# Onsite Testing for Nonlinear Analysis of an Earthquake Damaged Historical Church in Italy
## 4 Finite Element Model
### Sensitivity Analysis  
The global dynamic behavior is considered the reference experimental result for the model updating; thus, the numerical eigenvalues analysis of the FEM was used to select the most influential parameters for the dynamic behavior. The relationship between the experimental dynamic behavior and the model parameters depends on the heterogeneity of the materials, including the possibility of unidentified damage (i.e., crack, loss of material/detachment, loose interlocking and microcracks), and the interaction among the structural elements, in particular, at the foundation level. In addition, the complexity of the model increases the number of parameters; for these reasons, the initially examined parameters were chosen based on engineering judgment and the already demonstrated influence of the boundary conditions [18; 28]. First, the relevant parameters were estimated according to the results of the testing campaign and the gathered information on the construction stages of the church [8]: the moduli of elasticity of the north and south exterior walls, the interior south masonry and the Palazzo della Provincia were obtained from the double flat jack tests; the interior North masonry was assumed with the same [PERSON]'s modulus of the interior south side because it is the same type of structural element and the same period of construction. The dome and foundations moduli of elasticity were hypothesized as the average of the experimental moduli of the north and south exterior and interior masonry, while for the damaged masonry, which interests mainly the dome and part of the drum with the triumphal arch, it was considered a modulus of elasticity equivalent to one fifth of the dome. The modulus of elasticity of the facade was assumed as the average of the moduli of the north and south exterior masonries. The stiffness of the base springs was estimated through the [PERSON]'s formulation considering a modulus of subsoil reaction equal to 36,000 kN/m\({}^{3}\) as suggested by [29] for medium dense soils. As already considered in paragraph 3, a masonry density of 18 kN/m\({}^{3}\)[15] was assumed and the sensitivity of the dome's masonry was analyzed separately (\(\rho_{1}\)) and the masonry of the rest of the church (\(\rho_{2}\)). The parameters are reported in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline North exterior wall & E1 & 2800 & MPa \\ South exterior wall & E2 & 4600 & MPa \\ North internal masonry & E3N & 1800 & MPa \\ South internal masonry & E3S & 1800 & MPa \\ Dome & E4 & 2750 & MPa \\ Damaged masonry & E5 & 550 & MPa \\ Facade & E6 & 2750 & MPa \\ Palazzo della Provincia & E7 & 1600 & MPa \\ Foundations & E8 & 4833 & MPa \\ Soil & K & 6405 & N/mm \\ Dome density & \(\rho_{1}\) & 18 & kN/m\({}^{3}\) \\ Masonry density & \(\rho_{2}\) & 18 & kN/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model parameters influencing the dynamic behavior. The evaluation of the relevant parameters was quantified with the sensitivity coefficient (2) according to [30], the sensitivity analysis considered an increment of 10% from the reference values of Table 2. In Equation (2):  
\[S_{i,j}=100\times\frac{X_{j}}{R_{i}^{FEM}}\times\frac{\Delta R_{i}^{FEM}}{ \Delta X_{j}} \begin{array}{l}i=1\ \ldots\ M\ \ \ \ \ \ \ \ M\ \ \ \ \ \ M\ \ \ =\ Considered modes\\ j=1\ \ldots\ N\ \ \ \ \ N\ \ \ \ \ N\ \ =\ Considered parameters\end{array}  \tag{2}\]  
\(X_{i}\) represents the \(i\)th model parameter;  
\(R_{i}^{FEM}\) represents the \(i\)th output of the analysis (the modal frequency);  
\(\Delta X_{j}\) represents the variation of the model parameter (10% increment from the initial value);  
\(\Delta R_{i}^{FEM}\) represents the variation of the output of the analysis. The parameters with the highest sensitivity coefficient are those with major influence on the dynamic behavior.