Analysis of Equivalent Diaphragm Vault Structures in Masonry Construction Under Horizontal Forces

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Article Analysis of Equivalent Diaphragm Vault Structures in Masonry Construction under Horizontal Forces

Pier Silvio Marseglia †, Francesco Micelli * and Maria Antonietta Aiello † Department of Innovation Engineering, University of Salento, 73100 Lecce, Italy; [email protected] (P.S.M.); [email protected] (M.A.A.) * Correspondence: [email protected]; Tel.: +39-0832-297380 These authors contributed equally to this work. † Received: 6 August 2020; Accepted: 25 August 2020; Published: 3 September 2020

Abstract: In seismic areas, masonry construction is prone to brittle failures due to the mechanical behavior of the constituent materials and to the low capacity of force redistributions. The redistribution capacity is mainly due to the presence of horizontal connections upon the walls and to the stiffness of the roof, which is typically a vaulted structure. The modeling of the global behavior of a masonry building, taking into account the accurate stiffness of the vaults, is a major issue in seismic design and assessment. The complex geometry of the vaults can be considered as an equivalent plate, able to replicate the stiffness behavior and the force redistribution capacity of the real vault. In this study, the efforts of the authors are addressed to the definition of a plate, able to replace the vaulted surfaces in a global numerical model. The ideal diaphragm is considered as a generally orthotropic plate with the same footprint and the same thickness of the original vault. An extended parametric study was conducted in which the mechanical and geometrical parameters were varied, such as the vault thickness, its dimensions, the constraint conditions, and the possible presence of side walls. The results are presented and discussed herein, with the aim of providing useful information to the researchers and practitioners involved in seismic analyses of historical masonry construction.

Keywords: masonry; vault; stiﬀness; seismic; structural analysis; plate; FEM

1. Introduction The preservation of cultural heritage is particularly relevant in many European countries, such as Italy, where there are 47 UNESCO World Heritage sites (UNESCO). The high vulnerability of monumental masonry constructions was highlighted by seismic events that happened in the last few centuries in the European territory. Among them, those buildings with vaulted roofs represent a large number, and they were prone to sudden failures since they were designed and built to support only gravity loads. The observation of the post-earthquake damages showed that the vaults, domes, clocks towers, bell towers, steep roofs, gables, and apses of the churches exhibited collapses and extensive cracking. Some recent structural failures involving architectural heritage subjected to seismic forces are presented and discussed in [1–4]. With particular reference to a vaulted masonry structure, the damage caused by the 2012 Italian earthquake was simulated in [5]. Nonlinear static and dynamic analyses were carried out using a damage-plasticity constitutive law for masonry. All these studies revealed that the quality of the masonry plays a crucial role, as well as its texture. Under seismic loadings, the vaults often represent a weak link in the structural hierarchy because of their low stiﬀness in plane. In addition, they may cause out of plane collapses of the walls connected to them when the thrusts are not contained by proper ties or external buttresses. Nowadays, the seismic assessment of existing structures has become one of the most challenging tasks for structural civil engineers. The need to improve the safety levels of existing structures with respect to the actual technical codes,

Heritage 2020, 3, 989–1017; doi:10.3390/heritage3030054 www.mdpi.com/journal/heritage Heritage 2020, 3 990 especially under seismic forces, requires complex modeling for ancient vaulted masonry buildings. In this scenario, it is well known that the stiffness of the horizontal elements strongly influences the overall response of the structure [6]. For this reason, when masonry vaults are present as a floor system, knowledge of their seismic response is mandatory for predicting a reliable distribution of the horizontal forces. With this aim, the evaluation of an equivalent plane element replacing the vaulted floor seems an effective solution, as also presented in [7], where, for some vault typologies, isotropic or orthotropic membranes were studied as equivalent elements in the global analyses. The evaluation of an equivalent elastic stiffness was proposed to idealize these structural elements. In the present study, an extension of [7] is proposed by considering a similar approach, which was suggested also in [8,9]. In [7], a correlation is proposed between the axial and shear stiffness of a specific vault type with those of an equivalent plane membrane. The equivalence is an elastic one, since the Young’s modulus in the two principal directions, EX and EY; Poisson’s ratio, υ; and the shear modulus, GX,Y are assumed for comparison. In [8], the authors simplified the vaults by considering a full-rigid in-plane diaphragm getting a global stiffer behavior. In [9], the stiffness was considered by analyzing the local modes due to the mass concentrated in the center of mass of the diaphragm. On the other hand, in relation to the assessment and mitigation of seismic risk of the cultural heritage the Italian technical code [10], takes into account the possible presence of curved floors. In fact, referring to constructions with vaulted floors, it is suggested to adopt an equivalent slab with a suitable stiffness to be determined according to the type of the vault, its geometry, the characteristics of the materials, and the type of connection to the vertical structural elements. The utility of a such a simplified approach is clear when the mechanical characteristics of the vaults are studied by considering the nonlinear behavior and the geometrical parameters that may affect the equilibrium of the structural system or its sub-parts, as shown in the literature. A recent review study [11] presented a discussion about the different research approaches, followed by different authors, in studying cross vaults. The authors conclude that numerical methods, compared to experimental studies, can provide the tools for the quantification of the damages produced by external forces applied to cross vaults. A case study on cross-vaults is presented in [12], showing how to approach the damage assessment of vaulted masonry by using a tridimensional discretization in which the nonlinear behavior is assigned to the interfaces between adjoining elements. A similar task has been studied more recently in [13], where a new unified computational method is proposed for the structural safety assessment of masonry vaults, able to catch the different geometries and loading conditions. The authors illustrate an analytical validation by comparing the results of the proposed model with those available in the literature. A simplified model has been presented in [14], where the vaults are considered in forms of equivalent trusses. The equivalent stiffness of the trusses has been calibrated in order to obtain a simplified structural system able to show the same or a closely similar seismic response to the real one. Finite element analyses were used to model the real conditions, to be compared to the proposed simplified scheme. A parametric analysis has been illustrated in order to validate the results of the proposed approach with respect to those of the vaults when fully modeled. A detailed parametric study has been presented in [15], with the aim of providing exhaustive knowledge of the seismic response of cross vaults by individuating the failure mechanisms in the function of the design parameters. The results of Multiple Linear Regression are provided in order to help researchers and practitioners develop a quick assessment of the seismic capacity of cross masonry vaults. Very few are those studies studied and modeled an entire structural system considering the real vault’s geometry. A masonry chapel belonging to the German architectural heritage has been studied [16], considering the real geometry of the ribbed vaults with warped intersections that constitute the ancient roof. The authors underline how special methods of numerical structural analysis are needed to solve this kind of problem, and how difficult the geometrical modeling is in a FEM (Finite Element Modeling) environment when the geometry is not regular. In this case, an Operational Modal Analysis has also been used to experimentally determine the natural frequencies of the building from ambient vibration tests. These data were used to calibrate the FE model using genetic algorithms. Heritage 2020, 3 991 Heritage 2020, 3 FOR PEER REVIEW 3

Thecalibrated calibrated FE model FE model has has been been run run by by performing performing a anon non-linear-linear analysis, analysis, able able to to catch catch the load-bearingload-bearing capacitycapacity ofof thethe building.building. WithWith “ordinary”“ordinary” ancientancient constructions—i.e.,constructions—i.e., thosethose usedused asas civilcivil buildingsbuildings withwith vaultedvaulted roofs—itroofs—it isis notnot alwaysalways possiblepossible toto useuse sophisticatedsophisticated 3D3D analysesanalyses consideringconsidering the real geometrygeometry of the vaults.vaults. ThisThis becausebecause itit wouldwould requirerequire highlyhighly skilledskilled engineersengineers andand veryvery powerfulpowerful workstationsworkstations ableable toto runrun solidsolid modelsmodels withwith thousands thousands (or (or millions) millions) of of finite finite elements. elements. In In these these cases, cases, structural structural engineers engineers need need a solutiona solution that that would would allow allow them them to performto perform a global a global analysis analysis without without losing losing much much accuracy, accuracy, and and at the at samethe same time time without witho unsustainableut unsustainable costs, costs, also also in terms in terms of time. of time. InIn this this paper, paper the, the results results of an of extensive an extensive study study are presented, are presented, with the with aim of the providing aim of providing a simplified a approachsimplified for approach studying for the studying masonry the of masonry vaulted structures, of vaulted especiallystructures, under especially seismic under forces. seismic The resultsforces. willThe beresults provided will be and provided discussed, and considering discussed, theconsidering precious backgroundthe precious provided background by theprovided few scientific by the studiesfew scientific available studies nowadays, available also nowadays, showing aalso case showing study. a case study. ItIt isis remarkableremarkable toto rememberremember thatthat thethe scopescope ofof thisthis studystudy isis notnot thethe analysisanalysis ofof thethe collapsecollapse ofof singlesingle vaults,vaults, since since this this occurrence occurrence is governedis governed by equilibriumby equilibrium considerations consideration baseds based on kinematic on kinematic limit analysis.limit analysis. The approach The approach proposed proposed herein herein refersto refer thoses to vaults those without vaults without severe damage severe indamage the scenario in the ofscenario the global of the analysis global ofana thelysis entire of the building. entire building.

2.2. EquivalentEquivalent StiStiffnessffness andand MaterialMaterial PropertiesProperties InIn thisthis section,section, a a plate plate will will be be identified identified which which is is intended intended to to be be equivalent equivalent to to a masonry a masonry vault vault in termsin terms of stiof ffstiffness,ness, according according to to the the vault’s vault’ geometrys geometry and and the the mechanical mechanical properties properties ofof itsits materials.materials. TheThe target target is tois calibrate to calibrate the equivalentthe equivalent in-plane in- stiplaneffness stiffness as reported as in reported the following. in the This following. equivalence This wouldequivalen resultce would extremely result useful extremely when useful a global when analysis a global of theanalysis existing of the masonry existing structures masonry isstructures needed. Theis needed. equivalent Theplate equivalent elastic moduliplate elastic are obtained moduli asare proposed obtained in as [7 ]proposed by considering in [7] theby vaultedconsidering floor the as avaulted macro element,floor as a with macro appropriate element, wi constraintsth appropriate (Figure constraints1a,b). (Figure 1a,b).

(a) (b)

FigureFigure 1. 1.Conﬁguration Configuration of the of constraintsthe constraints and displacements and displacements for axial for and axial tangential and tangentialstiﬀness evaluation. stiffness (evaluationa) vaults with. (a) loadvaults and with constraints; load and (constraints;b) deformed (b) vault. deformed vault.

InIn Figure Figure1 a,1a, the the unstrained unstrained side-dimension side-dimension of of the the vaults vaults is is 2a2a,, andand thethe imposed imposed in-plane in-plane displacementdisplacement isis ∆Δuu,, which which isis assignedassigned inin orderorder toto evaluateevaluate thethe constraints’constraints’ reactionsreactions RiRi.. WithWith thethe obviousobvious meaningsmeanings ofof thethe usedused symbols,symbols, thethe equivalentequivalent longitudinallongitudinal andand tangentialtangential elasticelastic modulimoduli ofof thethe plateplate areare deﬁneddefined byby EquationsEquations (1)(1) and and (2), (2), respectively: respectively:

퐸푣 1 휎 1 P∑ 푅푖 (2⋅푎) Ev =1 σ ⋅ =1 ⋅ Ri ⋅ (2 a,) (1) 퐸= 퐸 =휀 퐸 (푠푣⋅2⋅푎) 훥푢· (1) E E · ε E · (sv 2 a) · ∆u 퐺푣 1 휏 1 · · 퐹 = ⋅ = ⋅ (2) 퐺 퐺 훾 퐺 푠푣 ⋅ (푢̅̅푥̅ + 푢̅̅푦̅)

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Gv 1 τ 1 F = = (2) G G · γ G · sv ux + uy · where 2a is the original side-dimension of the vault; sv is the vault thickness; ∆u is the imposed displacement, with its components in the x/y direction; E is the elastic modulus of the plate (Ex or Ey); σ is the normal stress that is generated in the x/y directions, due to ∆u, and ε is the corresponding strain; G is the shear elastic modulus of the plate; τ is the shear stress induced by the presence of the shear forces; γ is the corresponding distortion; F is the resultant of the shear stress applied to the main arches of the vault; Ev and Gv are the axial and shear elastic moduli of the vault, respectively. As it is known, the mechanical response of a specific vault type is related to several parameters, including the geometry, connections to other structural elements (i.e., ties), in-plane dimensions, thickness of the masonry surface, condition of constraints, and overlying filling materials. In this study, the contribution due to the filling materials is disregarded. In this simplification, it may be considered as a simple gravity load. Different types and geometries of vaults and constraint configurations were considered, corresponding to the most common problems in real practice in Mediterranean countries. In all cases, an in-plane square geometry of the vault was assumed, even if this hypothesis does not affect the meaning of the results, and can be easily removed by considering two different dimensions of the vault sides. T barrel vaults and cross vaults were studied, which are commonly found all over Europe and elsewhere. Moreover, two typical masonry vaults, commonly used in Southern Italy (Province of Lecce, Puglia) were also considered and studied. In this last case, many details about the vault geometry are presented in the following section to provide a clear reading of the structural problems and how this kind of vault was derived from the most common vaulted shapes. The structural analyses were performed using the software ANSYS v.14 [17]. FEM analyses were carried out considering the vaults as linear elastic systems with the hypothesis of a homogeneous solid. By performing the FEM, the element “solid186” was used from the ANSYS library, which is defined by 20 nodes in its most regular shape (prism with a rectangular base), and through a minimum of 10 nodes in the form suitable for irregular geometries (tetrahedral shape with a triangular base). Table1 shows the mechanical properties of the masonry assumed for the analysis.

