Parametric Analysis on Local Mechanisms of Masonry Churches in Teramo (Italy)

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Article Parametric Analysis on Local Mechanisms of Masonry Churches in Teramo (Italy)

1, 1, , 2, Generoso Vaiano † , Antonio Formisano * † and Francesco Fabbrocino † 1 Department of Structures for Engineering and Architecture, University of Naples “Federico II”, 80125 Naples, Italy; [email protected] 2 Department of Engineering, Pegaso University, 80125 Naples, Italy; [email protected] * Correspondence: [email protected] These authors contributed equally to this work. † Received: 6 March 2020; Accepted: 31 March 2020; Published: 1 April 2020

Abstract: Local collapse mechanisms related to the out-of-plane response of walls are commonly observed in existing masonry buildings subjected to earthquakes. In such structures, the lack of proper connections among orthogonal walls and between walls and ﬂoors does not allow a global box-type behaviour of the building to develop, which would be governed by the in-plane response of walls. In this paper, parametric linear kinematic analyses on the main local mechanisms of masonry churches were performed with the aim to evaluate the corresponding horizontal load multipliers. This study was conducted on 12 masonry churches, located in Teramo (Italy) and aﬀected by the 2016 Central Italy earthquake, whose main out-of-plane collapse mechanisms, namely facade overturning, vertical bending, corner overturning and roof gable wall overturning, have been analysed. For each mechanism, parametric analysis was carried out on varying heights and thicknesses of walls. Firstly, the acceleration values activating the considered mechanisms were calculated in order to conduct checks prescribed by the current Italian standard. Subsequently, on the basis of the obtained results, simple analytical procedures to determine load collapse multiplier for each mechanism were drawn. Finally, ranges of suitable values of both the thickness and height of walls were found in order to always satisfy seismic checks.

Keywords: masonry churches; load collapse multiplier; local mechanisms; parametric analysis; seismic checks

1. Introduction Local mechanisms are generally characterized by macro-blocks, separated by a number of cracks, where relative motions are activated during an earthquake [1,2]. For these mechanisms, there is a wealth of literature references aimed at calculating the ultimate load factors by means of limit analysis, especially with kinematic approaches considering frictional behaviour [3–12]. In the last few decades, researchers have also developed analytical methods in the field of finite element (FE) analyses to examine ordinary masonry constructions [13–17] or have implemented large scale and detailed numerical investigations to study monumental constructions like churches [18–33]. The evaluation of potential local mechanisms is one of the most commonly used approaches to study the vulnerability of masonry buildings. According to the European [34] and Italian Codes [35–37], the global and local mechanisms can be studied by using different investigation types, namely linear static, non-linear static, linear kinematic and non-linear kinematic analyses. With regard to linear static analyses, the reference seismic action at ultimate limit state (ULS) is reduced through a behaviour factor to allow for a check in the elastic field. This implicitly takes into account the further displacement capacities once the limit resistance is reached before the structure attains the ULS. It is emphasized that

Heritage 2020, 3, 176–197; doi:10.3390/heritage3020011 www.mdpi.com/journal/heritage Heritage 2020, 3 177 the application of this method in the case of historic buildings can be problematic due to the difficulty of defining an appropriate behaviour factor. With reference to non-linear static analyses, they consist of the evaluation of the structure seismic behaviour at ULS through the representation of a proper force–displacement curve. In particular, the capacity displacement at ULS is compared with that required by the earthquake (demand displacement), evaluated in spectral terms. Such an analysis can be performed with a model that represents the overall structure behaviour or through models of substructures (macro-elements). The analysis consists of applying both gravitational loads and a system of horizontal forces proportional to either structural masses or the vibration first mode. In the case of masonry churches, the variety of geometries and constructive systems makes it impossible to define an accurate distribution of static forces equivalent to the earthquake. Instead, it is easier to use the limit analysis for the collapse kinematics, assigning to each possible kinematic mechanism an increasing displacement configuration. This method is called non-linear kinematic analysis and allows one to evaluate the construction displacement capacity after the mechanism activation. Nevertheless, this method is not easily implemented for large scale or parametric analyses due to large computational efforts. Contrary, where seismic analysis is based on the separate evaluation of different local mechanisms, both for global and local analysis it is easier to use linear kinematic analysis based on the kinematic theorem which, as defined in [37], consists of calculating the horizontal load multiplier activating each possible kinematic. This method is preferred for the result reliability and for its simplicity of use due to a well-established behaviour factor, assumed equal to two for masonry churches. The aim of this paper is therefore focused on the linear kinematic approaches applied to masonry churches. Linear kinematic analysis is based on the determination of the system resistance with regard to the horizontal acceleration activating the local mechanism. This approach is in terms of force and a behaviour factor is introduced to decrease the demand depending on the ductility features estimated for the masonry structure [35]. This analysis consists of quantifying the seismic collapse factor α, which is the horizontal load multiplier activating the local mechanism of a given macro-element. In fact, it is possible to consider the masonry structures as composed of different macro-elements involved by kinematic mechanisms. In particular, a macro-element is a constructive and recognizable part of the structure, which has a fully describable behaviour under seismic actions. The collapse multiplier α is obtained by applying the principle of virtual works, equalizing the total work performed by external forces to that of the internal forces. In this study, local collapse mechanisms based on the out-of-plane behaviour of masonry walls are evaluated in 12 masonry churches located in the Italian city of Teramo. The main local mechanisms considered are facade overturning, vertical bending, corner overturning and roof gable wall overturning. For each mechanism, parametric analyses are conducted with varying height and thickness as geometrical features of walls, with the final aim to find useful and simple theoretical relationships to calculate the related collapse multiplier.

2. The Case Studies The case studies herein examined are represented by 12 churches located in Teramo and its districts (Figures1 and2). These churches were chosen as case studies since they were investigated by some of the authors in order to evaluate their usability after the 2016 Central Italy earthquake. Moreover, they belong to the same territory and are placed a few kilometres away from each other, so are subjected to very similar spectral accelerations. They are similar constructions, generally characterised by one single-storey, small dimensions and architectural conﬁguration, the latter mostly made of one nave (only in two cases are there more than one nave) and an apse. Five of the 10 churches are also provided with a bell tower. In general, the load bearing vertical structures of all churches are made of chaotic rough stones, which are covered in most cases by timber trussed and gable roofs. In a few cases, the presence of a cross vault roof and a barrel one is found. The investigated churches are:

