A variational approach for an inverse dynamical problem for composite beams

Abstract

This paper deals with a problem of nondestructive testing for a composite system formed by the connection of a steel beam and a reinforced concrete beam. The small vibrations of the composite beam are described by a differential system where a coupling takes place between longitudinal and bending motions. The motion is governed in space by two second order and two fourth order differential operators, which are coupled in the lower order terms by the shearing, k, and axial, µ, stiffness coefficients of the connection. The coefficients k and µ define the mechanical model of the connection between the steel beam and the concrete beam and contain direct information on the integrity of the system. In this paper we study the inverse problem of determining k and µ by mixed data. The inverse problem is transformed to a variational problem for a cost function which includes boundary measurements of Neumann data and also some interior measurements. By computing the Gateaux derivatives of the functional, an algorithm based on the projected gradient method is proposed for identifying the unknown coefficients. The results of some numerical simulations on real steel-concrete beams are presented and discussed.

A non-destructive method for damage detection in steel-concrete structures based on finite eigendata

“…In order to prove that eigenvalues and eigenfunctions of T() given in (18) admit convergent series expansions in the perturbation parameter in a neighbourhood of the unperturbed operator T ¼ T(0), we need some preliminary results. Before presenting these results, it is convenient to introduce some notation and definitions which will be used in the sequel.…”

“…the corresponding quantities of the unperturbed and perturbed problem, respectively. Assume first that the unperturbed eigenvalue problem (defined by (18) with ¼ 0) has only simple eigenvalues, that is…”

“…Recalling that the perturbed operator A() is given by (19) and inserting (23), (24) in (18), equating term by term we obtain the following boundary value problems for the perturbation quantities v j and u j :…”

“…The encouraging results found in [18], and the appreciable sensitivity to damage of flexural mode shapes experimentally detected in [14], suggest that variational methods based on both frequency and eigenfunction data may be useful to detect damage in steelconcrete composite structures. In this article we investigate this direction of research.…”

“…An extension of the above result was obtained in [17], where measurements are taken simultaneously at both the ends of the beam. The full coupled vibrational problem, which includes both flexural and axial motions, was examined in [18]. A variational procedure based on time-history measurements taken at one end and, possibly, at some interior portions of the beam was proposed and used for the identification of the shearing and axial stiffness of the connection.…”