Table 1. Mechanical properties of the masonry used in the FEM analyses.

Compressive Strength Young’s Modulus E Tangential Elastic Masonry Density 3 fm (MPa) (MPa) Modulus G (MPa) (kg/m )

3.5 1000 fm 0.4 E 1600 · ×

The material properties used in the analyses of this study were taken considering the real applications in the ﬁeld of natural limestone masonry. This choice was made in respect of the materials that are used in the typical vaults presented in the following. The values were provided by an extensive experimental campaign that the authors performed on an existing historical building located in the city of Lecce, Italy [18]. The value of the Young’s modulus was taken according to EC6 [19], even if respectfully to the experimental outcomes. The value of the elastic modulus in accordance with EC6 was chosen as the necessary value for the numerical analyses; in fact, the analysis is carried out in the elastic range and the equivalent modulus is expressed as a percentage of the initial value. This approach is preferred, since if using a lower elastic modulus value corresponding to cracked elements, the percentage value would be the same.

3. Vault Geometry and Construction Details The cases of T barrel and cross vaults (also sometimes known as double barrel vaults or groin vaults) will not be discussed herein, since they are well known in the literature. The special vaults typical of the Southern Puglia will be presented due to their unique shape. These vaults can be grouped into two families, even if both of them can be considered as derived from the most common cross vaults. Heritage 2020, 3 993

In Southern Italy, and in particular in the Puglia region, complex vaults shapes are widespread, derived from the cross vault. They are commonly referred as “leccesi” vaults. These vaults are typically made HeritageHeritage 20 202020, ,3 3 FOR FOR PEER PEER REVIEW REVIEW 55 according to a rigorous sequence of masonry blocks, which are cut and assembled after the execution of the abutments and walls. The construction of the vault starts with that of the supporting elements assembledassembled afterafter thethe execexecutionution ofof thethe abutmentsabutments andand walls.walls. TheThe constructionconstruction ofof thethe vaultvault startsstarts withwith (called “hung” and “nails”), thus a speciﬁc procedure is followed as described in [20]. Two types of thatthat of of the the supporting supporting elements elements (called (called ““hunghung”” and and “nails” “nails”),), thus thus a a specific specific procedure procedure is is followed followed as as leccesi vaults are known, which are called “edge vaults” and “square vaults”, as shown in Figure2. describeddescribed in in [19]. [19]. Two Two types types of of leccesi leccesi vaults vaults are are known, known, which which are are called called “ed “edgege vaults” vaults” and and “square “square The edge vaults are also called “star vaults” due to their typical aspect. vaults”,vaults”, as as shown shown in in Figure Figure 2. 2. T Thehe edge edge vaults vaults are are also also called called “star “star vaults” vaults” due due to to their their typical typical aspect. aspect.

RR1 OO OO 1 RR11

MM HH JJ

((aa)) ((bb))

FigureFigureFigure 2. 2. 2. Leccesi LeccesiLeccesi vaults: vaults: vaults: ( (a (a)a) edge) edge edge vault, vault, vault, ( (b (bb)) )square square square vault. vault. vault.

ReferringReferringReferring toto aa typical typical historical historicalhistorical building,building, thethe structural structuralstructural unit unitunit can cancan be bebe considered consideredconsidered to toto be bebe composed composedcomposed by bybythe the the vault vault vault and and and the the the vertical vertical vertical bearing bearing bearing elements elements elements constraining constraining constraining the the the vaults, vaults, vaults, as as shownas shown shown in in Figurein FigureFigure3. 3. 3.

((aa)) ((bb))

FigureFigureFigure 3. 3. 3. Main MainMain geometrical geometrical geometrical paramet paramet parameters:eersrs: :( (a (a)a) square) square square vault, vault, vault, ( (b (bb)) )edge edge edge vault vault vault.. .

ReferringReferringReferring to to to the the the simple simple simple square square square roof roof roof of of side ofside side 2a 2a (Figure (Figure 2a (Figure 3), 3), which 3which), which is is a a typical istypical a typical subsystem subsystem subsystem of of the the ofroofs roofs the underunderroofs under study study, study,, andand to to and the the to drawings drawings the drawings illustrated illustrated illustrated in in FigureFigure in Figuresss 44––747,– , 7 the the, the main main main geometric geometric geometric parameters parameters parameters that that characterizecharacterizecharacterize thethe construction construction formform areare the thethe height heightheight of ofof the thethe poles polespoles (H); (H);(H); the thethe distance distancedistance of ofof the thethe keystone keystonekeystone from fromfrom the thethe sourcesource planeplane (ZO);(ZO); thethe portionportion ofof barrelbarrel vault,vault, (f)(f) (Figures(Figures 44 andand 7)7); ; thethe ZZR1R1 dimensiondimension ofof thethe R1R1 pointpoint of of the the nails nails (terminal (terminal part part of of the the abutme abutments);nts); Z Zmm (M) (M) dimensions dimensions (edge (edge vault) vault) and and Z Zj jdimensions dimensions (J(J and and H H symmetrical symmetrical with with respect respect to to the the diagonal diagonal in in the the square square vault); vault); Z Zf,f ,which which is is the the distance distance from from thethe intersectionintersection bbetweenetween thethe nailnail andand thethe capcap toto thethe plantplant ofof thethe source;source; 2a2a andand 22aa′′, , respectivelyrespectively, , areare thethe totaltotal sizesize ofof thethe vaultvault andand thethe sizesize ofof thethe vaultvault atat thethe pointpoint wherewhere thethe capcap isis supportedsupported onon thethe

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source plane (ZO); the portion of barrel vault, (f) (Figures4 and7); the Z R1 dimension of the R1 point of the nails (terminal part of the abutments); Zm (M) dimensions (edge vault) and Zj dimensions (J and H symmetrical with respect to the diagonal in the square vault); Zf, which is the distance from the intersection between the nail and the cap to the plant of the source; 2a and 2a0, respectively, are the totalHeritage size 2020 of, 3 the FOR vault PEER and REVIEW the size of the vault at the point where the cap is supported on the hangers,6 respectively (Figure5a); the same meaning is assumed for the dimensions b and b of the edge vaults hangers, respectively (Figure 5a); the same meaning is assumed for the dimension0 s b and b′ of the (Figure5b). The hangers represent the support base of the cap and the nails. edge vaults (Figure 5b). The hangers represent the support base of the cap and the nails.

(a)

(b)

FigureFigure 4.4. Main geometrical parameters ofof thethe vaults’vaults’ toptop crosscross section:section: ((aa)) squaresquare vault,vault, ((bb)) edgeedge vault.vault.

AsAs itit cancan bebe observedobserved fromfrom thethe picturespictures inin FigureFigure5 5 the the edge edge vault vault can can be be seen seen as as derived derived from from a a crosscross vault,vault, wherewhere thethe centralcentral portionportion is obtained by the intersection of a cross vault and an ellipsoid. TheThe squaresquare vaultvault isis characterizedcharacterized byby thethe angles,angles, whichwhich areare emptiedemptied fromfrom thethe constructionconstruction materialmaterial oror realizedrealized asas samesame asas thethe cornerscorners ofof aa quarterquarter ofof aa pavilionpavilion vault.vault. InIn thisthis typetype ofof vault,vault, thethe diagonaldiagonal axisaxis coincidescoincides withwith thatthat ofof thethe diagonaldiagonal ofof anan equivalentequivalent squaresquare vault.vault. The result is that thethe planplan ofof thethe squaresquare vaultvault isis anan eight-pointeight-point star, star, while while the the plan plan of of the the edge edge vault vault is is a a four-point four-point star. star. Figures Figures6 and 6 and7 show7 show the the projection projection in thein the horizontal horizontal plan plan and and in the in verticalthe vertical plan plan of an of edge an edge vault vault and a and square a sq vault,uare respectively.vault, respectively. These drawings These drawings are those are used those during used theduring construction the construction phases, wherephases, the where master the builders master arebuilders asked are to checkasked theto check imposed the geometryimposed geometry at each stage at each of the stage construction of the construction process. process.

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(a)

(b)

Figure 5. (a) PlantPlant of of a a square square vault, vault (b,) ( plantb) plant of an of edge an edge vault. vault.

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(a()a ) (b()b )

FigureFigureFigure 6. 6. 6.Geometrical GeometricalGeometrical parameters parametersparameters for forfor the thethe construction constructionconstruction of ofof a asquarea squaresquare v vault:ault:vault: (a (()aa )plant) plant,plant, (,b ((b)b )cro) crosscrossss section section.section. .

(a()a ) (b()b )

FigureFigureFigure 7. 7. 7.Geometrical GeometricalGeometrical parameters parametersparameters for forfor the thethe construction constructionconstruction of ofof an anan edge edgeedge vault: vault:vault: (a (()aa )plant;) plant;plant; (b ((b)b )cross) crosscross section section.section. .

TheTheThe cap capcap of ofof the thethe vault vaultvault can cancan be bebe considered consideredconsidered as asas a a aportion portionportion of ofof an anan ellipsoid ellipsoid ellipsoid with with with the the the following following following equation: equation: equation: 2 2 2 2 2 2 푥 푥 2 푦푦2 푧 푧2 x y z (3()3 ) 2 2+++ 2 2+++ 2 2===111 (3) 훼훼α2 훽훽β2 훾γ훾2 wherewhere α αrefer refers tos to the the measurements measurements on on the the x xaxis, axis, β βrefer refers tos to the the measurements measurements on on the the y yaxis, axis, and and γ γ where α refers to the measurements on the x axis, β refers to the measurements on the y axis, and γ referrefers tos to the the measurements measurements on on the the z zaxis. axis. The The regions regions of of the the nails nails are are mathematically mathematically described described with with refers to the measurements on the z axis. The regions of the nails are mathematically described with thethe same same equations equations of of a abarrel barrel vault. vault. In In particular, particular, in in the the edge edge vaults vaults these these regions regions can can be be considered considered the same equations of a barrel vault. In particular, in the edge vaults these regions can be considered asas the the intersection intersection of of two two-barrel-barrel vaults vaults with with perpendicul perpendicularar longitudinal longitudinal directions. directions. The The construction construction as the intersection of two-barrel vaults with perpendicular longitudinal directions. The construction geometrygeometry of of the the square square vault vault is isknown known when when the the positions positions of of the the points points Q, Q, M, M, J, J,H, H, F ,F R, 1R, 1and, and R 2R, 2are, are geometry of the square vault is known when the positions of the points Q, M, J, H, F, R1, and R2, determined.determined. The The same same conclusion conclusion is isvalid valid for for the the edge edge vault vault when when the the positions positions of of P, P, M, M, R 1R, 1and, and R 2R 2 are determined. The same conclusion is valid for the edge vault when the positions of P, M, R1, and R2 areare set. set. T heThe different different values values of of α, α, β, β, and and γ γare are those those corresponding corresponding to to the the positions positions of of these these points. points. In In are set. The different values of α, β, and γ are those corresponding to the positions of these points. In the thethe construction construction field field, the, the expert expert master master builders builders set set these these points points by by means means of of a graphicala graphical construction construction construction field, the expert master builders set these points by means of a graphical construction mademade in in the the field. field. Thus Thus, the, the method method which which is istypically typically followed followed in in the the field field is isan an empirical empirical one. one. made in the field. Thus, the method which is typically followed in the field is an empirical one. FromFrom a amore more rigorous rigorous perspective, perspective, assuming assuming a asquare square shape shape of of the the plant plant of of the the vaults vaults (as (as it itoften often is isin in real real cases), cases), the the values values of of α, α, β, β, and and γ γfollow follow the the equations equations reported reported herein. herein.

