# Bayesian Updating for Random Tensile Force Identification of Ancient Tie Rods Using Modal Data
###### Abstract  
The rods play a crucial role in civil engineering, particularly in controlling lateral thrusts in arches and vaults, and enhancing the structural integrity of masonry buildings, both historic and contemporary. Accurately assessing the tensile axial forces in tie rods is challenging due to the limitations of existing methodologies. These methodologies often rely on indirect measurements, computational models, and optimization procedures, resulting in single-point solutions and neglecting both modeling and measurement uncertainties. This study introduces a novel Bayesian updating framework to effectively address these limitations. The framework aims to accurately identify the structural parameters influencing tie rod behavior and estimate uncertainties using natural frequencies as references. A key innovation lies in the mathematical formulation of Bayesian updating, which is founded upon the definition of computational models integrating uncertain updating parameters and latent random variables derived from a rigorous sensitivity analysis aimed at quantifying the impact of the updating parameters on the natural frequencies. Notably, the application of Bayesian updating to the structural identification problem of ancient tie rods represents a significant advancement. The framework provides a comprehensive description of the uncertainties associated with computational models, offering valuable insights for practitioners and researchers alike. Moreover, the results of the sensitivity analysis serve as a valuable tool for setting up inverse problems geared towards accurately identifying tensile axial forces. and dimensions, actual length (i.e., length of the tie beam including the anchorage lengths in the side walls), bending stiffness, and boundary conditions used to describe the tie rod insertion length into the lateral walls [6; 7; 8; 9]. Since direct measurements of the unknowns can be challenging on tie rods in existing constructions, many indirect methods based on in-situ non destructive tests have been developed. These methods can be classified in static, mixed static/dynamic, and fully dynamic methods [7]. Static methods for the axial force estimation are based on measurements of displacements and/or strains and flexural displacements and/or curvatures measured at a few sections of tie rods affected by concentrated static loads [10; 11]. Mixed methods consist of an analytical procedure based on static measurements of vertical displacements caused by concentrated loads and dynamic measurements of the first vibration frequency [12]. Fully dynamic methods are based on the identification of tensile axial force and all the problem unknowns measuring the tie rods vibration frequencies and, in a few cases, the associated vibration modes, via experimental modal analysis [2; 13; 14], operational modal analysis [9; 15], and acoustic measurements [16]. Fully dynamic methods are typically used in the literature since static and mixed static/dynamic methods require small displacement measurements and significant load application often challenging for the heights at which tie rods are placed, especially in historic masonry structures. Using dynamic methods, the axial load and the other unknowns are identified by solving a non-linear inverse problem aimed at minimizing the distance between the numerical and measured modal parameters. Analytical methods were assessed in [17; 18; 19]. The problem was solved for a slender beam constrained by two sets of elastic rotational end springs using one natural frequency and one flexural mode shape as a target in [17; 18]. These methods do not require the knowledge of the effective length of the beam but only the bending stiffness and the mass density. The axial force as well as the flexural stiffness of the end constraints are identified under the assumption of infinite translational stiffness at the beam ends. The problem was solved for an Euler-Bernoulli beam constrained at the end by both rotational and vertical springs accounting for bending stiffness in [19].