Seismic Vulnerability Assessment and Simplified Empirical Formulation

geosciences

Article Seismic Vulnerability Assessment and Simpliﬁed Empirical Formulation for Predicting the Vibration Periods of Structural Units in Aggregate Conﬁguration

Nicola Chieffo 1,* , Antonio Formisano 2 , Giovanni Mochi 3 and Marius Mosoarca 1

1 Department of Architecture and Urbanism, Politehnica University of Timi¸soara,Traian Lalescu Street, 300223 Timi¸soara,Romania; [email protected] 2 Department of Structures for Engineering and Architecture, School of Polytechnic and Basic Sciences, University of Naples “Federico II”, P.le V. Tecchio, 80125 Naples, Italy; [email protected] 3 Department of Civil and Environmental Engineering, University of Perugia, 06123 Perugia, Italy; [email protected] * Correspondence: [email protected]

Abstract: The present research aims at investigating the vibration period of structural units (SUs) of a typical masonry aggregate located in the historical center of Mirandola, a municipality in the province of Modena. The clustered building consists of eighteen SUs mutually interconnected to each other, which are characterized by solid brick walls and deformable floors. First of all, non-linear static analyses are performed by adopting the 3Muri software focusing on two distinct modelling techniques concerning the analyzed SUs in isolated and clustered configurations. Congruently to the procedure adopted, in order to evaluate a reliable seismic structural response of the SUs arranged in Citation: Chieffo, N.; Formisano, A.; aggregate conditions, the contribution in terms of stiffness and mass derived from adjacent buildings Mochi, G.; Mosoarca, M. Seismic is considered. The analysis results are represented in terms of risk factor, stiffness, and ductility. Vulnerability Assessment and Simplified Empirical Formulation for Secondly, the eigenvalue analysis is faithfully developed to identify the main vibration modes of the Predicting the Vibration Periods of investigated SUs by proposing an empirical formulation, that allows for predicting the vibration Structural Units in Aggregate period of structural units placed in aggregate configuration starting from the corresponding isolated Configuration. Geosciences 2021, 11, ones. Finally, fragility functions are derived for both the heading and intermediate SUs to point out 287. https://doi.org/10.3390/ the expected damages under earthquakes with different intensities. geosciences11070287 Keywords: masonry aggregates; non-linear static analysis; eigenvalue analysis; risk factor; ductility; Academic Editors: Sabina Porfido vibration period; fragility curves and Jesus Martinez-Frias

Received: 4 June 2021 Accepted: 7 July 2021 1. Introduction Published: 9 July 2021 The seismic safety of existing masonry buildings represents, as it is known, one

Publisher’s Note: MDPI stays neutral of the main priorities for safeguarding historical centers of countries belonging to the with regard to jurisdictional claims in Mediterranean area [1,2]. Generally, over the centuries, masonry constructions were published maps and institutional affil- considered as the predominant construction type, considering both the easy finding of the iations. basic material and the simple execution phases of the artworks. However, awareness of the architectural and structural development that characterized the masonry constructions throughout the various eras is not only an essential prerequisite for the design of new buildings, but also constitutes an essential technical knowledge to clearly intervene on the existing building heritage with maintenance, consolidation, upgrading or seismic Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. adaptation interventions [3]. This article is an open access article Nowadays, in most Italian centers, it is easy to find a built heritage of historic masonry distributed under the terms and structures without an adequate safety level to resist the increasingly frequent and disastrous conditions of the Creative Commons seismic actions. This structural inadequacy generates a drastic increase in terms of global Attribution (CC BY) license (https:// vulnerability and, therefore, of the seismic risk of entire urbanized sectors. Thus, seismic creativecommons.org/licenses/by/ vulnerability represents one of the main elements to be taken into consideration for the 4.0/). protection of historic buildings [4,5].

Geosciences 2021, 11, 287. https://doi.org/10.3390/geosciences11070287 https://www.mdpi.com/journal/geosciences Geosciences 2021, 11, 287 2 of 23

As it is well known, vulnerability can be understood as the propensity of buildings to suffer a certain level of damage due to a seismic event. The methods for assessing seismic vulnerability suggested by the design code require a wide and articulated knowledge of the essential typological and structural requirements (types of connection between walls, types of floors and roofs, etc.) of construction [6–8]. Since these existing buildings are frequently grouped in aggregate configuration into urban centers, the intrinsic vulnerability factors are difficult to be identified in advance, since the construction matrix of SUs is the living representation of the evolution of a set of design techniques relating to different time periods. Moreover, the complex structural articulation makes these buildings mutually interacting, so to further aggravate their susceptibility at damage under seismic phenomena [9–11]. Buildings erected in aggregate conditions are often built according to traditional prac- tice with very variable types of vertical structure (heterogeneous or multi-leaf masonry walls) and deficient construction details (bad connections between orthogonal walls and between walls and floors, variation in the thickness of the walls along with the building height), which implicitly involve behavioral deficiencies in terms of stability and safety in areas characterized by considerable seismicity. Therefore, such types of clustered constructions have a structural behavior that cannot be identified a priori to predict the possible failure scenario [12–14]. Nonetheless, several factors influence their seismic performance, mainly depending on the interactions among the individual SUs. In particular, the presence of not significantly effective connections between the different SUs favors the occurrence of local collapse mechanisms in different cases. Jointly, an aggravation of the kinematic stability conditions is given by the presence of construction irregularities (e.g., orthogonal walls not well connected) and/or geometric (e.g., buildings with different heights), which are the main causes of the activation of out-of-plane collapses [15,16]. The interactions between the contiguous structural cells must be appropriately considered in the study of the vulnerability of the whole aggregate since the dynamic response of each SU is strongly influenced by the mutual interaction between the structural bodies involved in the seismic scenario [17–19]. The capacity offered by the single SU against the earthquake may differ considerably from the capacity of the entire aggregate, especially in the case of deformable floors and walls with poor mechanical characteristics. Such constructions, characterized by a strongly non-linear response, can inevitably lead to evaluation errors, that could compromise the result of the seismic evaluation [20,21]. For this reason, it is essential to consider a simplified structural model to take into account all the intrinsic peculiarities of the SUs, that can more correctly estimate their vulnerability level. More generally, the dynamic characterization, which is obtained through stochastic modal identification methods in the frequency domain where the source of the exciting impulse is the environmental vibration of the SUs, would be effective for an accurate cognitive investigation of the health state of buildings [22,23]. However, this task is not always possible or easy to be implemented. Therefore, simplified procedures for simulated modal identification analysis of the inspected buildings are developed. In this regard, many mechanical models are analyzed in several scientific papers [4,5,9], where the uncertainties related to vulnerability factors are often considered to quantify the seismic response of the entire aggregate and to set up appropriate risk mitigation procedures [24,25]. Thus, based on these considerations, the present study aims to focus on the seismic response of a masonry aggregate composed of eighteen SUs located in the historical center of Mirandola, a municipality within the province of Modena. This clustered building, affected by the earthquake which occurred in Emilia-Romagna in 2012, is made up of SUs representative of the building classes (solid brick walls with deformable floors) present in the examined urban area. The main objective of the present research is to study the influence of the vibration period of the structural units on the global response of the whole aggregate. Moreover, non-linear static analyses are performed to evaluate the seismic behavior of the SUs in both isolated and aggregate conditions. Geosciences 2021, 11, x 3 of 25

the examined urban area. The main objective of the present research is to study the influence of the vibration period of the structural units on the global response of the whole aggregate. Moreover, non-linear static analyses are performed to evaluate the seismic behavior of the SUs in both isolated and aggregate conditions. Geosciences 2021, 11, 287 From the acquired results, it is observed that the behavior offered in an aggregate3 of 23 condition considerably increases the expected seismic response of SUs, providing the main engineering demand parameters (EDPs), such as the maximum base shear and displacements, are greater than those of the corresponding isolated SUs. Furthermore, From the acquired results, it is observed that the behavior offered in an aggregate through mechanical simulation, it is detected how the vibration period decreases in ag- condition considerably increases the expected seismic response of SUs, providing the main gregate configurations due to the increase in both the stiffness and mass of single SUs engineering demand parameters (EDPs), such as the maximum base shear and displacements,deriving are from greater the adjoining than those clustered of the corresponding buildings. To this isolated purpose, SUs. a Furthermore, simplified mathemat- through mechanicalical formulation simulation, is provided it is detected for predicting how the the vibration vibration period period decreases of aggregate in aggregate SUs confined, con- figurationsstarting from due that to theof the increase same instructural both the cells stiffness in isolated and mass configuration. of single SUs Finally, deriving a set from of thefragility adjoining curves clustered are derived buildings. to estimate To this the purpose, propensity a simplified at the damage mathematical of the formulationSUs investi- isgated. provided for predicting the vibration period of aggregate SUs confined, starting from that of the same structural cells in isolated configuration. Finally, a set of fragility curves are2. Framework derived to estimateof the Study the propensity at the damage of the SUs investigated. 2.1. Historical Centre of Mirandola 2. Framework of the Study The historic center of Mirandola developed over the centuries was based on a casual 2.1.urban Historical process. Centre The first of Mirandola “Forma Urbis”, apparently casual, responds to settlement rules predisposedThe historic to the center creation of Mirandola of roads, developed public spaces over thenecessary centuries for was civic based life, onand a casualpublic urbanbuildings, process. where The the first bureaucratic “Forma Urbis”, activities apparently of the entire casual, community responds were to settlement carried out. rules predisposedThe historic to the center creation of Mirandola of roads, is public the repr spacesesentation necessary of the for spatial civic configuration life, and public of buildings,urban interventions where the developed bureaucratic since activities the Middle of the Ages. entire The community city was wereerected carried as a fortified out. city, Thein which historic the center conformation of Mirandola of the is histor the representationical settlement of is the well spatial structured configuration and orga- of urbannized. interventionsThe pole of the developed urban layout since is the the Middle Pico Castle, Ages. which The city stands was erectedon the hill as aof fortified the Fa- city,vorita, in whicha connecting the conformation element with of the the historical adjacent settlementvillages and is wellthe conformation structured and of organized. the main Theroads. pole The of urban the urban configuration layout is theof Mirandola Pico Castle, was which conceived stands onby theincorporating hill of the Favorita,the most aimportant connecting buildings element (houses, with the churches, adjacent villagesand palaces) and the formed conformation around the of the Pico main Castle, roads. to Thecreate urban a planimetric configuration asset of with Mirandola new architectu was conceivedral elements. by incorporating In general, the the most buildings important are buildingsarranged in (houses, a clustered churches, configuration and palaces) on the formed main path, around presenting the Pico a Castle, free front to create for the a planimetricexternal view asset and with access new and architectural an opposite elements. front that In overlooks general, the buildingsrelevant area are (Figure arranged 1) in[26]. a clustered configuration on the main path, presenting a free front for the external view and access and an opposite front that overlooks the relevant area (Figure1)[26].

