# Seismic Analysis of Slender Monumental Structures: Current Strategies and Challenges
## 4 Structural Analysis Methods  
In the light of the above considerations, it is clear that the inherent characteristics of slender monumental structures considerably affect their behavior under horizontal loads. Most of these structures were built in ancient times, in the absence of specific seismic codes, thus they were essentially conceived to withstand static vertical loads. In recent times, national and international standards [50; 51; 52] have imposed the evaluation of their seismic performance, encouraging the use of sophisticated nonlinear analysis methods [6]. However, in spite of the latest computational developments, structural analysis in earthquake engineering remains a complex task because the activated structural behavior is typically nonlinear and uncertain, and the input data concerning the ground motions are random and unpredictable [3; 53]. The accuracy of the analysis method is of crucial importance, since a conservative approach may lead to unnecessary expensive interventions [54] (likely with a loss of cultural value), while a non-conservative approach may leave buildings exposed to excessive risk. In the framework of the seismic analysis of slender structures belonging to the cultural heritage, linear and non-linear methods, static or dynamic, can be used for their global assessment [7; 55; 56; 57], while limit analysis methods can be adopted to evaluate predefined mechanisms [51; 58; 59]. The main differences among the various global analysis procedures concern the assumptions about the behavior of the structure, the modelling of the seismic action and the computation of the structural response to that action. Linear analysis methods assume the structural behavior is linear elastic and the seismic demand is computed through the q-factor approach (defined as seismic behavior factor, in Europe, or seismic response modification factor, in the USA), namely by introducing an elastic force reduction factor which accounts for the typology, energy dissipation capacity and regularity of the structure, where the latter aspect is rather arguable for ancient masonry constructions. In the case of linear static analysis or \"lateral force method\", the seismic \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{3}{c}{**Before**} & \multicolumn{3}{c}{**After**} & \multirow{2}{*}{\(\Delta_{f}\)} & \multicolumn{3}{c}{**Before**} & \multicolumn{3}{c}{**After**} & \multirow{2}{*}{\(\Delta_{k}\)} \\ \cline{2-2} \cline{5-10}  & \(f\) **(Hz)** & & & & & & & & & & \\ \hline
1 st & 2.15 & 1.85 & 2.56 & 0.21 & +19.3 & 2.68 & 219.5 & 1.25 & 0.13 & \(-\)53.3 \\
2 nd & 2.58 & 1.05 & 2.76 & 0.30 & +6.7 & 1.71 & 94.0 & 1.35 & 0.17 & \(-\)21.0 \\
3 rd & 4.98 & 0.69 & 7.15 & 0.27 & +43.7 & 2.05 & 65.3 & 1.20 & 0.14 & \(-\)41.3 \\
4 th & 5.74 & 1.56 & 8.86 & 0.47 & +54.4 & 2.40 & 24.3 & 1.31 & 0.13 & \(-\)45.7 \\
5 th & 6.76 & 1.13 & 9.21 & 0.21 & +36.1 & 2.14 & 31.7 & 1.16 & 0.12 & \(-\)45.6 \\
6 th & 7.69 & 2.94 & 15.21 & 2.24 & +97.9 & 2.33 & 56.0 & 2.54 & 0.24 & \(+\)9.1 \\
7 th & 8.98 & 1.21 & 16.91 & 1.40 & +88.3 & 2.30 & 46.4 & 1.49 & 0.23 & \(-\)35.1 \\ Avg & – & 1.49 & – & 0.73 & +49.5 & 2.23 & 76.8 & 1.47 & 0.17 & \(-\)40.3 \({}^{*}\) \\ \hline \hline \end{tabular}  
* Average value of damping calculated only for negative differences.  
\end{table} 
Table 2: Dynamic response of Mogadouro Tower before and after rehabilitation. Figure 7: Overview of the Mogadouro tower before, during and after the rehabilitation works. action may be modelled through equivalent lateral forces proportional to the inertial masses and distributed according to the main vibration mode of the structure in the examined direction, whereas, in the case of linear dynamic analysis or \"response spectrum analysis\", the effects induced by the seismic action, represented by the elastic response spectrum, are computed--and then combined--for all vibration modes with effective modal mass larger or equal to 5% and activating at least 85% of the overall mass.