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HeritageFrom 2020, 3 a FOR more PEER rigorous REVIEW perspective, assuming a square shape of the plant of the vaults (as it often9 is in real cases), the values of α, β, and γ follow the equations reported herein. CaseCase of of the the edge edge vault: vault: 2 푧z2 훾22 = 푅R1 1 γ = 푥 2, (4)(4) 1−( 푅x1) 2 1 훽 R1 − β 푧2 ∙푥2 2 푀 2푅1 2 2푧푀− z2 x 2 2z2 푧푅M1 · R1 훼 = M 22 , (5) −푧 zR α2 = [1−( 푀 ) ] 1 (5) 푧푅1 2 zM 1 z 훼2 =−훽2.R 1 (6) 2 = 2 Case of the square vault: α β . (6)

Case of the square vault: 푥 2 훼2 = 푅1 , 푧2 2 (7) 2 1−xR푅11 α = 훾2 (7) z2 1 R1 푡′ = 1 − tan− ⁡γ(2휏), (8) 2 2 2 2 2 2 푥퐻t0 =푦퐻1 tan(τ2) 2 푥퐻 +푦퐻 (8) 푧퐻−[푧푅1∙( 2+ 2−)] 푧퐻−[푧푅1∙( 2 )] 2 푥푅1 푦푅2 푥푅1 훾 = x2 2 2 y 2 = 2 x2 2+y . 2 (9) 2 2 푥퐻 H 푦+퐻 H 2 푥퐻2 +푦퐻 H H zH [1z−R(1 2+2 2)] 2 z[H1−( zR1 2 )] 2 − 푥·푅1xR1 푦푅2 yR2 − 푥푅1· xR1 γ2 = = (9) 2 2 2 2 xH yH xH +yH 1 2 + 2 1 2 − xR1 yR2 − xR1

For common building geometries and for a square plant, zR1 = zR2 = a; xR1 = yR2 = a0 t0 = For common building geometries and for a square plant, zR1 = zR2 = a; xR1 = yR2 = a′∙t′ = a·cos(30°)· t′; a cos(30◦) t0; zM = a/2; zH = zJ = a/2; xH = a0 + h0; yH = a0. zM· = a/2; zH· = zJ = a/2; xH = a′ + h0; yH = a′. ForFor the the 3D modeling of of the the vault, vault, dimensions dimensions were were assigned assigned in order in order to create to create a numerical a numerical model, close to real cases; h0 = 500 mm, τ = 22◦, and we use values derived from (4)–(9) model, close to real cases; h0 = 500 mm, τ = 22°, and we use values derived from (4)–(9) AccordingAccording to to the the analytical analytical equations equations,, the the surfaces surfaces of of the the vaults vaults were were defined deﬁned and and represented represented throughthrough the the Auto Autodeskdesk Inventor Inventor software, software, as as shown shown in in Figure Figure 8.8.

(a) (b)

FigureFigure 8. 8. TridimensionalTridimensional geometrical geometrical models models of of the the studied studied vaults: vaults: (a ()a )square square vault vault,, (b (b) )edge edge vault vault..

WithWith the the purpose purpose of of performing performing a aparametric parametric analysis, analysis, the the different different types types of of vault vault were were analyzed analyzed byby varying varying the the in in-plane-plane dimension dimensionss (2a (2a oror 2b2b).). Other Other variables variables were were considered considered in in the the parametric parametric analyses:analyses: the the constraints constraints configuration, configuration, the the thickness thickness of of the the side side walls walls (s (ps),p), and and the the thickness thickness of of the the vaultvault (sv).). The di differentfferent constraint constraint configurations configurations are shown are shown in Figure in 9Figure. The positions9. The positions of the constraints of the constraintsare strongly are related strongly to therelated distributions to the distributions of horizontal of horizontal forces in seismic forces conditions.in seismic conditions.

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(a) (b)

FigureFigure 9. 9. BoundaryBoundary conditions: conditions: different different constraints constraints at at the the abutments abutments and and at atthe the kidney kidneys.s. (a) vaults constrained in the abutment plane with and unitary displacement on the opposite side; (b) vaults Possibleconstrained constraint in the abutment conditions plane to are responses shown to in the Figure shear stress 9. The of the different roofs; (c) vaults configurations constrained of in the constraintsthe kidneys in the plane different with andpositions unitary of displacement the vault strongly on the oppositeinfluence side; the ( ddistribution) vaults constrained of the horizontal in the forceskidneys that are plane generated to responses under to theearthquakes. shear stress Figure of the roofs. 9a shows the condition in which the vaults are constrained by supports or in correspondence with the abutment plane, and this corresponds to the lowerPossible limit of constraintthe distribution conditions capacity are shown of the inhorizontal Figure9. actions. The di ff erentWith configurationsreference to Figure of the 1, constraints the vault isin supported the different on positions one side of andthevault a unitary strongly displacement influence the isdistribution applied to of the the opposite horizontal side. forces Figure that are9b representsgenerated under the case earthquakes. of the maximum Figure 9 distributiona shows the capacity condition of in the which horizontal the vaults actions are constrained, in which the by stiffnesssupports of or the in correspondencestructural system with increases the abutment as the cover plane, is and constrained this corresponds on the plane to the lowerpassing limit through of the thedistribution kidneys. In capacity Figure of9c,d the, the horizontal responses actions. to the shear With referencestress of the to roofs Figure are1, theconsidered vault is supportedin accordance on withone sideFigure and 2. aRespectively unitary displacement, the constraints is applied are applied to the opposite on the level side. toFigure the shutter9b represents and to the the kidneys. case of the maximumThese different distribution constraint capacity conditions of the horizontal have been actions, applied in to which the different the stiffness cases of theof numerical structural analysis,system increases as reported as the in the cover following. is constrained on the plane passing through the kidneys. In Figure9c,d, the responsesIn total, four to the different shear stress types of of the vaults roofs are have considered been studied in accordance using numerical with Figure analys2. Respectively,is—namely, crossthe constraints vault, barrel are applied vault, edge on the vault, level and to the square shutter vault. and to Two the differentkidneys. dimensions of the vaults’ thicknessThese (cap diff regions)erent constraint have been conditions considered: have s beenv = 0.8 applied mm (thin to the vaults) different and casessv = 200 of mm numerical (thick analysis,vaults). Tashe reported values that in the were following. set by the authors reflect a wide range of real practical applications. For each vault In configuration, total, four di twofferent different types ofboundary vaults have condi beentionsstudied were assumed using numerical in terms analysis—namely,of the side walls: vaultscross vault, constrained barrel at vault, their edge abutments vault, withand square side walls vault., and Two vaults diff erent constrained dimensions at their of abutments the vaults’ withoutthickness considering (cap regions) the side have walls. been considered: sv = 0.8 mm (thin vaults) and sv = 200 mm (thick vaults). The values that were set by the authors reflect a wide range of real practical applications. 4.For Equivalent each vault Stiffness configuration, Results two different boundary conditions were assumed in terms of the side In this section, the computation of the equivalent stiffness starts by showing the example of the barrel vaults, since this type of vault does not have symmetry in the two horizontal directions, as

Heritage 2020, 3 999 walls: vaults constrained at their abutments with side walls, and vaults constrained at their abutments without considering the side walls.

4. Equivalent Stiffness Results In this section, the computation of the equivalent stiffness starts by showing the example of the Heritage 2020, 3 FOR PEER REVIEW 11 barrel vaults, since this type of vault does not have symmetry in the two horizontal directions, as found infound the otherin the types. other Intypes. civil In buildings, civil buildings they were, they typically were typically used as used roof foras roof the basementfor the basement and ground and floor,ground having floor, ahaving span that a span is typically that is typically between between 3 and 5 3 m. and In 5 order m. In to order evaluate to evaluate the equivalent the equivalent elastic tensileelastic tensile modulus, modulus, it is necessary it is necessary to consider to consider the orthotropic the orthotropic behavior behavior of the of barrel the barrel vault vault in the in two the horizontal directions, due to the lack of symmetry. The two terms (E /E) and (E /E) represent the two horizontal directions, due to the lack of symmetry. The two termsV (ELV/E)L andV (ETV/E)T represent ratiothe ratio between between the the equivalent equivalent elastic elastic modulus modulus and and the the original original elastic elastic modulus modulus in in thethe longitudinallongitudinal direction and in the transverse direction, respectively. In the longitudinal direction L, the ratio (E /E) direction and in the transverse direction, respectively. In the longitudinal direction L, the ratio (EVV/E)LL can be easily considered as A /A, which is the ratio between the area of the resistant cross section of the can be easily considered as Av v/A, which is the ratio between the area of the resistant cross section of vaultthe vault and the and area the of area the of resistant the resistant cross section cross sectionof the equivalent of the equivalent plate. This plate. last term This can last be term computed can be as A = 2R s according to the dimensions illustrated in Figure 10, which reports the parameters that computed· as A = 2R·s according to the dimensions illustrated in Figure 10, which reports the characterizeparameters that the geometrycharacterize and the the geometry cross section and the of a c barrelross section vault. of a barrel vault.

Figure 10.10. Transversal cross section of a barrel vault.vault.

By assuming the symbols of Figure 1010,, for the barrel vault the following equations cancan be written inin orderorder toto computecompute thethe areaarea ofof thethe crosscross sectionsection andand describedescribe thethe geometry:geometry: 2 2 " 푠2 2# 푠 훼 퐴 = 휋 [(푅 +s ) − (푅s − α) ] ∙ = 푅푠훼 (10) A푣v = π R + R = Rsα , (10) 2 2 − − 2 ·22π 2휋 푅−푓! R f (11) 훼 =α2=푎푟푐표푠2arcos ( − ) . (11) R푅 According to EquationEquation (9)(9) thethe ratioratio ofof thethe elasticelastic modulimoduli isis writtenwritten as:as: 퐸 퐴 푅푠훼 휋 푣 푣 (Ev ) =Av = Rsα| = π, (12) 퐸 퐿= 퐴 = 2푅∙푠 훼=휋 = 2 (12) E L A 2R s α=π 2 which shows that the ratio of the elastic moduli is independent· from the length of the vault, resulting whichin a constant shows thatvalue the equal ratio to of π/2 the elastic= 1.57. moduliThis means is independent that Ev is 157% from therespect length to ofthe the longitudinal vault, resulting elastic in amodulus constant E value. equal to π/2 = 1.57. This means that Ev is 157% respect to the longitudinal elastic modulusIn orderE. to compute the longitudinal stiffness along the transverse direction of the vault, a unitaryIn orderdisplacement to compute has the been longitudinal imposed stionﬀ theness free along side the of transverse the vault directionby considering of the vault, the other a unitary side displacementconstrained by has a simple been imposed support on (Figure the free 11 sidea). With of the the vault same by hyp consideringothesis, the the tangential other side stiffness constrained has bybeen a simple computed. support In (Figurethis case 11, a).a simple With the support same hypothesis, has been applied the tangential on the sticonstrainedﬀness has been regions computed. of the Invault, this considering case, a simple a self support-balanced has beensystem applied of forces on theapplied constrained to the free regions regions of the that vault, is able considering to produce a a shear distortion (Figure 11b).

