the seismic assessment of historical masonry structures · the seismic assessment of historical...

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Abstract The seismic assessment and subsequent rehabilitation of historical masonry structures constitutes a contemporary issue in most seismic regions around the world and particularly in Europe where monumental buildings represent a significant cultural heritage which must be preserved. In most cases seismic retrofitting projects of monumental structures have been performed without first proceeding to a reliable assessment of the actual structural conditions. This apparent paradox is mainly due to difficulties related the numerical prediction of the seismic behaviour of monumental structures. Refined finite element numerical models, such as the smeared cracked and discrete crack finite element models, able to predict the complex non-linear dynamic mechanical behaviour and the degradation of the masonry media, need sophisticated constitutive laws and significant computational cost. As a consequence, these methods are nowadays unsuitable for practical application and extremely difficult to apply to large structures. Many authors developed simplified or alternative methodologies that, with a reduced computational effort, should be able to provide numerical results that can be considered sufficiently accurate for engineering practice purposes. However, most of these methods are based on too simplified hypotheses that make these approaches inappropriate for monumental buildings.

In this paper a new discrete-modelling approach, conceived for the seismic assessment of a monumental structure, is applied to a benchmark historical basilica church for which results are available in the literature. The proposed numerical strategy represents the first introduction of the discrete-element approach able to represent masonry structures with curved geometry and, at the same time, to simulate both the in-plane and the out of plane response of masonry media.

Keywords: macro-element, discrete element, monumental structures, push-over analysis.

Paper 0123456789 The Seismic Assessment of Historical Masonry Structures S. Caddemi, I. Caliò, F. Cannizzaro and B. Pantò Dipartimento Ingegneria Civile e Architettura University of Catania, Italy

Civil-Comp Press, 2014 Proceedings of the Twelfth International Conference on Computational Structures Technology, B.H.V. Topping and P. Iványi, (Editors), Civil-Comp Press, Stirlingshire, Scotland.

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1 Introduction Historical Monumental Structures represent a high percentage of existing constructions in many seismically prone areas around the world and particularly in Europe. A large number of Historical buildings belongs to our cultural heritage and encompass the art of building at the time of realization. However, in most cases seismic retrofitting projects have been performed without first proceeding to a reliable assessment of the actual structural conditions. This apparent paradox is mainly due to difficulty related the numerical prediction of the seismic behaviour of monumental structures. The nonlinear dynamic behaviour of URM building subjected to earthquake loadings is governed by a complex interaction between the in-plane and out-of-plane response of masonry. The different behaviour of masonry structures, compared to ordinary concrete and steel buildings, requires ad hoc algorithms capable of reproducing the nonlinear behaviour of masonry media and providing reliable numerical simulations [1]. Refined finite element numerical models, such as the smeared cracked and discrete crack finite element models, able to predict the complex non-linear dynamic mechanical behaviour and the degradation of the masonry media, require sophisticated constitutive laws and a huge computational cost. As a consequence these methods are nowadays not suitable for practical applications and extremely difficult to apply to large structures. On the other hand, in order to estimate the seismic vulnerability of an existing building and to assess whether the structure requires a seismic upgrade, a structural engineer needs simple and efficient numerical tools, whose complexity and computational demand must be appropriate for practical engineering purposes. For these reasons, in the last six decades, many authors have developed simplified or alternative methodologies for predicting the nonlinear seismic behaviour of URM building. A common limitation of the existing simplified approaches, currently used in practical engineering, is the basic assumption of in-plane behaviour of masonry walls making these approaches unsuitable for monumental buildings.

In this paper, an original modelling approach for the simulation of the nonlinear behaviour of masonry historical structures under static and seismic loadings is considered. The proposed approach is based on the concept of macro-element discretization [2] and has been conceived with the aim of capturing the nonlinear behaviour of an entire building by means of an assemblage of several discrete-elements which can be characterised by different level of complexity according to the role of the element in the global model.

