# Onsite Testing for Nonlinear Analysis of an Earthquake Damaged Historical Church in Italy
## 4 Finite Element Model
### Semi-Automated Model Updating  
The model updating is an iterative method, which aims at tuning up the physical parameters of a model until it reproduces the experimental data within a tolerated error. Figure 14: Average sensitivity coefficient: (**a**) considering the first three modes of vibration, (**b**) volumes of the groups respect the overall volume of the model. The model updating procedure relies on the minimization of two quantities: the relative frequency error (\(Df_{ij}\)), defined in (3) and the modal assurance criterion (\(MAC_{ij}\)) [21], defined in (4). \[Df_{ij}=\frac{f_{num,i}-f_{exp,j}}{f_{exp,j}}\times 100 \tag{3}\]  
\(f_{num,i}\) is the numerical frequency of the model and \(f_{exp,i}\) the experimental frequency for the \(i\)th mode of vibration. \[MAC_{ij}\left(\left\{\varphi_{num,i}\right\},\left\{\varphi_{exp,j}\right\} \right)=\frac{\left|\left\{\varphi_{num,i}\right\}^{T}\left\{\varphi_{exp,j} \right\}\right|^{2}}{\left(\left\{\varphi_{num,i}\right\}^{T}\left\{\varphi_{ num,i}\right\}\right)\left(\left\{\varphi_{exp,j}\right\}^{T}\left\{\varphi_{exp,j} \right\}\right)} \tag{4}\]  
\(\varphi_{num,i}\) is the numerical modal vector of the model and \(\varphi_{exp,i}\) is the experimental modal vector for the \(i\)th mode of vibration. In this work, the Douglas-Reid method was implemented to optimize the structural parameters of the FEM. The optimization is performed through an algorithm implemented in MATLAB [31] to minimize the objective function defined in (5) considering the first three modes of vibration (\(M=3\)). \[f(x)=\sum_{i=1}^{M}\biggl{[}(Df_{ii})^{2}+\biggl{(}\frac{1-\sqrt{MAC_{ii}}}{ MAC_{ii}}\biggr{)}\biggr{]} \tag{5}\]  
The experimental modal results obtained from the OMA were compared with the related output values obtained from the numerical eigenvalues analysis. The Douglas-Reid method uses an approximated model to interpolate with a quadratic function the structural parameters within a range of coordinate points representing the possible combination of parameters. The error reduction is performed by selecting the proper structural parameters from the set range of values using a batch analysis procedure. The range of values was defined based on the reliability of the data and the variability of the typology of parameter. The moduli of elasticity of the north external wall, the south external wall and the south internal masonry estimated with the double flat jack tests and confirmed by the sonic tomography (southeast pillar) were set within a range of variation of \(\pm 20\%\) since the results of the local tests are extended to a large volume of masonry, and it is reasonable to consider a scatter of the parameter form the reference experimental value. The moduli of elasticity estimated as the average of the experimental data (north internal masonry and dome) were set within a range of variation of \(\pm 30\%\) due to the lack of information on the interested elements. A uniform density for both the church and its dome has been assumed, with a range of variation of \(\pm 6\%\) because 1900 kg/m\({}^{3}\) is considered the highest reasonable density for the brick masonry in question. On the other hand, the soil structure interaction was refined by dividing the spring supports in two independent groups (north and south) because of the different soil slope and further improved by disconnecting the stiffness of the point springs in the horizontal direction from the vertical direction. The four independent parameters representing the soil structure interaction were set within a range of variation of \(\pm 20\%\). The initial model parameters and their updated values are presented in Table 3.  
The model updating yielded to a reduction of the average frequency error of the first four modes of 2% (from 10.44% to 8.54%) as shown in Figure 15.  
The modal deformations also returned an improvement, the gap between the numerical and the experimental modes decreased on average of 6.8% as reported in Figure 16 with the green columns.