Heritage 2020, 3 1000 self-balanced system of forces applied to the free regions that is able to produce a shear distortion (Figure 11b). Heritage 2020, 3 FOR PEER REVIEW 12

(a) (b)

FigureFigure 11. 11Schemes. Schemes for for the the computation computation ofof stistiffnessffness in the the transverse transverse direction direction for for a abarrel barrel vault vault.. (a) barrel vaults constrained in the abutment plane with and unitary displacement on the opposite side;The (sameb) barrel computation vaults constrained has been in theperformed abutment considering plane to responses the presence to the shear of side stress walls of the, as roofs; shown in Figure(c) barrel11c,d. vaultsThis because, with side in wall the in case the abutmentof barrel vaults, plane with the andabsence unitary of displacementsymmetry, due on theits oppositeshape, might be stronglyside; (d) influenced barrel vaults by constrained the presence in the of abutment transverse plane elements with side (side wall walls) to responses that are to the able shear to reduce stress the displacements.of the roofs. The results shown in the following strongly highlight the role of the transverse wall in increasing the stiffness of the barrel vaults in the transverse direction. TheFigures same 12 computation and 13 show has the beenconstraint performed conditions considering and applied the presence forces for of the side square walls, vaults as shown and inedge Figure vaults 11,c,d. respectively. This because, in the case of barrel vaults, the absence of symmetry, due its shape, might be strongly influenced by the presence of transverse elements (side walls) that are able to reduce the displacements. The results shown in the following strongly highlight the role of the transverse wall in increasing the stiffness of the barrel vaults in the transverse direction. Figures 12 and 13 show the constraint conditions and applied forces for the square vaults and edge vaults, respectively. In order to make the reading and analysis of the results simpler, they will be shown by referring to dimensionless geometrical dimensions. A standard side length of 4 m has been chosen as a reference for the dimensionless ratio. With this assumption, the variable arif should be considered as the ratio of the side dimension of the vaults (2a) in meters, and 4 m: arif = 2a/4 [m/m]. The thickness of the side walls, when considered, is indicated as sp, which will be used to represent the variables sp/arif and sp/2a, useful to plot the equivalent stiffness in the presence of side walls with different thicknesses. In the following subsections, the results will be illustrated and discussed in relationships with the single variable that was considered in the parametric study.

4.1. Influence of the Thickness of the Vaults and Presence of Side Walls on the Stiffness of the Vaults (a) (b) The presence of side walls is meaningful only for barrel vaults. The results related to this kind of vault showFigure that12. Schemes the presence for the ofcomputation side walls of is the a relevant longitudinal variable (a) and for tangential the transversal (b) stiffness stiff ofness a of the vaults, as expected from theory. This is validsquare for both vault the. global axial and tangential stiffness of the vault. Another important parameter that strongly influences the lateral stiffness of the vault is the thickness of the cap region. As specified above, two different values of the vaults’ thickness (sv) were tested: 80 mm and 200 mm. The diagrams in Figure 14 show clearly how the equivalent longitudinal

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(a) (b)

Figure 11. Schemes for the computation of stiffness in the transverse direction for a barrel vault.

Heritage 2020, 3 1001 The same computation has been performed considering the presence of side walls, as shown in Figure 11c,d. This because, in the case of barrel vaults, the absence of symmetry, due its shape, might stibeff stronglyness (Figure influenced 14a) and by tangential the presence stiff nessof transverse (Figure 14 elementsb) of the vaults(side walls) decrease that with are theable respective to reduce side the dimensionsdisplacements. (2a )The when results varying shown the in thickness the following of the capstrongly region hig (shlightv). Only the the role tangential of the transverse stiffness ratiowall ofin theincreasing cross vault the stiffness with the of lowest the barrel thickness vaults has in the an increasingtransverse trenddirection. for side dimensions higher than 4 m.Figures In all the 12 other and cases,13 show the the trend constraint is a descending conditions one. and This applied may beforces due for to the the additional square vaults stiff nessand providededge vaults by, respectively. the membrane effect generated by the shape of the vault.

(a) (b)

FigureFigure 12. 12Schemes. Schemes for for the the computation computation ofof the longitudinal longitudinal (a) (a )and and tangential tangential (b) ( bstiffness) stiﬀness of a of a Heritagesquare 2020, vault.3 FOR PEER REVIEW square vault. 13

(a) (b)

FigureFigure 13.13 SchemesSchemes for for the the computation computation of the of thelongitudinal longitudinal (a) and (a) andtangential tangential (b) stiffness (b) stiﬀ ofness an ofedge an edge vault. vault.

In order to make the reading and analysis of the results simpler, they will be shown by referring to dimensionless geometrical dimensions. A standard side length of 4 m has been chosen as a reference for the dimensionless ratio. With this assumption, the variable arif should be considered as the ratio of the side dimension of the vaults (2a) in meters, and 4 m: arif = 2a/4 [m/m]. The thickness of the side walls, when considered, is indicated as sp, which will be used to represent the variables sp/arif and sp/2a, useful to plot the equivalent stiffness in the presence of side walls with different thicknesses. In the following subsections, the results will be illustrated and discussed in relationships with the single variable that was considered in the parametric study.

4.1. Influence of the Thickness of the Vaults and Presence of Side Walls on the Stiffness of the Vaults The presence of side walls is meaningful only for barrel vaults. The results related to this kind of vault show that the presence of side walls is a relevant variable for the transversal stiffness of the vaults, as expected from theory. This is valid for both the global axial and tangential stiffness of the vault. Another important parameter that strongly influences the lateral stiffness of the vault is the thickness of the cap region. As specified above, two different values of the vaults’ thickness (sv) were tested: 80 mm and 200 mm. The diagrams in Figure 14 show clearly how the equivalent longitudinal stiffness (Figure 14a) and tangential stiffness (Figure 14b) of the vaults decrease with the respective side dimensions (2a) when varying the thickness of the cap region (sv). Only the tangential stiffness ratio of the cross vault with the lowest thickness has an increasing trend for side dimensions higher than 4 m. In all the other cases, the trend is a descending one. This may be due to the additional stiffness provided by the membrane effect generated by the shape of the vault.

Heritage 2020, 3 1002 Heritage 2020, 3 FOR PEER REVIEW 14 Heritage 2020, 3 FOR PEER REVIEW 14 9% STIFFNESS VAULT without walls 9% STIFFNESS VAULT 8% without walls 8% 7% 7% 6% 6% 5% 5% Gv/G 4%

Gv/G 4% 3% 3% 2% 2% 1% 1% 0% 3 3.5 4 4.5 5 0% 2a [m] 3 3.5 4 4.5 5 2a [m] (a) (b)

cross sv = 20 cm(a; ) edge sv = 20 cm; square sv = 20 cm; (b) barrel sv = 20 cm;

crosscross s vs =v =8 20cm cm; ; edgeedge s vs =v =8 20cm cm; ; squaresquare sv s =v =8 20cm cm; ; barrelbarrel sv s=v 8= cm20 cm; ;

cross sv = 8 cm; edge sv = 8 cm; square sv = 8 cm; barrel sv = 8 cm; FigureFigure 1 14.4. LongitudinalLongitudinal (a) (a )and and tangential tangential (b) (b )stiffness stiffness of of the the vaults: vaults: influence influence of of the the type, type, thickness thickness,, andandFigure in in-plane-plane 14. Longitudinal dim dimensionsensions (a)in in theand the absence absencetangential of of side side(b) stiffnesswalls. walls. of the vaults: influence of the type, thickness, and in-plane dimensions in the absence of side walls. FigureFigure 1155aa,b,b illustrate illustrate the the diagrams diagrams of of the the vaults vaults’’ stiffness stiffness in in the the presence presence of of side side walls, walls, consideringconsideringFigure different1 di5ffa,erentb illustrate thickness thickness the values values diagrams of of the the ofside side the walls. walls. vaults The The’ stiffness results results are arein relatedthe related presence to to those those of simulations simulations side walls, ininconsidering which which the the differentside side dimensions dimensions thickness of of valuesthe the vaults vaults of the are are side arifa rif= walls. 4= m4, m, byThe byvarying results varying th aree thestiffness related stiffness toof thosethe of theside simulations side walls walls sp. Thesinp. which stiffening The stitheffening side effect dimensions e ffprovidedect provided ofby the the by vaults side the walls sideare a wallsrifcan = 4 be m can ,easily by be varying easilyquantified quantifiedthe stiffness in terms in of termsof the the side ofEv/E thewalls ratio,Ev s/Ep . whichratio,The stiffening whichis increased is effect increased with provided respect with respecttoby the the results side to the walls of results Figure can of be1 Figure4 aeasily. The 14 trendquantifieda. The appears trend in termsto appears be linear of tothe beor E pseudov linear/E ratio, or- linearpseudo-linearwhich in is allincreased cases. in all Higher with cases. respect Highervalues to ofvalues the the results G ofv/G the ofratioG Figurev/G wereratio 1 4found werea. The found intrend the in appearspresence the presence to of be side linear of sidewalls or walls pseudo too, too,as- shownaslinear shown in allFigure in Figurecases. 15 bHigher15, comparedb, compared values to of toFigure the Figure G 1v4/G 14b. ratiob.In Inthese thesewere cases cases,found, the thein trends the trends presence are are not not linearof linear side and andwalls differ di fftoo,er for foras thetheshown case case inof of Figurethe the barrel barrel 15b vaults,,vaults, compared which which to show showFigure an an 1 4inversion inversionb. In these of of casesthe the trend trend, the trendsfor for the the arecase case not ssvv/ /alineararifrif == 0.0200.020. and. differ for the case of the barrel vaults, which show an inversion of the trend for the case sv/arif = 0.020.

(a) (b) cross sv/arif = 0.05;(a ) edge sv/arif = 0.05; square sv/arif = 0.05; (b) barrel sv/arif = 0.05;

crosscross sv/ sav/rifa rif= =0.02; 0.05; edgeedge sv/ sav/rifa rif= 0.02;= 0.05; squaresquare sv/ sav/rifa =rif 0.02; = 0.05; barrelbarrel sv/ sav/rifa rif= 0.02;= 0.05;

cross sv/arif = 0.02; edge sv/arif = 0.02; square sv/arif = 0.02; barrel sv/arif = 0.02; Figure 15. Longitudinal (a) and tangential (b) stiffness of the vaults: influence of the side walls’ FigureFigure 15. 15Longitudinal. Longitudinal (a) and(a) and tangential tangential (b) stithickness (b)ﬀness stiffness of. the vaults:of the vaults: inﬂuence influence of the side of walls’the side thickness. walls’ thickness.

Heritage 2020, 3 1003 Heritage 2020, 3 FOR PEER REVIEW 15

AllAll the the curves curves in Figuresin Figure 14s and 14 15and refer 15 torefer cases to in cases which in the which vaults the are vaults constrained are constrained at the abutments, at the asabutments, is commonly as is found commonly in real cases.found In in the real following cases. In diagrams the following of Figure diagrams 16, different of Figure constraint 16, schemesdifferent areconstraint also considered schemes in are the also presence considered of side in walls. the presence The results of show side that walls. the The presence results of show constraints that the at thepresence abutments of constraints provides at a the higher abutments stiffness, provides as expected. a higher A distiffnessfference, as of expected. about 20% A isdifference found for of all about the vault20% is types found in for terms all the of the vault longitudinal types in terms stiffness. of the The longitudinal differences stiffness. in terms The of tangentialdifferences sti inff nessterms are of muchtangential higher, stiffness as illustrated are much in Figurehigher, 16 asb. illustrate In thesed cases, in Figure differences 16b. Inof these up tocases, almost differences 40% were of found up to foralmost barrel 40% vaults were for found a side fordimension barrel vaults2a = 4 mfor (common a side dimension cases in civil 2a buildings).= 4 m (common This means cases that in civil an investigationbuildings). This in order means to that verify an the investigation real static scheme in order of to the verify vaults the is real mandatory static scheme when theof the accuracy vaults ofis themandatory structural when model the is accuracy a fundamental of the structura issue. l model is a fundamental issue.

