It has become apparent from the recent work by Choi et al. [3] on the nonlinear beam deflection problem, that analysis of the integral operator $mathcal{K}$ arising from the beam deflection equation on linear elastic foundation is important. Motivated by this observation, we perform investigations on the eigenstructure of the linear integral operator $mathcal{K}_l$ which is a restriction of $mathcal{K}$ on the finite interval [$-l,l$]. We derive a linear fourth-order boundary value problem which is a necessary and sufficient condition for being an eigenfunction of $mathcal{K}_l$. Using this equivalent condition, we show that all the nontrivial eigenvalues of $mathcal{K}l$ are in the interval (0, 1/$k$), where $k$ is the spring constant of the given elastic foundation. This implies that, as a linear operator from $L^2[-l,l]$ to $L^2[-l,l]$, $mathcal{K}_l$ is positive and contractive in dimension-free context.