FigureFigure 1.1. Historical-urbanHistorical-urban developmentdevelopment ofof thethe citycity ofof MirandolaMirandola [[26].26].

Nowadays, the the new new buildings, buildings, the the union union of ofthe the villages, villages, and and the theorganization organization of new of newneighborhoods neighborhoods have havekept unchanged kept unchanged the primordial the primordial planimetric planimetric configuration configuration of the city of the(Figure city 2). (Figure Actually,2). Actually, the historical the historical center is centercharacterized is characterized mainly by mainly masonry by buildings, masonry buildings, erected in aggregate configuration, in which the coexistence of construction heterogeneities, sometimes incongruous, has led to the manifestation of specific seismic vulnerability factors.

2.2. Case Study Aggregate The historic center of the city of Mirandola consists of 43 urban areas consisting of buildings erected in the aggregate conﬁguration, as speciﬁed in the Recovery Plan of the city approved in July 2001. The characterization of the present built-up heritage highlights how the size of urban areas is connected to the evolutionary process of the city. Although Mirandola did not Geosciences 2021, 11, x 4 of 25 Geosciences 2021, 11, 287 4 of 23

erected in aggregate configuration, in which the coexistence of construction heterogenei- haveties, sometimes a radial growth, incongruous, as hashappened has led to forthe othermanifestation historic Italianof specific cities, seismic it is noted vulnerability that the dimensionalfactors. variation of the building aggregates is related to the transition from the ﬁrst expansion phase of the city (quadrangular plan) to the second one (octagonal plan) [26].

(a) (b)

FigureFigure 2. HistoricalHistorical compartment compartment and and urban urban plan plan of ofthe the city city with with the themain main roads roads and andsub-urban sub-urban borough borough (a,b) ([26];a,b)[ (c26,d];) (stylesc,d) styles of architectural of architectural buildings. buildings.

2.2. CaseGlobally, Study theAggregate buildings located in the historic center are mainly characterized by constructionThe historic heterogeneity center of the (solid city brickof Mirandol walls)a with consists the presenceof 43 urban of semi-rigidareas consisting and/or of deformablebuildings erected ﬂoors. in This the buildingaggregate type configuration, represents the as mostspecified common in the construction Recovery Plan typology of the ofcity the approved historical in centers July 2001. of the Emilia-Romagna Region. ConcerningThe characterization the case study of the building, present built-up it is an existing heritage masonry highlights row how aggregate the size composed of urban ofareas 18 SUsis connected mutually interactingto the evolutionary with each proc otheress (Figure of the3 ).city. So, theAlthough structural Mirandola units identiﬁed did not

Geosciences 2021, 11, x 5 of 25

have a radial growth, as has happened for other historic Italian cities, it is noted that the dimensional variation of the building aggregates is related to the transition from the first expansion phase of the city (quadrangular plan) to the second one (octagonal plan) [26]. Globally, the buildings located in the historic center are mainly characterized by construction heterogeneity (solid brick walls) with the presence of semi-rigid and/or deformable floors. This building type represents the most common construction typology of the Geosciences 2021, 11, 287 historical centers of the Emilia-Romagna Region. 5 of 23 Concerning the case study building, it is an existing masonry row aggregate composed of 18 SUs mutually interacting with each other (Figure 3). So, the structural units identified as SU1 and SU18 (see Figure 3b), located on the external part of the aggregate, as SU1 and SU18occupy (see the Figure head3 position;b), located instead, on the the external 16 internal part cells of the are aggregate, placed in an occupy intermediate posi- the head position;tion. On instead, average, the 16the internal floor number cells are of placed SUs vari ines an from intermediate two to four position. above Onground with an average, the ﬂooraverage number inter-story of SUs height varies of from 2.80 two m. The to four vertical above structures, ground with characterized an average by solid brick inter-story heightwalls, of have 2.80 m.an Theaverage vertical thickness structures, ranging characterized from 0.24 bym to solid 0.50 brick m. The walls, horizontal have structures an average thickness ranging from 0.24 m to 0.50 m. The horizontal structures are mainly are mainly made of wooden elements with upper double planking. made of wooden elements with upper double planking.

(a)

(b)

FigureFigure 3. Identiﬁcation3. Identification of theof the case case study study aggregate aggregate (a) ( anda) and view view along along the the main main street street (b). (b).

Along the clustered building façades, which are oriented in the longitudinal direction, the absence of steel tie-rod elements is noticed. This means that, in this direction, the mutual connection between the various SUs is completely entrusted to the compressive contact actions between the walls of adjacent units. Regarding the physical condition of the examined aggregate, the main view along Francia Corta Street does not show, globally, a marked deterioration of the materials. There are evident spots of humidity and partial detachment of the plaster, which do not alter the building’s static. Similarly, the façade facing Dei Quartieri Street does not show signs of structural decay or cracks. Geosciences 2021, 11, 287 6 of 23

The mechanical properties of the clustered building structural elements were assumed according to the indications prescribed by Table C8.5.I of the NTC18 [27]. Due to the absence of accurate on-site test procedures, the mechanical characteris- tic of the masonry was suitably reduced by assuming a conﬁdence factor, CF, equal to 1.35, which corresponds to a level of knowledge LC1 (limited knowledge), based on the historical-critical analysis, the complete geometric survey and limited investigations on the construction details (§ C8.5.2 of the Italian standard explicative Circular [27]) of the examined building, as well as limited tests on the mechanical characteristics of the materials (§ C8.5.3 of the Italian standard explicative Circular [27]). Under these circumstances, for mechanical parameters (fm, τ0, fv0) the minimum values of the intervals reported in Table C8.5.I of the Italian standard [27] were considered, while for the elastic moduli (E and G), the average values provided by the above-mentioned table were taken into account. Based on these considerations, the following mechanical parameters for solid brick masonry are depicted in Table1:

Table 1. Basic mechanical properties of masonry walls used for inspected SUs [27].

Mechanical Parameter CF (LC1) −2 Average compressive strength fm 2.6 Nmm −2 Shear strength τ0 0.05 Nmm Average shear strength f 0.13 Nmm−2 v0 1.35 Young modulus E 1500 Nmm−2 Tangential elasticity modulus G 500 Nmm−2 Speciﬁc weight w 18.0 KNm−3

3. Seismic Vulnerability Assessment 3.1. Empirical Method To estimate the seismic vulnerability of buildings aggregate, it was adopted a specific vulnerability form proposed in [4,28,29], which is based on Benedetti and Petrini’s vulnerability index method. The proposed vulnerability form, widely recognized internationally, was appropriately conceived for historical aggregate buildings by adding—to the original form—five new additional parameters, which take into account the effects of the mutual interaction among SUs under seismic action. In particular, the new parameters were added to the 10 basic parameters used in the past to survey the buildings considered in isolated configuration [4,28,29] (Table2).

Table 2. Vulnerability form for historical aggregate buildings.

Vulnerability Class Parameters Weight ABCD 1. Organization of vertical structures 0 5 20 45 1 2. Nature of vertical structures 0 5 25 45 0.25 3. Location of the building and type of foundation 0 5 25 45 0.75 4. Distribution of plan resisting elements 0 5 25 45 1.5 5. In-plane regularity 0 5 25 45 0.5 6. Vertical regularity 0 5 25 45 0.8 7. Type of floor 0 5 25 45 0.8 8. Roofing 0 15 25 45 1 9. Details 0 0 25 45 0.25 10. Physical conditions 0 5 25 45 1 11. Presence of adjacent building with different height −20 0 15 45 1 12. Position of the building in the aggregate −45 −25 −15 0 1.5 13. Number of staggered floors 0 15 25 45 0.5 14. Structural or typological heterogeneity among S.U. −15 −10 0 45 1.2 15. Percentage difference of opening areas among adjacent façades −20 0 25 45 1 Geosciences 2021, 11, x 7 of 25

4. Distribution of plan resisting elements 0 5 25 45 1.5 5. In-plane regularity 0 5 25 45 0.5 6. Vertical regularity 0 5 25 45 0.8 7. Type of floor 0 5 25 45 0.8 8. Roofing 0 15 25 45 1 9. Details 0 0 25 45 0.25 10. Physical conditions 0 5 25 45 1 11. Presence of adjacent building with different height −20 0 15 45 1 12. Position of the building in the aggregate −45 −25 −15 0 1.5 13. Number of staggered floors 0 15 25 45 0.5

14.Geosciences Structural2021, 11or, 287typological heterogeneity among S.U. −15 −10 0 45 1.2 7 of 23 15. Percentage difference of opening areas among adjacent façades −20 0 25 45 1

