# Bayesian Updating for Random Tensile Force Identification of Ancient Tie Rods Using Modal Data
###### Abstract
The axial force and the four boundary stiffnesses are determined using both vibration frequencies and corresponding vibration modes as reference data. Minimization optimization functions were proposed in [2; 7; 13; 20]. The axial force identification procedure was assessed using as a reference structural system a simply supported Euler-Bernoulli beam with a uniform section hinged at both ends with rotational springs, or resting on a Winkler-type foundation over a length corresponding to the portion of the tie rods constrained into the wall, subjected to constant axial force. The characteristics equation of beam motion is used to estimate the approximate numerical solution of the unknowns (i.e., tensile axial force and stiffnesses of boundaries), solving an iterative optimization procedure aimed at minimizing a suitable error function representing the distance between the measured and the model predicted modal parameters. In particular, the first measured three modal frequencies of the tie rods are used to estimate the axial tensile force, the bending stiffness of the section, and the stiffness of the rotational springs in [2; 20]. The axial load, the elastic Winkler bed characteristics, representing the contact between stonework and the tie-rod, and the error function are estimated using as many natural frequencies that can be identified from the measurements in [7; 13]. The Finite Element (FE) model updating technique was described in [21; 22; 23]. Different types of mathematical formulations were used to solve the inverse updating problem. A traditional iterative procedure based on the manual tuning of the unknowns FE model parameters (i.e., axial force and stiffness of the elastic springs) in order to match the first two experimentally identified vibration frequencies and associated mode shapes is used in [22]. A genetic algorithm procedure based on a fitness function fed by a simple FE model is developed in [23]. The first four vibration modes and vibration frequencies are used to update the FE model and to determine the axial force, the stiffnesses of the rotational springs at the beam ends, the bending stiffness, and the effective length of the tie rod in [21]. All these methods are used to estimate the axial force and the other problem unknowns using beams with a uniform cross section, constant mass density, and elastic modulus as a reference structural model. In practice, ancient metallic tie rods are often not characterized by a uniform cross section, containing irregularities and/or discontinuities [24]. Furthermore, the mass density and the elastic properties of the ancient metal constituting the ties may also vary along the actual length of ties for the different manufacturing and not standardized procedures [25]. Moreover, all the above discussed methods are set in a deterministic environment and do not account for the uncertainty on the physical and mechanical properties inherent to tie rods in existing buildings, especially the historical ancient ones. Uncertainties are related to the effective length of the tie rods since the anchorages are usually hidden inside the arches or vault imposts. Uncertainties are related to the boundary conditions and, in particular, to the elastic stiffnesses of the springs (rotational and/or elastic Winkler bed springs) used to reproduce the insertion length of the tie rods inside the lateral walls. In all the methods assessed in the literature, these stiffnesses are assumed to be symmetric. Uncertainties are obviously related to the mechanical properties of the ties (i.e., elastic moduli, mass densities, and axial tensile force). A recent attempt to investigate the influence of uncertain parameters on the tie rod's axial load estimates by means of vibration frequencies is carried out using the General Polynomial Chaos expansion method in [6].