Two-sided eigenvalue bounds for the Euler-Bernoulli beam

TL;DR

This paper introduces new guaranteed lower bounds for eigenvalues of the Euler-Bernoulli beam with variable stiffness, enhancing the accuracy of eigenvalue estimates crucial for buckling analysis.

Contribution

It develops a novel method for obtaining guaranteed lower bounds for eigenvalues, applicable to both constant and variable stiffness beams, including nonlinear models.

Findings

The method provides reliable lower bounds that complement standard upper bounds.

Numerical experiments confirm the convergence and effectiveness of the bounds.

The approach is efficient for piecewise constant stiffness and adaptable to nonlinear models.

Abstract

We derive novel guaranteed lower bounds for eigenvalues of the Euler-Bernoulli beam with variable bending stiffness. While the standard finite element Rayleigh-Ritz method automatically yields upper bounds, we obtain lower bounds by employing interpolation error estimates with the explicitly known value of the associated constant. This approach is especially efficient and easy to apply for piecewise constant bending stiffness. For general variable material parameters, we obtain guaranteed lower bounds through an auxiliary beam-bending problem. The first eigenvalue is of primary interest in applications because it represents the critical load that causes buckling of the beam. Our method is, however, suitable also for the higher buckling modes. In addition, it can be applied to the physically more relevant nonlinear Gao beam model with piecewise constant bending stiffness, which has the same first eigenvalue as the classical Euler--Bernoulli beam. The presented numerical experiments illustrate the performance of the proposed eigenvalue bounds, demonstrating their convergence rates.