## 5 The Step towards Full Dynamic Seismic Assessment Tools
### Code Based Assessment According to the Italian Building Code  
Once the load multipliers and mechanisms of the case studies have been computed, a force-based and equivalent static displacement-based assessment was conducted, using the procedure defined in the Italian building code [16; 17]. The code-based assessments were performed for the so-called life-safety limit state (SLV) in the code, which typically corresponds to a return period of 475 years. Regarding the force-based assessment, the code defines the acceleration capacity as the spectral acceleration of mechanism activation \(a_{0}^{*}\), which is computed by using the following equation:  
\[a_{0}^{*}=\frac{\widetilde{\lambda}g}{e^{*}CF} \tag{9}\]  
Figure 10: (**a**) Load multiplier vs. Iterations and comparison with the reference value arising by using a non-associative friction solution [41]. (**b**) Geometry representation of the failure mechanism at the last iteration. Figure 9: (**a**) Geometry configuration; (**b**) failure mechanism obtained by using non-associative friction solution for load configuration given Reference [41]. where \(\widetilde{\lambda}\) is the load multiplier that activates the rocking mechanism (Section 2.2), \(g\) is the acceleration due to gravity, \(CF\) is the confidence factor (which is here assumed equal to 1.00) and \(e^{*}\) is the participating mass fraction related to the effective participating mass \(M^{*}\). The latter two parameters are computed using the following equations:  
\[M^{*}=\frac{(\sum_{i=1}^{n}W_{i}\delta_{x,i})^{2}}{g\sum_{i=1}^{n}W_{i}\delta_ {x,i}^{2}} \tag{10}\]  
\[e^{*}=\frac{gM^{*}}{\sum_{i=1}^{n+m}W_{i}} \tag{11}\]  
where \(W_{i}\) and \(\delta_{x,i}\) are defined in Section 2.1. The corresponding acceleration demand is equal to the PGA \(a_{g}\) multiplied by the soil factor \(S\), in order to get the maximum acceleration expected on the site, divided by the behavior factor \(q\), which is typically equal to 2 for masonry buildings. Here, the objective is to determine the PGA \(a_{g,FB}\) for which this safety criterion is violated, which is found by setting the demand and capacity equal to each other, resulting in:  
\[a_{g,FB}=\frac{qa_{0}^{*}}{S}=\frac{q\widetilde{\lambda}g}{Se^{*}CF} \tag{12}\]  
In the case of the displacement-based procedure, the ultimate displacement capacity \(d_{SLV}^{*}\) is defined as 40% of the displacement \(d_{0}{}^{*}\), which would cause the structure to overturn. Specifically, \(d_{0}{}^{*}\) is computed as the spectral displacement, which corresponds to the null value of the spectral acceleration. Thus:  
\[d_{SLV}{}^{*}=0.4d_{0}{}^{*} \tag{13}\]  
The corresponding displacement demand \(\Delta_{d}\) is then computed by evaluating the elastic spectral response acceleration \(S_{e}\) at the equivalent secant period of the mechanism \(T_{SLV}\) with the latter calculated at \(d_{SLV}{}^{*}\). The acceleration \(a_{SLV}{}^{*}\), corresponding to this displacement, is obtained from the capacity curve generated for each mechanism (see Figure 11). The control points adopted to generate these curves were selected to coincide with the centroid of the macro-blocks. As Figure 11 illustrates, the curves are linear for both mechanisms and are defined by \(a_{0}^{*}\) and \(d_{0}^{*}\), where \(d_{0}^{*}\) is the overturning displacement of the structure, as defined above. The displacement demand \(\Delta_{d}\) can thus be calculated as:  
\[\Delta_{d}=\frac{T_{s}^{2}}{4\pi^{2}}gS_{e}(T_{s}) \tag{14}\]  
where:  
\[T_{s}=1.68\pi\sqrt{\frac{d_{SLV}{}^{*}}{a_{SLV}{}^{*}}} \tag{15}\]  
In the present study, the displacement demand is computed by means of the 5% damped elastic response spectrum, as defined in Eurocodes [42]. Note that Eurocode8 recommends two different design spectrums-Type 1 for high and moderate seismicity regions and Type 2 for low seismicity regions and near field earthquakes. The Type 1 spectrum was used for these analyses as the church is located in Italy, which is a zone more prone to moderate magnitude earthquakes at closer distances.