The vulnerability index, IV, was evaluated for each SU as the weighted sum of the class Theselected vulnerability (for each of index, the 15 IV parameters, was evaluated listed for in Table each SU2) multiplied as the weighted by the sumrespective of the weightclass selected [28]. It (foris possible each of to the notice 15 parameters how these listedparameters in Table are2) distributed multiplied into by the four respective decreas- ingweight vulnerability [28]. It is possibleclasses (A, to noticeB, C, D), how whose these weight parameters expresses are distributed the influence into fourof the decreas- single parametering vulnerability on the global classes vulnerability (A, B, C, D), whoseof the building. weight expresses the inﬂuence of the single parameterThe vulnerability on the global index, vulnerability IV, was evaluated of the building. by adopting the following Equation (1): The vulnerability index, IV, was evaluated by adopting the following Equation (1): 15 ⋅ I=Vii15 S W (1) IV = ∑i=1Si · Wi (1) i=1 For an easy assessment of the expected vulnerability, the vulnerability range was as- sumedFor in an the easy interval assessment (0 ÷ 1) ofby the adopting expected a normalization vulnerability, theof the vulnerability vulnerability range indexes was throughassumed the in following the interval mathematical (0 ÷ 1) by adopting Equation a (2): normalization of the vulnerability indexes through the following mathematical Equation (2): 15  ⋅  I-(SVmini15 W) V=IV−( ∑i=1Smin · Wi) I 15 =  (2) V =  i 1  (2) I  15 ()()SW-SW⋅⋅    max i min i  ∑i=1[(Smax · Wi) − (Smin · Wi)] i=1 Thus, based on these conditions, the distribution of the vulnerability indices concern- Thus, based on these conditions, the distribution of the vulnerability indices concerning the analyzed SUs is plotted in Figure 4. ing the analyzed SUs is plotted in Figure4.

FigureFigure 4. 4. VulnerabilityVulnerability index distribution for the SUs of the case study aggregate.

signiﬁcantly inﬂuences the global vulnerability. Subsequently, the typological vulnerability curves were derived to characterize the expected damage of the analyzed SUs varying the macroseismic intensity [30]. Mathemati- cally, the curves were derived according to Equation (3): I − + 6.25 · V − 13.1 µ = 2.5 · 1 + tan h EMS 98 I (3) D Q

From the above-presented equation, the vulnerability curves depend on three main factors, that are the normalized vulnerability index (VI), the hazard, expressed in terms of macroseismic intensity (I), and a ductility factor, Q, which was herein assumed as equal to 2.3 and describes the ductility of typological classes of buildings examined [30]. Geosciences 2021, 11, x 8 of 25

From the acquired vulnerability scenario, it appears that the estimated vulnerability is below the average threshold of 0.5. As it can be seen, the variability of seismic vulnerability is mainly influenced by the in-elevation interaction between adjacent SUs. This issue is related to the presence of adjacent buildings with different heights, which, consequently, significantly influences the global vulnerability. Subsequently, the typological vulnerability curves were derived to characterize the expected damage of the analyzed SUs varying the macroseismic intensity [30]. Mathemat- ically, the curves were derived according to Equation (3): I +6.25⋅ V -13.1 ⋅ EMS-98 I μ=2.51+tanhD  (3) Q From the above-presented equation, the vulnerability curves depend on three main Geosciences 2021, 11, 287 factors, that are the normalized vulnerability index (VI), the hazard, expressed in terms8 ofof 23 macroseismic intensity (I), and a ductility factor, Q, which was herein assumed as equal to 2.3 and describes the ductility of typological classes of buildings examined [30]. Therefore, the mean typological vulnerability curves presented in Figure 5 were plotted to Therefore,estimate the the expected mean typological level of damage vulnerability for the curvesbuildings’ presented stock analyzed. in Figure 5Moreover, were plot- theted other to estimate two curves the expected(VI + 2σ; V levelI − 2σ of) identify damage the for upper the buildings’ and lower stock bounds analyzed. of the statistical Moreover, rangethe other of the two expected curves damage (VI + 2[31].σ;V I − 2σ) identify the upper and lower bounds of the statistical range of the expected damage [31].

(a) (b)

FigureFigure 5. 5. VulnerabilityVulnerability curves curves for for the the case case study study aggregate aggregate (a ()a )and and statistical statistical variation variation of of the the vulnerability vulnerability index index for for each each macroseismicmacroseismic intensity intensity (b (b).). It is worth noting that in Figure5b, the fixed vulnerability range highlights how It is worth noting that in Figure 5b, the fixed vulnerability range highlights how var- varying the hazard threshold is, expressed in terms of macroseismic intensity, and the ying the hazard threshold is, expressed in terms of macroseismic intensity, and the vul- vulnerability indexes progressively increase with a maximum expected damage level equal nerability indexes progressively increase with a maximum expected damage level equal to D4 (partial collapse, VI = 0.48 and µD = 4.19) to D4 (partial collapse, VI = 0.48 and µD = 4.19) 3.2. Mechanical Analyses 3.2. Mechanical Analyses The numerical analyses were performed using the 3Muri software [32] based on the FrameThe Macro-Elements numerical analyses (FME) were theory. performed In this approach,using the 3Muri it was assumedsoftware that[32] masonrybased on wallsthe Frameare considered Macro-Elements as a set of(FME) single-dimensional theory. In this macro-elements approach, it was (columns, assumed beams, that andmasonry nodes) wallssuitably are considered interconnected as a set to eachof single-dimen other. The strengthsional macro-elements criteria of deformable (columns, elements beams, and were nodes)assumed suitably based interconnected on EN 1998-3 provisionsto each other. [33 The], which strength have criteria established, of deformable as maximum elements drifts werefor shearassumed and based flexural on collapse EN 1998-3 failures, provisions the values [33], ofwhich 0.4% have and 0.6%,established, respectively. as maximum Another driftsparameter for shear to be and considered flexural forcollapse the global failures, seismic the structuralvalues of response0.4% and is 0.6%, the damping, respectively. which Anotherconditions parameter the dissipation to be considered of the seismic for energy the global input, seismic limiting structural the forced response vibrations is of the the damping,structure. which More conditions in particular, the thisdissipation parameter of the is based seismic on energy intrinsic input, mechanisms limiting the concerning forced the building material, the structural type, and the type of soil systems, that could be properly defined as the percentage of the total vibration energy lost in a loading cycle. From a standard point of view, NTC18 [27] provides a reference design spectrum assuming a damping value of 5% and suggesting a formulation to modify the spectrum when the effective damping assumes different values. In the case under study, the analyses were carried out along the two main global directions of buildings, X and Y, assuming a design spectrum damping of 5% for the evaluation of the seismic demand, Dmax. These analyses were interrupted at a 20% decay of the maximum shear resistance according to the NTC18 indications [27]. The main topic considered for the reference case study building was to estimate the influence of the mutual interaction among adjacent structural units. The reference case study clustered structural cells are illustrated in Figure6, where the aggregate configuration effect was modelled taking into account the half part of the units contiguous to the reference one accounting for their contribution in terms of both stiffness and mass. Geosciences 2021, 11, x 9 of 25

vibrations of the structure. More in particular, this parameter is based on intrinsic mechanisms concerning the building material, the structural type, and the type of soil systems, that could be properly defined as the percentage of the total vibration energy lost in a loading cycle. From a standard point of view, NTC18 [27] provides a reference design spectrum assuming a damping value of 5% and suggesting a formulation to modify the spectrum when the effective damping assumes different values. In the case under study, the analyses were carried out along the two main global directions of buildings, X and Y, assuming a design spectrum damping of 5% for the evaluation of the seismic demand, Dmax. These analyses were interrupted at a 20% decay of the maximum shear resistance according to the NTC18 indications [27]. The main topic considered for the reference case study building was to estimate the influence of the mutual interaction among adjacent structural units. The reference case study clustered structural cells are illustrated in Figure 6, where the aggregate configuration effect was modelled Geosciences 2021 11 , , 287 taking into account the half part of the units contiguous to the reference one accounting 9 of 23 for their contribution in terms of both stiffness and mass.

(a) SU 1 (head) (b) SU 2 (intermediate) (c) SU 3 (intermediate)

(d) SU 4 (intermediate) (e) SU 5 (intermediate) (f) SU 6 (intermediate)

Geosciences 2021, 11, x 10 of 25

(g) SU 7 (intermediate) (h) SU 8 (intermediate) (i) SU 9 (intermediate)

(l) SU 10 (intermediate) (m) SU 11 (intermediate) (n) SU 12 (intermediate)

(o) SU 13 (intermediate) (p) SU 14 (intermediate) (q) SU 15 (intermediate)

(r) SU 16 (intermediate) (s) SU 17 (intermediate) (t) SU 18 (head)

Figure 6.FigureThe 6. The macro-element macro-element modelsmodels of of the the investig investigatedated SUs in SUsthe clustered in the conditions clustered (pink conditions) and the adjacent (pink) buildings and the adjacent (brown). buildings (brown). In addition, the behavior of the above-presented SUs was also analyzed in isolated configurations to appreciate the differences in terms of stiffness, ductility, ultimate displacement, and base shear concerning the case of aggregate constructions [4]. To this purpose, four different distributions of seismic forces (±X, ±Y) were considered, so as to simulate 288 analyses (144 for each configuration, namely aggregate and isolated). The results in terms of capacity curves are presented in Figure 7.

(a) (b)

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(l) SU 10 (intermediate) (m) SU 11 (intermediate) (n) SU 12 (intermediate)

(o) SU 13 (intermediate) (p) SU 14 (intermediate) (q) SU 15 (intermediate)

(r) SU 16 (intermediate) (s) SU 17 (intermediate) (t) SU 18 (head) Geosciences 2021, 11, 287 10 of 23 Figure 6. The macro-element models of the investigated SUs in the clustered conditions (pink) and the adjacent buildings (brown).

