Abstract
Masonry structures, although classically suitable to withstand gravitational loads, are sensibly vulnerable if subjected to extraordinary actions such as earthquakes, exhibiting cracks even for events of moderate intensity compared to other structural typologies like as reinforced concrete or steel buildings. In the last half-century, the scientific community devoted a consistent effort to the computational analysis of masonry structures in order to develop tools for the prediction (and the assessment) of their structural behavior. Given the complexity of the mechanics of masonry, different approaches and scales of representation of the mechanical behavior of masonry, as well as different strategies of analysis, have been proposed. In this paper, a comprehensive review of the existing modeling strategies for masonry structures, as well as a novel classification of these strategies are presented. Although a fully coherent collocation of all the modeling approaches is substantially impossible due to the peculiar features of each solution proposed, this classification attempts to make some order on the wide scientific production on this field. The modeling strategies are herein classified into four main categories: block-based models, continuum models, geometry-based models, and macroelement models. Each category is comprehensively reviewed. The future challenges of computational analysis of masonry structures are also discussed.
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References
Como M (2013) Statics of historic masonry constructions. Springer, Berlin
Hendry AW (1998) Structural masonry. Macmillan Education, London
Page A (1981) The biaxial compressive strength of brick masonry. Proc Inst Civ Eng 71(3):893–906
Page A, Samarasinghe W, Hendry A (1982) The in-plane failure of masonry. A review. Proc Br Ceram Soc 30:90
Page A (1983) The strength of brick masonry under biaxial tension-compression. Int J Mason Constr 3(1):26–31
Magenes G, Calvi GM (1997) In-plane seismic response of brick masonry walls. Earthq Eng Struct Dyn 26(11):1091–1112
Calderini C, Cattari S, Lagomarsino S (2009) In-plane strength of unreinforced masonry piers. Earthq Eng Struct Dyn 38(2):243–267
Beyer K (2012) Peak and residual strengths of brick masonry spandrels. Eng Struct 41:533–547
Petry S, Beyer K (2014) Influence of boundary conditions and size effect on the drift capacity of URM walls. Eng Struct 65:76–88
Messali F, Rots J (2018) In-plane drift capacity at near collapse of rocking unreinforced calcium silicate and clay masonry piers. Eng Struct 164:183–194
D’Altri AM, de Miranda S, Castellazzi G, Sarhosis V (2018) A 3D detailed micro-model for the in-plane and out-of-plane numerical analysis of masonry panels. Comput Struct 206:18–30
Facconi L, Minelli F, Vecchio FJ (2018) Predicting uniaxial cyclic compressive behavior of brick masonry: new analytical model. J Struct Eng 144(2):04017213
Sassoni E, Mazzotti C, Pagliai G (2014) Comparison between experimental methods for evaluating the compressive strength of existing masonry buildings. Constr Build Mater 68:206–219
Kržan M, Gostič S, Cattari S, Bosiljkov V (2015) Acquiring reference parameters of masonry for the structural performance analysis of historical buildings. Bull Earthq Eng 13(1):203–236
Esposito R, Messali F, Ravenshorst GJP, Schipper HR, Rots JG (2019) Seismic assessment of a lab-tested two-storey unreinforced masonry Dutch terraced house. Bull Earthq Eng 17(8):4601–4623. https://doi.org/10.1007/s10518-019-00572-w
Messali F, Esposito R, Jafari S, Ravenshorst G, Korswagen P, Rots JG (2018) A multiscale experimental characterization of dutch unreinforced masonry buildings. In: Proceedings of the 16th European conference on earthquake engineering, 16ECEE
Borri A, Castori G, Corradi M, Speranzini E (2011) Shear behavior of unreinforced and reinforced masonry panels subjected to in situ diagonal compression tests. Constr Build Mater 25(12):4403–4414
Lumantarna R, Biggs DT, Ingham JM (2014) Compressive, flexural bond, and shear bond strengths of in situ New Zealand unreinforced clay brick masonry constructed using lime mortar between the 1880s and 1940s. J Mater Civ Eng 26(4):559–566
McCann D, Forde M (2001) Review of NDT methods in the assessment of concrete and masonry structures. NDT & E Int 34(2):71–84
Bosiljkov V, Bokan-Bosiljkov V, Strah B, Velkavr J, Cotič P (2010) Review of innovative techniques for the knowledge of cultural assets (geometry, technologies, decay). PERPETUATE (EC-FP7 project), Deliverable D6
Borri A, Corradi M, Castori G, De Maria A (2015) A method for the analysis and classification of historic masonry. Bull Earthq Eng 13(9):2647–2665
Tondelli M, Rota M, Penna A, Magenes G (2012) Evaluation of uncertainties in the seismic assessment of existing masonry buildings. J Earthq Eng 16(sup1):36–64
