On positiveness and contractiveness of the integral operator arising from the beam deflection problem on elastic foundation

TL;DR

This paper proves that the integral operator from a beam deflection problem on an elastic foundation has no nontrivial eigenvalues outside a specific interval, clarifying its spectral properties.

Contribution

It provides a complete proof regarding the positiveness and contractiveness of the integral operator associated with the beam deflection problem.

Findings

No nontrivial eigenvalues outside (0,1/k)

Spectral properties of the integral operator clarified

Implications for stability analysis of the beam

Abstract

We provide a complete proof that there are no nontrivial eigenvalues of the integral operator $\mathcal{K}_l$ outside the interval $(0,1/k)$. $\mathcal{K}_l$ arises naturally from the deflection problem of a beam with length $l$ resting horizontally on an elastic foundation with spring constant $k$, while some vertical load is applied to the beam.