Eigenfunction Solution for Beam on Elastic Foundation

A solution for the end problem of a rectangular beam resting on a simple elastic foundation is obtained as a series expansion in the eigenfunctions of the system. For a beam aligned with the ξ-axis, the eigenfunctions are of the form e^γξf(y), where γ is one of the complex eigenvalues. The eigenvalue equation is determined by requiring continuity of the normal displacement and pressure at each point of the beam-foundation interface and seeking a nontrivial solution. In order to evaluate the accuracy and limitations of several approximate beam theories, the eigenvalue predicted by each of these theories is compared with the first eigenvalue of the exact solution. It is shown that the approximate theories give adequate accuracy if the beam modulus, E, exceeds the foundation stiffness, k, and that all tend to the same result as E/k → ∞. From a comparison of the first and second eigenvalues of the exact solution, it is found that the first mode (corresponding to beam behavior) ceases to dominate the higher modes for E/k < 1. Thus the approximate theories are necessarily restricted to E/k > 1 since they predict only the first mode.