Recommendations for the analysis, conservation and structural restoration of architectural heritage, International scientific committee for analysis and restoration of structures of architectural heritage, ratified as ICOMOS document by the General Assembly in Zimbabwe (2003)
Eurocode 8 - Design of structures for earthquake resistance. Part 3: Assessment and retrofitting of buildings, Brussels, Belgium (2005)
Direttiva del Presidente del Consiglio dei Ministri 9 febbraio 2011. Valutazione e riduzione del rischio sismico del patrimonio culturale con riferimento alle Norme tecniche per le costruzioni di cui al D.M. 14/01/2008
Guide for the structural rehabilitation of heritage buildings, prepared by CIB commission W023—Wall structures (2010)
Cattari S, Lagomarsino S, Bosiljkov V, D’Ayala D (2015) Sensitivity analysis for setting up the investigation protocol and defining proper confidence factors for masonry buildings. Bull Earthq Eng 13(1):129–151
Haddad J, Cattari S, Lagomarsino S (2019) Sensitivity and preliminary analyses for the seismic assessment of Ardinghelli Palace. In: Structural analysis of historical constructions. Springer, Berlin, pp 2412–2421
Chiozzi A, Grillanda N, Milani G, Tralli A (2018) UB-ALMANAC: an adaptive limit analysis NURBS-based program for the automatic assessment of partial failure mechanisms in masonry churches. Eng Fail Anal 85:201–220
Castellazzi G, D’Altri A, Bitelli G, Selvaggi I, Lambertini A (2015) From laser scanning to finite element analysis of complex buildings by using a semi-automatic procedure. Sensors 15(8):18360–18380
Castellazzi G, D’Altri AM, de Miranda S, Ubertini F (2017) An innovative numerical modeling strategy for the structural analysis of historical monumental buildings. Eng Struct 132:229–248
Korumaz M, Betti M, Conti A, Tucci G, Bartoli G, Bonora V, Korumaz AG, Fiorini L (2017) An integrated terrestrial laser scanner (TLS), deviation analysis (DA) and finite element (FE) approach for health assessment of historical structures. A minaret case study. Eng Struct 153:224–238
D’Altri AM, Milani G, de Miranda S, Castellazzi G, Sarhosis V (2018) Stability analysis of leaning historic masonry structures. Autom Constr 92:199–213
Cerone M, Croci G, Viskovic A (2000) The structural behaviour of colosseum over the centuries. In: More than two thousand years in the history of architecture
Macchi G, Ruggeri G, Eusebio M, Moncecchi M (1993) Structural assessment of the leaning tower of pisa. In: IABSE reports. IABSE Internationa Association for Bridge, pp 401–401
DeJong MJ, Belletti B, Hendriks MA, Rots JG (2009) Shell elements for sequentially linear analysis: lateral failure of masonry structures. Eng Struct 31(7):1382–1392
Rots JG, Belletti B, Invernizzi S (2008) Robust modeling of RC structures with an “event-by-event” strategy. Eng Fract Mech 75(3–4):590–614
Reddy JN (2004) An introduction to nonlinear finite element analysis. Oxford University Press, Oxford
Clough RW, Penzien J (2003) Dynamics of Structures, (revised). Computers and Structures, Inc., Berkeley, Calif
Heyman J (1966) The stone skeleton. Int J Solids Struct 2(2):249–279
Angelillo M (ed) (2014) Mechanics of masonry structures. Springer, Vienna
Huerta S (2001) Mechanics of masonry vaults: The equilibrium approach. SAHC 2001
Giuffrè A, Carocci C (1993) Statica e dinamica delle costruzioni murarie storiche. pp 539–598
Marmo F, Rosati L (2017) Reformulation and extension of the thrust network analysis. Comput Struct 182:104–118
Chiozzi A, Milani G, Tralli A (2017) A genetic algorithm NURBS-based new approach for fast kinematic limit analysis of masonry vaults. Comput Struct 182:187–204
Bauer S, Lackner R (2015) Gradient-based adaptive discontinuity layout optimization for the prediction of strength properties in matrix-inclusion materials. Int J Solids Struct 63:82–98
Lourenço PB (2002) Computations on historic masonry structures. Progress Struct Eng Mater 4(3):301–319
Roca P, Cervera M, Gariup G, Pela’ L (2010) Structural analysis of masonry historical constructions. Classical and advanced approaches. Arch Comput Methods Eng 17(3):299–325
Lagomarsino S, Cattari S (2015) PERPETUATE guidelines for seismic performance-based assessment of cultural heritage masonry structures. Bull Earthq Eng 13(1):13–47
Page AW (1978) Finite element model for masonry. J Struct Div 104(8):1267–1285
Dolatshahi KM, Yekrangnia M (2015) Out-of-plane strength reduction of unreinforced masonry walls because of in-plane damages. Earthq Eng Struct Dyn 44(13):2157–2176
Minga E, Macorini L, Izzuddin BA (2018) A 3D mesoscale damage-plasticity approach for masonry structures under cyclic loading. Meccanica 53(7):1591–1611
D’Altri AM, Messali F, Rots J, Castellazzi G, de Miranda S (2019) A damaging block-based model for the analysis of the cyclic behaviour of full-scale masonry structures. Eng Fract Mech 209:423–448. https://doi.org/10.1016/j.engfracmech.2018.11.046
Lotfi HR, Shing PB (1994) Interface model applied to fracture of masonry structures. J Struct Eng 120(1):63–80
Rots J (1991) Numerical simulation of cracking in structural masonry. Heron 36(2):49–63