The basic element, initially developed for the simulation of the in-plane response, is constituted by an articulated quadrilateral with four rigid edges and four hinged vertices connected by two diagonal nonlinear springs; each of the rigid edges can be connected to other elements by means of discrete distributions of nonlinear springs with limited tension strength. This plane discrete-element has been implemented in the computer code 3DMacro [3], and allow the simulation of masonry structures in which the out-of-plane response can be ignored [4-7]. The basic plane-element is intended for the simulation of the response of masonry walls in their own plane since the out-of-plane response is not taken into account. In order to overcome this

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significant restriction, common to almost all the simplified approaches, the plane macro-element has been upgraded by introducing a third dimension and the needed additional degrees-of-freedom, for the simulation of the out-of-plane response [8]. Nevertheless, in order to simulate the seismic response of monumental masonry constructions, in many cases it is also necessary to model the nonlinear behaviour of structural elements with curved geometry, such as arches, vaults, domes whose role is fundamental both in the local and global behaviour of the structures. With the aim to simulate the nonlinear behaviour of masonry structures with curved geometry the macro-element has been further upgraded [9].

The computational cost of the proposed numerical approach is greatly reduced, compared to a traditional nonlinear finite element modelling and the calibration of the model and the interpretation of the numerical results are simple and unambiguous, since based on a straightforward fiber discretization.

In this paper the proposed discrete element method is applied to a benchmark historical masonry building, already investigated by Mele et al. [11]. Namely some different sub-structural systems, typical of a Basilica plan church, have been investigated by using the proposed discrete element method approach. The results, expressed in terms of collapse mechanisms and push-over curves, show a satisfactory agreement with those obtained by Mele et al. through nonlinear finite element simulations.

2 The proposed macro-elements approach The approach here adopted is based on macro-elements of different complexity according to the corresponding role in the global model. Namely, plane, spatial and shell macro-elements have been conceived; in the following their capability in modelling monumental masonry buildings is briefly shown.

2.1 The plane element

The basic element of the proposed approach has a simple mechanical scheme. It is represented by an articulated quadrilateral constituted by four rigid edges connected by four hinges and two diagonal nonlinear springs [7]. Each side of the panel can interact with other panels, elements or supports by means of a discrete distribution of nonlinear springs, denoted as interface. Each interface is constituted by n nonlinear springs, orthogonal to the panel side, and an additional longitudinal spring which controls the relative motion in the direction of the panel edge. In spite of its great simplicity, such a basic mechanical scheme is able to simulate the main in-plane failures of a portion of masonry wall subjected to horizontal and vertical loads [7]. Each element exhibits three degrees-of-freedom, associated to the in-plane rigid-body motion, plus a further degree-of-freedom needed to describe the in-plane deformability. The deformations of the interfaces are associated to the relative motion between corresponding panels, therefore no further Lagrangian parameters must be introduced in order to describe their kinematics.

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void

void

(a) (b)

Figure 1: Masonry wall and corresponding macro-element discretization.

Since the proposed modelling approach is suitable for describing the behaviour of a plane wall loaded in its own plane, a three-dimensional masonry building can be modelled as an assemblage of plane walls. As a consequence, in this simplified approach the behaviour of the wall in the out-of-plane direction is not considered. In Figure 1 it is shown how a simple masonry wall can be modelled by means of the proposed modelling approach. Namely, Figure 1b refers to a basic scheme composed by 12 panels and is characterised by 48 degrees of freedom only.

The effectiveness of the simulation of the nonlinear behaviour relies on a suitable choice of the mechanical parameters of the model, inferred by equivalence between the masonry wall and a reference continuous model, characterised by simple but reliable constitutive laws. This equivalence relies on a straightforward fiber calibration procedure, and is based only on the main mechanical parameters which characterise the masonry according to an orthotropic homogeneous medium [7]. It is worth noticing that, in the proposed approach each macro-element inherits the plane geometrical properties of the modelled masonry portion, as a consequence, contrary to the simplified models based on equivalent frame element approach, there is no need to define an effective dimension of the structural element.

2.2 The spatial element The three-dimensional macro-element represents the spatial upgrade of the plane macro-element described in the previous section. In particular, three additional degrees-of-freedom have been considered for the description of the out-of-plane kinematics and further nonlinear springs have been introduced in the interfaces in order to account for the three-dimensional mechanical behaviour, Figure 2.

The kinematics of the spatial macro-element is governed by 7 degrees-of-freedom able to describe both the rigid body motions and the in-plane deformability. Each 3D-interface possesses m rows of n longitudinal (i.e. perpendicular to the planes of the interface) NLinks. Consequently, each of the planes of the interface discretised, similarly to what is done in classical fiber models, in m×n sub-areas (Figure 2b). Each longitudinal spring of the 3D interface represents a strip of masonry with base area equal to that of the sub-area which pertains to it. The fiber discretization is selected according to the desired level of accuracy of the nonlinear response. The

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3D-interface possesses also further sliding springs. These are required to control the relative displacement of the panels perpendicular to their plane. To this purpose, two sliding NLinks, perpendicular to the plane of the panel and contained in the plane of the interface, have been provided (Figure 2c). These NLinks control the out-of-plane sliding mechanisms and the torsion around the axis perpendicular to the plane of the interface [8].