1. St. John, placed in the historic centre of Teramo and built in 1736. Heritage 2020, 3 178 Heritage 2020, 3 178 2. St. Anastasio, located in Poggio Cono, a small district of Teramo. It was built in the 17th century and nowadays it is unusable because of damages provoked by the 2009 April 6th 2. St.earthquake. Anastasio, located in Poggio Cono, a small district of Teramo. It was built in the 17th century 3.and Holy nowadays Mary of it Carmine, is unusable located because in Cavuccio, of damages a district provoked of Teramo, by the 2009 built April from 6th 1832 earthquake. to 1837. 3. 4.Holy St. MaryNicola, of Carmine,located in located Cavuccio, in Cavuccio, a district a districtof Teramo. of Teramo, It was built built from in 1832the to16th 1837. century, 4. St.presumably Nicola, located in 1551. in Cavuccio, Nowadays, a district the church of Teramo. is unusable. It was built in the 16th century, presumably 5.in St. 1551. Catherine Nowadays, of Alexandria, the church located is unusable. in the historic centre of Teramo. The church is opened to 5. St.the Catherine people ofonly Alexandria, few days locatedper year in during the historic the celebration centre of Teramo. of the Saint The Catherine’s church is opened festivity. to the It peoplewas built only fewaround days the per 9th year century during an thed it celebrationis considered of theone Saint of the Catherine’s oldest and festivity.the smallest It was church built aroundin Abruzzo. the 9th century and it is considered one of the oldest and the smallest church in Abruzzo. 6. 6.St. St. Luca, Luca, located located in in Teramo Teramo and and built built in thein the 14th 14th century. century. Currently, Currently, the the building building is not is not used used for religiousfor religious services, services, but it is but rather it is used rather occasionally used occasionally as a chapel as anda chapel as a place and foras a art place exhibitions. for art 7. St.exhibitions. Mary De Preadiis, located in Castagneto, a district of Teramo. It was built between the 10th 7.and St. the Mary 11th De century Preadiis, and located it is considered in Castagneto, to be one a district of the oldestof Teramo. churches It was in Teramo. built between the 8. St.10th Michael and Archangel,the 11th century located and in it Magnanella, is considered a districtto be one of Teramo,of the oldest and builtchurches in the in 19th Teramo. century.

9. 8.St.St. Francis Michael of Assisi, Archangel, placed located in Villa in Vomano, Magnanella, a district a district of Teramo. of Teramo, It was builtand inbuilt 1937. in the 19th century. 10. St. John In Pergulis, located in Valle San Giovanni, a district of Teramo, which was probably built 9. St. Francis of Assisi, placed in Villa Vomano, a district of Teramo. It was built in 1937. in the 16th century. 10. St. John In Pergulis, located in Valle San Giovanni, a district of Teramo, which was probably 11. Most Holy Salvatore is in Frondarola, a district of Teramo. It was built around 1330, but the built in the 16th century. structure which we can see today was renovated in the 19th century and restored in the second 11. Most Holy Salvatore is in Frondarola, a district of Teramo. It was built around 1330, but the half of the 20th century. structure which we can see today was renovated in the 19th century and restored in the 12. St. Stephan, located in Rapino, a district of Teramo. It was built around the 11th century and it second half of the 20th century. was renovated in 1932. 12. St. Stephan, located in Rapino, a district of Teramo. It was built around the 11th century and Tableit was1 shows renovated the main in 1932. geometrical dimensions of each church. In addition, both the height and the thicknessTable 1 shows of façade the main walls geometrical of each church dimensions are reported of each in Table church.2, where In addition, the average both andthe height maximum and thevalues thickness of height, of façade together walls with of theeach average church andare reported minimum in values Table 2, of where thickness the foraverage facades and of maximum examined valuesecclesiastic of height, constructions, together with are also the average provided. and minimum values of thickness for facades of examined ecclesiastic constructions, are also provided.

Figure 1. Location of churches investigated in the city of Teramo. Figure 1. Location of churches investigated in the city of Teramo.

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Church name External view Internal View Plan layout

1. Saint John

2. Saint Anastasio

3. Holy Mary of Carmine

4. Saint Nicola

5. Saint Catherine of Alexandria

6. Saint Luca

7. Saint Mary de Praediis

Figure 2. Cont.

Heritage 2020, 3 180 Heritage 2020, 3 180

8. Saint Michael Archangel

9. Saint Francis of Assisi

10. Saint John in Pergulis

11. Most Holy Salvatore

12. Saint Stephen

FigureFigure 2. External 2. External and and internal internal views views and and plan plan layouts layouts of churches of churches investigated investigated in the in citythe city of Teramo. of Teramo.

Table 1. Geometrical dimensions of studied churches.

Hall Apse Bell Tower Average Average Major Minor Major Minor Estimated Church height height yes/no side [m] side [m] side [m] side [m] height [m] [m] [m] St. John 16.50 14.30 10.00 - no - St. Anastasio 14.70 7.20 6.70 7.50 3.50 7.15 yes 10.00 Holy Mary of 11.30 6.20 7.20 6.45 5.00 7.00 yes 15.00 Carmine St. Nicola 11.00 4.60 3.50 - no - St. Catherine 13.40 6.50 7.50 - no - St. Luca 8.00 4.00 6.00 - no - St. Mary de 14.50 9.00 5.00 3.00 1.50 5.00 no - Praediis

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Table 1. Geometrical dimensions of studied churches.

Hall Apse Bell Tower Church Major Minor Average Major Minor Average Estimated yes/no Side [m] Side [m] Height [m] Side [m] Side [m] Height [m] Height [m] St. John 16.50 14.30 10.00 - no - St. Anastasio 14.70 7.20 6.70 7.50 3.50 7.15 yes 10.00 Holy Mary of 11.30 6.20 7.20 6.45 5.00 7.00 yes 15.00 Carmine St. Nicola 11.00 4.60 3.50 - no - St. Catherine 13.40 6.50 7.50 - no - St. Luca 8.00 4.00 6.00 - no - St. Mary de 14.50 9.00 5.00 3.00 1.50 5.00 no - Praediis St. Michael Arch. 16.20 5.30 5.00 - no - St. Francis of Assisi 11.40 7.00 7.80 5.85 5.80 6.00 yes 13.00 St. John In Pergulis 14.90 9.30 8.00 - yes 12.90 Most Holy Salvatore 10.70 7.20 8.00 - yes 18.00 St. Stephan 14.40 5.25 5.85 3.40 5.25 3.50 yes 15.00

Table 2. Main geometrical features of walls of studied churches.