InIn addition, addition, the the behavior behavior of of the the above-pr above-presentedesented SUs SUs was was also also analyzed analyzed in in isolated isolated configurationsconﬁgurations to appreciate thethe differences differences in in terms terms of of stiffness, stiffness, ductility, ductility, ultimate ultimate displace- dis- placement,ment, and baseand base shear shear concerning concerning the case the case of aggregate of aggregate constructions constructions [4]. [4]. ToTo this this purpose, purpose, four four different distributions of seismic forces ((±X,±X, ±Y)±Y) were were consid- consid- ered,ered, so so as as to to simulate simulate 288 288 analyses analyses (144 (144 for for each each configuration, conﬁguration, namely namely aggregate aggregate and and isolated).isolated). The The results results in in terms terms of of capaci capacityty curves curves are are presented presented in in Figure Figure 77..

(a) (b)

Figure 7. Capacity curves in X (a) and Y (b) directions for each SU in aggregate and isolated conditions.

From a careful analysis of the results, it is possible to notice how the global behavior

of the structure in both configurations, namely aggregate and isolated, showed a capacity variability in the two analysis directions. In particular, in the longitudinal direction (X-direction), an average increment of 48% was noted for SUs placed in aggregate configurations in comparison to the isolated ones. Furthermore, in terms of displacement ductility, a decrease of 55% was obtained in the aggregate configuration since, in this case, the confinement effect offered by the contiguous structural units on the reference one limits the expected displacements compared to the isolated cases. However, this limitation also depends on the torsional phenomena, which are more critical for the external structural cells. The torsional effects arose from the addition of aliquots of structural mass passing from the isolated configuration to the aggregate one, causing a non-uniform distribution of the in-plane resistant elements. Thus, the more pronounced eccentricity between centroid and stiffness center triggered a premature rupture of more wall panels. Conversely, in direction Y, the aggregate configuration denotes an increase of 41% in terms of both resistance and displacement capacity. Subsequently, to understand the variation of the global response between the examined structural configurations (aggregate and isolated), the main capacity parameters (stiffness, strength, ductility, and ultimate displacement) were plotted and correlated to each other. The results are depicted in Figure8. From the achieved results, it is possible to note the variability of the monitored structural parameters in the two analysis directions. First of all, by analyzing the maximum stiffness (see Figure8a) in X-direction, it can −1 −1 be seen that Kagg = 375,000 daNcm and Kiso = 190,000 daNcm . So, passing from aggregate to isolated configurations, a stiffness reduction of −97% was attained. Similarly, in the Y-direction, Kagg. exhibits an average increase of two times compared to the value provided by Kiso. In terms of maximum shear strength (see Figure8b), it was observed how in the ±X and ±Y-directions, Vagg, globally, had a noticeable reduction compared to the corresponding Viso. This result is due to the fact that, in aggregate configuration, the seismic load is distributed on a higher number of shear-resistant wall areas. Regarding the displacement ductility (see Figure8c), it was observed that in both analysis directions, the points cloud is quite homogeneous for 1 < µ < 10, while µ > 10 is reached for the heading structural units. Finally, it is noted how in ±X-directions, the ultimate displacements, Geosciences 2021, 11, x 11 of 25

Figure 7. Capacity curves in X (a) and Y (b) directions for each SU in aggregate and isolated conditions.

From a careful analysis of the results, it is possible to notice how the global behavior of the structure in both configurations, namely aggregate and isolated, showed a capacity variability in the two analysis directions. In particular, in the longitudinal direction (X- direction), an average increment of 48% was noted for SUs placed in aggregate configurations in comparison to the isolated ones. Furthermore, in terms of displacement ductility, a decrease of 55% was obtained in the aggregate configuration since, in this case, the confinement effect offered by the contiguous structural units on the reference one limits the expected displacements compared to the isolated cases. However, this limitation also depends on the torsional phenomena, which are more critical for the external structural cells. The torsional effects arose from the addition of aliquots of structural mass passing from the isolated configuration to the aggregate one, causing a non-uniform distribution of the in-plane resistant elements. Thus, the more pronounced eccentricity between centroid and stiffness center triggered a premature rupture of more wall panels. Conversely, in direc- Geosciences 2021, 11, 287 tion Y, the aggregate configuration denotes an increase of 41% in terms of both resistance11 of 23 and displacement capacity. Subsequently, to understand the variation of the global response between the examined structural configurations (aggregate and isolated), the main capacity parameters d , associated with the aggregate conﬁguration are reduced concerning the isolated ones (stiffness,u strength, ductility, and ultimate displacement) were plotted and correlated to (see Figure8d). Conversely, in ±Y-directions, the behavior is reversed for the aggregate each other. The results are depicted in Figure 8. conﬁgurations providing an increase of about 1.5 times with respect to the isolated cases.

Geosciences 2021, 11, x 12 of 25

(a) (b)

First of all, by analyzing the maximum stiffness (see Figure 8a) in X-direction, it can be seen that Kagg = 375,000 daNcm−1 and Kiso = 190,000 daNcm−1. So, passing from aggregate to isolated configurations, a stiffness reduction of −97% was attained. Similarly, in the Y- direction, Kagg. exhibits an average increase of two times compared to the value provided by Kiso. In terms of maximum shear strength (see Figure 8b), it was observed how in the ±X and ±Y-directions, Vagg, globally, had a noticeable reduction compared to the corresponding Viso. This result is due to the fact that, in aggregate configuration, the seismic load is distributed on a higher number of shear-resistant wall areas. Regarding the displacement ductility (see Figure 8c), it was observed that in both analysis directions, the points cloud is quite homogeneous for 1 < µ < 10, while µ > 10 is reached for the heading structural units. Finally, it is noted how in ±X-directions, the ultimate displacements, du, associated with the aggregate configuration are reduced concerning the isolated ones (see (cFigure) 8d). Conversely, in ±Y-directions, the behavior is reversed(d) for the aggregate con- FigureFigurefigurations 8. 8.TheThe main providing main EDPs EDPs evaluated evaluatedan increase for for aggregate of aggregate about and1.5 and timesisolated isolated with SUs. SUs. respect to the isolated cases. Furthermore, to take into account the effective degree of confinement provided by theFrom SUsFurthermore, adjacent the achieved to tothe takeresults, reference into it accountis structural possible the to ce effective notells, the the most degreevariability significant of confinement of the three monitored configurations, provided struc- by turalthenamely parameters SUs adjacent(i) the headin to the the cell two reference confined analysis structural ondirections. one side cells, only the (SU most 1), significant (ii) intermediate three configurations, cell confined namelyon three (i) sides the head(SU 7) cell and confined (iii) intermediate on one side confined only (SU on 1), two (ii) intermediate sides (SU 13), cell were confined extrapo- on threelated sidesfrom (SUthe considered 7) and (iii) intermediateportions of the confined building on aggregate two sides (Figure (SU 13), 9). were extrapolated from the considered portions of the building aggregate (Figure9).

(a) SU 1 (head) (b) SU 7 (inter. 3 side) (c) SU 13 (inter. 2 side)

FigureFigure 9.9. NumericalNumerical modelmodel ofof thethe referencereference casecase studystudy SUs:SUs: ((aa)) headhead SU,SU, ((bb)) intermediateintermediate SUSU conﬁnedconfined onon 33 sides,sides, andand ((cc)) intermediateintermediate SUSU conﬁnedconfined onon 22 sides.sides.

TheseThese conﬁgurationsconfigurations werewere compared,compared, inin termsterms ofof capacitycapacity curves,curves, withwith thethe analogousanalogous SUsSUs placedplaced inin isolatedisolated morphologymorphology conditions,conditions, asas reportedreported inin FigureFigure 10 10..

(a) (b) Figure 10. Capacity curves of the examined SUs in X- (a) and Y- (b) directions.

The achieved results show a difference in terms of behavior in the two analysis directions. In particular, in X-direction, SU 7 offers the highest strength and stiffness since it is confined by other structural cells, but it has reduced displacements compared to the

Geosciences 2021, 11, x 12 of 25

First of all, by analyzing the maximum stiffness (see Figure 8a) in X-direction, it can be seen that Kagg = 375,000 daNcm−1 and Kiso = 190,000 daNcm−1. So, passing from aggregate to isolated configurations, a stiffness reduction of −97% was attained. Similarly, in the Y- direction, Kagg. exhibits an average increase of two times compared to the value provided by Kiso. In terms of maximum shear strength (see Figure 8b), it was observed how in the ±X and ±Y-directions, Vagg, globally, had a noticeable reduction compared to the corresponding Viso. This result is due to the fact that, in aggregate configuration, the seismic load is distributed on a higher number of shear-resistant wall areas. Regarding the displacement ductility (see Figure 8c), it was observed that in both analysis directions, the points cloud is quite homogeneous for 1 < µ < 10, while µ > 10 is reached for the heading structural units. Finally, it is noted how in ±X-directions, the ultimate displacements, du, associated with the aggregate configuration are reduced concerning the isolated ones (see Figure 8d). Conversely, in ±Y-directions, the behavior is reversed for the aggregate configurations providing an increase of about 1.5 times with respect to the isolated cases. Furthermore, to take into account the effective degree of confinement provided by the SUs adjacent to the reference structural cells, the most significant three configurations, namely (i) the head cell confined on one side only (SU 1), (ii) intermediate cell confined on three sides (SU 7) and (iii) intermediate confined on two sides (SU 13), were extrapolated from the considered portions of the building aggregate (Figure 9).

(a) SU 1 (head) (b) SU 7 (inter. 3 side) (c) SU 13 (inter. 2 side) Figure 9. Numerical model of the reference case study SUs: (a) head SU, (b) intermediate SU confined on 3 sides, and (c) intermediate SU confined on 2 sides.

