1. Introduction
The conservation and the safety of art collections have been gaining increasing interest in the academic community [
1,
2,
3,
4,
5,
6]. Artworks exhibited within museums are safer than those located outdoors since they are protected against some of the possible threats, such as vandalism, blasts [
7,
8], traffic, thermal variations and environmental actions [
9,
10]. However, at the occurrence of earthquakes, the building hosting the museum can increase the actions affecting the artworks [
11,
12] or, at least, amplify the seismic acceleration at the ground level. This amplification of seismic actions can especially occur for museums consisting of monumental buildings, which hardly respect the current safety standards.
Seismic performance—and safety—of artworks largely varies depending on their properties [
13,
14,
15,
16,
17], such as dimension, shape, material, age, and state of conservation. The dynamic response of a standing object can be described on the basis of the following behaviors: sticking, sliding, rocking and free flight [
18]. Sticking describes the stationarity of the object at the occurrence of the excitation, and it does not present any concern for the object’s safety. Sliding is the lateral displacement of the object across its base surface, and it occurs when the inertial force overcomes the friction between the contact surfaces. It can be dangerous for the artworks’ safety, since they can abruptly bump into other objects or the room’s walls. The rocking consists of the pivoting around the object’s base edge. It is very common for slender objects, having relevant height-to-width ratios, especially for high values of friction between the object’s foot and the bottom surface. It is the most dangerous dynamic response of artworks, since it can induce their overturning, which is often associated with severe damage. Finally, the free flight, considered to be a limit scenario, occurs when the object completely loses contact with its base, becoming airborne [
19,
20].
This work is focused on the combined occurrence of rocking and sliding. The considered case study is the artwork “Marzocco”, the stone masterpiece by Donatello, which is the heraldic symbol of Florence, currently exhibited at the National Museum of Bargello. It follows the investigation presented in Azzara et al. 2023 [
21] on Marzocco, where its seismic performance was checked by means of a simplified approach, consisting of comparing the threshold displacement for overturning to the horizontal displacement of the statue found through the floor spectra provided by the Italian Technical Code [
22]. In this work, the seismic response of the artwork is represented through a time-history analysis, where the seismic input consists of an ensemble of seven ground motions, whose average spectrum closely fits the elastic spectrum provided by the Italian Technical Code [
22] for the location’ soil type. Since the statuary complex is located on the first floor of the building, the seismic action experienced by Marzocco can consistently differ from that at the ground level. Therefore, the dynamic analysis has been performed with reference to the seismic inputs both at the ground and at the floor levels, to check the difference between these two conditions.
The seismic action on the first floor has been found accounting for the transfer-effect of the building by means of a simplified SDOF model having the same frequency as the building, i.e., T = 0.35 s. Such frequency, in turn, has been found by the Authors in a former investigation, through dynamic monitoring by means of seismic motions, as described in [
21].
The seismic performance of Marzocco has been checked with reference to both rocking and sliding. The most probable dynamic response of the case study depends on the consistency of the connection between its components to the corresponding support [
23,
24], which is difficult to predict without specific information about the artwork setup. As regards Marzocco, it appears to be cemented to its base through mortar. This connection between sculptures and their pedestals was quite common in past centuries, and it presents high friction. For this reason, overturning has been considered to be the main collapse mechanism of the case study, and a proper structural model, allowing the detachment between the statue and its support, has been assumed for the numerical analysis. Such modeling is not the most suitable for representing the material degradation, which has not been studied in the current research.
The artwork consists of a stone statue placed over a marble base and a further marble pedestal; therefore, three overturning mechanisms have been considered, respectively involving the statue only (Model A), the statue and the base (Model B), and the entire complex, i.e., the statue, the base, and the pedestal (Model C).
The dynamic response of the case study, represented through the three models, has been checked in terms of rocking angle and sliding displacement. The comparison between the dynamic response of the three models and the corresponding thresholds showed Model A (overturning of the statue only) to be the most compatible with the overturning collapse. Therefore, Model A has been selected to perform a further analysis, where a 2-dimensional seismic input has been assumed. In terms of the rocking, the response provided by the latter analysis has been very similar to that induced by the 1-dimensional seismic input. However, the lateral displacement due to sliding was largely increased by the two-dimensional input, affecting the seismic safety of the case study.
The obtained results show that Marzocco has a good seismic performance, and it does not overturn for earthquakes compatible with the seismic hazard of the area. The analysis shows the importance of considering floor accelerations, which are consistently larger than those acting at the ground level, inducing much larger seismic responses. The performed analysis leads to assessing the role of the various assumptions, such as the selection of the spectra, the choice of the structural model and the two-directionality of the seismic input.
6. Conclusions
In this work, the dynamic response of the statue “Marzocco” by Donatello to seismic excitation has been checked. The possible collapse mechanism of the artwork, consisting of the statue, its base and its pedestal, has been investigated through three different models, which assume, respectively, the overturning of the statue only, of the statue with its base, and of the entire complex.
The seismic response of the three models has been found with reference to an ensemble of seven ground motions, spectrum compatible with the elastic response spectrum provided by the current Italian Technical Code for the Life Safety limit state (return period equal to 712 years) and a soil type B, whose PGA is equal to 0.1764 g.
The ensemble of ground motions represents the seismic input at the ground. Since the room where Marzocco is exhibited is located on the first level, the seven ground motions have been filtered through a simplified SDOF system having the same frequency as the first level of the building. The seismic response of the artwork has been found both using the original ensemble, representing the seismic ground excitation, and the arranged one, representing the floor excitation.
The performed analyses provided the following evidence:
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The seismic performance of the case study fully complies with the limit conditions both for rocking and sliding.
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The floor excitation provides a dynamic response much larger (over 1000 times) than the ground excitation. Such amplification is largely affected by the frequency content of each ground motion. Anyway, the effect of the input propagation played by the building cannot be neglected.
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The response of the case study is largely affected by the considered ground motion (both the ground and the floor ones): some of the ground motions induce a maximum displacement 6 times larger than the mean displacement of the ensemble.
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The three models provide different predictions of the dynamic response of the artwork. Namely, the most significant, i.e., “dangerous”. The collapse mechanism is the overturning of the statue only (Model A).
With reference to Model A, a further investigation has been carried out, consisting of a two-dimensional analysis. The rocking obtained in this latter analysis was similar to that found in the previous analysis; the sliding, instead, was much larger than that provided by the 1d input. As a consequence, the Safety Index, found as the ratio between the capacity of the system and its dynamic response, changes very much, going from 1d to 2d seismic input.
The work offered an exhaustive investigation of “Marzocco”, facing its possible collapse mechanisms, overturning and sliding. It explains the importance of a careful assumption of the seismic input, which must take into account the exact position of the artwork within the exhibition building. Furthermore, the work shows a consistent parametric analysis of the main choices affecting the analysis, and it provides a comprehensive assessment of the seismic safety of “Marzocco”.