Discrete Macro-Element Method (DMEM)

The Discrete Macro-Element Method has been proposed by a research group of the University of Catania (Caliò et al. 2004) according to an original approach within the framework of a discrete element formulation strategy. Such an approach is based on the subdivision of the structure under consideration into several macro-portions; then, after a homogenization of the mechanical properties of the components (mortar and units), each macro-portion is regarded as an equivalent continuum whose mechanical properties can be assumed as isotropic or orthotropic depending on the masonry texture. The next step is the discretization by means of a mesh of macro-elements chosen according to the macro-portion that has to be modelled. Figure 1 reports a qualitative subdivision of a dome by means of several macro-portions that, according to a macro-element strategy, will be represented by shell macro-elements. In this approach, each macro-element, that is not rigid, interacts with the adjacent elements through nonlinear distributed zero-thickness interfaces. The nonlinear behaviour of the structure is captured through an assemblage of macro-elements, characterized by different levels of complexity according to the role played in the global model. The degrees of freedom needed to describe the macro-elements’ kinematics are those strictly related to the rigid body motion plus a single degree of freedom governing the element deformability. In the following subsections, a brief description of the different macro-elements introduced so far is reported.

The basic 2D macro-element

The basic 2D macro-element is a plane quadrangular element endowed with four degrees of freedom, Figure 2.

The 2D macro-element, firstly proposed in 2004 (Caliò et al. 2004), has been conceived for the simulation of the nonlinear response of masonry walls in their own plane. The element can be regarded as an articulated quadrilateral of rigid beams connected by four hinges, leading to a kinematics governed by four degrees of freedom only. Zero-thickness interfaces govern the interaction with the adjacent elements, while the element deformability is conveniently ruled by a single diagonal nonlinear link. The kinematics of the mechanical scheme, after a proper calibration procedure of the nonlinear links, is capable of simulating the main in-plane collapse failure modes of a masonry panel: flexural failure, diagonal shear failure and sliding shear failure (Caliò et al., 2012a). In spite of its simplicity, the assemblage of these elements allows the simulation of the global nonlinear response of masonry buildings also in the presence of openings, allowing a geometrically consistent simulation of the masonry walls in their own plane. Each macro-element exhibits three degrees of freedom associated with the in-plane rigid body motion, plus the additional degree of freedom, needed for the description of the in-plane shear deformability. The deformations of the interfaces are related to the relative motion between corresponding panels; therefore, no further Lagrangian parameter has to be introduced to describe their kinematics. The adopted model has the advantage of interacting with the adjacent elements along the whole perimeter, thus allowing the possibility of using different mesh discretizations as highlighted in the following paragraphs. The numerical approach has been validated by several researchers (Marques and Lourenço, 2011) and it has been implemented in the software 3DMacro (Caliò et al., 2012b) currently used for research and practical applications. The geometric consistency of the elements also allows an efficient simulation of infilled frame structures. Figure 3 reports an example of an infilled frame model by means of a hybrid approach in which the beams are modelled as frame elements and the infill is modelled by means of a mesh of plane macro-elements.

The 3D macro-element

The 2D macro-element allows the simulation of a masonry wall in its own plane but ignores the out-of-plane response. To overcome this significant restriction, a third dimension and the relevant needed additional degrees of freedom have been introduced in a 3D macro-element (Pantò et al., 2017a; Pantò, 2007; Caddemi et al., 2014).

Figure 4 reports the 3D macro-element (Pantò et al., 2017a; Pantò, 2007; Caddemi et al., 2014), obtained as the extension to the space of the plane element described in the previous paragraph. The kinematics of the spatial macro-element is governed by 7 degrees-of-freedom, able to describe the in- and out-of-plane rigid body motions of the quadrilateral and the in-plane shear deformability. The interaction of the spatial macro-element with the adjacent elements or the external supports is ruled by 3D-interfaces. Each 3D-interface possesses m rows of n orthogonal (i.e., perpendicular to the planes of the interface) nonlinear links. Consequently, each interface is discretised, similarly to what is done in classical fibre models, into m×n sub-areas (Figure 4b). The 3D interfaces are endowed with additional shear-sliding springs (Figure 4a), required to control the in-plane and out-of-plane sliding mechanisms and the torsion around the axis perpendicular to the plane of the interface. The number of NLinks adopted in the 3D-interfaces is selected according to the desired level of accuracy of the nonlinear response. A detailed description of the mechanical calibration of the spatial macro-element and its numerical and experimental validation is reported in Pantò et al. 2017. This model has also been applied for the simulation of infilled frame structures accounting for the in- and the out-of-plane behaviour of infills (Pantò et al. 2018).

