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

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

which is a restriction of

on the finite interval [

]. We derive a linear fourth-order boundary value problem which is a necessary and sufficient condition for being an eigenfunction of

. Using this equivalent condition, we show that all the nontrivial eigenvalues of

are in the interval (0, 1/

), where

is the spring constant of the given elastic foundation. This implies that, as a linear operator from

to

,

is positive and contractive in dimension-free context.