Rots JG (1997) Structural masonry: an experimental/numerical basis for practical design rules. AA Balkema, Leiden
Lourenço PB, Rots JG (1997) Multisurface interface model for analysis of masonry structures. J Eng Mech 123(7):660–668
Sandoval C, Arnau O (2016) Experimental characterization and detailed micro-modeling of multi-perforated clay brick masonry structural response. Mater Struct 50(1):34
Calderón S, Sandoval C, Arnau O (2017) Shear response of partially-grouted reinforced masonry walls with a central opening: testing and detailed micro-modelling. Mater Des 118:122–137
Senthivel R, Lourenço P (2009) Finite element modelling of deformation characteristics of historical stone masonry shear walls. Eng Struct 31(9):1930–1943
Oliveira DV, Lourenço PB (2004) Implementation and validation of a constitutive model for the cyclic behaviour of interface elements. Comput Struct 82(17–19):1451–1461
Gambarotta L, Lagomarsino S (1997) Damage models for the seismic response of brick masonry shear walls. Part I: the mortar joint model and its applications. Earthq Eng Struct Dyn 26(4):423–439
Alfano G, Sacco E (2006) Combining interface damage and friction in a cohesive-zone model. Int J Numer Methods Eng 68(5):542–582
Parrinello F, Failla B, Borino G (2009) Cohesive–frictional interface constitutive model. Int J Solids Struct 46(13):2680–2692
Formica G, Sansalone V, Casciaro R (2002) A mixed solution strategy for the nonlinear analysis of brick masonry walls. Comput Methods Appl Mech Eng 191(51–52):5847–5876
Malomo D, Pinho R, Penna A (2018) Using the applied element method for modelling calcium silicate brick masonry subjected to in-plane cyclic loading. Earthq Eng Struct Dyn 47(7):1610–1630
Casolo S (2000) Modelling the out-of-plane seismic behaviour of masonry walls by rigid elements. Earthq Eng Struct Dyn 29(12):1797–1813
Orduña A (2017) Non-linear static analysis of rigid block models for structural assessment of ancient masonry constructions. Int J Solids Struct 128:23–35
Baraldi D, Cecchi A (2016) Discrete approaches for the nonlinear analysis of in plane loaded masonry walls: molecular dynamic and static algorithm solutions. Eur J Mech-A/Solids 57:165–177
Baraldi D, Cecchi A (2017) A full 3D rigid block model for the collapse behaviour of masonry walls. Eur J Mech-A/Solids 64:11–28
Macorini L, Izzuddin B (2011) A non-linear interface element for 3D mesoscale analysis of brick-masonry structures. Int J Numer Methods Eng 85(12):1584–1608
Chisari C, Macorini L, Amadio C, Izzuddin B (2015) An inverse analysis procedure for material parameter identification of mortar joints in unreinforced masonry. Comput Struct 155:97–105
Chisari C, Macorini L, Amadio C, Izzuddin BA (2018) Identification of mesoscale model parameters for brick-masonry. Int J Solids Struct 146:224–240
Minga E, Macorini L, Izzuddin B (2018) Enhanced mesoscale partitioned modelling of heterogeneous masonry structures. Int J Numer Methods Eng 113(13):1950–1971
Zhang Y, Macorini L, Izzuddin BA (2016) Mesoscale partitioned analysis of brick-masonry arches. Eng Struct 124:142–166
Aref AJ, Dolatshahi KM (2013) A three-dimensional cyclic meso-scale numerical procedure for simulation of unreinforced masonry structures. Comput Struct 120:9–23
Wilding BV, Dolatshahi KM, Beyer K (2017) Influence of load history on the force-displacement response of in-plane loaded unreinforced masonry walls. Eng Struct 152:671–682
Dolatshahi KM, Aref AJ (2016) Multi-directional response of unreinforced masonry walls: experimental and computational investigations. Earthq Eng Struct Dyn 45(9):1427–1449
Dolatshahi KM, Nikoukalam MT, Beyer K (2018) Numerical study on factors that influence the in-plane drift capacity of unreinforced masonry walls. Earthq Eng Struct Dyn 47(6):1440–1459
Kuang JS, Yuen Y (2013) Simulations of masonry-infilled reinforced concrete frame failure. Proc Inst Civ Eng: Eng Comput Mech 166(4):179
Miglietta PC, Bentz EC, Grasselli G (2017) Finite/discrete element modelling of reversed cyclic tests on unreinforced masonry structures. Eng Struct 138:159–169
Sarhosis V, Bagi K, Lemos JV, Milani G (2016) Computational modeling of masonry structures using the discrete element method. IGI Global, Pennsylvania
Cundall PA, Strack OD (1979) A discrete numerical model for granular assemblies. Geotechnique 29(1):47–65
Cundall PA (1980) UDEC-A generalised distinct element program for modelling jointed rock., tech. rep., Cundall (Peter) Associates Virginia Water (England)
Çaktı E, Saygılı Ö, Lemos JV, Oliveira CS (2016) Discrete element modeling of a scaled masonry structure and its validation. Eng Struct 126:224–236
Papantonopoulos C, Psycharis I, Papastamatiou D, Lemos J, Mouzakis H (2002) Numerical prediction of the earthquake response of classical columns using the distinct element method. Earthq Eng Struct Dyn 31(9):1699–1717
Bui T, Limam A, Sarhosis V, Hjiaj M (2017) Discrete element modelling of the in-plane and out-of-plane behaviour of dry-joint masonry wall constructions. Eng Struct 136:277–294
