# Bayesian Updating for Random Tensile Force Identification of Ancient Tie Rods Using Modal Data
## 5 Bayesian Solution with Dynamic Data
### Low-Tensioned Short Tie Rod  
Following the mathematical formulation of the Bayesian updating framework in Section 2 and according to the results of the GSA in Section 4, for the low-tensioned short tie rod six updating parameters are considered (i.e., \(\boldsymbol{\xi}=\boldsymbol{\Theta}\in\mathbb{R}^{6}\)): Young modulus, \(E\), mass density, \(\rho\), cross-section transversal dimension, \(a\), left and right spring stiffness, \(k_{l}\) and \(k_{r}\), and axial tensile force, \(N\). The effective length, \(l\), of the tie is considered as a known measurable quantity. Zero left truncated Gaussian distributions are assumed as prior distributions of the uncertain parameters, whose mean and coefficient of variation, \(cov\), are reported in Table 2.  
The results of the Bayesian updating framework (i.e., posterior distributions of the updating parameters, \(\Theta_{j}\)) are reported in Figure 4. It is worth noting that the updating framework is able to modify the probability distributions of the updating parameters \(E\), \(\rho\), and \(a\), reducing the \(cov\) with respect to the prior distributions. The opposite behavior is obtained for the two spring stiffnesses, \(k_{l}\) and \(k_{r}\), and the tensile axial force, \(N\), with a \(cov\) of the posterior distributions greater than that of the prior distributions. These results were expected considering the results of GSA in Figure 3. When low-tensioned short tie rods are of interest, the most sensitive parameters on the QoIs are the Young modulus, \(E\), the mass density, \(\rho\), and the transversal square cross-section dimension, \(a\). The left and right spring stiffnesses, \(k_{l}\) and \(k_{r}\), Sobol indices assume near zero values in all cases, with the exception of the first natural frequency, and the tensile axial force, \(N\), Sobol indices assume values different from zero only when the first two natural frequencies are considered. However, it is worth observing that all the posterior distribution mean values, \(\mu_{\Theta}  
the true values (i.e., actual values used to simulate the reference dataset, **D**), showing the accuracy of the Bayesian updating framework to give useful information on the single-point solution of the structural identification problem. In order to further validate the proposed methodology, Figure 4g reports the comparison between the natural frequencies, \(f_{i}^{sim}\), used as reference dataset **D** and the natural frequencies, \(f_{i}\), obtained from the computational model driven by the posterior mean value of the updating parameters, \(\mu_{\Theta_{j}^{up}}\). A good agreement between the reference natural frequencies and those predicted by the updated numerical model can be observed with a percentage relative difference (i.e., \(1-f_{i}^{sim}/f_{i}\)) always lower than 1%. Furthermore, Figure 4h shows the ratio between the posterior mean value of the updated parameters, \(\mu_{\Theta_{j}^{up}}\), and the true value, \(\Theta_{j}^{true}\), of the computational model input parameters used to obtain the reference dataset, **D**; a ratio equal to 1 indicates a perfect match. Values of the posterior mean value, \(\mu_{\Theta_{j}^{up}}\), of approximately 1.05 times the true value, \(\Theta_{j}^{true}\), are obtained for the Young modulus, \(E\), and the equivalent stiffness of the two lateral springs, \(k_{l}\) and \(k_{r}\); values of the posterior mean, \(\mu_{\Theta_{j}^{up}}\), of approximately 0.95 times the true value, \(\Theta_{j}^{true}\), are obtained for the mass density, \(\rho\), and the cross-section transversal dimension, \(a\), value of posterior mean value, \(\mu_{\Theta_{j}^{up}}\), of approximately 0.87 times the true value, \(\Theta_{j}^{true}\), is obtained for the tensile axial force, \(N\). The obtained results indicate that the computational model driven by the posterior mean value of the updating parameters, \(\Theta_{j}\), provide a good representation of the structural system and that the tensile axial force value estimation is accurate.