# Onsite Testing for Nonlinear Analysis of an Earthquake Damaged Historical Church in Italy
## 4 Finite Element Model  
The structural behavior of the church was studied with a tridimensional finite element model of the construction developed with the software Midas Fea NX [22]. The model is based on a three-dimensional geometrical model provided by Studio Da Gai. The foundations have been included in the model considering past geotechnical investigations. The model was processed to create a mesh of 302,138 solid tetrahedron elements. The horizontal structures, such as the wooden roofs and the reinforced concrete slabs, were modeled as lumped masses with rigid diaphragms while the more deformable and critical groin and barrel vaults in the sacristy and presbytery were accurately modeled considering constant thicknesses. The model was subdivided in nine element groups based on the architectural features, the material characteristics, the historical phases, and the identified damage (Figure 12). The soil-structure interaction was modeled with 2177 elastic three-dimensional point spring elements positioned at the base of the foundation and 11,658 elastic 2D point spring elements positioned on the lateral surfaces of the foundation walls considering as initial value a stiffness ratio of 1 to 10 between the horizontal and vertical directions (as considered by [PERSON] for a similar church [23]).  
Particular attention was paid to the ovoidal dome. The Italian Baroque architecture of the dome origins from the models of the ancient Rome are described by [PERSON] [24]; during the XVII century, the dome evolved towards complex geometries with elliptical and ovoidal plans, a drum at the base and a lantern on top [25]. The dome was typically composed of two shells interconnected by ribs and surrounded by buttresses [26]. The geometrical survey reported a dome with varying thickness from 250 to 80 cm; therefore, a double shell dome with six radial ribs (in correspondence with the stuccos shown in Figure 2) was assumed and the internal cavity filled with light backing materials to transmit the lateral thrust of the intrados layer to the six buttresses was assumed. The passive resistance of the backing material (in the lower part of the cavity) was modeled with compression-only elastic springs, which link the intrados layer to the spandrels. The longitudinal crack was modeled by assigning a weaker and softer material to the region interested by damage (the  
Figure 11: Modal assurance criterion between the experimental modal shapes, in yellow the detected dependencies between the modes.  
damaged masonry was assumed with one fifth of the mechanical parameters of the dome). Figure 13 shows a section cut of the model of the dome.  
Figure 12: Section cut of the finite element model: (**a**) south side, (**b**) north side.  
Figure 13: Section cut of the double shell ovoidal dome: the damaged masonry is shown in red and the compression-only spring elements are shown in blue.  
The study aimed to focus on the structural behavior of the church; for this reason, the constraining effect of the annex Palazzo della Provincia was modeled by applying pinned condition to the boundary nodes of the east side.  
The model was first analyzed in the elastic domain. In this initial phase, the material parameters (Young's modulus and mass density) and the stiffness of the point spring elements were defined with the information collected during the onsite testing campaign and then tuned up based on the global dynamic parameters resulting from the OMA through the semi-automated model updating method of Douglas-Reid [27]. This process, including a preliminary study on the most sensitive structural parameters, is described in detail in the following paragraphs.