Sarhosis V, Sheng Y (2014) Identification of material parameters for low bond strength masonry. Eng Struct 60:100–110
Lemos JV (2007) Discrete element modeling of masonry structures. Int J Archit Herit 1(2):190–213
Tóth AR, Orbán Z, Bagi K (2009) Discrete element analysis of a stone masonry arch. Mech Res Commun 36(4):469–480
Simon J, Bagi K (2016) Discrete element analysis of the minimum thickness of oval masonry domes. Int J Archit Herit 10(4):457–475
Forgács T, Sarhosis V, Bagi K (2017) Minimum thickness of semi-circular skewed masonry arches. Eng Struct 140:317–336
Lengyel G (2017) Discrete element analysis of gothic masonry vaults for self-weight and horizontal support displacement. Eng Struct 148:195–209
Foti D, Vacca V, Facchini I (2018) DEM modeling and experimental analysis of the static behavior of a dry-joints masonry cross vaults. Constr Build Mater 170:111–120
Shi G-H (1992) Discontinuous deformation analysis: a new numerical model for the statics and dynamics of deformable block structures. Eng Comput 9(2):157–168
Thavalingam A, Bicanic N, Robinson J, Ponniah D (2001) Computational framework for discontinuous modelling of masonry arch bridges. Comput Struct 79(19):1821–1830
Jean M (1999) The non-smooth contact dynamics method. Comput Methods Appl Mech Eng 177(3–4):235–257
Moreau JJ (1988) Unilateral contact and dry friction in finite freedom dynamics. In: Nonsmooth Mech Appl. Springer, Berlin, pp 1–82
Rafiee A, Vinches M (2013) Mechanical behaviour of a stone masonry bridge assessed using an implicit discrete element method. Eng Struct 48:739–749
Rafiee A, Vinches M, Bohatier C (2008) Application of the nscd method to analyse the dynamic behaviour of stone arched structures. Int J Solids Struct 45(25–26):6269–6283
Lancioni G, Gentilucci D, Quagliarini E, Lenci S (2016) Seismic vulnerability of ancient stone arches by using a numerical model based on the non-smooth contact dynamics method. Eng Struct 119:110–121
Beatini V, Royer-Carfagni G, Tasora A (2017) A regularized non-smooth contact dynamics approach for architectural masonry structures. Comput Struct 187:88–100
Sarhosis V, Lemos J (2018) A detailed micro-modelling approach for the structural analysis of masonry assemblages. Comput Struct 206:66–81
Munjiza AA (2004) The combined finite-discrete element method. Wiley, Hoboken
Smoljanović H, Živaljić N, Nikolić Ž (2013) A combined finite-discrete element analysis of dry stone masonry structures. Eng Struct 52:89–100
Smoljanović H, Živaljić N, Nikolić Ž, Munjiza A (2018) Numerical analysis of 3D dry-stone masonry structures by combined finite-discrete element method. Int J Solids Struct 136:150–167
Smoljanović H, Nikolić Ž, Živaljić N (2015) A combined finite-discrete numerical model for analysis of masonry structures. Eng Fract Mech 136:1–14
Ali SS, Page AW (1988) Finite element model for masonry subjected to concentrated loads. J Struct Eng 114(8):1761–1784
Petracca M, Pelà L, Rossi R, Zaghi S, Camata G, Spacone E (2017) Micro-scale continuous and discrete numerical models for nonlinear analysis of masonry shear walls. Constr Build Mater 149:296–314
Addessi D, Sacco E (2016) Nonlinear analysis of masonry panels using a kinematic enriched plane state formulation. Int J Solids Struct 90:194–214
Serpieri R, Albarella M, Sacco E (2017) A 3D microstructured cohesive-frictional interface model and its rational calibration for the analysis of masonry panels. Int J Solids Struct 122:110–127
Baggio C, Trovalusci P (1998) Limit analysis for no-tension and frictional three-dimensional discrete systems. J Struct Mech 26(3):287–304
Baggio C, Trovalusci P (2000) Collapse behaviour of three-dimensional brick-block systems using non-linear programming. Struct Eng Mech 10(2):181
Ferris M, Tin-Loi F (2001) Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints. Int J Mech Sci 43(1):209–224
Sutcliffe D, Yu H, Page A (2001) Lower bound limit analysis of unreinforced masonry shear walls. Comput Struct 79(14):1295–1312
Orduña A, Lourenço PB (2005) Three-dimensional limit analysis of rigid blocks assemblages. Part I: Torsion failure on frictional interfaces and limit analysis formulation. Int J Solids Struct 42(18–19):5140–5160
Orduña A, Lourenço PB (2005) Three-dimensional limit analysis of rigid blocks assemblages. Part II: load-path following solution procedure and validation. Int J Solids Struct 42(18–19):5161–5180
Gilbert M, Casapulla C, Ahmed H (2006) Limit analysis of masonry block structures with non-associative frictional joints using linear programming. Comput Struct 84(13–14):873–887
Portioli F, Casapulla C, Cascini L, D’Aniello M, Landolfo R (2013) Limit analysis by linear programming of 3D masonry structures with associative friction laws and torsion interaction effects. Arch Appl Mech 83(10):1415–1438