(a) (b) (c)

Figure 2: (a) The spatial macro-element: (b) the element with the representation of

the orthogonal NLinks for the simulation flexural behaviour; (c) the element with the representation of the transversal and diagonal NLinks for the simulation of shear and torsional behaviour.

2.3 The spatial element for shell modelling The discrete element, conceived to model shell masonry structures, represents a further upgrade of the previously described spatial element. Its core is still constituted by a plane irregular articulated quadrilateral, however the orientation and the dimensions of the four rigid layer edges are related to the shape and to the thickness of the portion of curved masonry element that has to be represented, Figure 3, as it will highlighted in the following.

(a) (b)

Figure 3: Masonry portion and its equivalent mechanical model without the NLink

representations.

n1

n2

n4

n3

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A diagonal spring simulates the in-plane deformability of the element in its own medium plane (Figure 4c), while the interfaces (Figure 4b) govern the interaction with the adjacent elements or the external supports. As mentioned before, these interfaces are in general skew, relative to the medium plane of the element, and their kinematics is still ruled by a discrete number of nonlinear springs which incorporates the element properties according to an original fiber calibration strategy. Each quadrilateral is defined by the geometric coordinates of his vertices, the four normal vectors to the surface and the thickness in the same points (Figure 3a).

(a) (b) (c)

Figure 4: (a) the shell macro-element ‘quad’. (b) the orthogonal NLinks for the simulation of membrane and flexural behaviour; (c) transversal and diagonal NLinks for the simulation of in-plane and torsional behaviour

The kinematics of the spatial macro-element is still governed by 7 degrees-of-

freedom only for the description of both the rigid body motions and the in-plane shear deformability. 2.2.1 Notes on calibration procedures The calibration procedure is based on the assumption that the behaviour of a continuously curved surface can be adequately represented by flat elements, Figure 5. In the sub-division of an arbitrary shell into flat elements generally both triangular and quadrilateral elements should be used. For the sake of conciseness, here only the quadrilateral shape element is described in more detail. Each element must be representative of the corresponding finite portion of the shell cut out by plane sections which are located at the edges of the irregular quadrilateral and whose orientations and thicknesses are associated to the actual represented shell.

The membrane and the bending behaviour of the element is governed by the nonlinear links orthogonal to the rigid layer edges, while three additional transversal links control both the in-plane and out-of plane sliding shear and the twisting of two adjacent layers. A single diagonal nonlinear link governs the in-plane nonlinear shear behaviour of the quadrilateral element. The orthogonal NLinks that govern the membrane and flexural response are calibrated by means of a fiber modeling

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approach. The calibration procedures of each nonlinear link are based on the mechanical characterization of masonry and the geometric properties of the elements. Masonry, considered as a continuous homogeneous solid, can be modelled considering different constitutive laws for each fundamental behaviour: membrane, bending, shear, sliding, and torque.

(a) (b)

Figure 5: (a) Quadrilateral portion of masonry structures; (b) Flat element representation.

(a)

(b)

Figure 6: Representation of the generic fiber corresponding to two opposite Nlinks.

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The discrete model has been conceived as a fiber, and each element of the mechanical scheme simulates the behavior of a corresponding strip. Each orthogonal link is calibrated in order to be equivalent to the corresponding portion of masonry which represents, according to the pertinent volume, Fig. 6a. Similar to the procedure adopted for the plane element [7], the calibration process consists of two phases. In the first phase the mechanical properties of two links, which correspond to the fibers of each of the two elements connected by the interface, are calculated. In the second phase the two links in series will be condensed in a single equivalent nonlinear link. Each of the two links can be calibrated considering a nonlinear elastic-plastic beam with a variable section, Fig. 6b. The transversal links govern the in-plane and the out-of-plane sliding of the shell element according to a Mohr-Coulomb law. In the elastic range the diagonal link is calibrated by imposing an energy equivalence with a continuous reference elastic model represented by a deformable plate with variable thickness subjected to the same displacement field of the discrete element. The current yielding forces are associated to the reaching of the limits of tensile or compressive stresses in the reference continuous model. Further details on the calibration approach can be found in [9].

