# Seismic Analysis of Slender Monumental Structures: Current Strategies and Challenges
## 3 Dynamic Amplification and Higher Modes  
In the of case of slender monumental historical structures, mostly built using masonry, the problems associated with the accurate modelling of the material internal structure add up to the uncertainties related to their dynamic behavior during earthquakes. Although apparently fixed to the naked eye, each structure does experience small oscillations that are dependent on its physical and mechanical properties, like mass, stiffness and energy dissipation, as well as on its boundary conditions. These small movements allow one to obtain vibration modes, i.e., different patterns in which the system tries to oscillate naturally, described by specific dynamic features, i.e., frequencies, mode shapes and damping ratios. Vibration modes can be tracked over time in a continuous or intermittent way for manifold objectives, including the calibration of realistic numerical models, the validation of strengthening interventions or the assessment of the structural conditions in pre- and post-seismic scenarios [29, 30, 31, 32, 33, 34, 35, 36, 37]. Characterizing ancient masonry constructions from a dynamic point of view allows a better interpretation of their structural behavior, which is complex and rather diverse, especially during earthquakes [19, 38]. The most common procedure to identify the dynamic properties of historical structures is through operational modal analysis [30, 39, 40, 41, 42], a dynamic identification technique for estimation that requires only records of the structural response to freely available ambient excitations (like wind, traffic, microtremos and human walking), thereby allowing one to take into account the true operational and boundary conditions of the system. With the input load being unknown, modal parameters are identified from output-only data by applying suitable stochastic modal identification techniques, either in the frequency domain and/or in the time domain [30]. Among the main crucial aspects to consider in the seismic assessment of slender monumental structures is the considerable dynamic amplification that can occur if relevant frequencies of the structure are in the range of the predominant frequencies of the ground motion [43, 44, 45]. Indeed, the application of a dynamic force to a vibrating system causes oscillations that will tend to increase in amplitude whenever the frequency of the applied force coincides or approaches one of the system's natural frequencies, leading to the so-called resonance phenomenon. As an example, Figure 4 and Table 1 show the response measured by the strong motion recorders installed in the Church of the Jeronimos Monastery in Lisbon, Portugal (one located at the base of the structure near the chancel, and the other one installed on the wave extrados) during the 6-magnitude earthquake occurred on 12 February 2007. A dynamic amplification occurred in both transversal and vertical directions; indeed, the peaks of the frequency content of the measured response spectra fell exactly within the range of natural frequencies estimated for the church. This phenomenon affected the vibration response of the structure, but the estimated frequency downshifts did not exceed the confidence limits established around each predicted value, meaning that only minor damage took place during the earthquake, see [46, 47] for details. In the worst-case scenario, dynamic amplification can originate from an extreme structural response, with large vibrations triggering important nonlinear phenomena and even partial or global collapses (Figure 5). Figure 4: Seismic response of the Church of Jerónimos Monastery, Portugal (adapted from [46,47]): (**a**) exterior view; (**b**) location of strong motion recorders (A1: chancel base; A2: nave vault extrados); (**c**,**d**) acceleration response at the base and at the top of the nave during the earthquake; (**e**) measured response spectra.