# Onsite Testing for Nonlinear Analysis of an Earthquake Damaged Historical Church in Italy
## 5 Nonlinear Analysis
### Nonlinear Dynamic Analysis  
Nonlinear time-history analysis was performed using the three synthetic accelerators generated from the elastic design spectra according to the Italian Code [15] at the ultimate limit State (SLV) with the software SIMQKE [35] reported in Figure 21, where SLV EQX, EQY and EQZ are the spectra of the accelerograms in the three principal directions. The accelerograms were applied in the three directions, for a total duration of 10 s due to the cumbersome of the computer runs. First, the three control points are defined based on the expected maximum drifts: TL (top of the lantern), BL (base of the lantern) and BD (base of the dome). The moduli of the relative acceleration are monitored over the time history analysis as reported in Figure 22a. The maximum acceleration results at 8.28 s in node TL positioned at 35.6 m height. At this time step, the displacements of the three control points are linearly proportional to the height of the structure, thus no relevant damage appears yet, as shown in Figure 22b. However, the acceleration of node TL results 3.4 times higher than BD and the acceleration of BL results 1.5 times higher than BD. This means that the lantern is subject to higher accelerations than the rest of the structure because of the dynamic amplification, this effect yields to the formation of local collapse mechanism with the persisting of the earthquake. The structural response of the model is analyzed in Figure 23, considering the longitudinal direction (X) by comparing the displacement of the control point TL with the seismic coefficient (\(\alpha_{\mathrm{x}}\)) and the corresponding maximum principal strain distribution (tensile deformation). The time history highlights the residual translation of control point TL towards the negative X direction (facade side), the principal strain status during the maximum positive displacement (6.4 mm) in Figure 23a shows reasonable homogeneous distribution due to the small seismic excitation. Starting from the maximum seismic coefficient (0.24) in Figure 23b, the critical portions of masonry are identified in correspondence of the central part of the facade and the dome along the longitudinal direction. At the maximum seismic coefficient in the negative direction (\(-\)0.23) shown in Figure 23c, the principal strains increase by one or two orders of magnitude and all the potential collapse mechanisms emerge (roof of Palazzo della Provincia, central portion of the south wall, roof of the sacristy on the southeast side, south-west diagonal crack of the dome) which are confirmed by the last plot during the maximum negative displacement (\(-\)54 mm) shown in Figure 23d, corresponding to the maximum concentrations of principal strains as a response to the maximum seismic coefficient in the negative direction and the accumulated damage. Figure 21: Horizontal and vertical elastic response spectra in Macerata with synthetic generated accelerograms. Figure 22: (**a**) Time history acceleration of the control points, (**b**) deformed shape and relative total acceleration at 8.28 s and (**c**) relative total acceleration and total displacement of the control points along the height of the structure. The nonlinear dynamic response is also analyzed considering the transversal direction (Y) in Figure 24. Along the transversal direction, control point TL performs large oscillations around the initial position highlighting higher deformation capacity than the longitudinal direction without any evident residual displacements. The distribution of maximum tensile principal strain during the maximum seismic coefficient in the negative direction (\(-0.16\)) and the consequent maximum displacement (\(63\) mm) in Figure 24a,b shows the presence of the already mentioned potential collapse mechanisms, in addition to the warning due to stress and strain concentration at the base of the two facade towers.