# Seismic Analysis of Slender Monumental Structures: Current Strategies and Challenges
## 4 Structural Analysis Methods
However, time-dependent response and loading history are not available with this type of analyses. Among the main issues concerning the seismic behavior of slender monumental structures is the influence of the vertical stresses induced by gravity loads, whose values are often of the same order of magnitude as the ultimate compressive strength of the material [7]. This aspect, combined with the very low tensile strength of masonry, means that the non-linear behavior of these structures can be triggered even by a moderate increase in the stress level in the case of earthquakes, thus the use of inelastic procedures such as non-linear static or dynamic analyses seems far more adequate for their assessment. In what concerns non-linear static analysis, viz. the classical \"pushover analysis\", the seismic force is applied to the structure as an incremental static lateral loading, usually assuming a first-modal or mass-proportional force distribution, and results are plotted in the form of a capacity curve (base shear vs. roof top horizontal displacement). Pushover analysis represents one of the most rational analysis methods for practical applications, but it was originally devised for single-mode dominated structures. To account for multiple modes contribution, several proposals have emerged in the literature in the last two decades and the recourse to such methods is explicitly allowed by the EC8 Part 3 [60]. One of these methods is the \"modal pushover analysis\" (MPA) introduced by [61] that combines classical pushover and response spectrum approaches. The effects induced by the seismic action are computed by combining the results of \(N\) pushover analyses, where \(N\) is the number of modes chosen based on the amount of activated mass, carried out assuming an invariant inertia force distribution for each mode. Despite the conceptual simplicity and computational attractiveness, MPA relies on the same approximations of all multi-mode methods, such as the superposition of effects and the modal combination rules, which are valid for elastically responding structures but rather questionable in the inelastic range. It follows that the non-linear dynamic analysis or \"time history analysis\" seems to be a more suitable option for the seismic assessment of masonry structures. Here, the seismic input is represented by spectrum-compatible accelerograms, whose appropriate selection is important to account for the variability associated with different input characteristics in terms of amplitude, energy and frequency content. According to the Italian [50; 51] and European [52] codes, at least seven time-history analyses need to be carried out to embrace a wide range of responses, and mean output values are to be considered [62]. Incremental dynamic analysis methods employing multiple non-linear dynamic analyses under scaled ground motion records have also been developed in order to force the structure all the way from elasticity to final global dynamic instability and to have a comprehensive evaluation of the seismic performance of structures [63]. Yet, their application to special constructions, such as ancient monumental structures, is still in its infancy. Considering the peculiar behavior of slender masonry structures, not all analysis methods result adequate. In this regard, Figure 8 displays the results obtained from different types of analysis carried out for the seismic assessment of the Qutb Minar tower [64], the highest monumental structure in India and one of the tallest masonry towers in the world (Figure 9). It is interesting to note that the nonlinear static and dynamic analyses provide quite a different response of the structure to earthquakes.