An improved solution for beam on elastic foundation using quintic displacement functions

Abstract

The theory of beam on elastic foundation is a simple and popular analytic approach for computing the response of laterally loaded piles. For a laterally loaded pile with constant subgrade reaction coefficient, an analytical solution can be easily deduced based on the theory of beam on elastic foundation when the load distribution and boundary condition are simple. However, when the subgrade reaction coefficient increases linearly with the depth or when the constraint condition is complex, an approximate solution can only be obtained by numerical method. At present, the node-spring simulation method and the modifying stiffness matrix method are two main solution methods for beam on elastic foundation with a nonuniform distribution of subgrade reaction coefficient, but a large number of elements are necessary for obtaining a sufficient calculation accuracy. Based on the Winkler elastic foundation model, an improved Finite Element (FE) method for the laterally loaded pile on an elastic foundation with a linearly distributed modulus of the subgrade reaction is proposed. A quintic displacement function is proposed as an approximate solution, and the weighted residual method is used for solving differential equations. The corresponding element stiffness matrix and nodal force vector are derived, and a more accurate nodal displacement, element internal force and displacement distribution can then be obtained by employing fewer elements. Three beams on elastic foundations under different boundaries and loading conditions are taken as typical examples to compare the difference of the calculation accuracy between the improved method and the node-spring simulation method. A laterally loaded pile is analyzed by the improved method, and the numerical results show that two elements for one soil layer can provide a sufficient calculation accuracy.