## Advantages of "consistent" approach over lumped methods for analysis of beams on elastic foundations

It is common to analyze laterally loaded piles by treating the problem as a beam on an elastic foundation. In general, the finite element method (FEM) is used to solve the governing differential equation, in preference to the more popular finite difference approaches. The "consistent" approach, where the governing differential equation is discretized using a more rigorous Galerkin formulation, is demonstrated by Griffth (1989) to be considerably more accurate than a ‘lumped’ approach where the lumped springs are applied at the node positions. The improvement is particularly noticeable when the finite element discretization becomes coarser. It is, therefore, possible that only fewer elements are required to obtain the accurate solutions.

This approach has been adopted by our PileLAT and PileGroup programs to analyze the behavior of piles under lateral loading. The stiffness matrix is evaluated at the element level and formed based on Galerkin formulation. This is different from the conventional finite difference approach where the formulation is developed at the node level and therefore is more efficient mathematically. Arc-length technique is also adopted to solve the equilibrium equations with the capability of capturing nonlinear load path, especially for softening load-displacement behavior. Those are two most important features which differentiate our products from the conventional finite difference method based programs.

The following figure shows the comparison analysis results of the single pile under lateral loading as output from PileLAT program. Comparison results between PileLAT and Hetenyi’s closed-form equations (Hetenyi 1946) are shown. The results from two separate runs with different element numbers (30 elements and 100 elements) based on PileLAT are plotted against the results from the closed-form equations. PileLAT program based on the consistent approach essentially predicts exactly the same results as Hetenyi’s closed-form equations and the results are not sensitive to the number of elements adopted in the analysis. 