(a) (b) cross K edge K square K barrel K cross T + W edge T + W square T + W barrel T + W

FigureFigure 16.16. AxialAxial ((a)a) andand tangentialtangential ((b)b) stistiffnessﬀness forfor vaultsvaults boundedbounded atat thethe abutmentsabutments inin thethe presencepresence ofof sideside wallswalls and at the kidneys. T T ++ WW = =boundedbounded at atthe the abutments abutments in inthe the presence presence of side of side walls; walls; k = kbounded= bounded at the at thekidneys. kidneys.

TheThe previousprevious diagramsdiagrams showshow thatthat thethe presencepresence ofof sideside wallswalls contributescontributes toto providingproviding closed-boxclosed-box behaviour,behaviour, whichwhich isis thethe structuralstructural responseresponse desireddesired byby thethe designers.designers. TheThe presencepresence ofof aa sideside wallwall producesproduces more more benefits benefits than than the the constraints constraints at theat the kidneys kidneys in termsin terms of the of globalthe global vault’s vaul stit’ffsness. stiffness. In this In sense,this sense the capacity, the capacity to redistribute to redistribute the horizontal the horizontal forces for theforces vaults for with the vaultsside walls with and side constraints walls and at theconstraints abutments at appearsthe abutments to be three appears times to that be three of the times same that vaults of withthe same constraints vaults atwith the constraints kidneys without at the sidekidneys walls without (for the side case walls of sv =(for200 the mm). case For of example,sv = 200 mm for). edge For vaults,example, the fo stir ffedgeness vaults values,, the in termsstiffness of thevalues, capacity in terms of horizontal of the capacity force redistribution, of horizontal goforce from redistribution, 18.78% with constraintgo from 18. at78% the kidneyswith constraint to 41.86% at inthe the kidneys presence to of41. side86% walls in the for presence a span 2a of= side3 m. walls The same for a values span 2a go = from 3 m. 13.89% The same to 30.22% values for go a spanfrom 2a13.89%= 5 m, to which 30.22% is for more a span common 2a = 5 in m real, which cases. is Formore the common edge vaults, in real these cases. diff Forerences the edge are 59% vaults and, these 32% fordifferences the cases are of 2a59%=3 and m and 32%2a for=5 the m, cases respectively. of 2a =3 Form and barrel 2a =5 vaults, m, respectively. the differences For arebarrel 58% vaults and 25%, the fordifferences the cases are of 2a58%= 3and m and25%2a for= 5the m, cases respectively. of 2a = 3 m and 2a =5 m, respectively. InIn allall cases,cases, asas expected,expected, thethe stistiffnessffness ratioratio ofof thethe vaultsvaults decreasesdecreases withwith anan increaseincrease inin thethe spanspan ((2a2a).). OnlyOnly forfor the the cases cases of of the the tangential tangential sti stiffnessffness of of the the square square vaults, vaults, bounded bounded at the at the abutments abutments in the in presencethe presence of side of side walls, walls, there there is a kneeis a knee point point (2a =(2a4.2 = m).4.2 m). In thisIn this case, case the, the tangential tangential stiff stiffnessness seems seems to slightlyto slightly increase increase after after the the knee knee point. point.

4.2. Comparison of the Structural Response in Absence of Side Walls In this section, the results will be presented in terms of the longitudinal and tangential stiffness for those cases in which the thickness of the vault cap is 200 mm (more common than thin vaults); the vaults are constrained at the abutments, as is common in real cases; the vaults are constrained at

Heritage 2020, 3 1004

4.2. Comparison of the Structural Response in Absence of Side Walls In this section, the results will be presented in terms of the longitudinal and tangential stiﬀness for those cases in which the thickness of the vault cap is 200 mm (more common than thin vaults); Heritagethe vaults 2020, are 3 FOR constrained PEER REVIEW at the abutments, as is common in real cases; the vaults are constrained16 at the kidneys; side walls are not present. In Figure 17a,b, the stiﬀness ratios are plotted with reference to the kidneys; side dimensions side walls of theare vaults.not present. In Figure 17a,b, the stiffness ratios are plotted with reference to the side dimensions of the vaults.

(a) (b) cross K edge K square K barrel K cross T edge T square T barrel T

Figure 17. 17. AxialAxial ( a(a)) and and tangential tangential (b) sti(b)ﬀ nessstiffness without without sidewalls side whenwalls varyingwhen varying the constraint the constraint positions. positionT = constraintss. T = constraints at the abutments; at the abutments; k = constraints k = constraints at the kidneys. at the kidneys.

From the the plots plots,, it it is is evident evident that that the the same same decreasing decreasing trend trend is isfound found as asthe the span span 2a increases,2a increases, as expected.as expected. The The behaviour behaviour of the of edge the edge and cross and crossvaults vaults seems seems to be overlapped to be overlapped for those for cases those in cases which in thewhich span the is spanequal is to equal or higher to or than higher 4.0 than m. This 4.0 m.was This found was for found both forthe bothaxial theand axial tangential and tangential stiffness ratios.stiﬀness ratios.

5. Mechanical Mechanical Response Response of of Vault Vaultss and Equivalent Plate Systems In this section,section, thethe results results of of the the FEM FEM analyses analyses will will be shown,be shown, comparing comparing the vaultsthe vaults modeled modeled with withtheir geometry,their geometry, with equivalent with equivalent plate systems plate systems in which in the which reduced the modulireduced were moduli computed were computed according accordingto Equations to (1)Equations and (2). (1) The and structural (2). The scheme structural used scheme for the used computation for the computation of the seismic of response the seismic was responseconceived was as aconceived structural as roof a structural supported roof by supported four columns, by four as shown columns, in the as shown cross section in the cross of Figure section 18. of FigureFor the18. computation of the excited mass, the ﬁlling materials that are placed around the cap of the vault were considered with a density of 10 kN/m3. The columns were considered with a square cross section. For those cases in which the vault has been modelled as an equivalent plate, all the loads where considered as nodal forces at the top of the columns. In the ﬁrst stage, the vaults were loaded with a distributed load fi, as shown in Figure 18, with a resultant value of 1 kN. The reaction at the base of the columns is called F1 with reference to the applied F = Σ fi.

Figure 18. Structural scheme for the computation of the seismic response of the vaults.

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the kidneys; side walls are not present. In Figure 17a,b, the stiffness ratios are plotted with reference to the side dimensions of the vaults.

(a) (b) cross K edge K square K barrel K cross T edge T square T barrel T

Figure 17. Axial (a) and tangential (b) stiffness without side walls when varying the constraint positions. T = constraints at the abutments; k = constraints at the kidneys.

From the plots, it is evident that the same decreasing trend is found as the span 2a increases, as expected. The behaviour of the edge and cross vaults seems to be overlapped for those cases in which the span is equal to or higher than 4.0 m. This was found for both the axial and tangential stiffness ratios.

5. Mechanical Response of Vaults and Equivalent Plate Systems In this section, the results of the FEM analyses will be shown, comparing the vaults modeled with their geometry, with equivalent plate systems in which the reduced moduli were computed according to Equations (1) and (2). The structural scheme used for the computation of the seismic Heritageresponse2020 ,was3 conceived as a structural roof supported by four columns, as shown in the cross section1005 of Figure 18.

FigureFigure 18. 18Structural. Structural scheme scheme for for the the computation computation of of the the seismic seismic response response of of the the vaults. vaults.

The side of the square cross section of the column is represented by B, while the height of the columns is indicated by H. In the numerical simulations, the range of B was contained between 0.3 and 2 m, while H varied from 1.8 to 4.00 m. The span of the vault will be indicated as arif, which corresponds to the side 2a in the previous sections. The capacity of the lateral force redistribution of the structural roof (vault/plate) has been evaluated in terms of the F1/F ratio, which is correlated to the ratios (B/arif) and (H/arif). The numerical results were also obtained for the limit cases F1/F = 0.5, which means stiff plate, and F1/F = 1.0, which means that the plate has zero stiffness. The parametric analyses were performed by varying the dimensions of the columns (B and H), assuming span values are in accordance with those cases of real engineering practice. In Figure 19, the results of the numerical analyses are shown for the edge vault with side walls, where sv = 0.2 m, and the equivalent plate, obtained using Equations (1) and (2). In Figure 20, the force redistribution ratio (F1/F) is plotted with reference to the dimensionless geometrical ratio (B/arif). The label v refers to the case of the numerical simulations of vaults; the label p refers to cases of the numerical simulations of equivalent plates; the label DE refers to the case of deformable plates (zero stiffness); the label S indicates the cases of stiff plate. Figure 20a refers to the case of the absence of side walls, while Figure 20b refers to the presence of side walls. In both cases, the solution with equivalent plate falls in the range between the limit cases of infinitely stiff and deformable plates. This confirms that the equivalent plate provides a realistic solution in terms of the lateral force redistribution. The condition of H/arif = 0.45 (corresponding to a high stiffness of the columns with respect to the stiffness of the roof), as expected, shows that F1 prevails with respect to the forces transmitted by the roof to the columns placed on the opposite side. This trend is more evident when the values of the B/arif ratio increase (stiff columns). The presence of the side walls has an important role in the distribution of the lateral force F. The fraction of the whole horizontal action between the piers becomes more efficient, with differences of up to 10%. Similar results (not all shown in the present work) can be deduced for the cross, barrel, and square vaults; in the case of the barrel vault, the influence of the side walls has a higher incidence (about 29%). Heritage 2020, 3 FOR PEER REVIEW 17

For the computation of the excited mass, the filling materials that are placed around the cap of the vault were considered with a density of 10 kN/m3. The columns were considered with a square cross section. For those cases in which the vault has been modelled as an equivalent plate, all the loads where considered as nodal forces at the top of the columns. In the first stage, the vaults were loaded with a distributed load fi, as shown in Figure 18, with a resultant value of 1 kN. The reaction at the base of the columns is called F1 with reference to the applied F = Σ fi. The side of the square cross section of the column is represented by B, while the height of the columns is indicated by H. In the numerical simulations, the range of B was contained between 0.3 and 2 m, while H varied from 1.8 to 4.00 m. The span of the vault will be indicated as arif, which corresponds to the side 2a in the previous sections. The capacity of the lateral force redistribution of the structural roof (vault/plate) has been evaluated in terms of the F1/F ratio, which is correlated to the ratios (B/arif) and (H/arif). The numerical results were also obtained for the limit cases F1/F = 0.5, which means stiff plate, and F1/F = 1.0, which means that the plate has zero stiffness. The parametric analyses were performed by varying the dimensions of the columns (B and H), assuming span values are in accordance with those cases of real engineering practice. In Figure 19, the results of the numerical analyses are shown for the edge vault with side walls, where sv = 0.2 m, and the equivalent plate, obtained using Equations (1) and (2). In Figure 20, the force redistribution ratio (F1/F) is plotted with reference to the dimensionless geometrical ratio (B/arif). The label v refers to the case of the numerical simulations of vaults; the label p refers to cases of the numerical simulations of equivalent plates; the label DE refers to the case of deformable plates (zero stiffness); the label S indicates the cases of stiff plate. Figure 20a refers to the case of the absence of side walls, while Figure 20b refers to the presence of side walls. In both cases, the solution with equivalent plate falls in the range between the limit cases of infinitely stiff and deformable plates. This confirms that the equivalent plate provides a realistic solution in terms of the lateral force redistribution. The condition of H/arif = 0.45 (corresponding to a high stiffness of the columns with respect to the stiffness of the roof), as expected, shows that F1 prevails with respect to the forces transmitted by the roof to the columns placed on the opposite side. This trend is more evident when the values of the B/arif ratio increase (stiff columns). The presence of the side walls has an important role in the distribution of the lateral force F. The fraction of the whole horizontal action between the piers becomes more efficient, with differences of up to 10%. Similar results (not all shown in the present work) can be deduced for the cross, barrel, and square vaults; in the case of the barrel vault, the influenceHeritage 2020 of, 3the side walls has a higher incidence (about 29%). 1006

Heritage 2020, 3 FOR PEER REVIEW(a) (b) 18 F1/F = 51.9%

Figure 19. RedistributionRedistribution of of the the lateral lateral fo forcesrces in in the the edge vaults and the equivalent plate with side walls (s v == 0.20.2 m). m). (a (a) )Edge Edge vault vault with with the the application application of of horizontal horizontal force, force, ( (bb)) deformation deformation edge edge vault vault with the application of horizontal force, force, ( cc)) equivalent equivalent plate plate model model with with horizontal horizontal force force application, application, (d) deformation equivalent plate model with horizontal force application.