Portioli F, Casapulla C, Gilbert M, Cascini L (2014) Limit analysis of 3D masonry block structures with non-associative frictional joints using cone programming. Comput Struct 143:108–121
Milani G (2008) 3d upper bound limit analysis of multi-leaf masonry walls. Int J Mech Sci 50(4):817–836
Milani G, Beyer K, Dazio A (2009) Upper bound limit analysis of meso-mechanical spandrel models for the pushover analysis of 2d masonry frames. Eng Struct 31(11):2696–2710
Milani G, Zuccarello F, Olivito R, Tralli A (2007) Heterogeneous upper-bound finite element limit analysis of masonry walls out-of-plane loaded. Comput Mech 40(6):911–931
Cavicchi A, Gambarotta L (2006) Two-dimensional finite element upper bound limit analysis of masonry bridges. Comput Struct 84(31–32):2316–2328
Abdulla KF, Cunningham LS, Gillie M (2017) Simulating masonry wall behaviour using a simplified micro-model approach. Eng Struct 151:349–365
Zhai C, Wang X, Kong J, Li S, Xie L (2017) Numerical simulation of masonry-infilled rc frames using xfem. J Struct Eng 143(10):04017144
Del Piero G (1989) Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials. Meccanica 24(3):150–162
Maier G, Nappi A (1990) A theory of no-tension discretized structural systems. Eng Struct 12(4):227–234
Angelillo M (1994) A finite element approach to the study of no-tension structures. Finite Elements Anal Des 17(1):57–73
Alfano G, Rosati L, Valoroso N (2000) A numerical strategy for finite element analysis of no-tension materials. Int J Numer Methods Eng 48(3):317–350
Cuomo M, Ventura G (2000) A complementary energy formulation of no tension masonry-like solids. Comput Methods Appl Mech Eng 189(1):313–339
Lucchesi M, Padovani C, Pasquinelli G (2000) Thermodynamics of no-tension materials. Int J Solids Struct 37(45):6581–6604
Bruggi M (2014) Finite element analysis of no-tension structures as a topology optimization problem. Struct Multidiscip Optim 50(6):957–973
Bruggi M, Taliercio A (2015) Analysis of no-tension structures under monotonic loading through an energy-based method. Comput Struct 159:14–25
Bruggi M, Taliercio A (2018) Analysis of 3D no-tension masonry-like walls. In: European mechanics society ESMC 2018
Hillerborg A, Modéer M, Petersson P-E (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concret Res 6(6):773–781
Rots JG, De Borst R (1987) Analysis of mixed-mode fracture in concrete. J Eng Mech 113(11):1739–1758
Dragon A, Mroz Z (1979) A continuum model for plastic-brittle behaviour of rock and concrete. Int J Eng Sci 17(2):121–137
Løland K (1980) Continuous damage model for load-response estimation of concrete. Cem Concret Res 10(3):395–402
Lubliner J, Oliver J, Oller S, Onate E (1989) A plastic-damage model for concrete. Int J Solids Struct 25(3):299–326
Lee J, Fenves GL (1998) Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 124(8):892–900
Lotfi H, Shing P (1991) An appraisal of smeared crack models for masonry shear wall analysis. Comput Struct 41(3):413–425
Toti J, Gattulli V, Sacco E (2015) Nonlocal damage propagation in the dynamics of masonry elements. Comput Struct 152:215–227
D’Altri AM, Castellazzi G, de Miranda S (2018) Collapse investigation of the Arquata del Tronto medieval fortress after the 2016 Central Italy seismic sequence. J Build Eng 18:245–251
Bartoli G, Betti M, Vignoli A (2016) A numerical study on seismic risk assessment of historic masonry towers: a case study in San Gimignano. Bull Earthq Eng 14(6):1475–1518
Castellazzi G, D’Altri AM, de Miranda S, Chiozzi A, Tralli A (2018) Numerical insights on the seismic behavior of a non-isolated historical masonry tower. Bull Earthq Eng 16(2):933–961
Valente M, Milani G (2016) Seismic assessment of historical masonry towers by means of simplified approaches and standard FEM. Constr Build Mater 108:74–104
Betti M, Vignoli A (2011) Numerical assessment of the static and seismic behaviour of the basilica of Santa Maria all’Impruneta (Italy). Constr Build Mater 25(12):4308–4324
Milani G, Valente M (2015) Failure analysis of seven masonry churches severely damaged during the 2012 Emilia-Romagna (Italy) earthquake: Non-linear dynamic analyses vs conventional static approaches. Eng Fail Anal 54:13–56
Fortunato G, Funari MF, Lonetti P (2017) Survey and seismic vulnerability assessment of the baptistery of san giovanni in tumba (Italy). J Cultural Heritage 26:64–78
Elyamani A, Roca P, Caselles O, Clapes J (2017) Seismic safety assessment of historical structures using updated numerical models: the case of Mallorca cathedral in spain. Eng Failure Anal 74:54–79
Betti M, Galano L (2012) Seismic analysis of historic masonry buildings: the vicarious palace in Pescia (Italy). Buildings 2(2):63–82
Tiberti S, Acito M, Milani G (2016) Comprehensive fe numerical insight into finale emilia castle behavior under 2012 emilia romagna seismic sequence: Damage causes and seismic vulnerability mitigation hypothesis. Eng Struct 117:397–421