Church Thickness [m] Height [m] Average Height [m] Maximum Height [m] St. John 1.0 10.0 St. Anastasio 1.0 6.5 Holy Mary of Carmine 0.6 7.2 6.7 10.0 St. Nicola 0.5 3.5 St. Catherine 1.0 7.5 St. Luca 0.5 6.0 Minimum Thickness [m] Average Thickness [m] St. Mary de Praediis 0.5 5.0 St. Michael Archangel 0.5 5.0 St. Francis of Assisi 0.5 7.8 0.5 0.7 St. John In Pergulis 0.7 8.0 Most Holy Salvatore 0.6 8.0 St. Stephan 1.0 5.8

3. Local Collapse Mechanisms

3.1. Description Local collapse mechanisms analysed in the current study are facade overturning, vertical bending, corner overturning and roof gable wall overturning. For each mechanism, the collapse load multiplier α is evaluated. In order to estimate the horizontal load multiplier acting on macro-elements, the following parameters should be taken into account [35]:

W , weight of the i-th macro-element; − i F , thrust vertical component of arches or vaults in correspondence of the i-th floor; − Vi F , thrust horizontal component of arches or vaults in correspondence of the i-th floor; − Hi P , floor weight of the i-th level; − Si P , i-th load transmitted overhead to the j-th macro-element; − Vij P, load transmitted by roof beam; − N, generic vertical load acting on the macro-element top; − PH, static thrust transmitted by the top floor; − P , i-th static thrust component transmitted by the top floor on the j-th macro-element; − Hij T , action of metallic tie rods at the i-th floor; − i s , wall thickness; − i h , vertical distance from the application point of the action transmitted by the floor and/or the − i metallic tie rod of i-th floor to the hinge of the i-th macro-element; Heritage 2020, 3 182

h , vertical distance from the action application point transmitted by the floor of the i-th level to − Pi the base hinge; x , horizontal distance between the centroid of the i-th wall and the overturning point; − Gi y , vertical distance between the centroid of the i-th wall and the wall base; − Gi d, horizontal distance of generic vertical load transmitted on the macro-element; − d , horizontal distance of vertical load transmitted on wall in correspondence of the i-th floor; − i d , horizontal distance of the i-th vertical load applied to the wall top; − ij a , horizontal distance of load transmitted by floor on wall in correspondence of the i-th floor; − i h , vertical distance of arches or vaults thrust at the i-th floor; − Vi d , horizontal distance of arches or vaults thrust at the i-th floor. − Vi 3.2. Façade Overturning The façade overturning mechanism is activated by the out-of-plane rotation of the façade wall, which results in its detachment from the roof (Figure3a). The multiplier of the horizontal loads acting on macro-elements is evaluated as follows:

s W + FV dV + PS d + T h FH hV PH h α = · 2 · · · − · − · , (1) W y + FV hV + P h · G · S· The causes responsible for the activation of this mechanism are no constraint at the top, no connection of the facade to the orthogonal walls, absence of metallic tie rods, deformable ﬂoors and presence of non-counteracted thrusts. When the mechanism is activated, vertical cracks on masonry, out-of-thumb of walls and beams extraction from ﬂoors can be observed. This mechanism can involve walls of one or more levels, as well as the entire thickness of the wall or only some part of it.

3.3. Vertical Bending Mechanism This mechanism is activated by the formation of a horizontal cylindrical hinge, which divides the wall into two blocks. It is described by the reciprocal rotation of each block around an axis (Figure3b). The horizontal load multiplier acting on macro-elements is evaluated by the following equation:

E α = , (2) h1 W yG + FV hV + PS hp + (W2 yG + FV2hV2) 1 1 1 1 1 2 h2

where

W1 h1 W2 E = s1 + FV1hV1 + (W2 + Ps2 + N + FV2)s2 s2 + Ps2a2 + Nd + FV2dV2 FH2hV2 + Ps1a1 FH1hV1 + ThP (3) 2 h2 2 − − The causes responsible for the activation of this mechanism are the lack of connection of the wall to orthogonal ones, excessive wall slenderness, masonry poor quality, horizontal localized thrusts and intermediate ﬂoors badly connected to the walls. When this mechanism is activated, out-of-thumb of walls, horizontal and vertical cracks and beams extraction from ﬂoors can be observed. This mechanism can involve walls of one or more levels, as well as the entire thickness of the wall or only a part of it.

3.4. Corner Overturning This mechanism is activated by a rigid rotation of a wedge, delimited by fracture surfaces with diagonal patterns in the corners of walls (Figure3c), with respect to a hinge placed at the wall base. The multiplier of horizontal loads acting on macro-elements is evaluated as follows:

E α = , (4) WyG + FVhV + (P + PV1+PV2)h Heritage 2020, 3 183

where E = Wx + FVdV + PdP + P d + PV d + T0 + T0 h F0 hV PH + P0 + P0 h, (5) G V1 1 2 2 1 2 − H − H1 H2 This mechanism is frequent in buildings with non-counteracted thrust corresponding with corners, due to loads transmitted by sloped roofs. The kinematic mechanism is deﬁned by the rotation of the macro-element around an axis perpendicular to the vertical plane with an angle of 45◦.

3.5. Roof Gable Overturning This mechanism is activated by the out-of-plane overturning of the top part. The resulting crack is represented by an almost horizontal lesion (Figure3d). Horizontal multiplier of the loads acting on macro-elements is evaluated by the following equation:

s P Heritage 2020, 3 (W1 + W2) 2 cosβ + w + P(dPcosβ + w) + i,j PVij dijcosβ + w 183 α = P , (6) W1xG1 + W2xG2 + PxP + i,j PVijxPVij () ()∑, ()  = , (6) The kinematic mechanism is usually caused by the∑, cyclic hammering action of the roof ridge beam. In fact,The thekinematic presence mechanism of large is ridgeusually beams caused causes by the acyclic signiﬁcant hammering thrust action to the of the churches roof ridge wall top part, determiningbeam. In fact, the the detachmentpresence of large of cuneiform ridge beams macro-elements. causes a significant thrust to the churches wall top part, determining the detachment of cuneiform macro-elements.

(b)

(d)

FigureFigure 3. Calculation 3. Calculation schemes schemes of of the the main main collapse collapse mechanisms: facade facade overturning overturning (a), vertical (a), vertical bending (b), corner overturning (c) and roof gable wall overturning (d). bending (b), corner overturning (c) and roof gable wall overturning (d). 3.6. Safety Checks

According to the Italian standards [36,37], the spectral acceleration (α*0) which activates the mechanism is defined by the following equation [35]:

∙∑ ∗ = , (7) ∗ where: - M* is the participant mass, which can be evaluated considering the virtual displacement of application points of different weights (Pi). Indicated by (n+m) the number of Pi and by δxi the virtual horizontal displacement of i-th weight Pi application point, the participating mass is defined as

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3.6. Safety Checks

* According to the Italian standards [36,37], the spectral acceleration (α 0) which activates the mechanism is deﬁned by the following equation [35]:

P + α n m P 0· i=1 i α0∗ = , (7) M∗ FC where:

M* is the participant mass, which can be evaluated considering the virtual displacement of − application points of diﬀerent weights (Pi). Indicated by (n+m) the number of Pi and by δxi the virtual horizontal displacement of i-th weight Pi application point, the participating mass is deﬁned as Pn+m 2 ( = Piδx,i) M = i 1 , (8) ∗ Pn+m 2 g i=1 Piδx,i

is the gravity acceleration; − FC is the confidence factor; − α is the horizontal multiplier of loads. − 0 In the case of either an isolated element or a building portion fixed to the ground, the seismic check is satisfied if the spectral acceleration (α*0) activating the mechanism fulfils the following relationship:

ag(pVR) S α∗ · = α∗ , (9) 0 ≥ q 0,min where:

ag is the site acceleration, which is a function of both the exceeding probability referred to the − ULS and the reference life, as defined in [36]; S is an amplification coefficient, given by the product between the stratigraphic amplification − coefficient and the topographic one, both defined in [36]; q is the behaviour factor [35,36], which can be assumed as being equal to 2.0. − Instead, if the local mechanism involves a portion of the construction placed at a certain height, it should be considered that the absolute acceleration of the building portion involved by the kinematic mechanism is generally amplified with respect to the ground acceleration. An acceptable approximation consists of verifying, in addition to Equation (9), the following relationship:

Sg(T1) Ψ(Z) g α∗ · · = α∗ , (10) 0 ≥ q 0,min where:

Sg (T ) is the spectral acceleration at the ULS calculated in correspondence with the period T ; − 1 1 T is the vibration first period of the entire structure in the considered direction; − 1 Ψ (Z) is the first vibration mode displacement in the considered direction at a certain height, − normalized to that at the building top; it can be assumed as Z/H, H being the entire structure height and Z the distance between the building foundation and the centroids of blocks affected by the mechanism; γ is the modal participation coefficient; − q is the behaviour factor [35,36], which can be assumed as being equal to 2.0. − Heritage 2020, 3 185

In this case, both checks described in Equations (9) and (10) must be satisﬁed. When checks performed according to the above formulas are not satisﬁed, a strengthening intervention must be designed and performed.

4. Parametric Analyses

4.1. Premise The goal of parametric analyses is to identify the horizontal loads multiplier (α) trend by varying two geometrical features, namely thickness and height, of church walls. Exploiting Table3, a range of variable values of height and thickness can be identiﬁed for the execution of the two following parametric analyses types:

Type A: average height is ﬁxed and thickness is varied from the minimum value of 0.30 meters − (minimum thickness of walls detected in other churches of Teramo not presented in the current work) to two meters, each time with an increase of 10 centimetres. Type B: average thickness is ﬁxed and the total height is changed from the minimum value of − three meters to the maximum height of 14 meters (maximum value detected in other churches of Teramo not presented in the current work), with an increasing step of 50 centimetres.

HeritageWith 2020 reference, 3 to the coverage, timber (gable, truss and hipped) and masonry (barrel vaults and185 cross vaults) roofs are examined. InIn order to to obtain obtain unfavourable unfavourable results, results, a p aoor-quality poor-quality masonry masonry (disordered (disordered masonry masonry with w=19 with wKN/m= 193 KN) is /considered.m3) is considered.

4.2.4.2. FaçadeFaçade OverturningOverturning TheThe consideredconsidered mechanismmechanism isis thethe samesame ofof aa freefree wallwall subjectedsubjected toto overturning,overturning, butbut obviouslyobviously thethe complexitycomplexity levellevel isis muchmuch greatergreater andand dependsdepends onon thethe construction construction geometricgeometric characteristics.characteristics. InIn thisthis section,section, thethe simplesimple façadefaçade overturningoverturning withoutwithout connectionsconnections toto thethe adjacentadjacent perpendicularperpendicular wallswalls isis illustrated.illustrated. Furthermore, Furthermore, the the total total overturning overturning involv involvinging the the entire entire wall wall is considered. is considered. Obviously, Obviously, the thekinematic kinematic mechanism mechanism also also involves involves the the vaults vaults and/or and/or the the roofs roofs supported supported by by the the wall wall that overturns, causingcausing itsits collapse.collapse. FigureFigure4 4 shows shows the the geometrical geometrical features features used used for for parametric parametric analyses analyses types types A A andand B.B.

Figure 4. Geometrical features (dimensions in meters) of the facade for overturning mechanism analysis. Figure 4. Geometrical features (dimensions in meters) of the facade for overturning mechanism analysis. In parametric analysis type A, the church height is ﬁxed to the value of 6.70 meters (average height, see Table2). Figure5a shows the parametric analysis results. It is observed that loads collapse In parametric analysis type A, the church height is fixed to the value of 6.70 meters (average multiplier increases in two cases; that is, when either thickness increases or the roof weight is reduced. height, see Table 2). Figure 5a shows the parametric analysis results. It is observed that loads collapse For each analysis, safety checks as described in Section 3.6 are carried out. The analysis results are multiplier increases in two cases; that is, when either thickness increases or the roof weight is visible in Figure5b–e, where the limit value of α for the check satisfaction is represented with a horizontal reduced. red line, so that values below this line correspond to cases of unsatisﬁed checks. In particular, it is For each analysis, safety checks as described in Section 3.6 are carried out. The analysis results found that the limit values of α are 0.184 for trussed roofs, 0.166 for gable roofs, 0.167 for cross vault are visible in Figure 5b–e, where the limit value of α for the check satisfaction is represented with a roofs and 0.163 for barrel vault roofs. horizontal red line, so that values below this line correspond to cases of unsatisfied checks. In particular, it is found that the limit values of α are 0.184 for trussed roofs, 0.166 for gable roofs, 0.167 for cross vault roofs and 0.163 for barrel vault roofs. In the case considered, checks are not satisfied for thickness values lower than 1.10 meters for trussed roofs, lower than 1.30 meters for gable roofs and cross vault roofs and lower than 1.40 meters for barrel vault roofs. In these situations, a strengthening intervention must be designed in order to avoid the overturning mechanism of facades under earthquakes.

(a)

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In order to obtain unfavourable results, a poor-quality masonry (disordered masonry with w=19 KN/m3) is considered.

4.2. Façade Overturning The considered mechanism is the same of a free wall subjected to overturning, but obviously the complexity level is much greater and depends on the construction geometric characteristics. In this section, the simple façade overturning without connections to the adjacent perpendicular walls is illustrated. Furthermore, the total overturning involving the entire wall is considered. Obviously, the kinematic mechanism also involves the vaults and/or the roofs supported by the wall that overturns, causing its collapse. Figure 4 shows the geometrical features used for parametric analyses types A and B.

Figure 4. Geometrical features (dimensions in meters) of the facade for overturning mechanism analysis.

