Variational setting for cracked beams and shallow arches

TL;DR

This paper develops a rigorous mathematical framework for modeling cracked beams and shallow arches, introducing a novel operator to handle boundary conditions at cracks and justifying an efficient eigenvalue computation method.

Contribution

It introduces a new linear operator to incorporate crack boundary conditions and provides a mathematical foundation for the Modified Shifrin's method for eigenvalue problems.

Findings

Mathematically justified the Modified Shifrin's method.

Designed a linear operator to handle boundary conditions at cracks.

Established a comprehensive Hilbert space framework.

Abstract

We develop a rigorous mathematical framework for the weak formulation of cracked beams and shallow arches problems. First, we discuss the crack modeling by means of massless rotational springs. Then we introduce Hilbert spaces, which are sufficiently wide to accommodate such representations. Our main result is the introduction of a specially designed linear operator that "absorbs" the boundary conditions at the cracks. We also provide mathematical justification and derivation of the Modified Shifrin's method for an efficient computation of the eigenvalues and the eigenfunctions for cracked beams.