Degli Abbati S, D’Altri AM, Ottonelli D, Castellazzi G, Cattari, S, de Miranda S, Lagomarsino S (2019) Seismic assessment of interacting structural units in complex historic masonry constructions by nonlinear static analyses. Comput Struct 213:51–71. https://doi.org/10.1016/j.compstruc.2018.12.001
Pelà L, Aprile A, Benedetti A (2009) Seismic assessment of masonry arch bridges. Eng Struct 31(8):1777–1788
Zampieri P, Zanini MA, Modena C (2015) Simplified seismic assessment of multi-span masonry arch bridges. Bull Earthq Eng 13(9):2629–2646
Saloustros S, Pelà L, Cervera M (2015) A crack-tracking technique for localized cohesive-frictional damage. Eng Fract Mech 150:96–114
Saloustros S, Pelà L, Cervera M, Roca P (2018) An enhanced finite element macro-model for the realistic simulation of localized cracks in masonry structures: A large-scale application. Int J Archit Herit 12(3):432–447
Rots J, Messali F, Esposito R, Jafari S, Mariani V (2016) Computational modeling of masonry with a view to groningen induced seismicity. In: 10th international conference on Structural Analysis of Historical Constructions, SAHC, pp 13–15
Lourénço PB, De Borst R, Rots JG (1997) A plane stress softening plasticity model for orthotropic materials. Int J Numer Methods Eng 40(21):4033–4057
Lourenço PB, Rots JG, Blaauwendraad J (1998) Continuum model for masonry: parameter estimation and validation. J Struct Eng 124(6):642–652
Lopez J, Oller S, Onate E, Lubliner J (1999) A homogeneous constitutive model for masonry. Int J Numer Methods Eng 46(10):1651–1671
Berto L, Saetta A, Scotta R, Vitaliani R (2002) An orthotropic damage model for masonry structures. Int J Numer Methods Eng 55(2):127–157
Pelà L, Cervera M, Roca P (2011) Continuum damage model for orthotropic materials: application to masonry. Comput Methods Appl Mech Eng 200(9–12):917–930
Pelà L, Cervera M, Roca P (2013) An orthotropic damage model for the analysis of masonry structures. Constr Build Mater 41:957–967
Pelà L, Cervera M, Oller S, Chiumenti M (2014) A localized mapped damage model for orthotropic materials. Eng Fract Mech 124:196–216
Reyes E, Gálvez J, Casati M, Cendón D, Sancho J, Planas J (2009) An embedded cohesive crack model for finite element analysis of brickwork masonry fracture. Eng Fract Mech 76(12):1930–1944
Milani G, Casolo S, Naliato A, Tralli A (2012) Seismic assessment of a medieval masonry tower in northern Italy by limit, nonlinear static, and full dynamic analyses. Int J Archit Herit 6(5):489–524
Pantò B, Cannizzaro F, Caddemi S, Caliò I (2016) 3d macro-element modelling approach for seismic assessment of historical masonry churches. Adv Eng Softw 97:40–59
Pantò B, Caliò I, Lourenço P (2018) A 3D discrete macro-element for modelling the out-of-plane behaviour of infilled frame structures. Eng Struct 175:371–385
Anthoine A (1995) Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int J Solids Struct 32(2):137–163
Cavalagli N, Cluni F, Gusella V (2011) Strength domain of non-periodic masonry by homogenization in generalized plane state. Eur J Mech-A/Solids 30(2):113–126
Taliercio A (2014) Closed-form expressions for the macroscopic in-plane elastic and creep coefficients of brick masonry. Int J Solids Struct 51(17):2949–2963
Stefanou I, Sab K, Heck J-V (2015) Three dimensional homogenization of masonry structures with building blocks of finite strength: a closed form strength domain. Int J Solids Struct 54:258–270
Milani G (2011) Simple lower bound limit analysis homogenization model for in-and out-of-plane loaded masonry walls. Constr Build Mater 25(12):4426–4443
Sacco E, Addessi D, Sab K (2018) New trends in mechanics of masonry. Meccanica 53(7):1565–1569
Bertolesi E, Milani G, Casolo S (2018) Homogenization towards a mechanistic rigid body and spring model (HRBSM) for the non-linear dynamic analysis of 3D masonry structures. Meccanica 53(7):1819–1855
Petracca M, Pelà L, Rossi R, Oller S, Camata G, Spacone E (2016) Regularization of first order computational homogenization for multiscale analysis of masonry structures. Comput Mech 57(2):257–276
Leonetti L, Greco F, Trovalusci P, Luciano R, Masiani R (2018) A multiscale damage analysis of periodic composites using a couple-stress/Cauchy multidomain model: application to masonry structures. Compos Part B: Eng 141:50–59
Pietruszczak S, Niu X (1992) A mathematical description of macroscopic behaviour of brick masonry. Int J Solids Struct 29(5):531–546
Briccoli Bati S, Ranocchiai G, Rovero L (1999) A micromechanical model for linear homogenization of brick masonry. Mater Struct 32(1):22–30
Masiani R, Trovalusci P (1996) Cosserat and Cauchy materials as continuum models of brick masonry. Meccanica 31(4):421–432
Stefanou I, Sulem J, Vardoulakis I (2008) Three-dimensional Cosserat homogenization of masonry structures: elasticity. Acta Geotechnica 3(1):71–83
Cecchi A, Sab K (2002) A multi-parameter homogenization study for modeling elastic masonry. Eur J Mech-A/Solids 21(2):249–268
Cecchi A, Sab K (2007) A homogenized reissner-mindlin model for orthotropic periodic plates: application to brickwork panels. Int J Solids Struct 44(18–19):6055–6079
