Interpolation-Element for Beam on Winkler Elastic Foundation

Abstract

On the basis of the deformation characteristics of Euler and Timoshenko beams on Winkler elastic foundation, the displacement shape functions were created. Lagrange interpolation function was used for the deformation in the axial direction, and Hermite interpolation function was applied in the bending direction. By applying the principle of minimum potential energy, energy integral and variation, the stiffness matrixes and equivalent load arrays of Euler and Timoshenko beams were derived. The model of elastic foundation beam was built by using elements constructed in this paper. And the beam deformation was calculated and compared with theoretical solution. The results show that the difference between deformations resulted from interpolation function and theoretical solution is small. With a smaller mesh size, the results obtained from interpolation function are approaching those from the theoretical solution. It illustrates that the application of polynomial function as the displacement shape function of elastic foundation beam could meet the requirement of accuracy and the elements in this paper could be applied in practice.