Analytical solution for laterally loaded long piles based on Fourier–Laplace integral

Piles are frequently used to support lateral loads. Elastic solutions based on the Winkler foundation model are widely used to design laterally loaded piles at working load. This paper reports a simplified analytical solution for laterally loaded long piles in a soil with stiffness linearly increasing with depth. Based on a Fourier–Laplace integral, a power series solution for small depth and a Wentzel–Kramers–Brillouin (WKB) asymptotic solution for large depth are derived. By using this analytical solution, the deflection and bending moment profiles of a laterally loaded pile can be obtained through simple calculation. The proposed power series solution is exact for infinitely long piles. Numerical examples show that this solution agrees well with other existing methods on predicting the deflection and bending moment of laterally loaded piles. The WKB asymptotic solution developed in this study has never been introduced before. The simplified analytical solution obtained in this study provides a better approach for engineers to analyze the responses and design of laterally loaded long piles.