The shell macro-element for modelling curved geometry

The 3D macro-element (Pantò et al., 2017a) allows the simulation of the in-plane and out-of-plane behaviour of plane masonry walls. However, historical structures are often characterised by the presence of curved geometry structures whose role in the global and local response cannot be ignored. Aiming at modelling curved geometry, a more general shell macro-element for modelling arches, vaults, domes and masonry arch bridges has been introduced. The shell macro-element is characterized by four rigid layer edges whose orientation and dimension is now associated with the shape of the element and to the thickness of the portion of structure to be modelled, Figure 5. The in-plane shear deformability is still governed by a single degree of freedom related to a diagonal spring placed along one of the diagonals of the quadrilateral. The plane interfaces rule the interaction with the adjacent elements or the external supports. However, due to the irregular geometry, these interfaces are in general skew with respect to the medium plane of the element. Curved surfaces are therefore modelled under the assumption that the behaviour of a continuously curved surface can be adequately represented by flat macro-elements. Each quadrilateral is geometrically defined by the coordinates of its vertices, the four normal vectors to the surface and the thicknesses at these points, Figure 6.

The most significant novelties of the improved shell element can be summarized in the following features:

interfaces are no longer orthogonal to the plane of the element, thus allowing to follow the curved geometry of the structure;
the thickness can be linearly variable at each interface;
the shape of the element can be represented by a generic quadrangular element.

In spite of the complications due to the curved geometry, the model keeps the original simplicity and computational cost. Its kinematics is still ruled by seven degrees of freedom (six rigid body motion degrees of freedom and one associated with the in-plane shear deformability). The irregular geometry implies that each link corresponds to a prismatic fiber, whose cross-sectional area varies with a parabolic trend, Figure 7.

The nonlinear sliding links are three in each interface (Figure 5b): one along the axis of the interface (in-plane sliding link) and two orthogonal to the axis and still lying on the plane of the interface (out-of-plane sliding links). The calibration strategy follows the same philosophy of the spatial regular model. Since those links have to simulate the occurrence of sliding along the bed joints, their nonlinear behaviour is closely affected by friction phenomena and the yielding domain accounts for the influence of the normal force acting on the interface. In the subdivision of an arbitrary shell into flat elements, both triangular and quadrilateral elements should generally be used (see Figure 1). The triangular elements are assumed to be rigid in their own plane and are therefore characterized by six degrees of freedom only. A detailed description of the mechanical characterization of this non-trivial shell discrete element is outside the purpose of the present chapter that is oriented to a methodological description of this computationally effective approach aiming at demonstrating its suitability for practical applications devoted to the structural assessment of existing masonry structures.

References

Caddemi S, Caliò I, Cannizzaro F, Occhipinti, G, & Pantò, B 2015, 'A parsimonious discrete model for the seismic assessment of monumental structures', in Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing, eds J. Kruis, Y. Tsompanakis, and B. H. V. Topping (Stirlingshire, UK: Civil-Comp Press), Paper 82.

Caddemi, S, Caliò, I, Cannizzaro, F, & Pantò, B 2013, 'A new computational strategy for the seismic assessment of infilled frame structures', in Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, eds B. H. V. Topping and P. Iványi (Stirlingshire, UK: Civil-Comp Press), Paper 77. Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License.

