Numerical formulation and implementation of Euler-Bernoulli pile elements considering soil-structure-interaction responses

Pile serves as an essential component for transferring loads from superstructures to the soil ground. Proper consideration of soil-structure interaction (SSI) responses is crucial in evaluating the pile capacity and deflections under external loads. In the current design practices, semi-empirical or linear-elastic analyses are utilized, which are often overly conservative and unable to properly consider the nonlinear SSI responses. Thus, this paper presents a new Euler-Bernoulli pile element for robustly and efficiently simulating the piles considering the SSI responses. The nonlinear springs distributed continuously along with the pile are directly integrated into the element formulations. A quasi-analytical solution based on the Gauss-Legendre method is utilized in the summation processes of the total potential energy to significantly simplify the mathematical expressions and ease difficulties in programming. The element tangent stiffness matrix and secant relations are respectively formulated for predicting the displacement and eliminating the accumulative errors in a Newton-Raphson incremental-iterative numerical procedure. Because a pile might exhibit large deflections in soft soils, the kinematic motion description using the Updated Lagrangian (UL) approach is developed where the equilibrium conditions are determined by referring to the last configurations. Finally, several examples are provided to validate the accuracy and efficiency of the proposed method.