3 Numerical application The assessment of seismic behaviour of historical monumental structure still represents a challenging problem which involves the contribution of many research groups around the world. Due to the difficulties to define three dimensional global nonlinear finite element models many researchers proposed alternative and simplified procedures for the assessment of the seismic vulnerability of a monumental structures. An interesting approach is that proposed by Mele et al. [10,11] which is based in a two steps analysis method. In the first step the overall structure is analysed in the linear range through a complete and refined 3D model, with the aim of characterising the linear static and dynamic behaviour, defining the corresponding internal force distribution among the single elementary parts and identifying the weak points of potential failure in the building. Secondly single structural elements are extracted from the 3D context and analysed in the linear and nonlinear range through refined 2D models, with the aim of defining the major structural properties which can be used for a simplified assessment of the seismic behaviour of the whole building. Consistently to this approach Mele et al. [11] investigated the seismic behaviour of a basilica type church, the S. Ippolisto Martire Church, located in Atripalda (Avellino, Italy), built between 1584 and 1612 on a previous basilica of the IV century AD. As highlighted by the authors in the same paper [11].This church features the presence of very definite structural typologies formed by the assemblage of quite repetitive structural schemes, that in the referenced paper have been called ‘macro-elements’. In the following this structural schemes will be identified as ‘sub-structural systems’ since in the previous paragraph, and in many papers in the literature, the term macro-element has been used at a different scale.

The basilica church consists of some structural elements, typical in this building category: nave and aisles connected by nave arcades, facade, transept, crossing and

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chancel. The detailed nonlinear analysis of each sub-structural system can provide fundamental information on the global seismic behaviour of the structure, at the same time allows an identification of the more vulnerable resistant mechanisms which contribute to the seismic performance of a specific monumental structure. In their study [11] Mele et al. firstly identified the main sub-structural systems which characterise the considered basilica plan church then performed nonlinear push-over analysis of all the sub-structural systems with the aim to provide a seismic vulnerability assessment. In order to obtain a numerical validation of the nonlinear static analyses they compared the results of the push-over analyses to the ultimate loads derived from limit analysis.

Plan of the church

T4 T4 T4 T4T5

T3T2

L2

L2

Figure 7: Schematic view of the considered plan church.

In this paper, with the aim to proceed to a first validation of the proposed discrete element approach, some of the results obtained Mele et al on some sub-structural system, are compared with those obtained by using the proposed discrete element method approach, that has been implement in a structural code, ‘HiStrA’ (Historical Structure Analysis), devoted the seismic assessment of monumental structures. Although the objective of the proposed approach is to provide full 3D nonlinear structures in this context the performed analyses will be limited to the nonlinear simulations of sub-structural systems.

The detailed description of the case study, both from a mechanical and geometrical point of views, is reported in the referenced paper [11]. In Figure 8 the geometrical layouts of the investigated sub-structural systems and the corresponding meshes, adopted for the discrete element simulations, are reported. Consistently to the nomenclature already adopted by Mele et al., the sub-structural systems have been identified by the labels L2, T2, T3, T4, T5, in which the capital letters L and T identify the Longitudinal and Transversal directions respectively.

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Element T3440 quad elements22 rigid elements3212 DOFsW 5507 KN

396 quad elements18 rigid elements2880 DOFsW 4461 KN

113

5

66

03

40

486 quad elements - rigid elements3402 DOFsW 5574 KN

Element T5

170 600190

515

325 600 190 170

375

12

53

10

32

5

100

07

95

81

018

5

60 305 240 895 605

420

Element T4


Element T2

149

53

00

10

60

143

5

755 745 755

475 1300 475 475 1300 475

14

95

605

890

300

365 390 365390745

970 785

115 130

265

265

550 335

435295

440

106

0

775

142

0

120 430 430430430430 120 120 120 120960 735 170

275

560

325 550 550 550

Element L2


Figure 8: The considered sub-structural systems: Geometrical layouts and mesh

discretization adopted in the discrete element simulations.

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T2

T3

T4

T5

Figure 9: Failure mechanisms corresponding to nonlinear static analyses associated

to horizontal in-plane mass-proportional load distributions for the transversal sub-structural systems.

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The plan position of the sub-structural systems is represented in the schematic plan view of Figure 7.

The applications here reported are relative to push-over analyses associated to the application of constant vertical loads, deriving from their own weight, and increasing horizontal in-plane mass-proportional loadings. The mechanical characteristics of the structural elements and their geometry have been derived by the data reported in the referenced paper [11]. In the performed analyses it has been assumed that each single structural-element is free from the remaining part of the structure, the role of each sub-structural in the global behaviour the church is at the moment a work in progress.