Caddemi, S, Caliò, I, Cannizzaro, F, & Pantò, B 2014, 'The seismic assessment of historical masonry structures', in Proceedings of the Twelfth International Conference on Computational Structures Technology, eds B. H. V. Topping and P. Iványi (Stirlingshire, UK: Civil-Comp Press), Paper 78.

Caddemi, S, Caliò, I, Cannizzaro, F, Chacara, C, D’Urso, D, Liseni, S, Lourenço, PB, Occhipinti, G, Pantò, B, & Rapicavoli, D 2018, ‘An original discrete macro-element method for the analysis of historical structures’, in Proceedings of the 16^th European Conference on Earthquake Engineering, Thessaloniki (Greece), 18-21 June 2018.

Caddemi, S, Caliò, I, Cannizzaro, C, D’Urso, D, Occhipinti, G, Pantò, B, & Rapicavoli, D 2019, ‘3D Discrete Macro-Modelling Approach for Masonry Arch Bridges’, in IABSE Symposium 2019, Guimarães (Portugal), 27-29 March 2019.

Caliò, I, Marletta, M, & Pantò, B 2004, ‘Un semplice macro-elemento per la valutazione della vulnerabilità sismica di edifici in muratura', atti dell’XI congresso nazionale l’Ingegneria Sismica in Italia, Genova 2004. (in Italian)

Caliò, I, Cannizzaro, F, D’Amore, E, Marletta, M, & Pantò, B 2008, 'A new discrete-element approach for the assessment of the seismic resistance of composite reinforced concrete-masonry buildings', in AIP Conference Proceedings, 1020 (PART 1); 2008 Jun 24–27; Reggio Calabria, pp. 832–839.

Caliò, I, Cannizzaro, F, & Marletta, M 2010, 'A discrete element for modeling masonry vaults' Advanced Materials Research, vol. 133-134, pp. 447–452, doi:10.4028/www.scientific.net/AMR.133-134.447.

Caliò, I, Cannizzaro, F, Pantò, B, & Rapicavoli, D 2015, 'HiStrA (historical structure analysis)', in HISTRA s.r.l (Catania, Italy). Release 17.2.3; April 2015. Available at: http://www.grupposismica.it

Caliò, I, Marletta, M, & Pantò, B 2005, 'A simplified model for the evaluation of the seismic behaviour of masonry buildings', in Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing, ed. B. H. V. Topping (Stirlingshire: Civil-Comp Press), 195.

Caliò, I, Marletta, M, & Pantò, B 2012a, 'A new discrete element model for the evaluation of the seismic behaviour of unreinforced masonry buildings' Engineering Structures, vol. 40, pp. 327–338, doi:10.1016/j.engstruct.2012.02.039.

Caliò, I, Cannizzaro, F, Marletta, M, & Pantò, B 2012b, '3DMacro: A 3D Computer Program for the Seismic Assessment of Masonry Buildings', Catania, Italy: Gruppo Sismica s.r.l.

Caliò, I, & Pantò, B 2014, 'A macro-element modelling approach of infilled frame structures', Computers & Structures, vol. 143, pp. 91–107, doi:10.1016/j.compstruc.2014.07.008.

Cannizzaro, F 2010, 'The Seismic Behaviour of Historical Buildings: A Macro-Element Approach' PhD Thesis in Structural Engineering, in Italian, University of Catania.

Cannizzaro, F, & Lourenço, PB 2017, 'Simulation of shake table tests on out-of-plane masonry buildings. Part (VI): discrete element approach', International Journal of Architectural Heritage, vol. 11, pp. 125–142.

Cannizzaro, F, Pantò, B, Caddemi, S, & Caliò, I 2018, ‘A Discrete Macro-Element Method (DMEM) for the nonlinear structural assessment of masonry arches’, Engineering Structures, vol. 168, pp. 243-256, doi: 10.1016/j.engstruct.2018.04.006.

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Chácara, C, Cannizzaro, F, Pantò, B, Caliò, I, Lourenço, PB 2018, ‘Assessment of the dynamic response of unreinforced masonry structures using a macro-element modeling approach’, Earthquake Engineering & Structural Dynamics, vol. 47(12), pp. 2426-2446.

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