In parametric analysis type A, the church height is fixed to the value of 6.70 meters (average height, see Table 2). Figure 5a shows the parametric analysis results. It is observed that loads collapse multiplier increases in two cases; that is, when either thickness increases or the roof weight is reduced. For each analysis, safety checks as described in Section 3.6 are carried out. The analysis results are visible in Figure 5b–e, where the limit value of α for the check satisfaction is represented with a Heritagehorizontal2020, 3red line, so that values below this line correspond to cases of unsatisfied checks. In186 particular, it is found that the limit values of α are 0.184 for trussed roofs, 0.166 for gable roofs, 0.167 for crossIn the vault case roofs considered, and 0.163 checks for barrel are notvault satisﬁed roofs. for thickness values lower than 1.10 meters for trussedIn the roofs, case lower considered, than 1.30 checks meters are for not gable satisfied roofs andfor thickness cross vault values roofs lowe andr lower than 1.10 than meters 1.40 meters for fortrussed barrel roofs, vault lower roofs. than In these1.30 meters situations, for gable a strengthening roofs and cross intervention vault roofs must and lower be designed than 1.40 in meters order to for barrel vault roofs. In these situations, a strengthening intervention must be designed in order to avoid the overturning mechanism of facades under earthquakes. avoid the overturning mechanism of facades under earthquakes.

Heritage 2020, 3 186 (a)

(b) (c)

(d) (e)

FigureFigure 5. 5.Type TypeA A parametricparametric analysisanalysis results for facade overturning mechanism mechanism ( (aa)) and and design design checks checks forfor trussed trussed ( b(b),), gable gable ( c(c),),cross cross vaultvault ((dd)) andand barrelbarrel vaultvault ((ee)) roofs.roofs.

InIn parametric parametric analysisanalysis typetype B,B, thethe churchchurch thicknessthickness is fixed ﬁxed to the value value of of 0.70 0.70 meters meters (average (average thickness,thickness, see see Table Table2 ).2). Figure Figure5 a5a shows shows that that the the collapse collapse multiplier multiplier increases increases whenwhen heightheight decreasesdecreases andand roof roof weight weight reduces. reduces. ForFor eacheach analysis,analysis, the designdesign checks according according to to the the Italian Italian standards standards (see (see SectionSection 3.6 3.6)) areare carriedcarried out.out. The limit values of of αα areare 0.172 0.172 for for trussed trussed roofs, roofs, 0.171 0.171 for for gable gable roofs, roofs, 0.1600.160 for for cross cross vault vault roofsroofs andand 0.1560.156 forfor barrelbarrel vaultvault roofs.roofs. From the analysis analysis results, results, it it is is noticed noticed that that forfor a a thickness thickness value value equalequal toto 0.700.70 meters,meters, mostmost checkschecks are not satisﬁed.satisfied. For For this this reason, reason, checks checks are are repeated considering a minimum thickness value of 1.40 meters. The design check results considering different roof types are shown in Figure 6b–e, where the red horizontal line represents the limit safety value. From the achieved results, it is observed that checks are not satisfied for height values lower than 8 meters for trussed roofs, 7.50 meters for gable roofs, 7.40 for cross vault roofs and 7.00 meters for barrel vault roofs. Therefore, in all these cases, an appropriate strengthening intervention should be planned in order to avoid the mechanism activation.

(a)

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(b) (c)

(d) (e)

Figure 5. Type A parametric analysis results for facade overturning mechanism (a) and design checks for trussed (b), gable (c), cross vault (d) and barrel vault (e) roofs.

In parametric analysis type B, the church thickness is fixed to the value of 0.70 meters (average thickness, see Table 2). Figure 5a shows that the collapse multiplier increases when height decreases Heritageand roof2020 weight, 3 reduces. For each analysis, the design checks according to the Italian standards (see 187 Section 3.6) are carried out. The limit values of α are 0.172 for trussed roofs, 0.171 for gable roofs, 0.160 for cross vault roofs and 0.156 for barrel vault roofs. From the analysis results, it is noticed that repeatedfor a thickness considering value aequal minimum to 0.70 thickness meters, most value chec of 1.40ks are meters. not satisfied. The design For checkthis reason, results checks considering are direpeatedﬀerent roof considering types are a shownminimum in Figurethickness6b–e, value where of 1.40 the meters. red horizontal The design line check represents results the considering limit safety value.different From roof the types achieved are shown results, in Figure it is observed 6b–e, where that the checks red horizontal are not satisﬁed line represents for height the limit values safety lower thanvalue. 8 metersFrom the for achieved trussed roofs,results, 7.50 it is meters observed for gablethat checks roofs, are 7.40 not for satisfied cross vault for height roofs andvalues 7.00 lower meters forthan barrel 8 meters vault for roofs. trussed Therefore, roofs, 7.50 in allmeters these for cases, gabl ane roofs, appropriate 7.40 for cross strengthening vault roofs intervention and 7.00 meters should befor planned barrel vault in order roofs. to Therefore, avoid the in mechanism all these cases, activation. an appropriate strengthening intervention should be planned in order to avoid the mechanism activation.

Heritage 2020, 3 187

(a)

(b) (c)

(d) (e)

FigureFigure 6. 6.Type Type B B parametric parametric analysisanalysis results for facade facade overturning overturning mechanism mechanism (a ()a and) and design design checks checks forfor trussed trussed ( b(b),), gable gable ( c(c),), cross cross vaultvault ((dd)) andand barrel vault ( (ee)) roofs. roofs.

4.3.4.3. Vertical Vertical Bending Bending MechanismMechanism VerticalVertical bending bending of of aa continuouscontinuous wallwall strip is herein considered. considered. This This strip strip belongs belongs to tothe the wall wall portionportion included included betweenbetween two two subsequent subsequent floors, ﬂoors, wh whichich is subjected is subjected to the to kinematic the kinematic mechanism. mechanism. For Forthis this reason, reason, required required parameters parameters should should be beevaluated evaluated considering considering a vertical a vertical masonry masonry strip strip with with unitaryunitary width width under under loadsloads derivingderiving from the influence inﬂuence area area acting acting on on it. it. In In this this way, way, the the analysis analysis considers all the possible positions of the cylindrical hinge along the wall height and indicates the one which the minimum value of the collapse multiplier corresponds to. In the case under study, a wall constrained to the top subjected to a mechanism affecting its whole thickness is hypothesized. For parametric analysis, timber, cross vault and barrel vault roofs are considered. In parametric analysis type A, the church height is fixed to the value of 6.70 meters. Figure 6a shows that collapse multiplier increases when both thickness and roof weight increase. Checks results are shown in Figures 7b–d, where it is noticed that they are not satisfied only in the case of timber roof for thickness values lower than 0.50 meters. From the analyses carried out, it is achieved that the limit values of α are 0.199 for timber roofs, 0.133 for cross vault roofs and 0.11 for barrel vault roofs.

