Spline finite element updating of a reinforced concrete beam

An updating procedure based on measured Frequency Response Function (FRF) data is proposed to correct the numerical parameters of the continuous B-spline model of a reinforced concrete beam. The beam model coefficients (e.g. flexural stiffness, mass density, damping ratio, joint stiffness) and displacement field are described as continuous parametric functions by means of B-spline shape functions. The Euler-Bernoulli kinematic assumption is considered but the rotatory inertia effect is taken into account as well. Moreover, the B-spline beam model employs only transversal degrees of freedom (dofs). The updating algorithm is based on the minimization of an objective function, expressed as a nonlinear least squares problem considering the difference between the model-based and the experimentally measured response, at the same frequency. The minimization is solved by using a truncated linear Taylor series of the objective functions and an iterative formulation. The use of FRF data measurements as input provides a large amount of input data in the frequency domain. Nevertheless, this problem can generally be ill conditioned, especially when the number of updating parameters is increased. Any singularity of the problem is addressed by the singular value decomposing the system matrix resulting at each solution iteration. An example is reported dealing with a reinforced concrete beam considered within the Bri.Vi.Di. research project, supported by the Italian Ministero dell'Università e della Ricerca (MIUR) under the "Progetti di Interesse Nazionale" (PRIN07) framework. The updating procedure was tested by adopting measured data as input. Results are illustrated and discussed.