E

Young Modulus

[MPa]

G Shear

Modulus [MPa]

c

Compressive strength [MPa]

t

Tensile strength

[KN/cm2]

o

Shear strength [MPa]

Friction

angle

W Unit

weigth [KN/m3]

1500 650 3 0.1/0.2/0.3 t 0.4 19

Table 1: Mechanical parameters assumed for the considered sub-structural systems

In the applications reported in the following an elastic-perfectly plastic behaviour

has been assumed for the nonlinear links. Since the lateral resistance of each sub-structural system is strictly dependent on the limit value assumed for the mechanical characteristics of both the limit tensile and shear strength. A sensitivity analysis has been performed, namely three different values of the limit tensile strength have been considered and the shear strength has been described according to a Mohr-Coulomb criteria in the cohesion o has been set equal to the tensile limit. The adopted mechanical parameters are reported in table 1, furthermore unlimited ductility has been assumed for all the Nlinks. The implementation of more realistic constitutive laws, which account for cyclic behaviour, stiffness and strength reduction and limited ductility, will allow a better simulation of the actual behaviour of both the sub-structural systems and the entire church. The numerical predictions here reported aim at evaluating the suitability of the proposed discrete element approach to be used as a further numerical tool for the seismic assessment of historical monumental structures.

In Figures 9 the failure mechanisms relative to the considered transversal structural element, corresponding to the value of the tensile limit t=0.2, are reported. In the same figure the relative distribution of stresses in the interfaces are represented by means of a chromatic scale in which the red and the blue colours correspond to tensile and compressive forces respectively.

Figure 9a refers to the transversal section of the triumphal arch, identified as T2. The sub-structural systems T3 and T4 constitute internal transversal elements while T5 identifies the facade. The collapse mechanism are in very good agreement with the damage scenario often observed during seismic events in many basilica churches and are close to the results of the nonlinear finite element numerical prediction provided by Mele et al. [11].

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Figure 10: Failure mechanisms corresponding to nonlinear static analyses associated to horizontal in-plane mass-proportional load distributions for the longitudinal sub-structural system.

The structural element L2 corresponds to the section of the church along the nave

arcade and represents a typical solution, given by the assemblage of columns supporting arches and vaults.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30 40 50

Cb

= V

b /

W

Top displacement [mm]

t=0,20 Mpa

t=0,30 Mpa

t=0,10 Mpa

0

0,05

0,1

0,15

0,2

0,25

0,3

0 10 20 30 40 50 60

Cb

= V

b /

W


t=0,20 Mpa

t=0,30 Mpa

t=0,10 Mpa

0

0,05

0,1

0,15

0,2

0,25

0,3

0 10 20 30 40 50 60 70 80 90

Cb

= V

b /

W


t=0,20 Mpa

t=0,30 Mpa

t=0,10 Mpa

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 10 20 30 40 50

Cb

= V

b /

W


t=0,20 Mpa

t=0,30 Mpa

t=0,10 Mpa

Figure 12: Transversal sub-structural systems: nonlinear force displacement curves for different values of the tensile limit strength.

T2 T3

T4 T5

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In the discrete element simulation a full 3D discretization has been considered, characterised by 737 quadrangular (shear deformable) elements and 38 triangular (rigid) elements. The discrete elements simulation is governed by 5457 degrees of freedom which can account for both the in-plane and the out of plane behaviour of the sub-structural system.

The failure mechanism is reported in figure 10, it can be observed a separation between the end wall on the left side and the sequence of arches on the right. This behaviour is due to the different lateral resistance of the two parts and to the law tensile resistance.

The results of the push-over analyses relative to the transversal sub-structural systems are reported in Figure 12 for three different values of the tensile strength. It can be observed the great influence of the value of the ultimate tensile strength on the ultimate load of the considered sub-structural systems. In figure 13 a comparison of the lateral resistance capacities of the considered transversal sub-structural systems is reported, for the value of the tensile strength t=0.2. The figure allows to identify how each structural scheme appears to be characterised by strong differences in terms of stiffness, ultimate loads displacement capacity.

0

500

1000

1500

2000

2500

3000

3500

0 10 20 30 40 50 60 70 80 90 100

Ba

se

sh

ea

r [k

N]


T4

T3

T2

T5

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30 40 50 60 70 80 90 100

Cb

= V

b /

W


T4

T3

T2

T5

Figure 13: Transversal sub-structural systems: nonlinear force displacement curves

for different values of the tensile limit strength.

