A pinned-pinned beam with and without a distributed foundation: A simple exact relationship between their eigenvalues

The body of this paper considers a pinned-pinned Bernoulli-Euler beam, from which the core natural frequencies and critical buckling loads corresponding to in-plane flexure, can be determined easily. The theory is then developed to yield an exact relationship between the static axial load in the beam and the frequency of vibration. This enables the core eigenvalues to be related exactly to their counterparts when the beam is additionally supported on a two parameter elastic foundation.The relationship is simple, exact and obviates the complex problems involved in solving the foundation problem using more traditional techniques. A number of illustrative problems are solved to confirm the accuracy and efficacy of the approach.