Abstract
Swelling forces from materials confined by structures can cause structural deformations and instability. Due to the complexity of interactions between expansive solid and solid-liquid equilibrium, the forces exerting on retaining structures from swelling are highly nonlinear. This work is our initial attempt to study a simplistic initial/boundary value problem based on the Euler-elastic beam theory and some swelling force model. In this paper, we study a nonlinear problem for the equation of a beam. The self-adjointness of the operator corresponding to the nonlinear problem for the Euler equations is proved. Two-sided estimates of the eigenvalues of the operator in question are established. Two-sided estimates of the eigenfunctions of the operator of the initial-boundary value problem for the beam equation are also obtained.