introduced a one dimensional finite element model, in which the soil was simulated as nonlinear springs and a beam column element for the pile. The Ramberg–Osgood model was used to define the soil behavior. Faruque and Desai (1982) implemented both numerical and geometric non-linearities in their three-dimensional finite element model. The Drucker-Prager plasticity theory was adopted to model the non-linear behavior of the soil. The researchers declared that the effect of geometric non-linearity can be crucial in the analysis of pile–soil interaction. Kumar (1992) investigated the behavior of laterally loaded single piles and piles group using a three-dimensional non-linear finite element modeling. Greimann et al. (1986) conducted a three dimensional finite element analysis to study pile stresses and pile–soil interaction in integral abutment bridges. The model accounted for both geometric and material nonlinearities. Nonlinear springs were used to represent the soil, and a modified Ramberg–Osgood cyclic model was used to obtain the tangential stiffness of the nonlinear spring elements. Kooijman (1989) presented a quasi three-dimensional finite element model. The rationale behind his model was that for laterally loaded piles, the effect of the vertical displacements was assumed to be negligible. Therefore, it was plausible to divide the soil into a number of interacting horizontal layers. For these layers an elastoplastic finite element discretization was used. The contact algorithm in this model was based on defining an interface element, which characterized the tangential and normal behavior of pile and soil contact. This simulated slip, debonding, and rebonding of the pile and the soil. Bijnagte et al. (1991) developed a three-dimensional finite element analysis of the soil-structure interaction. The model utilized an elastic-perfectly plastic theory implementing the Tresca and the Mohr–Coulomb failure criteria. That paper introduced recommendations for the design of piles and design values for thermal expansion coefficients. Arsoy et al. (1999) developed a plane strain finite element model with symmetry around the centerline of the bridge. The abutment was modeled using linear stress–strain criteria. The approach fill and the foundation soil were modeled using hyperbolic material properties. The loads applied on the model represent the loads reflected from the superstructure and the abutment. Ellis and Springman (2001) developed a plane strain FE model for the analysis of piled bridge abutments. The study used an equivalent sheet pile wall having the same flexural stiffness per unit width as the piles and soil that it replaced. Faraji et al. (2001) used a three dimensional FE model to study the effect of thermal loading on pile–soil-interaction. The authors relied on the p-y method to model the non-linear behavior of the soil. The soil pressure distribution on the abutment is typically nonlinear and varies with depth, amount, and mode of wall displacement. A small parametric study was conducted to study the effect of the level of soil compaction on the response of the composite pile–soil system. Rajashree and Sitharam (2001) developed a nonlinear finite element model of batter piles under lateral loading. In their model, the nonlinear soil behavior was modeled using a hyperbolic relation for static load condition and modified hyperbolic relation, including degradation and gap for cyclic load condition.

The research described in this paper presents a numerical investigation to study the composite pile–soil system. The objectives of this research are to: (1) analyze pile–soil interaction using the finite difference software LPILE 2012 and the finite element software Abaqus/Cae and SAP2000, (2) compare the bending moments and lateral displacements induced along the depth of the pile using the finite difference method and the finite element models, and (3) conduct a parametric study to determine the effect of relevant design parameters which include the soil modulus of elasticity, increasing the amount of clay surrounding the piles, and varying the number of soil springs on the pile induced bending moment and lateral displacements along its depth.

## 2 Bridge Description

## 3 Finite Difference Method Model

^{3}, cohesion of 47.85 kPa, and a strain factor ε

_{50}= 0.009 is considered as a realistic representation of the soil. The pile is oriented such that bending is about the weak axis.

## 4 Finite Element Models

Two 3D finite element models were developed of the pile–soil interaction using the finite element software Abaqus/Cae and SAP2000.

## 5 Loading

_{1}) and the lateral bending stress (S33) along the depth of the pile.

## 6 Comparison Between the FE Models and LPILE

## 7 Parametric Study

A parametric study was conducted to analyze the effect of crucial design parameters such as the variation in the magnitudes of modulus of elasticity, the amount of soil surrounding the pile, and the number of soil springs on the bending moment and lateral displacements induced along the depth of the pile.

### 7.1 Effect of Variation in Modulus of Elasticity

### 7.2 Effect of Variation in Amount of Soil Surrounding the Pile

### 7.3 Effect of Variation in Number of Soil Springs

### 7.4 Effect of Applying Axial Load

Kim and Jeong (2011) presented a study to investigate pile–soil interaction. They developed a series of 3D FE analyses. The analytical results and modeling methods that were used in this research were verified using results of field tests of large diameter laterally loaded piles in clay. The modulus of elasticity of soil ranges from 3 to 15 MPa. This range was covered in this research since the modulus of elasticity values used in this research ranges from 5 to 50 MPa. Lateral displacement and bending moment distribution versus pile depth was similar in trend to those determined in this research.