Heritage 2020, 3 188 considers all the possible positions of the cylindrical hinge along the wall height and indicates the one which the minimum value of the collapse multiplier corresponds to. In the case under study, a wall constrained to the top subjected to a mechanism affecting its whole thickness is hypothesized. For parametric analysis, timber, cross vault and barrel vault roofs are considered. In parametric analysis type A, the church height is fixed to the value of 6.70 meters. Figure6a shows that collapse multiplier increases when both thickness and roof weight increase. Checks results are shown in Figure7b–d, where it is noticed that they are not satisfied only in the case of timber roof for thickness values lower than 0.50 meters. From the analyses carried out, it is achieved that the limit Heritage 2020, 3 188 values of α are 0.199 for timber roofs, 0.133 for cross vault roofs and 0.11 for barrel vault roofs.

(a)

(b) (c)

(d)

Figure 7.7. Type AAparametric parametric analysis analysis results results for for vertical vertical bending bending mechanism mechanism (a) ( anda) and design design checks checks for timberfor timber (b), ( crossb), cross vault vault (c) and(c) and barrel barrel vault vault (d) roofs.(d) roofs.

In parametric analysisanalysis typetype B,B, thethe churchchurch thicknessthickness isis ﬁxedfixed toto thethe valuevalue ofof 0.700.70 meters.meters. Figure7 7aa shows thatthat thethe collapse collapse multiplier multiplier increases increases when when height height decreases decreases and and roof roof weight weight increases. increases. The limit The α valueslimit values of are of 0.186α are for0.186 timber for timber roofsand roofs 0.173 and for 0.173 cross for vault cross roofs vault and roofs barrel and vault barrel roofs. vault Design roofs. α checksDesign illustratedchecks illustrated in Figure in8 b–d,Figures where 8b–d, the where allowable the allowable value of valueis marked of α is withmarked a horizontal with a horizontal red line, arered notline, satisﬁed are not forsatisfied height for values height higher values than higher 9.00 metersthan 9.00 for meters timber for roofs, timber higher roofs, than higher 10 meters than for10 crossmeters vault for cross roofs vault and higher roofs and than higher 11.40 metersthan 11.40 for barrelmeters vault for barrel roofs. vault roofs.

Heritage 2020, 3 189 Heritage 2020, 3 189

(a)

(b) (c)

(d)

Figure Figure8. Type 8. B Typeparametric B parametric analysis analysis results resultsfor vertical for vertical bending bending mechanism mechanism (a) and (designa) and designchecks checksfor for timber timber(b), cross (b), vault cross (c vault) and (barrelc) and vault barrel (d vault) roofs. (d) roofs.

4.4. Corner4.4. CornerOverturning Overturning This mechanismThis mechanism involves involves the building the building corner overturning, corner overturning, which is whichgenerally is generallydetermined determined by the by roof elements’the roof elements’thrust. It thrust.is manifested It is manifested through throughthe rotation the rotationof a detachment of a detachment wedge wedgedelimited delimited by by fracturefracture surfaces surfaces in two in walls two orthogonal walls orthogonal to each to other. each other.Rotation Rotation happens happens around around a hinge a hingeplaced placed at at the wedgethe wedge base. Figure base. Figure9 shows9 showsthe geometrical the geometrical features features used for used parametric for parametric analyses analyses types A typesand B. A and TimberB. hipped Timber and hipped cross and vault cross roofs vault are considered roofs are considered as church coverage. as church Even coverage. if timber Even hipped if timber roofs hipped are notroofs detected are not in detected the studied in the churches, studied churches,since they since are theyboth are commonly both commonly used in used this instructural this structural typologytypology and strongly and strongly responsible responsible of the of activation the activation of such of sucha mechanism, a mechanism, they theyare taken are taken into intoaccount account to to evaluateevaluate the the load’s load’s collapse collapse multiplier multiplier evolution evolution as as the the roof weightweight decreases.decreases. For For this this kind kind of of analysis, analysis,a detachment a detachment wedge wedge height height is assumed is assumed to beto equalbe equal to 50to centimetres.50 centimetres. In parametricIn parametric analysis analysis type A, type the A,church the church height height is fixed is to ﬁxed the tovalue the valueof 6.70 of meters. 6.70 meters. Figure Figure 10a 10a showsshows that the that collapse the collapse multiplier multiplier increases increases when thickness when thickness increases increasesand roof weight and roof decreases. weight The decreases. α limit valuesThe limit of α values are 0.199 of forare timber 0.199for roofs timber and roofs0.204 andfor cross 0.204 vault for cross roofs. vault Design roofs. checks Design reported checks reportedin Figurein 10b,c Figure are 10notb,c satisfied are not for satisﬁed thickness for values thickness lower values than lower 0.60 meters than 0.60 both meters for timber both hipped for timber roofs hipped and crossroofs vault and crossroofs. vault This roofs.thickness This limit thickness value limit is obtained value is obtainedby the intersection by the intersection of curves of curveswith the with the α horizontalhorizontal red line red represen line representingting the allowable the allowable value valueof α. of .

Heritage 2020, 3 190

HeritageHeritage2020 2020, ,3 3 190190

Figure 9. Geometrical features of walls for corner overturning mechanism. Figure 9. Geometrical features of walls for corner overturning mechanism. Figure 9. Geometrical features of walls for corner overturning mechanism.

(a)

(b) (c)

FigureFigure 10. 10. TypeType A parametric A parametric(b) analysis analysis results results for for angle angle overturning overturning mechanism mechanism (a(c)( )anda ) and design design checks checks forfor timber timber hipped hipped (b) ( band) and cross cross vault vault (c) ( croofs.) roofs. Figure 10. Type A parametric analysis results for angle overturning mechanism (a) and design checks In Inparametricfor parametric timber hipped analysis analysis (b) andtype type cross B, B, the thevault church church (c) roofs. thicknes thickness s is fixedfixed toto thethe valuevalue of of 0.70 0.70 meters. meters. In In Figure Figure 11 a 11ait it is is noticed noticed that that the the collapse collapse multiplier multiplier increases increases when when both both height height and and roof roof weight weight decrease. decrease. In In fact, fact,a hipped a hippedIn parametric roof roof has has a loweranalysis a lower weight type weight thanB, the athan crosschurch a vaultcross thicknes one;vault whens one;is fixed thewhen roofto thethe weight value roof decreases,weightof 0.70 meters.decreases, the horizontal In Figurethe horizontalload11a it multiplier is noticedload increases, multiplierthat the collapse because increases, multiplier the stabilizingbecause increases bendingthe whenstabilizing moment both height decreases.bending and roofmoment Furthermore, weight decreases. decrease. if height In Furthermore,decreases,fact, a hipped the if height multiplierroof has decreases, aincreases lower theweight because multiplier than the a increases levercross armvault because reduces. one; when the The lever the limit roofarm values weightreduces. of α decreases,are The 0.169 limit forthe valuestimberhorizontal of roofsα are andload 0.169 0.167 multiplierfor fortimber cross roofsincreases, vault and roofs. 0.167 because In for Figure cross the 11 b,c,vaultstabilizing the roofs. results Inbending ofFigure design 11b,c,moment checks the areresults decreases. reported of designtogetherFurthermore, checks with are horizontalif reported height decreases, redtogether lines indicating withthe multiplier horizontal the threshold increases red lines value becauseindicating of the the theα leverfactor. threshold arm From reduces. value the verification of The the limitα α factor.results,values From itof is the observedare verification 0.169 thatfor timberresults, safety isroofs it achieved is observed and 0.167 not that for for heightsafety cross isvault values achieved roofs. higher not Inthan Figurefor height 9.00 11b,c, meters values the for resultshigher timber of α thanhippeddesign 9.00 meters roofschecks and forare higher timberreported thanhipped together 8.00 roofs meters with and forhorizontal hi crossgher than vault red 8.00 roofs.lines meters indicating for cross the vaultthreshold roofs. value of the factor. From the verification results, it is observed that safety is achieved not for height values higher 4.5.than Roof 9.00 Gable meters Wall for Overturning timber hipped roofs and higher than 8.00 meters for cross vault roofs. The horizontal bending mechanism is characterized by the identification of cuneiform macro-elements, which rotate through oblique cylindrical hinges. This mechanism is due to the absence of adequate connections between the top part of the wall and roof elements. The kinematic mechanism is, therefore, caused by the hammering action that the roof beam exerts on the top part of

