The capability of piles to withstand horizontal loads is a major design issue. The current research work aims to investigate numerically the responses of laterally loaded piles at working load employing the concept of a beam-on-Winkler-foundation model. The governing differential equation for a laterally loaded pile on elastic subgrade is derived. Based on Legendre-Galerkin method and Runge-Kutta formulas of order four and five, the flexural equation of long piles embedded in homogeneous sandy soils with modulus of subgrade reaction linearly variable with depth is solved for both free- and fixed-headed piles. Mathematica, as one of the world's leading computational software, was employed for the implementation of solutions. The proposed numerical techniques provide the responses for the entire pile length under the applied lateral load. The utilized numerical approaches are validated against experimental and analytical results of previously published works showing a more accurate estimation of the response of laterally loaded piles. Therefore, the proposed approaches can maintain both mathematical simplicity and comparable accuracy with the experimental results.

Pile foundations are frequently used, especially in weaker soils, to support various structures subjected to lateral loads such as high-rise buildings, communication towers, wind turbines, earth-retaining structures, bridges, tanks and offshore structures. Lateral loads owing to wind, wave, dredging, traffic and seismic events are considered significant on these structures since they are eventually transmitted to the piles [

Recently, many researchers used Legendre polynomials in different methods to construct various mathematical models. These methods can solve Lane-Emden type of differential equation [

This paper aims to present simple numerical methods to capture the behaviour of single piles under lateral working loads. First, mathematical formulation of the problem and the derivation of the governing differential equations are presented. Thereafter, two techniques are developed for numerical approximations of the derived equations. Then the effect of boundary conditions of the pile head on the behaviors of piles is studied as well. In order to examine the validity of the proposed numerical techniques, the obtained numerical results for deflection and bending moment along the laterally loaded piles are compared with results of previous studies. The proposed numerical techniques in the current study provide a better approach for structural designers to simply solve for the displacement and bending moment responses of laterally loaded piles. Consequently, the techniques can be easily applied in practice as an alternative approach to analyze and design laterally loaded long piles. It is worth noting that, the proposed solutions are applicable only for homogeneous soil with reaction modulus that varies linearly with depth. So, further investigation and analyses are needed for multi-layered soil with the modulus of subgrade reaction varies with any functions with the depth.

The model under examination consists of a pile that is assumed to be perfectly glued to the surrounding soils suggesting that there is no relative movement between the soil and the pile [

Assuming all forces acting horizontally and applying equilibrium conditions leads to

According to that,

Summation of moments about axis at end of element yields

The relation between bending moment and the shear force is given by

Differentiating

The basic moment-curvature relationship of elementary pile can be written as

The differential equation of a laterally loaded pile can take the form

Assuming a linear variation of soil stiffness with soil depth,

The solution of

Defining a dimensionless variable

By substituting

As shown in

Boundary conditions of infinitely long piles fixed at the bottom can be written as

The boundary conditions at the top of the pile depend on the circumstances of the lateral deflection, slope, bending moment and shear force. These are generalized into the following two categories.

The pile head is not restrained against rotation and translation. Therefore, the boundary conditions at the pile head can take the form

The pile head is completely restrained against rotation. The boundary conditions at the pile head are given by

Orthogonal systems play a vital role in performing mathematical analysis. This can be due to functions belonging to very general classes can be expanded in series of orthogonal functions e.g., Fourier-Bessel series, Fourier series, etc. Orthogonal polynomials of degree

Lemma [

To solve

So,

Moreover, the conditions of fixed-head pile are:

The solution of

Reducing

By substituting

In this case, the vector

The linear system in

By using the inverse linear transformation

In this case, the vector

By using the inverse linear transformation

The differential equation of laterally loaded pile,

If the pile top condition is a free-head one, then the system of equations given in

On the other hand, if the pile top condition is a fixed-head one, then the system of equations given in

The system of equations

To obtain local error estimates for adaptive step-size control effectively, consider two Runge-Kutta formulas of different orders

Considering ^{th} order formula) and ^{th} order formula).

The solution

The error between the two numerical solutions

In case of

The adapted step size is used to estimate the new values of

The performance and capability of the proposed methods to predict the behavior of laterally loaded piles in cohesion1ess soil have been demonstrated by comparing the obtained numerical results and the observed results from field experiments in full-scale lateral load tests reported by Cox. et al. [^{2} bending rigidity, was embedded in a deposit of submerged sand. The soil profile at the site is composed of a uniformly fine-graded sand with internal angle of friction, ^{3}. To investigate the soil properties below the ground surface, the Standard Penetration Test, SPT, was performed by [^{3}, [

The design of pile foundation under lateral loads is extensively bound to study both the pile head deflection and the maximum bending moment. Legendre-Galerkin method and Runge-Kutta formulas of order four and five were employed to solve the flexural equation of long piles embedded in homogeneous cohesionless soil with a modulus of subgrade reaction increases linearly with depth. In the numerical simulation, the pile is subjected to horizontal force ^{2} and 15000 kN/m^{3} respectively. In order to ensure that our results produce reliable pile deflection and bending moment not only at the pile head but also for the entire pile length, and the analysis is equally applicable for different pile head conditions, the obtained results are compared with the results of simplified solution proposed by Fayun et al. [

The plotted curves reveal complete overlaps between Legendre-Galerkin solution and Runge-Kutta solution. Furthermore, the comparison charts demonstrate a very good agreement between the numerical results estimated via Legendre-Galerkin and Runge-Kutta and the corresponding procedure introduced by Fayun et al. [

In the present study, Legendre-Galerkin and Runge-Kutta formulas of order four and five methods have been introduced to obtain simplified numerical approaches for understanding the behaviour of single piles against lateral loads. For the purpose of analysis and design of laterally loaded piles crossing sandy soil, simple expressions for the pile lateral deflection and bending moment can be evaluated. The procedure is programmed with the most computational software program Mathematica, which is considered as the world's leading computational software. The numerically computed pile responses are compared with the results from the full-scale lateral load tests. The proposed approaches are well validated. Moreover, these proposed approaches provide evidence that high precision can be achieved with a small amount of computational work. It has been found from the study that the Legendre-Galerkin solution almost coincides with the Runge-Kutta solution for both free-head and fixed-head piles. The suggested numerical expressions obtained in this study can be reasonably applied to analyze and design laterally loaded long piles conveniently. In addition, these techniques can also modify to design/analyze laterally loaded long piles in soil with the modulus of subgrade reaction in any functions of the depth. The proposed approaches capture the long pile behaviour. Furthermore, the proposed solutions are aimed at providing an effective and convenient method for engineers to predict the responses of the entire pile length under the applied lateral load.