Geosciences 2021, 11, 287 These configurations were compared, in terms of capacity curves, with the analogous12 of 23 SUs placed in isolated morphology conditions, as reported in Figure 10.

(a) (b)

Figure 10. Capacity curves of the examined SUs in X- (a) and Y- ((bb)) directions.directions.

The achieved achieved results results show show a adifference difference in in terms terms of ofbehavior behavior in inthe the two two analysis analysis directions.directions. In In particular, particular, in in X-direction, X-direction, SU SU 7 offers thethe highesthighest strength strength and and stiffness stiffness since since it itis is confined confined by by other other structural structural cells, cells, but bu it hast it reducedhas reduced displacements displacements compared compared to the to other the clustered units. The highest strength and stiffness is due to the large contribution offered in X-direction by masonry walls of the other three SUs working under earthquake. On the other hand, SU 7 exhibited the lowest ultimate displacement, since its seismic movements are strongly limited by the adjacent SUs, which have constrained on three sides (instead of one and two sides for SU 1 and SU 13, respectively) of the inspected structural cell. In contrast, SU 1 (head cell) has both strength and stiffness lower than those of the other SUs in the aggregate configurations. This is justified by the fact that the position of SU 1 (constrained on one side only) makes it more susceptible to damage, since the structural connection with contiguous units occurs only on one side by compressive contact, while the opposite free side of the SU is subjected to torsion phenomena, which cause an increment of the damage. However, as desirable, the SUs in the aggregate configuration provide shear and stiffness greater than the same units considered in the isolated configurations. Nevertheless, at the same time, the aggregate SUs show a reduction in terms of displacement due to the clustered effect given by the contiguous structural cells. In Y-direction, it can be seen how SU 13 in aggregate configuration has a higher stiffness and resistance compared to the ones of SU 1 and SU 7, respectively. Furthermore, the above-mentioned SU 13 provides an ultimate displacement approximately equal to three times higher than that of SU7 and 1.5 times greater than that of SU 1. Consequently, by comparing the same SUs with the analogous cells in isolated configuration, it was noted how the intermediate aggregate units (SU 7 and SU 13), even if they have a higher strength and stiffness, provide a reduced displacement capacity. Nonetheless, SU 1 showed the worst behavior in terms of strength, stiffness, and ultimate displacement. Moreover, a seismic check at the Ultimate Limit State (ULS), defined according to the Italian Code, NTC18 [27], was carried out. In particular, the seismic verification was estimated through the seismic index, ζ, which represents the ratio between the capacity acceleration (PGAC), evaluated at the ULS considered, and the corresponding acceleration demand, PGAD, related to the construction site. In particular, for ζ ≥ 1 of the analyzed SUs provided a good seismic response without reaching significant structural damages; conversely, for ζ < 1, a more or less damaged condition was reached and the SUs required appropriate retrofitting interventions [4]. The ζ-safety indexes for all examined cases are reported in Figure 11. Geosciences 2021, 11, x 13 of 25

other clustered units. The highest strength and stiffness is due to the large contribution offered in X-direction by masonry walls of the other three SUs working under earthquake. On the other hand, SU 7 exhibited the lowest ultimate displacement, since its seismic movements are strongly limited by the adjacent SUs, which have constrained on three sides (instead of one and two sides for SU 1 and SU 13, respectively) of the inspected structural cell. In contrast, SU 1 (head cell) has both strength and stiffness lower than those of the other SUs in the aggregate configurations. This is justified by the fact that the position of SU 1 (constrained on one side only) makes it more susceptible to damage, since the structural connection with contiguous units occurs only on one side by compressive contact, while the opposite free side of the SU is subjected to torsion phenomena, which cause an increment of the damage. However, as desirable, the SUs in the aggregate configuration provide shear and stiffness greater than the same units considered in the isolated configurations. Nevertheless, at the same time, the aggregate SUs show a reduction in terms of displacement due to the clustered effect given by the contiguous structural cells. In Y- direction, it can be seen how SU 13 in aggregate configuration has a higher stiffness and resistance compared to the ones of SU 1 and SU 7, respectively. Furthermore, the above- mentioned SU 13 provides an ultimate displacement approximately equal to three times higher than that of SU7 and 1.5 times greater than that of SU 1. Consequently, by comparing the same SUs with the analogous cells in isolated configuration, it was noted how the intermediate aggregate units (SU 7 and SU 13), even if they have a higher strength and stiffness, provide a reduced displacement capacity. Nonetheless, SU 1 showed the worst behavior in terms of strength, stiffness, and ultimate displacement. Moreover, a seismic check at the Ultimate Limit State (ULS), defined according to the Italian Code, NTC18 [27], was carried out. In particular, the seismic verification was estimated through the seismic index, ζ, which represents the ratio between the capacity acceleration (PGAC), evaluated at the ULS considered, and the corresponding acceleration demand, PGAD, related to the construction site. In particular, for ζ ≥ 1 of the analyzed SUs provided a good seismic response without reaching significant structural damages; conversely, for ζ < 1, a more or Geosciences 2021, 11, 287 less damaged condition was reached and the SUs required appropriate retrofitting13 inter- of 23 ventions [4]. The ζ-safety indexes for all examined cases are reported in Figure 11.

(a) (b)

FigureFigure 11. 11. SeismicSeismic risk risk index index,, ζζ, ,evaluated evaluated forfor allall structuralstructural conﬁgurationsconfigurations in in X X ( a(a)) and and Y Y ( b(b)) analysis analysis directions. directions.

The obtained results show clear indications regarding the seismic capacity of the buildings aggregate aggregate compared compared to the isolated buildings buildings one. one. In In particular, particular, in in X-direction, X-direction, itit can be seen that SU 1 (head) in aggregate configurationconfiguration has a seismic index slightly higher than than that that of of the the isolated isolated one one due due to tothe the additional additional mass mass contribution contribution offered offered by SU by 2.SU Nevertheless, 2. Nevertheless, the isolated the isolated SU 1 SUhas 1a hassafety a safety index indexlower lowerthan 1. than Moreover, 1. Moreover, almost almost all the all the SUs (isolated and in aggregate conditions) have a seismic index lower than 1, not satisfying the verification. This inconsistency is given due to the presence of in-elevation discontinuity, which alters the expected global behavior. In Y-direction, it is noted that almost all the SUs, except for SUs 13 and 14 (which have more than two floors above ground), have a safety index lower than 1. Conversely, the same SUs in isolated conditions have a seismic index higher than 1, since they are not affected by the constraint effect given by the adjacent SUs.

3.3. Sensitivity Analysis In compliance with the analyses introduced in Section 3.2, herein the main objective is to analyze the uncertainties relating to the models examined by means of a sensitivity analysis (SA). This analysis is performed for identifying the parameters having the greatest impact in the calculation phase. Specifically, the characteristics of the materials, the thick- nesses of the masonry, the stiffness of the floors are all factors affected by high uncertainties that can alter the achieved results. From an applicative point of view, an SA can be of two types: (i) knowledge sensitivity and (ii) improvement sensitivity. In the first case, it is possible to deepen the knowledge of the most significant structural parts by optimizing the on-site test procedures; in the second case the structural elements leading to the structural upgrading by optimizing expected costs are identified. Mathematically, the sensitivity indexes (IS), appropriately diversified according to the type of analysis considered, can be calculated as follows [32]:

αmean− αmin IS = knowledge sensitivity (4) αmean

αmean− αmax IS = improvement sensitivity (5) αmean

where αmean, αmin and αmax represent the mean, minimum and maximum safety index achieved during the non-linear calculation phase by combining, simultaneously, the para- metric variations of improved material properties (group G1) and retrofit interventions on floor types (group G2). In this case, this methodological procedure focused on the relevant structural schemes, i.e., those relating to SUs 1, 7 and 13, by adopting the 3Muri sensitivity post-processor [32]. In this perspective, the mechanical characteristics of the masonry, fm, τ0, fv0, E and G (group G1) were varied as a function of a possible enhancement intervention on the floor Geosciences 2021, 11, 287 14 of 23

Geosciences 2021, 11, x 15 of 25 systems (group G2) by assuming a variability interval equal to 10%. The elaborated results are presented in Figure 12.

FigureFigure 12. 12. SensitivitySensitivity analysis analysis for for the the clustered clustered configurations conﬁgurations examined. examined.