Heritage 2020, 3 191

the wall. For the roof gable wall overturning mechanism, timber trussed roofs are considered and the church height including the top part is ﬁxed to the value of 7.10 meters. Figure 12a shows the geometrical features of the roof gable wall used for parametric analysis. Analyses are carried out considering hinges without inclination in respect to the horizontal direction. The limit value of α is 0.176. Figure 12b shows that the collapse multiplier increases when thickness increases and checks are

Heritagealways 2020 satisﬁed., 3 191 (a)

( b ) (a) (c)

Figure 11. Type B parametric analysis results for angle overturning mechanism (a) and design checks for timber hipped (b) and cross vault (c) roofs.

4.5. Roof Gable Wall Overturning The horizontal bending mechanism is characterized by the identification of cuneiform macro- elements, which rotate through oblique cylindrical hinges. This mechanism is due to the absence of adequate connections between the top part of the wall and roof elements. The kinematic mechanism is, therefore, caused by the hammering action that the roof beam exerts on the top part of the wall. For the roof gable wall overturning mechanism, timber trussed roofs are considered and the church height including the top part is fixed to the value of 7.10 meters. Figure 12a shows the geometrical features of the roof (gableb) wall used for parametric analysis. Analyses (arc) e carried out considering hinges without inclination in respect to the horizontal direction. The limit value of α is 0.176. Figure 12bFigureFigure shows 11.11. Type thatType B the parametric B parametriccollapse analysis multiplier analysis results results increases forfor angle angle overturningwhen overturning thickness mechanism mechanism increases (a) (anda )and and design designchecks checks checks are always satisfied.forfor timber timber hipped hipped (b) ( band) and cross cross vault vault (c) (roofs.c) roofs.

4.5. Roof Gable Wall Overturning The horizontal bending mechanism is characterized by the identification of cuneiform macro- elements, which rotate through oblique cylindrical hinges. This mechanism is due to the absence of adequate connections between the top part of the wall and roof elements. The kinematic mechanism is, therefore, caused by the hammering action that the roof beam exerts on the top part of the wall. For the roof gable wall overturning mechanism, timber trussed roofs are considered and the church height including the top part is fixed to the value of 7.10 meters. Figure 12a shows the geometrical features of the roof gable wall used for parametric analysis. Analyses are carried out considering (a) hinges without inclination in respect to the horizontal direction. The limit value of α is 0.176. Figure (b) 12b shows that the collapse multiplier increases when thickness increases and checks are always satisfied.FigureFigure 12. 12.Roof Roof gable gable wall wall overturning overturning mechanism: mechanism: geometrical geometrical features features ( a()a) and and results results of of design design checkschecks (b ().b). 4.6. Final Remarks

The goal of parametric analyses is the determination, for each of the mechanisms analysed, of the limit values of both minimum thickness and maximum height of walls which satisfy seismic

(a) (b)

Figure 12. Roof gable wall overturning mechanism: geometrical features (a) and results of design checks (b).

Heritage 2020, 3 192 veriﬁcations. Parametric analyses results derived from considering all the roof types are presented in Table3.

Table 3. Limit values of geometrical dimensions of church walls satisfying seismic checks.

Collapse Mechanism Minimum Thickness [m] Maximum Height [m] Facade overturning 1.50 7.00 Vertical Bending 0.50 9.00 Corner overturning 0.60 7.50 Roof gable wall overturning 0.30 1.10

On the basis of the obtained results, simple equations based on wall thickness and height only which allow for a quick evaluation of horizontal loads collapse multiplier can be formulated. Referring to Figure5, the proposed equations are: Facade overturning mechanism—trussed roof (Figure 13a,b):

α = (0.155t + 0.002) (6.70/h) (11) · Vertical bending mechanism:

timber roof (Figure 14a): α = (0.297t + 0.70) (6.70/h) (12) · cross vaults roof (Figure 14b): α = (0.295t + 0.12) (6.70/h) (13) · barrel vaults roof (Figure 14c): α = (0.292t + 0.19) (6.70/h) (14) · Roof gable wall mechanism (Figure 15):

α = 1.35t 0.09 (15) − where α is the collapse load multiplier, t is the wall thickness and h is the church height. Equations (11) to (14) are composed by two aliquots: the first (as function of the thickness) is the trend line that approximates the curve, i.e., a linear straight line with a correlation coefficient of 1; the second (6.70/h) is instead the correlation based on the height variation. Contrary to this, Equation (15) is composed by the first aliquot only. Compared to the formulation obtained through the linear kinematic analysis shown in the Italian standard [35], the new equations herein proposed, that are of course valid only for the investigated churches, give slightly lower values, that allow one to operate in a more safe way. First, with reference to the façade overturning mechanism, Figure 13 shows the comparison between the kinematic analysis curve of Figure6b and that from the corresponding Equation (11) for the two wall thicknesses of 0.70 m and 1.40 m. The proposed formulation values overestimate those obtained by the kinematic analysis by 5.2% to 8.9%. Second, the comparison of curves dealing with the vertical bending mechanisms is depicted in Figure 14. In particular, the comparisons are between the kinematic analysis curve of timber roofs (Figure8b) and the curve from Equation (12) (Figure 14a), the kinematic analysis curve of cross vault roofs (Figure8c) and the curve from Equation (13) (Figure 14b), and the kinematic analysis curve of barrel vault roofs (Figure8d) and the curve from Equation (14) (Figure 14c). From the comparative study, it appears that the errors between the proposed formulations values and the kinematic analyses ones are in the following ranges: timber roof: from 1% to 19%; − cross vault roof: from 1.9% to 23.8%; − barrel vault roof from 1% to 31%. − In particular, it is observed that in all cases the errors increase as the church height augments. Finally, Figure 15 shows the roof gable overturning analysis, where it is noticed that the trend of Heritage 2020, 3 193 Heritage 2020, 3 193 EquationHeritage 2020 (15), 3 is a straight line with a correlation coefficient equal to one, which means that it perfectly193 HeritageEquationfits the 2020 kinematic (15), 3 is a straight analysis line results. with a correlation coefficient equal to one, which means that it perfectly193 fits the kinematic analysis results. Equation (15) is a straight line with a correlation coefficient equal to one, which means that it perfectly Equation (15) is a straight line with a correlation coefficient equal to one, which means that it perfectly fitsfits the the kinematic kinematic analysis analysis results. results.

