Force‐based derivation of exact stiffness matrix for beams onWinkler‐Pasternak foundation

Abstract

In this paper, the exact element stiffness matrix for a beam on Winkler‐Pasternak foundation and the fixed‐end force vector due to a linearly distributed load are alternatively derived based on the virtual force principle. The exact element flexibility matrix is at the core of the derivation of the exact element stiffness matrix and is formulated based on the exact force interpolation functions. The virtual force principle is employed to reveal the governing differential compatibility equations as well as the associated end‐boundary compatibility conditions. The exact force interpolation functions of the beam‐foundation system are derived based on the analytical solution of the governing differential compatibility equations of the problem for all combinations of foundation parameters and beam rigidity. This feature is unique to this paper. The matrix virtual force equation is employed to obtain the exact element flexibility matrix using the exact force interpolation functions. The so‐called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. Three numerical examples confirm the accuracy and the efficiency of the natural beam element on Winkler‐Pasternak foundation and show the deficiency of the widely used Winkler foundation model.