ByBy comparing comparing the the results results derived derived from from simulation, simulation, it itis ispossible possible to to note note how how globally, globally, forfor the the SU1, SU1, there there is isa ahigher higher sensitivity sensitivity associated associated with with group group G1 G1 than than the the other other one. one. Therefore,Therefore, the the improvement improvement of of the mechanical characteristicscharacteristics had had a a greater greater impact impact than than the theretrofitting retrofitting interventions interventions on on the the floor floor systems, systems, since since the the sensitivity sensitivity indices indices are are 21.87% 21.87% and and25.39%, 25.39%, respectively. respectively. On theOn contrary,the contrary, it is noted it is noted that for that SU7 for and SU7 SU13, and theSU13, second the second analysis analysisgroup (groupgroup G2)(group has G2) slightly has slightly higher sensitivityhigher sensitivity indices thanindices those than offered those by offered group by G1. groupThis highlightsG1. This highlights how both how the reinforcementboth the reinfo ofrcement the floor of structuresthe floor structures and the improvement and the im- provementof the characteristics of the characteristics of masonry of masonry are very ar importante very important for the for intermediate the intermediate units, units, whose whosebehavior behavior is influenced is influenced by the contiguousby the contiguous units. Finally, units. itFinally, was observed it was observed how the increasehow the in increasethe mechanical in the mechanical parameters parameters also increase also the increase overall seismic the overall safety seismic indexes safety (black indexes squares) (blackcompared squares) to the compared corresponding to the corresponding values in the phase values before in the interventions phase before (yellow interventions circles). (yellow3.4. Fragility circles). Assessment

3.4. FragilityIn this Assessment section, the fragility functions were analyzed for the reference case study SUs analyzed in Section 3.2. As it is known, fragility curves represent the probability of exceedingIn this section, a certain the damage fragility threshold functions varying were analyzed the intensity for the measurement, reference case IM, study generally SUs analyzed in Section 3.2. As it is known, fragility curves represent the probability of ex- represented by either the PGA or the spectral displacements, Sd [34,35]. The evaluation of ceedingthe fragility a certain curves damage was herein threshold estimated varying according the intensity to the methodologymeasurement, proposed IM, generally in [36 ], representedwhere four by damage either states,the PGA namely or the D1spectral (slight), displacements, D2 (moderate), Sd [34,35]. D3 (near The collapse), evaluation and ofD4-D5 the fragility (collapse), curves were was considered herein estimate as suggestedd according in [4]. Methodologically, to the methodology fragility proposed curves inwere [36], derived where accordingfour damage to Equation states, namely (6): D1 (slight), D2 (moderate), D3 (near collapse), and D4-D5 (collapse), were considered as suggested in [4]. Methodologically, fragility curves were derived according to Equation 1 (6):S P[DS|PGA] = Φ · · ln d (6) β S dDS 1 Sd P[DS|PGA]=Φ⋅⋅ ln  (6) where the operator Φ is the cumulative distributionβ function,S SdDS is the median displacement value associated with each damage threshold anddDSβ is the standard deviation of the log-normal distribution. This latter factor is a function of the structural ductility, µ, of where the operator Φ is the cumulative distribution function, SdDS is the median displace- the SDoF system, and it was evaluated as the natural logarithm of the ratio between the ment value associated with each damage threshold and β is the standard deviation of the ultimate displacement, d , and the corresponding yielding displacement, d , multiplied by log-normal distribution. Thisu latter factor is a function of the structural ductility,y µ, of the a correlation coefﬁcient of 0.45 [36]. Thus, the damage states and the standard deviations SDoF system, and it was evaluated as the natural logarithm of the ratio between the ulti- were identiﬁed, as reported in Table3: mate displacement, du, and the corresponding yielding displacement, dy, multiplied by a correlation coefficient of 0.45 [36]. Thus, the damage states and the standard deviations were identified, as reported in Table 3:

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Table 3. Damage levels, DSi, and the corresponding dispersion, β. Table 3. Damage levels, DSi, and the corresponding dispersion, β. Damage Thresholds DSi Damage Thresholds DSi D1D1 0.7·0.7dy ‧dy SlightSlight D2 1.5·d Moderate D2 1.5y ‧dy Moderate D3 0.5·(dy + du) Near collapse D3 0.5‧(dy + du) Near collapse D4, D5 du Collapse D4, D5 du Collapse StandardStandard Deviation Deviationβ β 0.45·ln(0.45µ‧)ln(µ)

Thus, based on these prerogatives, the fragility functions were plotted in Figure 13 forThus, the above-mentioned based on these case prerogatives, study SUs theexamined fragility in Section functions 3.2.were plotted in Figure 13 for the above-mentioned case study SUs examined in Section 3.2.

(a) SU 1 (b) SU 1

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(e) SU 13 (f) SU 13

FigureFigure 13. Fragility 13. Fragility curves curves for for the the examined examined structuralstructural units units in in X- X- (a (,ca,,ec), eand) and Y- ( Y-b,d (b,f,)d directions.,f) directions.

From the obtained results, it was observed how the probability of exceeding a certain damage threshold DSi (with i = 1, …, 4) is much more accentuated in the case of the clustered configuration. As explained in Section 3.2, the increment in terms of mass deriving from close SUs provides an increase in the seismic load, but a consequent reduction of the expected displacements. Considering SU 1 in X-direction, for a maximum displacement of 0.5 cm, the achieved damage thresholds D1 and D2, evaluated for aggregate and isolated configurations, are comparable to each other. Instead, starting from a target displacement of 1 cm, there is a difference in terms of the expected damage probability thresholds. In particular, SU 1 considered as isolated provided a lower probability of damage than the same SU placed in the masonry compound. Extending this consideration to the Y- direction), the same condition arises. This circumstance depends on the fact that the reference SU in the isolated condition shows an increase in the displacements (dy = 0.60 cm; du = 1.46 cm) compared to those of the clustered SU (dy = 0.23 cm; du = 1.44 cm) which has lower values of the expected damage thresholds. This condition was observed even for the other SUs analyzed. For an exhaustive comparison between aggregate and isolated SUs, in Table 4, the displacements (yielding and ultimate) achieved for the case study configurations are depicted.

Table 4. Target displacements, dy and du, for the examined SUs.

Isolated (X dir.) Aggregate (X dir.) Structural Units dy (cm) du (cm) dy (cm) du (cm) 1 0.22 1.56 0.22 1.10 7 0.51 2.21 0.30 0.69 13 1.00 2.50 0.48 1.53 Isolated (Y dir.) Aggregate (Y dir.) Structural Units dy (cm) du (cm) dy (cm) du (cm) 1 0.60 1.46 0.23 1.44 7 0.46 0.98 0.34 0.52 13 0.52 2.48 0.40 1.77

Finally, in Figure 14, the comparison in terms of damage levels between SU 1 (head SU, confined on one side) and SU 13 (intermediate SU, confined on two sides) is illustrated to point out the influence of structural performance on the fragility assessment.

Geosciences 2021, 11, 287 16 of 23

From the obtained results, it was observed how the probability of exceeding a certain damage threshold DSi (with i = 1, ... , 4) is much more accentuated in the case of the clustered configuration. As explained in Section 3.2, the increment in terms of mass deriving from close SUs provides an increase in the seismic load, but a consequent reduction of the expected displacements. Considering SU 1 in X-direction, for a maximum displacement of 0.5 cm, the achieved damage thresholds D1 and D2, evaluated for aggregate and isolated configurations, are comparable to each other. Instead, starting from a target displacement of 1 cm, there is a difference in terms of the expected damage probability thresholds. In particular, SU 1 considered as isolated provided a lower probability of damage than the same SU placed in the masonry compound. Extending this consideration to the Y-direction), the same condition arises. This circumstance depends on the fact that the reference SU in the isolated condition shows an increase in the displacements (dy = 0.60 cm; du = 1.46 cm) compared to those of the clustered SU (dy = 0.23 cm; du = 1.44 cm) which has lower values of the expected damage thresholds. This condition was observed even for the other SUs analyzed. For an exhaustive comparison between aggregate and isolated SUs, in Table4, the displacements (yielding and ultimate) achieved for the case study configurations are depicted.

Table 4. Target displacements, dy and du, for the examined SUs.

Isolated (X dir.) Aggregate (X dir.) Structural Units dy (cm) du (cm) dy (cm) du (cm) 1 0.22 1.56 0.22 1.10 7 0.51 2.21 0.30 0.69 13 1.00 2.50 0.48 1.53 Geosciences 2021, 11, x 18 of 25 Isolated (Y dir.) Aggregate (Y dir.) Structural Units dy (cm) du (cm) dy (cm) du (cm) The 1results show that 0.60 SU 1, which occupi 1.46es a heading 0.23position, provides 1.44 a higher probability7 of damage than 0.46 that of the corresp 0.98onding SU 13. 0.34It is worth highlighting 0.52 how the head13 unit, despite having 0.52 a lower number 2.48 of floors (therefore, 0.40 a lower mass 1.77 and stiffness) and a reduced lateral constraint offered by the adjacent cells, provided an expected damageFinally, level in greater Figure than 14, thethe comparisonSU 13 one, sinc ine terms the torsional of damage phenomena levels between limit the SU capacity 1 (head displacementsSU, confined on and, one consequently, side) and SU 13the (intermediate expected damage SU, confined is much on higher. two sides) is illustrated to point out the influence of structural performance on the fragility assessment.

(a) (b)

Figure 14. InfluenceInﬂuence of of the structural positions on on th thee fragility assessment of of inspected inspected SUs SUs in in ( (aa)) X X direction and and ( b))Y Y analysis directions.

4. Prediction of the Vibration Period 4.1. Modal Analysis Modal analysis was performed to point out the main vibration periods of the case study SUs evaluated in both configurations, namely clustered and isolated. The eigenvalue analysis was herein appropriately performed using the 3Muri software [32], taking into consideration the two translational vibration periods, T1 and T2, evaluated in the X- and Y-directions, respectively, and the in-plane rotational period, T3, associated to the vertical direction (Z). The modal analysis results are summarized in Figure 15.

(a) T1, Longitudinal (X) (b) T2, Transversal (Y)

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The results show that SU 1, which occupies a heading position, provides a higher probability of damage than that of the corresponding SU 13. It is worth highlighting how the head unit, despite having a lower number of floors (therefore, a lower mass and stiffness) and a reduced lateral constraint offered by the adjacent cells, provided an expected damage level greater than the SU 13 one, since the torsional phenomena limit the capacity displacements and, consequently, the expected damage is much higher.