(a) (b) (a) (b) Figure 13. Wall overturning mechanism of churches with trussed roof: comparison between kinematic Figure 13. Wall overturning mechanism of churches with trussed roof: comparison between kinematic analysis curve and analytical(a) formula one for thickness of 0.70 (a) and 1.40 m( b(b) ). analysis curve and analytical formula one for thickness of 0.70 (a) and 1.40 m (b). FigureFigure 13. 13. WallWall overturning overturning mechanism mechanism of of churches churches with with trussed trussed roof: roof: comparison comparison between between kinematic kinematic

analysisanalysis curve curve and and analytical analytical fo formularmula one one for for thickness thickness of of 0.70 0.70 (a (a) )and and 1.40 1.40 m m (b (b).).

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

(a) (b)

(c()c )

FigureFigure 14. 14. Vertical VerticalVertical bending bending mechanism: mechanism: comparison comparison comparison betw betw betweeneeneen kinematic kinematic kinematic analysis analysis analysis curve curve curve and and analytical analytical analytical formulaformulaformula one one one for for churches churches with with a a timber timber roof roof (a (),a ),cross ( crossc) vault vault roof roof (b (()b and)) andand barrel barrelbarrel vault vaultvault roof roofroof (c ). ((cc ).).

Figure 14. Vertical bending mechanism: comparison between kinematic analysis curve and analytical formula one for churches with a timber roof (a), cross vault roof (b) and barrel vault roof (c).

Figure 15. Roof gable wall overturning mechanism: comparison between kinematic analysis curve and FigureFigure 15.15. RoofRoof gablegable wall wall overturning overturning mechanism: mechanism: comparison comparison between between kinematic kinematic analysis analysis curve curve analytical formula one. andand analyticalanalytical formula formula one. one.

Figure 15. Roof gable wall overturning mechanism: comparison between kinematic analysis curve

and analytical formula one.

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5. Conclusions In this paper, local collapse mechanisms related to the out-of-plane wall response of masonry churches have been investigated. The performed research work has been conducted on 12 masonry churches located in Teramo (Italy), whose main local collapse mechanisms, namely facade overturning, vertical bending, corner overturning and roof gable wall overturning, have been inspected. Parametric analyses have been carried out through linear kinematic analyses by considering the variation of the most significant geometrical parameters of the walls of churches, namely thickness and height. For each local mechanism, the collapse load multiplier has been evaluated and the spectral acceleration activating that mechanism has been assessed in order to conduct the safety checks foreseen by the Italian standards. Two kinds of parametric analysis have been developed: in the first case (Type A), the average height has been fixed and the thickness has been augmented up to two meters, with an increasing step of 10 cm, while in the second case (Type B), the average thickness has been fixed and the height has been varied from 3 to 10 m, with an increasing step of 0.50 m. From the analyses performed, the following results are obtained:

1. Façade overturning mechanism: churches with trussed, gable, cross vault and barrel vault roofs have been considered. Type A analyses checks were not satisfied for thickness values lower than 1.10 m for trussed roofs, lower than 1.30 m for gable and cross vault roofs and lower than 1.40 m for barrel vault roofs. Meanwhile, type B analyses checks were not satisfied for height values lower than 8 m for trussed roofs, 7.50 m for gable roofs, 7.40 m for cross vault roofs and 7.00 m for barrel vault roofs. 2. Vertical bending mechanism: timber, cross vault and barrel vault roofs have been considered. Type A analyses checks were not satisfied only in the case of timber roof for thickness values lower than 0.50 m. Type B analyses checks were not satisfied for height values higher than 9.00 m for timber roofs, higher than 10 m for cross vault roofs and higher than 11.40 m for barrel vault roofs. 3. Corner overturning: hipped and cross vault roofs have been considered. Type A analyses checks were not satisfied for thickness values lower than 0.60 m both for timber hipped roofs and cross vault roofs. Type B analyses checks were not satisfied for height values higher than 9.00 m for timber hipped roofs and higher than 8.00 m for cross vault roofs. 4. Roof gable wall overturning: according to what was detected in situ, only timber trussed roofs were considered and the church height, including the top part, was fixed to the value of 7.10 m. Seismic checks were always satisfied.

In order to better highlight the study results, appropriate curves correlating the horizontal loads multiplier to the thickness (type A analysis) and the height (type B analysis) of church walls have been plotted. Finally, five equations are proposed on which a quick prediction of the horizontal multiplier loads can be done. These formulations give slightly higher values that allow one to operate in a safer way. The goal of these parametric analyses has been the determination, for a specific Italian area (district of Teramo) characterized by the same high seismicity and for each of the mechanisms analysed, of the limit values of both minimum thickness and maximum height of walls able to satisfy seismic verifications. In particular, the minimum thickness has been to be found variable from 0.30 m (roof gable wall overturning) to 1.50 m (facade overturning), while the maximum height has been framed in the interval from 1.10 (roof gable wall overturning) to 9.00 m (vertical bending). The proposed procedure can represent a very effective seismic analysis tool, since without making refined analyses but through the consultation of diagrams and equations only, it is possible to discover the masonry churches’ vulnerabilities towards local collapse mechanisms activated by earthquakes. In addition, by assessing the horizontal multiplier loads and the safety degree against the activation of possible local collapse mechanisms, it is possible to classify the churches according to the priority of upgrading interventions required. Heritage 2020, 3 195

The future development of this research will undoubtedly foresee the application of the proposed procedure to other church samples with diﬀerent geometrical characteristics and located in areas with diﬀerent seismic hazard levels. Furthermore, the procedure can also be extended to other monumental buildings, such as palaces and castles.

Author Contributions: Conceptualization, A.F.; Methodology, A.F.; Validation, A.F.; Investigation, G.V.; Data Curation, G.V.; Writing-Original Draft Preparation, G.V.; Writing-Review & Editing, F.F.; Supervision, A.F. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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