Figure 14 reports the results of the push-over analysis for the L2 sub-structural system, for three different values of the tensile strength. The horizontal force resultant is plotted vs. the displacements of three different target nodes P1, P2 and P3, whose positions are displayed in Figure 10. The target point P1 shows different displacement capacity of the target point P2 and P3 that are characterised by a similar response. This behaviour is better represented in Figure 14d in which a comparison for the curves corresponding to different target points, for the value of t=0.2 MPa, is reported. Also in this case the tensile strength has a great influence in the ultimate load capacity

15

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50

Cb

= V

b /

W


t=0,30 Mpa

t=0,30 Mpa

t=0,20 Mpa

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50

Cb

= V

b /

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t=0,10 Mpa

t=0,30 Mpa

t=0,20 Mpa

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50

Cb

= V

b /

W


t=0,10 Mpa

t=0,20 Mpa

t=0,30 Mpa

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50

Cb

= V

b /

W


P1

P2

P3

Figure 14: Longitudinal sub-structural systems: nonlinear force displacement

curves for different values of the tensile limit strength and for three different target points.

4 Conclusions In this paper an original modelling approach for the simulation of the nonlinear behaviour of monumental masonry buildings under static and seismic loadings is considered. The proposed approach is based on the concept of discrete-element discretization and has been conceived with the aim of capturing the nonlinear behaviour of a macro-portion of a masonry element and of an entire structure, by means of an assemblage of several discrete-elements. Each discrete-element can be characterised by different level of complexity according to the role of the element in the global model. With the aim to provide a first numerical validation the method has been applied to a benchmark historical masonry building for which results are available in the scientific literature. Namely some sub-structural systems of a typical basilica plan church have been analysed by means of nonlinear static analyses. The macro-elements introduced so far, enriches a larger computational framework, based on macro-element approach, devoted to the numerical simulation of the seismic behaviour of historical masonry structures.

P2

P3

P1

t=0.2 MPa

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References [1] Lourenço, P.B. (2002). Computations on historic masonry structures. Progr

Struct Eng Mater 2002;4(3):301–19. [2] Marques, R., Lourenço, P.B. (2011). Possibilities and comparison of structural

component models for the seismic assessment of modern unreinforced masonry buildings. Computers and Structures 89, 2079–2091.

[3] 3DMacro. Il software per le murature (3D computer program for the seismic assessment of masonry buildings). Gruppo Sismica s.r.l., Catania, Italy. Release 3.0, March 2014.. www.3dmacro.it

[4] Caliò, I., Cannizzaro, F., D’Amore, E., Marletta, M. and Pantò, B. (2008). A new discrete-element approach for the assessment of the seismic resistance of mixed masonry and reinforced concrete buildings. Seismic Engineering International Conference commemorating the 1908 Messina and Reggio Calabria Earthquake (MERCEA '08). Reggio Calabria, Italy, 8-11 July 2008.

[5] Caliò, I., Marletta, M., Pantò, B., (2008). A discrete element approach for the evaluation of the seismic response of masonry buildings. Proceedings of 14th World Conference of Earthquake Engineering, Beijing, China, 12-17October 2008.

[6] Caliò, I., Cannizzaro, F. and Marletta, M. (2010). A discrete element for modeling masonry vaults. Proceedings of 7th International Conference on Structural Analysis of Historical Constructions, Shanghai, China,6-8 October 2010.

[7] Caliò, I., Marletta, M. And Pantò, B. (2012). A new discrete element model for the evaluation of the seismic behaviour of unreinforced masonry buildings. Engineering Structures 40 (2012) 327–338.

[8] Pantò, B, (2007). “The seismic modeling of masonry structure, an innovative macro-element approach.” University of Catania, PhD Thesis in Structural Engineering, in italian.

[9] Cannizzaro, F, (2010). “The seismic behavior of historical buildings: a macro-element approach.” University of Catania, PhD Thesis in Structural Engineering, in italian.

[10] E. Mele, A. De Luca, “Behaviour and modelling of masonry church buildings in seismic regions”, Proceedings of Second International Symposium on Earthquake Resistant Engineering Strucures, ERES ’99, Catania, Italy, 1999.

[11] E. Mele, A. De Luca, A. Giordano, “Modelling and analysis of a basilica under earthquake loading”, Journal of Cultural Heritage 4 (2003) 355–367.

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