# Bayesian Updating for Random Tensile Force Identification of Ancient Tie Rods Using Modal Data
## 4 Global Sensitivity Analysis  
Sensitivity analysis is a process aimed at evaluating the contribution to the model response of each computational input parameter [46]. The sensitivity analysis methods can be distinguished into local and global methods depending on the selection of a proper uncertain parameter space [46,47]. Local Sensitivity Analysis (LSA) is based in partial derivatives or gradients, but these are dependent on the point of the parameter space for which the model is evaluated. Global Sensitivity Analysis (GSA) is based on variance decomposition of the output variance into a sum of the different components. Each component is a measure of the sensitivity of the computational model output response to variations of each input parameter taken alone or to variations of combinations of input parameters [48,49]. Among different methods available for GSA, the Sobol decomposition method is used in this paper [50,51]. [PERSON]'s method is based on decomposition of the computational model output variance into a sum of variances of the input parameters in increasing dimensionality. In particular, first-order Sobol indices allow one to estimate the contribution of the variance of each input parameter taken alone to the overall model output variance. The GSA of the computational model output, \(\hat{\mathcal{M}}\), in Figure 1 is evaluated considering five output response Quantity of Interest (QoIs), i.e., \(f_{i}\),\(i=1\),..., \(5\), natural frequencies. A set of six statistically independent uncertain computational model input parameters (\(\xi_{j}\), \(j=1\),..., \(6\)) are selected: Young modulus, \(E\), mass density, \(\rho\), transversal square cross-section dimension, \(a\), equivalent stiffness of left and right springs, \(k_{l}\) and \(k_{r}\), and tensile axial force, \(N\). For all the uncertain computational model input parameters, zero left truncated Gaussian distributions are selected, and their characteristics are summarized in Table 1. Different values of axial force, \(N\), distribution mean values are used in order to account for the full range of possible axial loads in real ties in monumental buildings. In particular, two different cases are distinguished: low-tensioned and high-tensioned beams. Low-tensioned beams are characterized by axial load mean values equal to 0.1, 0.2, and 0.3\(\times\) the axial load capacity of the tie, estimated assuming a material yield strength of N\({}_{k}=\) f\({}_{k}\)\(\times\) a\({}^{2}\), f\({}_{k}\) = 220 MPa. High-tensioned beams are characterized by axial load mean values equal to 0.4, 0.5, and 0.6\(\times\) the axial load capacity of the tie. Elastic modulus, \(E\), mass density, \(\rho\), and transversal square cross-section dimension, \(a\), mean values are selected using common literature case studies [6,26,27]. The mean values of the spring stiffnesses, \(k_{l}\) and \(k_{r}\), were determined through a preliminary parametric analysis, where the values of \(k_{l}\) and \(k_{r}\) were varied between the extreme conditions of pinned (\(k_{l}\) and \(k_{r}\) equal to 0) and fixed (\(k_{l}\) and \(k_{r}\) equal to \(10^{6}\) Nm) boundary conditions. Subsequently, an average value of \(4\times 10^{5}\) Nm was chosen, accompanied by a cov that adequately covers conditions near both pinned-pinned and fixed-fixed boundary conditions. Figure 2 illustrates the results obtained from this analysis. Since different results are expected in the case of short (i.e., length of tie rod lower than 5 m) and long (i.e., length of tie rod higher than 5 m) tie rods, two different tie rod lengths are considered: 5 m and 10 m. In all the considered cases,  
Figure 1: Scheme for axially loaded beam-like structure. the most significant variation of the natural frequencies due to variations of the spring stiffnesses occur near the condition of pinned-pinned beams (i.e., \(k_{i}\) and \(k_{r}\) equal to zero), with an initial rapid increase of the natural frequencies beyond which an almost constant value is obtained, demonstrated by a progressive flattening of all the surfaces.