Mistler M, Anthoine A, Butenweg C (2007) In-plane and out-of-plane homogenisation of masonry. Comput Struct 85(17–18):1321–1330
Drougkas A, Roca P, Molins C (2015) Analytical micro-modeling of masonry periodic unit cells-elastic properties. Int J Solids Struct 69:169–188
Cecchi A, Milani G, Tralli A (2005) Validation of analytical multiparameter homogenization models for out-of-plane loaded masonry walls by means of the finite element method. J Eng Mech 131(2):185–198
Kawa M, Pietruszczak S, Shieh-Beygi B (2008) Limit states for brick masonry based on homogenization approach. Int J Solids Struct 45(3–4):998–1016
De Buhan P, De Felice G (1997) A homogenization approach to the ultimate strength of brick masonry. J Mech Phys Solids 45(7):1085–1104
Zucchini A, Lourenço P (2002) A micro-mechanical model for the homogenisation of masonry. Int J Solids Struct 39(12):3233–3255
Zucchini A, Lourenço PB (2004) A coupled homogenisation-damage model for masonry cracking. Comput Struct 82(11–12):917–929
Wei X, Hao H (2009) Numerical derivation of homogenized dynamic masonry material properties with strain rate effects. Int J Impact Eng 36(3):522–536
Cecchi A, Sab K (2009) Discrete and continuous models for in plane loaded random elastic brickwork. Eur J Mech-A/Solids 28(3):610–625
Cavalagli N, Cluni F, Gusella V (2013) Evaluation of a statistically equivalent periodic unit cell for a quasi-periodic masonry. Int J Solids Struct 50(25–26):4226–4240
Milani G, Lourenço PB, Tralli A (2006) Homogenised limit analysis of masonry walls, Part I: Failure surfaces. Comput Struct 84(3–4):166–180
Milani G, Lourenço P, Tralli A (2006) Homogenization approach for the limit analysis of out-of-plane loaded masonry walls. J Struct Eng 132(10):1650–1663
Cecchi A, Milani G, Tralli A (2007) A reissner-mindlin limit analysis model for out-of-plane loaded running bond masonry walls. Int J Solids Struct 44(5):1438–1460
Cecchi A, Milani G (2008) A kinematic fe limit analysis model for thick english bond masonry walls. Int J Solids Struct 45(5):1302–1331
Godio M, Stefanou I, Sab K, Sulem J, Sakji S (2017) A limit analysis approach based on Cosserat continuum for the evaluation of the in-plane strength of discrete media: application to masonry. Eur J Mech-A/Solids 66:168–192
Milani G, Lourenço PB, Tralli A (2006) Homogenised limit analysis of masonry walls, Part II: structural examples. Comput Struct 84(3–4):181–195
Milani G, Lourenço P, Tralli A (2007) 3d homogenized limit analysis of masonry buildings under horizontal loads. Eng Struct 29(11):3134–3148
Casolo S (2004) Modelling in-plane micro-structure of masonry walls by rigid elements. Int J Solids Struct 41(13):3625–3641
Casolo S, Pena F (2007) Rigid element model for in-plane dynamics of masonry walls considering hysteretic behaviour and damage. Earthq Eng Struct Dyn 36(8):1029–1048
Silva LC, Lourenço PB, Milani G (2017) Nonlinear discrete homogenized model for out-of-plane loaded masonry walls. J Struct Eng 143(9):04017099
Papa E (1996) A unilateral damage model for masonry based on a homogenisation procedure. Mech Cohes-Friction Mater Int J Exp Model Comput Mater Struct 1(4):349–366
Luciano R, Sacco E (1997) Homogenization technique and damage model for old masonry material. Int J Solids Struct 34(24):3191–3208
Luciano R, Sacco E (1998) A damage model for masonry structures. Eur J Mech-A/Solids 17(2):285–303
Gambarotta L, Lagomarsino S (1997) Damage models for the seismic response of brick masonry shear walls. Part II: the continuum model and its applications. Earthq Eng Struct Dyn 26(4):441–462
Pietruszczak S, Ushaksaraei R (2003) Description of inelastic behaviour of structural masonry. Int J Solids Struct 40(15):4003–4019
Calderini C, Lagomarsino S (2006) A micromechanical inelastic model for historical masonry. J Earthq Eng 10(04):453–479
Zucchini A, Lourenço PB (2009) A micro-mechanical homogenisation model for masonry: application to shear walls. Int J Solids Struct 46(3–4):871–886
Sacco E (2009) A nonlinear homogenization procedure for periodic masonry. Eur J Mech-A/Solids 28(2):209–222
Marfia S, Sacco E (2012) Multiscale damage contact-friction model for periodic masonry walls. Comput Methods Appl Mech Eng 205:189–203
Massart TJ, Peerlings RHJ, Geers MGD (2007) An enhanced multi-scale approach for masonry wall computations with localization of damage. Int J Numer Methods Eng 69(5):1022–1059
Bacigalupo A, Gambarotta L (2010) Second-order computational homogenization of heterogeneous materials with periodic microstructure. ZAMM-J Appl Math Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 90(10–11):796–811
Bacigalupo A, Gambarotta L (2012) Computational two-scale homogenization of periodic masonry: Characteristic lengths and dispersive waves. Comput Methods Appl Mech Eng 213:16–28
Addessi D, Marfia S, Sacco E, Toti J (2014) Modeling approaches for masonry structures. Open Civil Eng J 8(1):288–300. https://doi.org/10.2174/1874149501408010288
