Elastic Solutions for Laterally Loaded Piles William Higgins, S.M.ASCE 1 ; Celio Vasquez 2 ; Dipanjan Basu, M.ASCE 3 ; and D. V. Griffiths, F.ASCE 4 Abstract: Laterally loaded piles are analyzed using the Fourier FEM. The analysis is performed for piles embedded in single-layer elastic soil with constant and linearly varying modulus and in two-layer elastic soil with constant modulus within each layer. The pile responses were ob- served to be functions of the relative stiffness of pile and soil, and of the pile slenderness ratio. Based on the analysis, equations describing pile head deflection, rotation, and maximum bending moment are proposed for flexible long piles and stubby rigid piles. These design equations are developed after plotting the pile responses as functions of pile-soil stiffness ratio and pile slenderness ratio. These plots can also be used as design charts. Design examples illustrating the use of the analysis are provided. DOI: 10.1061/(ASCE)GT.1943-5606.0000828. © 2013 American Society of Civil Engineers. CE Database subject headings: Piles; Lateral loads; Finite element method; Elasticity; Design. Author keywords: Pile; Lateral load; Finite element analysis; Elasticity; Design. Introduction Structures resting on piles are frequently subjected to horizontal forces from wind, traffic, and seismic activities. The horizontal forces acting on tall or heavy structures like high-rise buildings, bridge abutments, and earth-retaining structures are often of very large magnitude. Offshore structures like quays and harbors are also subjected to large lateral forces arising out of wind, waves, and ship berthing. The horizontal forces eventually get transmitted to the piles, which are analyzed considering a concentrated force and/or moment acting at the pile head. Even in structures where piles are used to resist vertical forces only, there may exist moments from load eccentricities caused by faulty construction. Consequently, proper analysis and design of piles subjected to lateral forces and moments is very important to ensure the stability and serviceability of various structures. Numerous research studies, both theoretical and experimental, have been performed on laterally loaded piles for more than six decades. The early theoretical works stem from the concept of re- presenting soil by discrete springs with the soil subgrade modulus as the spring constant. This approach was modified to account for plastic deformation of soil by incorporating nonlinearity in the soil springs (Matlock and Reese 1960; McClelland and Focht 1958). Further development of this method led to the well-known p-y method (Reese and Cox 1968; Matlock 1970; Reese et al. 1974, 1975). The continuum approach was also used for the analysis of laterally loaded piles. Poulos (1971a, b) applied an integral equation method of analysis while Banerjee and Davies (1978) used a similar boundary element algorithm. Sun (1994) and Basu et al. (2009) used variational principles to obtain analytical solutions for lateral pile displacements in elastic media. Guo and Lee (2001) assumed a stress field using the Fourier series and obtained a load transfer method for laterally loaded piles. These apart, the FEM (Desai and Appel 1976; Bhowmik and Long 1991; Bransby 1999; Hsiung and Chen 1997), finite elements coupled with Fourier series (Randolph 1981; Carter and Kulhawy 1992), the finite difference method (Klar and Frydman 2002; Ng and Zhang 2001), the boundary element method (Budhu and Davies 1988), and the upper-bound method of plasticity (Murff and Hamilton 1993) have been used to analyze laterally loaded piles. In this paper, the FEM coupled with Fourier techniques is used to analyze laterally loaded piles embedded in elastic continua. Piles with different lengths, flexibilities, and boundary conditions are considered. Subsurface profiles with constant and linearly varying moduli are assumed. Additionally, a two-layer profile is considered. A parametric study is performed in which the important variables governing the pile behavior are identified. Based on the study, de- sign equations are proposed using which pile deflection, slope, and bending moment can be calculated if the correct elastic soil modulus is available. Design examples are provided to illustrate the use of the analysis. Analysis Cylindrical piles with a lateral load F a and moment M a acting at the head are considered in this paper (Fig. 1). The pile is described by its radius r p , length L p , and Young’s modulus E p . The soil is described by its shear modulus G s and Poisson’s ratio y s . Three types of soil profiles are considered in this paper 1. Homogeneous soil in which G s remains spatially constant; 2. Heterogeneous soil in which G s increases linearly with depth from zero value at the ground surface; and 3. Two-layer soil with different values of G s that remain spatially constant within each layer (Fig. 2). 1 Former Graduate Student, Dept. of Civil and Environmental Engi- neering, Univ. of Connecticut, Storrs, CT 06269. E-mail: wthigginsiv@ gmail.com 2 Former Undergraduate Student, Dept. of Civil and Environmental Engineering, Univ. of Connecticut, Storrs, CT 06269. E-mail: celio .vasquez@uconn.edu 3 Assistant Professor, Dept. of Civil and Environmental Engineering, Univ. of Waterloo, Waterloo, ON, Canada N2L 1W7 (corresponding author). E-mail: dipanjan.basu@uwaterloo.ca 4 Professor of Civil Engineering, Division of Engineering, Colorado School of Mines, Golden, CO 80401. E-mail: d.v.griffiths@mines.edu Note. This manuscript was submitted on March 22, 2011; approved on September 4, 2012; published online on September 5, 2012. Discussion period open until December 1, 2013; separate discussions must be submitted for individual papers. This paper is part of the Journal of Geotechnical and Geoenvironmental Engineering, Vol. 139, No. 7, July 1, 2013. ©ASCE, ISSN 1090-0241/2013/7-1096–1103/$25.00. 1096 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / JULY 2013 J. Geotech. Geoenviron. Eng. 2013.139:1096-1103. Downloaded from ascelibrary.org by University of Waterloo on 01/17/14. Copyright ASCE. For personal use only; all rights reserved.