(a) (b)

(H/arif)DE (H/arif)p = 0.45 (H/arif)v = 1.0 (H/arif)v = 0.75

(H/arif)S (H/arif)v = 0.45 (H/arif)p = 1.0 (H/arif)p = 0.75

Figure 20. Diagrams of the redistribution of lateral forces in edge vaults and the equivalent plate with

side walls (sv = 0.2 m). (a) with side walls; (b) without side walls.

6. Dynamic Behavior of Edge Vaults and Equivalent Plate Systems A linear dynamic analysis (modal) has been implemented in the FEM code. The modal shapes and vibrational frequencies of the vaults and equivalent plates were studied and compared in order to verify how the mechanical equivalence is related to a similar dynamic response. The same solid models used for the static analyses previously described were used for the modal analyses. The study for the edge vault is presented herein. In Figure 21a, typical edge vaults, commonly found in real applications, are shown and compared to the equivalent plate system. The span between the columns is 4 m; the square cross section of the columns has dimensions of 700 × 700 mm; the height of the columns is 3.5 m. The thickness of the vault cap is 200 mm, with side walls that are present in all sides of the vault. The total mass of the vaulted structure was computed to be equal to 225 kN (22.5 ton); the mass of the side walls was computed to be equal to 100.8 kN (10.08 ton); the total mass of the

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Figure 19. Redistribution of the lateral forces in the edge vaults and the equivalent plate with side walls (sv = 0.2 m). (a) Edge vault with the application of horizontal force, (b) deformation edge vault Heritagewith2020 the, 3 application of horizontal force, (c) equivalent plate model with horizontal force application, 1007 (d) deformation equivalent plate model with horizontal force application.

(a) (b)

(H/arif)DE (H/arif)p = 0.45 (H/arif)v = 1.0 (H/arif)v = 0.75

(H/arif)S (H/arif)v = 0.45 (H/arif)p = 1.0 (H/arif)p = 0.75

Figure 20. Diagrams of the redistribution of lateral forces in edge vaults and the equivalent plate with Figure 20. Diagrams of the redistribution of lateral forces in edge vaults and the equivalent plate with side walls (sv = 0.2 m). (a) with side walls ; (b) without side walls. side walls (sv = 0.2 m). (a) with side walls; (b) without side walls.

6.6. DynamicDynamic BehaviorBehavior ofof EdgeEdge VaultsVaults andand EquivalentEquivalent PlatePlate SystemsSystems AA linearlinear dynamicdynamic analysisanalysis (modal)(modal) hashas beenbeen implementedimplemented inin thethe FEMFEM code.code. TheThe modalmodal shapesshapes andand vibrationalvibrational frequencies frequencies of of the the vaults vaults and and equivalent equivalent plates plates were were studied studied and and compared compared in orderin order to verifyto verify how how the mechanicalthe mechanical equivalence equivalence is related is related to a similar to a similar dynamic dyn response.amic response. The same The solid same models solid usedmodels for used the static for the analyses static analyses previously previously described described were used were for used the modal for the analyses. modal analyses. The study The for study the edgefor the vault edge is presentedvault is presented herein. In herein. Figure 21Ina, Figure typical 2 edge1a, typical vaults, edge commonly vaults,found commonly in real found applications, in real areapplications, shown and are compared shown and to compared the equivalent to the plateequivalent system. plate The system. span The between span between the columns the columns is 4 m; theis 4 square m; the cross square section cross of section the columns of the hascolumns dimensions has dimensions of 700 700 of mm;700 × the 700 height mm; ofthe the height columns of the is × 3.5columns m. The is thickness 3.5 m. The of thickness the vault of cap the is vault 200 mm, cap with is 200 side mm, walls with that side are walls present that inare all present sides of in the all vault.sides Theof the total vault. mass The of thetotal vaulted mass of structure the vaulted was computedstructure w toas be computed equal to 225 to be kN equal (22.5 ton);to 225 the kN mass (22.5 of ton); the sidethe mass walls of was the computed side walls to bewas equal computed to 100.8 to kN be (10.08 equal ton); to 100.8 the total kN mass(10.08 of ton); the fillingthe total at themass extrados of the wasfilling computed at the extrados to be equal was compute to 100 kNd to (10 be ton). equal The to 100 first kN modal (10 ton). shape The is relatedfirst modal to the shape translational is related vibration along one of the principal axes of the structure. Due to the symmetry of the structural system, the first and second modal shapes involve the same mass and the same frequencies. The third modal shape is related to rotational vibration, which describes a torsional behaviour of the system. The first (and the second, as also intended in the following) modal shape of the vault in the x (y)-direction involves 84% of the mass as excited in the first mode. The frequency is 4.96 Hz. For the equivalent plate system, in the same first (second) modal shape, the excited mass is 87%, while the vibrational frequency is 4.74 Hz. The rotational modal shape of the vaulted structure involves a negligible amount of mass (1%), with a frequency of 5.25 Hz. The rotational shape of the equivalent plate system involves 1% of the mass, with a frequency of 7.06 Hz. The results of the numerical simulations are summarized in Figure 21. In the absence of side walls, the same simulations were run. The vibrating mass of the first (second) modal shape was 84% and 78% of the total mass for the vaulted structure and for the equivalent plate, respectively. The third rotational modal shape involves less than 1% of the total mass. A parametric modal analysis was run considering the ratios B and H . The same case listed ari f ari f above for the edge vault with sv = 200 mm is shown herein, constrained at the abutments with side walls. The equivalent plate system was compared considering the same parameters. The results are Heritage 2020, 3 1008 illustrated in the form of diagrams in Figure 22. The curves refer to the vaulted system (v) and the equivalent plate (p); in all cases, they show an ascending branch in terms of the 1st vibrational frequency versus the increase in the B ratio. For high values of the B ratio, the frequency shows ari f ari f a peak. This means that, in those cases after the peak, the columns are very stiff and act as a rigid constraint for the roof. After the peak, the vibrational effect is related locally only to the roof system, without involving the support columns. The difference between the results obtained for the vault and the equivalent plate models, in terms of the vibrational frequency and the modal behavior of the structure, is shown in Table2 for the case of constraints at the abutments and in the presence of side walls, and in the Table3 for the case of constraints at the abutments and in the presence of side walls. When the constraints are at the kidneys, the stiffer behavior of the vault reduces the possibility of local modal deformations.

Table 2. Diﬀerence in terms of the vibrational frequency, for the case of constraints at the abutments and in the presence of side walls.

H B arif arif ∆ 1st Vibrational Frequency H = 0.45 0.15 < B < 0.43 ari f ari f <10% H = 0.75 0.2 < B < 0.5 ari f ari f <10% H = 1.0 0.2 < B < 0.5 ari f ari f <10% Modal behavior of the structure H = 0.45 B < 0.40 ari f ari f global deformation of the entire H = 0.75 B < 0.45 ari f ari f structure (no local modes) H = 1.0 B < 0.45 ari f ari f

Table 3. Diﬀerence in terms of the vibrational frequency, for the case of constraints at the kidneys and in the presence of side walls.

H B arif arif ∆ 1st Vibrational Frequency H = 0.45 0.20 < B < 0.32 ari f ari f <10% H = 0.75 0.25 < B < 0.5 ari f ari f <10% H = 1.0 0.33 < B < 0.58 ari f ari f <10% Modal behavior of the structure H = 0.45 B > 0.38 ari f ari f Local mode Heritage 2020, 3 FOR PEER REVIEW 20

= 0.75 0.25 < < 0.5 <10% = 1.0 0.33 < < 0.58 <10% Modal behavior of the structure Heritage 2020, 3 = 0.45 > 0.38 Local mode 1009

Ev/E = 22.49%; Gv/G = 29.56% f1 = 4.96 Hz; M1 = 84% f1 = 4.74 Hz; M1 = 87% (a) (b)

Ev/E = 22.49%; Gv/G = 29.56% f3 = 7.06 Hz; M3 < 1% f3 = 5.24 Hz; M3 < 1% (c) (d)

FigureFigure 21. 21. ResultsResults of of the the modal modal analysis analysis for for edge edge vaults vaults with with side side walls walls constrained constrained at at the the abutments abutments (s(sv v= =200200 mm); mm); (a ()a edge) edge vault: vault: 1st 1st vibrational vibrational frequency frequency and and vibrating vibrating mass; mass; (b (b) )model model equivalent equivalent plates:plates: 1st 1st vibrational vibrational frequency frequency and and vibrating vibrating mass; mass; (c ()c )edge edge vault: vault: 3rd 3rd vibrational vibrational frequency frequency and and vibratingvibrating mass; mass; (d (d) )model model equivalent equivalent plates: plates: 3rd 3rd vibrat vibrationalional frequency frequency and and vibrating vibrating mass. mass.

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(a) (b) (a) (b) Figure 22. Frequency of the first modal shape vs. (B/arif) for edge vaults with side wall constraints (W) Figure 22.22. FrequencyFrequency of of the the first first modal modal shape shape vs vs.. (B/a (B/arifrif) for) for edge edge vaults vaults with with side side wall wall constrain constraintsts (W) (W) (a) at the abutments (T), (b) at the kidneys (K). (sv = 200 mm) v = vaults; p = equivalent plate. ((aa)) atat thethe abutments (T) (T),, ( b)) at at the the kidneys kidneys (K) (K).. ( (ssvv == 200200 mm) mm) v v= =vaults;vaults; p = p equivalent= equivalent plate plate.. 7.7. CaseCase StudyStudy 7. Case Study AnAn entire entire ancient ancient building building was was studied studied and and analyzed analyzed by by applying applying the the criteria criteria of of equivalent equivalent An entire ancient building was studied and analyzed by applying the criteria of equivalent stistiffness,ffness, asasdescribed described before.before. TheThe masonrymasonry buildingbuilding waswas builtbuilt inin thethe 16th16th centurycentury inin SouthernSouthern ItalyItaly stiffness, as described before. The masonry building was built in the 16th century in Southern Italy inin aa lowlow seismicseismic area;area; the Italian Italian technical technical code code provides provides for for a aground ground acceleration acceleration peak peak of of (PGA) (PGA) ag in a low seismic area; the Italian technical code provides for a ground acceleration peak of (PGA) ag a=g 0.0= 0.055 g for g for the the site. site. This This value value is used is used for forthe the seismic seismic analys analysisis of the of thebuilding building.. The The building building has hastwo = 0.05 g for the site. This value is used for the seismic analysis of the building. The building has two two floors in addition to an underground floor, which was entirely excavated in the rock. The building floorsfloors in in addition addition to to an an underground underground floor floor, which, which was was entirely entirely excavated excavated in inthe the rock. rock. The The building building is is is irregular in terms of its mass distribution and stiffness in both the horizontal and vertical directions. irregularirregular in in terms terms of of its its mass mass distribution distribution and and stiffness stiffness in in both both the the horizontal horizontal and and vertical vertical directions. directions. Natural materials were used to build the walls (limestone blocks); at the ground floor, all the roofs are NaturalNatural materials materials were were used used to to build build the the walls walls (limestone (limestone blocks) blocks); at; atthe the ground ground floor floor, all, all the the roof roofs s made of vaulted structures. In Figure 23, an external view of the building is shown. areare made made of of vaulted vaulted structures. structures. In In Figure Figure 2 32,3 an, an external external view view of of the the building building is isshown. shown.

(a(a) ) (b()b )

Figure 23. Masonry building: south side (a), east side (b). Figure 2 23.3. MasonryMasonry building: building: south south side side (a ()a,), east east side side (b ()b. ).

