Citation: Arvan, P.A.; Arockiasamy, M. Energy-Based Approach: Analysis of a Laterally Loaded Pile in Multi-Layered Non-Linear Elastic Soil Strata. Geotechnics 2022, 2, 570–598. https://doi.org/10.3390/ geotechnics2030028 Academic Editor: Raffaele Di Laora Received: 3 June 2022 Accepted: 1 July 2022 Published: 7 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article Energy-Based Approach: Analysis of a Laterally Loaded Pile in Multi-Layered Non-Linear Elastic Soil Strata Prakash Ankitha Arvan 1, * and Madasamy Arockiasamy 2 1 Department of Civil, Environmental and Geomatics Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA 2 Center for Infrastructure and Constructed Facilities, Department of Civil, Environmental and Geomatics Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA; arockias@fau.edu * Correspondence: parvan2018@fau.edu Abstract: Several studies have been reported in published literature on analytical solutions for a laterally loaded pile installed in a homogeneous single soil layer. However, piles are rarely installed in an ideal homogeneous single soil layer. The present study describes a new continuum-based analysis or energy-based approach for predicting the pile displacement responses subjected to static lateral loads and moments considering the soil non-linearity. This analytical analysis treats the pile as an elastic Euler–Bernoulli beam and the soil as a three-dimensional (3D) continuum in which the non-linear elastic properties are described by a modulus degradation relationship. The principle of virtual work was applied to the energy equation of a pile–soil system in order to obtain the governing differential equation for the pile and soil displacements. An iterative procedure was adopted to solve the equations numerically using a finite difference method (FDM). The pile displacement response was obtained using the software MATLAB R2021a, and the results from the energy-based method were compared with those obtained from the field test data as well as the finite element analysis (FEA) based on the software ANSYS Workbench 2021R1. The present study investigated the effect of explicit incorporation of soil properties and layering through a parametric study in order to understand the importance of predicting appropriate pile displacement responses in a linear elastic soil system. The responses indicated that the effect of soil layers and their thicknesses, pile properties and the variation in soil moduli have a direct impact on the displacements of piles subjected to lateral loading. Hence, a proper emphasis has to be given to account for the soil non-linearity. Considering the effect of soil non-linearity, it is observed that the results obtained from the energy-based method agreed well with the field measured values and those obtained from the FEA. The results indicated a difference of approximately less than 7% between the proposed method and the FEA. The approach presented in this study can be further extended to piles embedded in multi-layered soil strata subjected to the combined action of axial loads, lateral loads and moments. Furthermore, the same approach can be extended to study the response of the soil to group piles. Keywords: piles; multi-layered soil; soil constitutive model; lateral load; pile displacement; varia- tional energy method; MATLAB R2021a; ANSYS Workbench 2021R1 1. Introduction The growing importance to analyze the structures, such as high-rise buildings, bridges, offshore platforms, etc., resting on pile foundations and acted upon by various horizonal forces (wind, wave, currents and seismic events) has led to various analysis methods over time. An extensive literature review has been conducted by the authors Moussa and Christou [1] who summarized and grouped the various analysis methods of laterally loaded single pile under static loading into four categories: (a) ultimate limit state method (ULS); (b) subgrade reaction approach; (c) finite element method (FEM); and (d) continuum method. Several researchers developed different types of the ultimate limit state (ULS) methods, Geotechnics 2022, 2, 570–598. https://doi.org/10.3390/geotechnics2030028 https://www.mdpi.com/journal/geotechnics