Geosciences 2021, 11, 287 17 of 23

The results show that SU 1, which occupies a heading position, provides a higher probability of damage than that of the corresponding SU 13. It is worth highlighting (a) (b) how the head unit, despite having a lower number of ﬂoors (therefore, a lower mass Figure 14. Influence of the structuraland stiffness) positions and on a reducedthe fragility lateral assessment constraint of inspected offered SUs by thein ( adjacenta) X direction cells, and provided (b) Y an analysis directions. expected damage level greater than the SU 13 one, since the torsional phenomena limit the capacity displacements and, consequently, the expected damage is much higher. 4. Prediction of the Vibration Period 4.1.4. Prediction Modal Analysis of the Vibration Period 4.1.Modal Analysisanalysis was performed to point out the main vibration periods of the case studyModal SUs evaluated analysis wasin both performed configurations, to point namely out the clustered main vibration and isolated. periods The of the eigen- case valuestudy analysis SUs evaluated was herein in both appropriately conﬁgurations, performed namely using clustered the and3Muri isolated. software The [32], eigenvalue taking intoanalysis consideration was herein the appropriately two translational performed vibration using periods, the 3Muri T1 and software T2, evaluated [32], taking in the into X- andconsideration Y-directions, the respectively, two translational and vibrationthe in-plane periods, rotational T1 and period, T2, evaluated T3, associated in the X-to andthe verticalY-directions, direction respectively, (Z). The modal and the analysis in-plane results rotational are summarized period, T3, associatedin Figure 15. to the vertical direction (Z). The modal analysis results are summarized in Figure 15.

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(a) T1, Longitudinal (X) (b) T2, Transversal (Y)

Figure 15. MainMain vibration periods of the examined SUs.

FromFrom the the comparison comparison presented, presented, it itwas was noti noticedced that that in X-direction, in X-direction, the thevibration vibration pe- riodsperiods provided provided by the by isolated the isolated configurations configurations are higher are higher compared compared to those to of those the aggre- of the gateaggregate ones. More ones. specifically, More specifically, the intermediate the intermediate SUs (n. SUs2, 8, (n.9, 13, 2, 8,15, 9, 17) 13, placed 15, 17) in placed isolated in positionsisolated positions reached vibration reached vibration periods higher periods than higher those than in thoseaggregate in aggregate configurations. configurations. AsAs it it is is seen seen in in Figure Figure 15a,15a, SU SU 13 13 had had a a vibration vibration period of T = 0.63 0.63 s, s, with with a a percentage percentage increaseincrease ofof moremore than than twice twice compared compared to theto the period period of the of aggregatethe aggregate SU. In SU. general, In general, as stated as statedin Section in Section 3.2, the 3.2, in-elevation the in-elevation regularity regulari andty the and presence the presence of adjacent of adjacent units, that units, reduce that reducethe modal the displacements,modal displacements, play an play important an import roleant in therole evaluation in the evaluation of the vibration of the vibration period. period.On the On other the hand, other inhand, Y-direction, in Y-direction, the results the results obtained obtained by analyzing by analyzing the twothe two structural structural configurations are comparable. As it is shown in Figure 15b, the SU 3 in isolated condition had a vibration period about three times lower than that of the same SU in aggregate configuration. This occurs since, in the aggregate configuration, the modal analysis takes into account the influence of the additional mass induced by the units adjacent to the reference one. Finally, in Figure 15c, the torsion mode was analyzed. Moreover, in this case, globally, the results obtained are comparable, except for both SUs 3 and 4, which show a reduction of the vibration period of 35% and 38%, respectively, in comparison to the values achieved for the same structural cells in clustered configurations.

4.2. Linear Regression Formula In a subsequent analysis phase, through a regression analysis, a mathematical law expressing the vibration periods of the aggregate SUs starting from those of the isolated ones was derived. This simplified methodology, even if it cannot replace a more accurate dynamic identification test, can be useful to predict the vibration periods of SUs of historical centers having morphological and structural features similar to those of buildings herein investigated. The achieved results are summarized in Figure 16.

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configurations are comparable. As it is shown in Figure 15b, the SU 3 in isolated condition had a vibration period about three times lower than that of the same SU in aggregate configuration. This occurs since, in the aggregate configuration, the modal analysis takes into account the influence of the additional mass induced by the units adjacent to the reference one. Finally, in Figure 15c, the torsion mode was analyzed. Moreover, in this case, globally, the results obtained are comparable, except for both SUs 3 and 4, which show a reduction of the vibration period of 35% and 38%, respectively, in comparison to the values achieved for the same structural cells in clustered configurations.

4.2. Linear Regression Formula In a subsequent analysis phase, through a regression analysis, a mathematical law expressing the vibration periods of the aggregate SUs starting from those of the isolated ones was derived. This simpliﬁed methodology, even if it cannot replace a more accurate Geosciences 2021, 11, x 20 of 25 dynamic identiﬁcation test, can be useful to predict the vibration periods of SUs of historical centers having morphological and structural features similar to those of buildings herein investigated. The achieved results are summarized in Figure 16.

(a) T1, Longitudinal (X) (b) T2, Transversal (Y)

Figure 16. LinearLinear regression relationships for estimation of the vibration periods of investigated SUs.

The regression formulations analyzed show how the determination coefﬁcientcoefficient (R(R22) is robustrobust since it assumed values greater than 0. 0.55 in all the cases examined. This coefﬁcientcoefficient implicitlyimplicitly takes takes into into account account all all variables variables asso associatedciated with with the increment the increment of both of bothmass, mass, stiff- nessstiffness and, and, the in-elevation the in-elevation interaction interaction among among structural structural units units examined. examined. The achieved mathematical formulations are reported in the following Equations (7)– (9): ⋅ T=αTagg. iso. (7) ⋅ T=βTagg. iso. (8) ⋅ T=θTagg. iso. (9)

where α, β, and θ are the empirical coefficients, estimated equal to 0.7905, 1.0732, 0.8867, respectively.

4.3. Derivation of Forecasting Empirical Formula Generally, the main technical regulations [27,33,37] allow identifying the fundamental vibration periods of buildings using empirical formulations base on two factors,

Geosciences 2021, 11, 287 19 of 23

The achieved mathematical formulations are reported in the following Equations (7)–(9): Tagg.= α · Tiso. (7)

Tagg.= β · Tiso. (8)

Tagg.= θ · Tiso. (9) where α, β, and θ are the empirical coefﬁcients, estimated equal to 0.7905, 1.0732, 0.8867, respectively.

Geosciences 2021, 11, x 4.3. Derivation of Forecasting Empirical Formula 21 of 25

Generally, the main technical regulations [27,33,37] allow identifying the fundamental vibration periods of buildings using empirical formulations base on two factors, namely a coefﬁcient,namely a coefficient, Ci, that takes Ci, that into takes account into account the type the of type construction, of construction, and the and height, the height, H, of the buildingH, of the (notbuilding exceeding (not exceeding 40 m), as 40 reported m), as reported in Equation in Equation (10): (10):

3/4 T T=C= C⋅· HH3/4 (10)(10) This formulation is based on the Rayleigh method [38], which was originally cali- This formulation is based on the Rayleigh method [38], which was originally calibrated brated on a class of American RC buildings that have different technological characteris- on a class of American RC buildings that have different technological characteristics from tics from those of the Italian heritage. However, this methodology was largely used for those of the Italian heritage. However, this methodology was largely used for the prediction the prediction of the predominant vibration periods of buildings when more detailed ofstructural the predominant analyses were vibration not carried periods out. of buildingsFurthermore, when in morethe case detailed of existing structural buildings, analyses werethis approach not carried would out. seem Furthermore, to overestimate in the th casee vibration of existing period, buildings, since the this slight approach variation would seem to overestimate the vibration period, since the slight variation of the functional of the functional parameter, Ci, dependent on the structural technology, is not able to parameter,cover all the C differenti, dependent seismic on therespon structuralses of the technology, real building is not stocks. able to cover all the different seismicHowever, responses the ofproposed the real formulation building stocks. is qualitatively ineffective for predicting the vi- brationHowever, period structural the proposed units placed formulation in aggregate is qualitatively configurations. ineffective For this forreason, predicting a mod- the vibrationification of period Equation structural (8), based units on the placed position in aggregate of the structural configurations. units in the Forbuilding this reason,com- a modificationpound, is herein of Equationprovided. (8),In particular, based on it the was position used a correlative of the structural coefficient, units C ini, evaluated the building compound,as the ratio between is herein the provided. percentage In of particular,the mass of itthe was single-cell used a and correlative the total mass coefficient, of the Ci, evaluatedwhole aggregate. as the ratioThe proposed between formulation the percentage is reported of themass in the of following the single-cell Equation and (11): the total mass of the whole aggregate. The proposed formulation is reported in the following M Equation (11): ⋅=3/4i ⋅ 3/4 T=Ci H H (11) 3/4 MTotMi 3/4 T = Ci · H = · H (11) MTot The utility of the present formulation is the simplicity of application for the prediction ofThe the utility vibration of the period present of clustered formulation units. is This the simplicity mathematical of application correlation for is not the dissim- prediction ofilar the from vibration the one period proposed of clustered in [27,33,37,39], units. Thisbut it mathematical has the advantage correlation of being is notfunctional dissimilar fromand more the one easily proposed usable for in estimating [27,33,37,39 the], pr butedominant it has the vibration advantage period of being of aggregate functional SUs. and moreThe proposed easily usable formula for estimatingprovided the the results predominant shown in vibration Figure 17. period of aggregate SUs. The proposed formula provided the results shown in Figure 17.

(a) (b)

Figure 17. Comparison of mechanical and empirical vibration periods (a) and (b) variation of the coefficient Ci for each Figure 17. Comparison of mechanical and empirical vibration periods (a) and (b) variation of the coefﬁcient Ci for each SUs. SUs.

First of all, in Figure 17a, the above-mentioned formulation is compared to the respective vibration periods obtained by numerical simulation in the two analysis directions. From the obtained results, it is clear that the proposed calibration is comparable with the periods deriving from the mechanical analysis. However, the variation of results (with a maximum average increase of 8% for the Y-direction and an average decrease of 1% for the X-direction) concerning those obtained from the mechanical analysis is essentially due to the position of the SUs. Furthermore, in Figure 17b, the variation of the correlation coefficient Ci, as a function of the vibration period of each SUs, is shown.