TheThe vaultsvaults at at the the ground ground floor floorfloor are areare barrel barrelbarrel vaults. vaults. vaults. The TheThe original originaloriginal roof roofroof at at atthe thethe first firstfirst floor floorfloor was waswas made mademade of of of woodwoodwood andand iron,iron, butbut it it was was demolished demolished after afterafter fire firefire events events events a a along long long time time time ago. ago. ago. Then Then,Then, the, thethe roof roofroof waswas was mademade made of of of reinforcedreinforcedreinforced concrete concreteconcrete slabs slabs slabs with with with T-beams. T T-beams.-beams. At At theAt the moment,the moment, moment, after aft aft aer globaler a aglobal global restoration restoration restoration of the of buildingofthe the building building which finishedwhichwhich finished finished in 2018, in in the 2018, 2018, roof the the is roof maderoof is is made of made glulam of of glulam glulam timber timber timber with awith reinforcedwith a reinforceda reinforced concrete concrete concrete stiffening stiffening stiffening plate plate at plate the extrados.atat thethe extrados.extrados. At the At At ground the the ground ground floor, thefloor floor walls, ,the the arewalls walls made are are ofmade made single of of single leaf single wall leaf leaf (200 wall wall mm (200 (200 thickness) mm mm thickness) thickness) or double or or leafdoubledouble wall leafleaf (300 wal wal mml l(300 (300 thickness), mm mm thickness), thickness), with limestone with with limestone limestone blocks that blocks blocks were that that produced were were produced produced by the quarry by by the the that quarry quarry was that close that towaswas the closeclose building. toto the the building. Inbuilding. the case In In ofthe the double case case of leafof double double walls, leaf leaf an walls, innerwalls, an non-structural an inner inner non non-structural-structural thickness thickness ofthickness 100 mm of of100 was 100 mmmm was was found, found, made made of of non non-structural-structural building building materials materials such such as as irregula irregular stonesr stones and and bricks bricks mixed mixed withwith earth. earth. This This was was made made by by ancient ancient handcrafts handcrafts to to increase increase the the stiffness stiffness of ofthe the walls walls and and to toprovide provide

Heritage 2020, 3 1011 found, made of non-structural building materials such as irregular stones and bricks mixed with earth. HeritageThis was 2020 made, 3 FOR by PEER ancient REVIEW handcrafts to increase the stiﬀness of the walls and to provide better thermal22 properties. In some cases, (ground ﬂoor) the single leaf walls were coupled with an external wall made betterof irregular thermal stones properties. without In any some structural cases, properties.(ground floor) The the volume single placed leaf walls underground were coupled was excavated with an externalfrom the wall rock, made with aof roof irregu madelar of stones a thick rockwithout plate. any structural properties. The volume placed undergroundThe structural was excavated model performed from the withinrock, with theSAP a roof 2000 made code, of whicha thick reproduces rock plate. the entire building, is illustratedThe structural in Figure model 24. performed within the SAP 2000 code, which reproduces the entire building, is illustrated in Figure 24.

Figure 24. Masonry building 3D model. Figure 24. Masonry building 3D model The material properties were extensively studied by performing experimental tests from samples thatThe were m availableaterial properties on site. More were than extensively 500 compression studied tests by were performing conducted experimental in accordance tests with from UNI samplesEN 1926:2007 that were on stone available cubes on with site. the More dimensions than 500 70 compression70 70 mm tests3. The were average conducted density in accordance of the stone × × withwas 1508UNI EN kg/ m1926:20073 and the on average stone cubes compression with the strengthdimensions was 70 8.86 × 70 MPa, × 70 withmm3. aThe standard average deviation density of of the2.58 stone MPa. was The 1508 elastic kg/m modulus3 and of the the ave stonerage was compression 3820 MPa. strength was 8.86 MPa, with a standard deviationIn the of case 2.58 ofMPa. double The leafelastic walls mod withulus an of inner the stone non-structural was 3820 MPa. core, the structural elements were takenIn asthe homogenized. case of double In leaf this walls process, with onlyan inner the structuralnon-structural parts core, were the considered, structural withelements the elasticwere takenmoduli as Ehomogenized.1 and E2, while In the this modulus process of, only the innerthe structural core was parts considered were considered, as E3 = 0. The with homogenized the elastic modulmodulusi E1 wasand computedE2, while the as: modulus of the inner core was considered as E3 = 0. The homogenized E1 s1 + E2 s2 modulus was computed as: E0 = · · (13) s1 + s2 + s3 where: 퐸1 ∙ 푠1 + 퐸2 ∙ 푠2 (12) E = elastic modulus of the i wall퐸0 leaf;= i 푠1 + 푠2 + 푠3 si = thickness of the i wall leaf; where1: and 2 refer to the structural leaves; 3 refers to the internal non-structural core. Ei = elastic modulus of the i wall leaf; Figures 25 and 26 show the plan of the buildings at the diﬀerent ﬂoors, as obtained from an si = thickness of the i wall leaf; 1exhaustive and 2 refer structural to the structural survey lea madeves; by the authors. The values of the elastic modulus used in the 3numerical refers to the simulations internal non are- reportedstructural close core to. each wall. Figures 25 and 26 show the plan of the buildings at the different floors, as obtained from an exhaustive structural survey made by the authors. The values of the elastic modulus used in the numerical simulations are reported close to each wall.

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Figure 25.25.. Plan of the buildings buildings—ground—ground ﬂoor.floor..

Figure 26.26.. Plan of the buildings buildings—ﬁrst—first ﬂoor.floor..

TheThe foundations of the building are placedplaced uponupon bedbed rock, made ofof thethe samesame limestonelimestone thatthat waswas usedused forfor thethe originaloriginal construction.construction. FromFrom aa globalglobal perspective,perspective, thethe vertical elements ofof the building were modelledmodelled as an equivalent frameframe structural system, system, in in which which the the columns columns are are continuous continuous elements elements that that form form the the foundations foundations to tothe the top. top. The horizontal The horizontal regions regions that connect that connect the vertical the vertical elements elements were modelled were modelled as beam elements as beam elementswhich are whichconnected are connectedto the vertical to the elements vertical through elements stiff through nodes,, with stiﬀ dimensionsnodes, with that dimensions are related that to the dimensions of the horizontal region and of the vertical element that generates the node. The underground region has been considered as a rigid constraint for the remaining part of the building.

Heritage 2020, 3 1013 are related to the dimensions of the horizontal region and of the vertical element that generates the

Heritagenode. The2020, underground3 FOR PEER REVIEW region has been considered as a rigid constraint for the remaining part of24 the building. As regards regards the the structural structural roof roof and and their their mass mass distributions, distributions, the following the following considerations considerations are valid. are valid.Due to Due the deformability to the deformability of the vaulted of the structuresvaulted structures (barrel vaults) (barrel and vaults) the impossibility and the impossibility of defining of a definstiff plateing a withstiff plate a unique with master- a unique= nodemaster in-= which node in the which total the mass total is consideredmass is considered [8], the seismic[8], the seismic masses weremasses distributed were distributed according according to the influence to the areas. influence For the areas. model For that the was model run with that the was equivalent run with plate the equivalentconfiguration, plate the configuration, same approach the was same followed. approach was followed. In the numerical model with vaulted structures, these are connected to the structural frame through rigid links that are placed at the quote of the kidneys, connecting all all the frames, as as seen in Figure 2727a. Figure 2277b shows the structural model obtained by computing the equivalent plates.

(a) (b)

Figure 2 27.7. StructuralStructural model with vaults ( a) and with equivalent plates ( b).).

The modal modal analysis analysis has has confirmed confirmed the thestrong strong irregularity irregularity of the of buildings the buildings in terms in of termsthe geometry of the geometryand stiffness and distribution. stiffness distribution. A total of 94.8%A total of of the 94. mass8% of was the excited mass was only excited at the 40thonly modalat the vibration40th modal in vibrationthe x-direction in the (long x-direction side). A(long total side). of 90.8% A total of theof 90. mass8% wasof the excited mass was only excited at the 40th only modal at the vibration40th modal in 1 vibrationthe y-direction in the (shorty-direction side). (short The first side). modal The first shape modal (Figure shape 28a), (Figure with a2 period8a), with T1 a= period0.311 s,T is= 0. related311 s, isto related the direction to the perpendiculardirection perpendicular to the barrel to the vaults; barrel in vaults this case,; in this 66.3% case of, 66. the3% total of the mass total is mass excited. is excited.Figure 28 Figureb is related 28b tois therelated second to themodal second shape modal along shapethe Y axis, along parallel the Y to axis, the vaults, parallel involving to the vaults, 32.4% involvingof the total 32. mass.4% of Inthe this total case, mass. the In periodthis case T2, the= 0.236 period s. T The2 = 0. third236 s. modal The third shape, modal with shape, a period withof a 3 Tperiod3 = 0.217 of T s,= is 0. also217 as, translationalis also a translational mode along mode the alongY axis, the involving Y axis, involving 36.4% of the36.4% total of mass.the total The mass. first Thethree first modes three represent modes represent the global the behaviorglobal behavior of the building.of the building. The cumulativeThe cumulative mass mass involved involved in the in Xthe-direction X-direction and and the Ythe-direction Y-direction is equal is equal to 69.12% to 69.12% and 67.85%, and 67. respectively.85%, respectively. The values The ofvalues the excited of the excitedmass in mass correspondence in correspondence with the with 4thmode the 4th and mode the and following the following is very low, is very since low, they since represent they represent the local vibrationsthe local vibrations of some parts of some of the parts building. of the The building. modal analysis The modal has confirmed analysis has the confirmed strong irregularity the strong of irregularitythe buildings of in the terms buildi ofngs the in geometry terms of and the stigeometffnessry distribution. and stiffness distribution. For the structural model implemented with with equivalent equivalent plates plates in in place place of of the the vaulted vaulted structures, structures, at the ground floorfloor the equivalent elastic properties were obtained for the plates according to EquatioEquationsns (1) and (2) inin the the longitudinal longitudinal direction direction (L) and the transvers transversee direction ( T), with respect to the axis axis of of the the barrel barrel vaults. vaults. The following The following values values were obtained were obtained and used and in the used numerical in the simulations: numerical v L v T v L (Esimulations:v/E)L = 5997.4%; (E /E) (E= v5997./E)T 4%;= 118.8%; (E /E) (G= v118./G)8%;L = 23.7%.(G /G) = 23.7%.

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(a(a) ) (b(b) ) (c(c) )

FigureFigure 28.2 288. .Results Results of of eigeneigen modalmodal analysis: analysis: first ﬁrstfirst modal modalmodal shapes shapesshapes ( (a(aa)),,) ,translation translationtranslation in inin the thethe X X direction direction (b(b (b)),,) , andand ( c(c) ) translation translation in in the the Y Y direction.direction direction. .

The ﬁrstfirst modal shape (Figure(Figure 2929a)a) ofof thethe equivalentequivalent model,model, withwith aa periodperiod TT11 == 0.302 s s,, is related The first modal shape (Figure 29a) of the equivalent model, with a period T1 = 0.302 s, is related toto thethe the directiondirection direction perpendicularperpendicular perpendicular to to to the the the barrel barrel barrel vaults; vaults vaults in; ;in thisin this this case, case case 65.8%, ,65. 65.8% of8% theof of the totalthe total total mass mass mass is excited. is is excited. excited. In this In In case,thisthis case case the, error,the the error error computed computed computed with with with respect respect respect to the to to vaultedthe the vaulted vaulted model model model was was negligible:was negligible: negligible: it was it it was 3%was in3% 3% terms in in terms terms of the of of periodthethe period period and and 1%and in1% 1% terms in in terms terms of the o of f excitedthe the excited excited mass. mass. mass. Figure Figure Figure 29b 29 is29b relatedb is is related related to the to to secondthe the second second modal modal modal shape shape shape for thefor for Y equivalentthethe equivalent equivalent model model model along along along the the theaxis, Y Y axis, axis, parallel paral parallel tolel theto to the the vaults, vaults, vaults, involving involving involving 47.9% 47. 47.9%9% of of theof the the total total total mass, mass, mass, with with with a perioda period T 2T=2 =0.244 0.244 s.s. InIn thisthis case,case, thethe errorerror inin termsterms of period was 4%, while for the mass it was 16%. 16%. a period T2 = 0.244 s. In this case, the error in terms of period was 4%, while for the mass it was 16%. EvenEven if if the the error error range range was was higher, higher, the the result result can can be be considered considered more more than than acceptable acceptable for for the the purpose purpose of global analysis. The third modal shape had a periodperiod TT3 = 0.0.219219 s s;; it is also a translationaltranslational mode of global analysis. The third modal shape had a period T3 3 = 0.219 s; it is also a translational mode alongalong thethe Y Y axis, axis, involving involving 19.0%19. 19.0%0% of of the the total total mass. mass. In In this this case case,case, ,the the maximum maximum error error was was in in terms terms of of thethe excitedexcited mass,mass, equalequal toto 17%,17%, whilewhile itit waswas was negligiblenegligible negligible inin termsterms terms ofof of thethe the period.period. period. Additionally, Additionally, inin the the casecase of ofof equivalent equivalentequivalent structure, structure, the the the values values values of of of the the the excited excited excited mass mass mass in in correspondence incorrespondence correspondence with with with the the 4th the 4th mode 4thmode mode and and andthethe following thefollowing following is is very very is very low, low,low, since since since they they they represent represent represent the the thelocal local local vibrations vibrations vibrations of of some ofsome some parts parts parts of of the ofthe thebuilding. building. building. In In Incorrespondencecorrespondence correspondence with with with thethe the 40th 40th 40th modal modal modal shapeshape shape,, ,95 95 95.0%.0%.0% of of of the the the total total massmass is is involved involvedinvolved in inin the the X X-direction-direction-direction (94.8%(94.(94.8%8% for for the the vaulted vaulted structure),struc structure),ture), and and 93.2%93. 93.2%2% of of the the total total mass mass is is involved involved in in the the Y Y--directiondirection-direction (90.8%(90. (90.8%8% for for thethe vaulted vaulted structure). structure).