## 8 Summary and Conclusions

The analysis of pile–soil interaction under lateral loading has always been a concern. A comparative study to analyze pile–soil interaction under lateral loading was conducted. A 2D finite difference method model was developed using LPILE, 2012. The soil was assumed to be stiff clay without free water with a unit weight of 2001.2 kg/m^{3}. The pile is oriented such that bending is about the weak axis. Two 3D finite element models were developed using the finite element software Abaqus/Cae and SAP 2000. In the 3D finite element model developed using Abaqus/Cae, both the pile and the soil were modeled using solid continuum elements (C3D8R) to account for the continuity of the soil. An elastic-perfectly plastic model was adopted for the pile. A Mohr–Coulomb failure criterion was defined for the clay. The clay was assumed to vary from soft to hard without free water. The contact behavior between the piles and the soil was defined using tangential and normal algorithms in ABAQUS/Cae. A rigid body motion was defined at the top of the pile by tying the degrees of freedom of the elements embedded in the pile cap (30.5 cm from the top of the pile) to a reference point at the centroid of the pile’s cross-section. Three boundary conditions were defined into the model: (1) the bottom of the pile was fixed to model its embedment into rock below a depth of 20.12 m from the top of the pile, (2) the exterior surface of the soil was fixed to model its confinement at its boundaries, and (3) a displacement of 2 cm was applied at the top of the pile while maintaining a zero slope in what simulates a guided fixation due to the embedment of the top of the pile into the concrete pile-cap for a distance of 30.5 cm. In the 3D finite element model developed using SAP2000, the pile was modeled using a continuum 3-D frame element while the soil was modeled using a number of nonlinear soil springs at predefined depth locations. The nonlinear soil properties were obtained using the p-y curves generated in LPILE at the predefined depth locations and modeled using the Plastic (Wen) link element available in SAP2000. A rigid body motion was defined at the top of the pile by assigning the proper degrees of freedom to the elements embedded in the concrete pile cap to maintain a zero slope in what simulates a guided fixation due to the embedment of the top of the pile into the pile-cap for a distance of 30.5 cm. The bottom of the pile was fixed to model its embedment into rock below a depth of 20.12 m from the top of the pile. Also, a displacement of 2 cm was applied at the top of the pile.

A parametric study was conducted to examine the effect of crucial design parameters such as the variation in the magnitudes of modulus of elasticity, the amount of soil surrounding the pile, and the number of soil springs on the bending moment and lateral displacements due to an applied lateral displacement of 2 cm at the top of the pile. The magnitude of the modulus of elasticity was varied to reflect a variation in the stiffness of the clay from soft to hard. As the magnitude of the modulus of elasticity increases, the discrepancy between the magnitudes of the bending moment and lateral displacements induced along the depth of the pile predicted by Abaqus/Cae and those obtained from LPILE is gradually reduced to reach a minimum value when the modulus of elasticity of the soil was assumed to be 20–25 MPa which indicates medium to stiff clay.

The effect of the amount of clay surrounding the pile on the induced bending moment and lateral displacement along the depth of the pile was studied in Abaqus/Cae. The pile–soil interaction model was compared to FD solutions for a single pile embedded in clay under a displacement of 2 cm. This is a convergence study to (1) establish the mesh density and (2) eliminate the effect of boundary condition by selecting the appropriate diameter of the soil medium around the pile. The results from FE and FD analyses showed that the discrepancy in the magnitudes of the bending moment and lateral displacements from both analyses was reduced with the increase in the amount of clay surrounding the pile. This indicates that increasing the amount of clay surrounding the piles reduces the induced bending moments and lateral displacements in the piles and thus increases its capacity to resist lateral loading. Therefore, the radius of the soil cylinder surrounding the pile was varied from 0.5 to 4 m to determine the most suitable soil diameter for analysis.

The effect of varying the number of soil springs on the induced bending moment and lateral displacement along the depth of the pile was examined using SAP2000. The results from SAP2000 were compared to those from FD solution by LPILE due to the effect of an induced displacement of 2 cm at the top of the pile. The number of nonlinear soil springs was varied between 7, 9, and 12 springs. Using a larger number of nonlinear soil springs showed a better agreement between bending moment and lateral displacement magnitudes obtained using SAP2000 and LPILE.

The results obtained from the FE models and FD solutions show that SAP2000 was capable of predicting the induced bending moments and lateral displacements along the depth of the pile more closely than Abaqus/Cae. The reason for that can be attributed to the nature of the soil definition in the finite element models. In SAP2000, the soil is defined as isolated springs, which is similar to the soil definition in LPILE, and the soil stiffness obtained from LPILE was used into SAP2000 which resulted in obtaining almost a perfect match for the bending moment and the lateral displacement curves. However, the soil definition in Abaqus/Cae is based on a soil continuum model which resulted in a discrepancy between the results obtained by LPILE and those calculated by Abaqus/Cae. Overall, the results of Abaqus/Cae are considered to be in a good agreement with those of LPILE.

Also, the effect of applying an axial load of 298 kN to the pile on the produced bending moment and lateral displacement along the depth of the pile due to the applied displacement of 2 cm at the top of the pile is minimal and can be neglected.

## 9 Recommendations

- 1.
An agreement between the results of LPILE, SAP2000, and Abaqus/Cae was achieved. It is recommended that a design engineer may use LPILE to predict pile–soil interaction.

- 2.
If SAP2000 is used, it is recommended that a design engineer may use the largest number possible of springs, similar to what is used in this study.

- 3.
It is recommended to investigate and compare the pile–soil interaction in a single pile against that of pile-bent subjected to axial and lateral loads. It will be important to study the effect of a wide range of important design parameters. This comparison will inform design engineers of the difference in pile–soil interaction between a single pile and a group of piles.

- 4.
It is recommended to design and conduct an experimental study to test a single pile in soft and stiff soil under the effect of axial and lateral loads.

## Declarations

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

## Authors’ Affiliations

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