Geosciences 2021, 11, 287 20 of 23

First of all, in Figure 17a, the above-mentioned formulation is compared to the respective vibration periods obtained by numerical simulation in the two analysis directions. From the obtained results, it is clear that the proposed calibration is comparable with the periods deriving from the mechanical analysis. However, the variation of results (with a maximum average increase of 8% for the Y-direction and an average decrease of 1% for the X-direction) concerning those obtained from the mechanical analysis is essentially due to the position of the SUs. Furthermore, in Figure 17b, the variation of the correlation

Geosciences 2021, 11, x coefﬁcient Ci, as a function of the vibration period of each SUs, is shown.22 of 25

As it is noticed, the coefﬁcient Ci presents a variability directly proportional to the mass variation of the single SUs compared to the total mass of the entire aggregate. In this As it is noticed, the coefficient Ci presents a variability directly proportional to the circumstance, SUs with a higher mass rate have a Ci proportional to the excited mass in terms ofmass expected variation frequencies.of the single SUs compared to the total mass of the entire aggregate. In this circumstance, SUs with a higher mass rate have a Ci proportional to the excited mass in Finally,theterms of expected proposed frequencies. method was suitably compared with similar formulations [27,33,37,39], which are basedFinally,on the the proposed above-mentioned method was suitab Equationly compared (8). Inwith particular, similar formulations Calvi et al. [39] assume [27,33,37,39], which are based on the above-mentioned Equation (8). In particular, Calvi a coefﬁcient Ci = 0.040, since the masonry structures have a considerable rigidity before et al. [39] assume a coefficient Ci = 0.040, since the masonry structures have a considerable cracking,rigidity while before EC8 cracking, [33] while and EC8 NTC08 [33] and [40 NTC08] adopt [40] adopt Ci = C 0.05i = 0.05 for for masonry build- building and the USB Americaning and the Code USB American [37] sets Code Ci [37]= 0.0488. sets Ci = The0.0488. results The results of theof the above-mentioned above-mentioned comparison are plottedcomparison in Figure are plotted 18. in Figure 18.

(a) (b) Figure 18. Comparison among different formulations to estimate vibration periods of investigated SUs (a) and period vs. Figure 18. Comparisonheight trends among obtained different from theoretical formulations relationships to and estimate mechanical vibration analyses (b periods). of investigated SUs (a) and period vs. height trends obtained from theoretical relationships and mechanical analyses (b). In Figure 18a, it can be seen that the period estimated by Equation (9) is comparable Inwith Figure that proposed18a, it can by Calvi be seenet al. [39] that and the the perioddesign codes estimated examinedby [33,37,40], Equation since the (9) is comparable correlative coefficients, Ci, are not dissimilar. However, when the mass of the single SUs with thatincreases proposed (see from by SU Calvi 12 to SU et al.16), [the39 ]correlations and the proposed design codesin the literature examined tend to [33 ,37,40], since the correlativeunderestimate coefﬁcients, the vibration C periodi, are since not they dissimilar. do not take However,into account the when mass thevariation mass of the single SUs increasesof the aggregate (see from structural SU 12cell.to Similarly, SU 16), in theFigure correlations 18b, the vibration proposed period, Ti in, is therelated literature tend to to the height of the SUs examined. underestimateFor heights the vibration between 4.0 period m and 7.5 since m, the they formulations do not proposed take into by account[27,33,37,39,40] the mass variation of the aggregateunderestimate structural the expected cell. vibration Similarly, period, wh inile Figure for heights 18b, ranging the vibration from 8.0 m period,to 12.0 Ti, is related to the heightm, the proposed of the SUsformulations examined. are comparable to each other. However, it seems that the Forliterature heights methods between are reliable 4.0 m in and predicting 7.5 m, the the vibration formulations periods. In proposedfact, for building by[ 27,33,37,39,40] heights of up to 8.0 m, the literature methods result in a safe condition, providing seismic underestimateforces to bethe adopted expected in the case vibration of linear an period,alyses (static while and dynamic) for heights greater rangingthan those from 8.0 m to 12.0 m,expected. the proposed Moreover, formulations for building heights are greater comparable than 8 m, the to methods each other. provide However, vibration it seems that the literatureperiods comparable methods to are the reliable numerical in ones. predicting the vibration periods. In fact, for building heights5. of Conclusions up to 8.0 m, the literature methods result in a safe condition, providing seismic forces to beThis adopted research inarticle the focused case of on linear the seismic analyses vulnerability (static assessment and dynamic) of a historical greater than those expected.center’s Moreover, structural forunits building using both heights empirical greater and mechanical than 8 approaches, m, the methods providing provide a vibration periodsmathematical comparable formulation to the numerical for predicting ones. their predominant vibration period. The case study aggregate, placed in the historic center of Mirandola in the Emilia-Romagna region of Italy, was made of eighteen SUs appropriately identified based on their architectural– 5. Conclusionsstructural features, which are representative of the most common typological class of Thisbuildings research found article in the inspected focused area. on the seismic vulnerability assessment of a historical center’s structural units using both empirical and mechanical approaches, providing a mathematical formulation for predicting their predominant vibration period. The case study aggregate, placed in the historic center of Mirandola in the Emilia-Romagna region Geosciences 2021, 11, 287 21 of 23

of Italy, was made of eighteen SUs appropriately identiﬁed based on their architectural– structural features, which are representative of the most common typological class of buildings found in the inspected area. First of all, the rapid screening of the seismic vulnerability was conducted according to an index-based method widely used for structural units in historical compounds. From the analysis performed, the following results were obtained:

• The estimated normalized vulnerability, VI, of SUs is lower than the mean threshold fixed to 0.5. Moreover, variability of the vulnerability indices of the estimated sample is mainly influenced by the in-elevation interaction between adjacent structural units (see parameter 11 of the survey form); • The presence of clustered buildings with different heights significantly influences the global vulnerability, since under seismic events such SUs are subjected to hammering effects; • The vulnerability range estimated varies from 0.32 < VI < 0.48 (medium vulnerability); • The typological vulnerability curves derived for the case study SUs show how as the hazard levels, expressed in terms of macroseismic intensity, increase, the expected damage augments as well until the maximum threshold, D4, is reached (partial collapse, VI = 0.48 and µD = 4.19). Secondly, non-linear static analyses were implemented to capture the difference between SUs in aggregate configurations and the same SUs considered as isolated structures. To this purpose, the 3Muri calculation software was used by adopting the macro elements modeling approach. The main outcomes were: • The structures in both configurations, namely aggregate and isolated, have a variable capacity in the two analysis directions X and Y. In particular, in the longitudinal direction (X-direction), an average strength increment of 48% was noted for the SUs placed in aggregate conditions with respect to the isolated ones; • In terms of displacement ductility, a decrease of 55% was obtained, since the SUs in a clustered configuration, given the confinement effect offered by the contiguous structural units, limit the expected displacements compared to the corresponding isolated cases. However, this limitation also depends on the torsion phenomena, which are more influent for the external structural cells; • In terms of stiffness, in X-direction, the isolated case provided a stiffness reduction of 402% compared to the aggregate one. Similarly, in the Y-direction, the aggregate case stiffness was averagely two times higher than the isolated case one; • In terms of maximum shear strength, it was found that in all analysis directions the strength of aggregate SUs is much higher (around three times) than the isolated SUs, since in the aggregate configuration the seismic load is distributed on a higher number of shear-resistant wall areas; • In terms of ductility, µ, it was observed that in both analysis directions 1 < µ <10 and µ > 10 was achieved for head SUs only; • In terms of the seismic index, ξ, almost all the SUs (isolated and in aggregate conditions) have a seismic index lower than 1, not satisfying the code provisions. This is due to the presence of in-elevation discontinuities, which alter the global seismic behavior. Thirdly, three different aggregate structural configurations were analyzed to provide suitable indications about the expected damage employing fragility curves. Overall, it was found that the structural units in aggregate configuration provide a greater probability of damage than the same SUs in isolated conditions. This depends on the fact that, in the clustered configuration, the examined units have generally reduced displacements since the elongated shape of the aggregate produces large displacements of the head SUs, which interrupt the analysis when the intermediate cells exhibit a scarce excursion in the plastic field. Finally, a simplified formulation for predicting the vibration period of SUs arranged in clustered configuration was provided. This empirical formulation, suitably calibrated Geosciences 2021, 11, 287 22 of 23

based on mechanical analysis, was compared with the main literature formulations. The main results achieved are discussed below:

• The coefﬁcient Ci used for the proposed calibration formula is intended as the ratio between the mass of the reference SU and the total mass of the aggregate; • The proposed calibration formula provides periods comparable with those deriving from the mechanical analysis. In particular, when compared to the 3Muri analysis results, it was observed how the empirical relation had a maximum average increase of 8% in Y-direction and an average decrease of 1% in X-direction; • The vibration period, Ti, predicted by literature and standard relationships, if evaluated for heights between 4.0 m < H < 7.5 m, underestimates the expected predominant period derived from the formulation herein proposed. Contrary, for heights ranging from 8.0 m to 12 m, the proposed formulation gives results comparable to the literature and standard ones.

6. Patents

Author Contributions: Conceptualization, A.F., N.C., G.M. and M.M.; methodology, N.C. and A.F.; software, N.C.; validation, A.F., N.C., G.M. and M.M.; formal analysis, N.C.; investigation, N.C. and A.F.; resources, A.F.; data curation, A.F. and G.M.; writing—original draft preparation, N.C. and A.F.; writing—review and editing, A.F., G.M. and M.M.; visualization, N.C.; supervision, A.F.; project administration, A.F. and G.M. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Data deriving from the current study can be provided to the readers based upon their explicit request. Acknowledgments: This work was developed under the financial support of the Italian Civil Protection Department within the ReLUIS-DPC 2019–2021 research project, which is gratefully acknowledged. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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