(a(a) ) (b(b) ) (c(c) ) Figure 29. Results of eigen modal analysis: first modal shape (a) translation X (b) and (c) translation Y. FigureFigure 29. 29. ResultsResults ofof eigeneigen modalmodal analysis:analysis: ﬁrstfirst modal shape (a) translation X ( b) and and ( (c)) translation translation Y Y..

DueDue toto the the geometrical geometrical irregularities, irregularities, the the seismic seismic input input along along the thethe two twotwo main mainmain directions directions,directions, ,X X and andand Y Y,, , generatesgenerates a a torque torque of of the the structure structure, structure, , which which causes causes flexuralflexural flexural moments moments in in the the direction direction orthogonal orthogonal tot oto the the seismic force. Thus, with a seismic input along the X-axis, a flexural moment M2 is computed in the seismicseismic force. force. Thus,Thus, withwith aa seismicseismic inputinput alongalong thetheX X-axis,-axis, a a flexural flexural moment moment M M22is is computed computed in in the the plane perpendicular to the seismic input, and a flexural moment M3 is computed in the plane of the planeplane perpendicular perpendicular toto the the seismic seismic input, input, and and a a flexural flexural moment moment M M33is is computed computed in in the the plane plane of of the the seismic input. The same conclusions are made for the shear forces V2 and V3, with the meaning of the seismicseismic input. input. The The same same conclusions conclusions are are made made for for the the shear shear forces forces V V2 2and and V V3,3, with with the the meaning meaning of of the the subscriptssubscripts 2 2 and and 3. 3. Along Along the the direction direction Y Y, ,the the flexural flexural moment moment and and shear shear forces forces are are computed computed too. too. In In thisthis case case, ,the the subscripts subscripts 2 2 and and 3 3 assume assume a a commuted commuted meaning. meaning.

Heritage 2020, 3 1015

Heritage 2020, 3 FOR PEER REVIEW 26 subscripts 2 and 3. Along the direction Y, the ﬂexural moment and shear forces are computed too. In this case, the subscripts 2 and 3 assume a commuted meaning. Figure 30 reports an example that illustrates a partial structural system of the building in order Figure 30 reports an example that illustrates a partial structural system of the building in order to show a comparison between the results of the numerical analyses for the vaulted and equivalent to show a comparison between the results of the numerical analyses for the vaulted and equivalent structure. This structural system was chosen, since it presented worse situations in terms of the error structure. This structural system was chosen, since it presented worse situations in terms of the error between the solutions of the vaulted model and that of the equivalent model. This representation between the solutions of the vaulted model and that of the equivalent model. This representation allows us to easily commence the comparison considering the results reported in Table 4, where the allows us to easily commence the comparison considering the results reported in Table4, where the internal forces values (moment and shear) can be analyzed. internal forces values (moment and shear) can be analyzed.

FigureFigure 30. 30.Sub-structural Sub-structural systemsystem shownshown forfor thethe comparisoncomparisonof ofnumerical numerical results. results.

InIn TableTable4 ,4 the, the results results of of the the vaulted vaulted models models are are reported reported in in terms terms of of the the ﬂexural flexural moments moments and and shearshear forces, forces, with with the the error error in in %, %, with with respect respect to to the the equivalent equivalent model model solved solved with with plates. plates.

TableTable 4. Numerical 4. Numerical results results and comparisonand comparison between between the vaulted the vaulted and equivalent and equivalent structural structural FEM models. FEM models. Quote Direction of V2 err% V3 err% M2 err% M3 ERR% Element # Element Quotem SeismicDirection Input of Text kNV2 err %% kNV3 %err% kN-mM2 %err% kN-mM3 ERR% % #98 m 0.00 Seismic ExInput Text 80.69kN 0%% 83.45kN 10%% 125.56kN-m 4%% 274.60kN-m 1%% − 9898 0. 2.7000 Ex 80.6980.69 0%0% 83.4583.45 10%−10% 102.24125.56 23%4% 57.87274.60 2%1% − − 9898 2. 0.0070 EyEx 23.4580.69 2%0% 174.5483.45 11%−10% 346.94102.24 24%−23% 46.7757.87 0%2% 9898 0. 2.7000 Ey 23.4523.45 2%2% 174.54174.54 11%11% 125.80346.94 14%24% 28.8746.77 1%0% − − 100 0.00 Ex 59.04 3% 34.99 12% 79.93 6% 209.38 3% 98 2.70 Ey 23.45 2% 174.54 − 11% 125.80 −14% 28.87 −1% 100 2.70 Ex 59.04 3% 34.99 12% 16.21 50% 51.19 4% 100 0.00 Ex 59.04 3% 34.99 − −12% 79.93 − 6% 209.38 3% 100 0.00 Ey 9.34 0% 120.40 7% 265.27 22% 46.61 2% 100 2.70 Ex 59.04 3% 34.99 −12% 16.21 −50% 51.19 4% 100 2.70 Ey 9.34 0% 120.40 7% 63.16 27% 27.64 8% − − 100124 0. 0.0000 ExEy 116.309.34 9%0% 99.65120.40 10%7% 120.23265.27 17%22% 442.7046.61 6%2% − − 100124 2. 2.7070 ExEy 116.309.34 9%0% 99.65120.40 10%7% 153.7363.16 6%−27% 138.6327.64 2%−8% − 124124 0. 0.0000 EyEx 29.95116.30 14%−9% 194.9299.65 6%10% 454.65120.23 6%17% 64.46442.70 5%−6% − − − 124124 2. 2.7070 EyEx 29.95116.30 14%−9% 194.9299.65 6%10% 78.81153.73 43%6% 18.98138.63 34%2% − − − − 124142 0. 0.0000 ExEy 106.4029.95 7%−14% 86.65194.92 8%−6% 123.16454.65 27%6% 478.5264.46 2%−5% 142 2.70 Ex 106.40 7% 86.65 8% 129.47 7% 226.66 9% 124 2.70 Ey 29.95 −14% 194.92 −6% 78.81 − −43% 18.98 − −34% 142 0.00 Ey 12.35 47% 338.73 5% 834.75 11% 21.47 50% 142 0.00 Ex 106.40 − 7% 86.65 − 8% 123.16 27% 478.52 − 2% 142 2.70 Ey 12.35 47% 338.73 5% 92.07 55% 16.37 31% 142 2.70 Ex 106.40 − 7% 86.65 − 8% 129.47 − −7% 226.66 − −9% 142 0.00 Ey 12.35 −47% 338.73 −5% 834.75 11% 21.47 −50% 142 2.70 Ey 12.35 −47% 338.73 −5% 92.07 −55% 16.37 −31% The direction of the seismic input is intended in the positive direction and reversed by 180◦. It is remarkable that, in the X-direction, the errors (% ratio between the results of the solid model respectThe to direction the plate of model) the seismic are less input than is those intended in the in Ythe-direction. positive direction This is due and to reversed the different by 180 impact°. It is theremarkable structural that irregularities, in the X-direction have in, thethe twoerrors orthogonal (% ratio between directions. the Alongresults the of theY-axis, solid there model are respect some elementsto the plate (beam model) 124 and are 142) less wherethan th theose di infference the Y- indirection the internal. This forces is due is significant. to the different impact the structural irregularities have in the two orthogonal directions. Along the Y-axis, there are some 8.elements Conclusions (beam 124 and 142) where the difference in the internal forces is significant. In the present work, the structural response of vaulted roofs was compared to the response of 8. Conclusions equivalent-plate structural systems. The equivalence between the vaults and the plates was assumed by consideringIn the present the work elastic, the stiff structuralness of the response respective of vaulted elements roofs along was the compared two principal to the response directions. of Diequivalentfferent types-plate of structural vaults were systems. considered, The equivalence including special-typebetween the vaults,vaults and which the are plates commonly was assumed used inby the considering south-eastern the regions elastic stiffness of Italy. Theof the simplified respective model elements proposed along herein the two allows princ usipal to replace directions. the vaultedDifferent roof types with of an vaults equivalent were plateconsider of theed same, including thickness special and-type the same vaults dimensions,, which are but commonly characterized used in the south-eastern regions of Italy. The simplified model proposed herein allows us to replace the vaulted roof with an equivalent plate of the same thickness and the same dimensions, but characterized by reduced elastic moduli Ex, Ey, and Gxy, which take into account the shape effect of

Heritage 2020, 3 1016

by reduced elastic moduli Ex, Ey, and Gxy, which take into account the shape effect of the vault. Using finite element analysis (FEM) conducted with the help of ANSYS software v.14, it was possible to evaluate the influence of the geometrical and mechanical parameters on the equivalent stiffness of the plate. The equivalent elastic moduli are expressed as a fraction of the modulus of the original material; the in-plane dimensions of the vaults varied from 3 to 5 m, the thicknesses of the vault were considered to be equal to 80 and 200 mm; different constraint conditions were also studied. On the basis of the obtained results, it appears that in all cases the typical leccesi vault, called a square vault, is the stiffest one; the behavior of the edge leccesi vault is similar to that of the cross vault. The constraint configuration significantly affects the in-plane stiffness; in particular, the highest stiffness values are obtained in the presence of side walls, often found in practical situations. The modal analyses run for the vaults and the equivalent plated systems revealed that the equivalence in terms of the excited mass and vibrational period is respected also for the dynamic properties. An entire building was studied and modelled in order to apply the proposed procedure, considering a real case of a heritage building with a high level of structural irregularity. In this particular case, the barrel vaults were replaced with equivalent plates. The results have shown that, in terms of the global response, the properties of both models were similar, without significant errors. The computation of the internal forces was accurate in most of the cases, even if unacceptable levels of error were found for two structural elements. In fact, for the first three modes, the vibration periods of the real system and of the simplified model evidenced differences of up to 4%; the differences in terms of the participant mass were equal to or less than 16%. The maps of the principal stresses, under seismic forces, show errors between the plate model and the vaulted model that are less than 10% for most of the elements. Only for some elements did the error reach 50%. Further research is needed in order to verify the accuracy of the plate-based model in place of vaulted complex 3D models. Further studies should compare the responses of the buildings modelled with real geometry and with a simplified strategy, considering non-linear static analysis. Different type of vaults should be investigated, assuming accurate constraint conditions, in order to compare the global response of the different numerical models.

Author Contributions: Conceptualization, F.M. and M.A.A.; Data curation, P.S.M.; Formal analysis, F.M.; Investigation, P.S.M.; Methodology, F.M.; Software, P.S.M.; Supervision, M.A.A.; Writing—original draft, F.M.; Writing—review & editing, P.S.M. and M.A.A. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The authors want to thank the support of the Municipality of Lecce and the RELUIS Project, which supported and continues to support research in the area of seismic engineering at UniSalento. Conﬂicts of Interest: The authors declare no conﬂict of interest

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