Salerno G, de Felice G (2009) Continuum modeling of periodic brickwork. Int J Solids Struct 46(5):1251–1267
Casolo S (2006) Macroscopic modelling of structured materials: relationship between orthotropic Cosserat continuum and rigid elements. Int J Solids Struct 43(3–4):475–496
Addessi D, Sacco E, Paolone A (2010) Cosserat model for periodic masonry deduced by nonlinear homogenization. Eur J Mech-A/Solids 29(4):724–737
De Bellis ML, Addessi D (2011) A Cosserat based multi-scale model for masonry structures. Int J Multisc Comput Eng 9(5):543
Addessi D, Sacco E (2012) A multi-scale enriched model for the analysis of masonry panels. Int J Solids Struct 49(6):865–880
Mercatoris B, Massart T (2011) A coupled two-scale computational scheme for the failure of periodic quasi-brittle thin planar shells and its application to masonry. Int J Numer Methods Eng 85(9):1177–1206
Petracca M, Pelà L, Rossi R, Oller S, Camata G, Spacone E (2017) Multiscale computational first order homogenization of thick shells for the analysis of out-of-plane loaded masonry walls. Comput Methods Appl Mech Eng 315:273–301
Brasile S, Casciaro R, Formica G (2007) Multilevel approach for brick masonry walls-part I: A numerical strategy for the nonlinear analysis. Comput Methods Appl Mech Eng 196(49–52):4934–4951
Brasile S, Casciaro R, Formica G (2007) Multilevel approach for brick masonry walls-part II: on the use of equivalent continua. Comput Methods Appl Mech Eng 196(49–52):4801–4810
Reccia E, Leonetti L, Trovalusci P, Cecchi A (2018) A multi-scale/multi-domain model for the failure analysis of masonry walls: a validation with a combined FEM/DEM approach. Int J Multisc Comput Eng 16:325–343
Greco F, Leonetti L, Luciano R, Blasi PN (2016) An adaptive multiscale strategy for the damage analysis of masonry modeled as a composite material. Compos Struct 153:972–988
Lloberas-Valls O, Rixen D, Simone A, Sluys L (2012) Multiscale domain decomposition analysis of quasi-brittle heterogeneous materials. Int J Numer Methods Eng 89(11):1337–1366
Quagliarini E, Maracchini G, Clementi F (2017) Uses and limits of the equivalent frame model on existing unreinforced masonry buildings for assessing their seismic risk: a review. J Build Eng 10:166–182
Augenti N (2006) Seismic behaviour of irregular masonry walls. In: Proceedings of the 1st European conference on earthquake engineering and seismology
Berti M, Salvatori L, Orlando M, Spinelli P (2017) Unreinforced masonry walls with irregular opening layouts: reliability of equivalent-frame modelling for seismic vulnerability assessment. Bull Earthq Eng 15(3):1213–1239
Calderoni B, Cordasco EA, Sandoli A, Onotri V, Tortoriello G (2015) Problematiche di modellazione strutturale di edifici in muratura esistenti soggetti ad azioni sismiche in relazione all’utilizzo di software commerciali. Atti del XVI convegno ANIDIS. L’Aquila, Italia
Dolce M (1991) Schematizzazione e modellazione degli edifici in muratura soggetti ad azioni sismiche. L’Industria delle costruzioni 25(242):44–57
Lagomarsino S, Penna A, Galasco A, Cattari S (2013) Tremuri program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings. Eng Struct 56:1787–1799
Moon FL, Yi T, Leon RT, Kahn LF (2006) Recommendations for seismic evaluation and retrofit of low-rise URM structures. J Struct Eng 132(5):663–672
Parisi F, Augenti N (2013) Seismic capacity of irregular unreinforced masonry walls with openings. Earthq Eng Struct Dyn 42(1):101–121
Parisi F, Lignola GP, Augenti N, Prota A, Manfredi G (2013) Rocking response assessment of in-plane laterally-loaded masonry walls with openings. Eng Struct 56:1234–1248
Lagomarsino S, Camilletti D, Cattari S, Marino S (2018) In plane seismic response of irregular URM walls through equivalent frame and finite element models. In: Recent advances in earthquake engineering in Europe: 16th European conference on earthquake engineering-Thessaloniki 2018. Springer, pp 123–151
Siano R, Roca P, Camata G, Pelà L, Sepe V, Spacone E, Petracca M (2018) Numerical investigation of non-linear equivalent-frame models for regular masonry walls. Eng Struct 173:512–529
Tomaževič M (1978) The computer program POR. Report ZRMK
Calderoni B, Marone P, Pagano M (1987) Modelli per la verifica statica degli edifici in muratura in zona sismica. Ingegneria sismica 3:19–27
Magenes G, Fontana A (1998) Simplified non-linear seismic analysis of masonry buildings. Proc Br Masonry Soc 8:190–195
Kappos AJ, Penelis GG, Drakopoulos CG (2002) Evaluation of simplified models for lateral load analysis of unreinforced masonry buildings. J Struct Eng 128(7):890–897
Roca P, Molins C, Marí AR (2005) Strength capacity of masonry wall structures by the equivalent frame method. J Struct Eng 131(10):1601–1610
Penelis GG (2006) An efficient approach for pushover analysis of unreinforced masonry (URM) structures. J Earthq Eng 10(03):359–379
Belmouden Y, Lestuzzi P (2009) An equivalent frame model for seismic analysis of masonry and reinforced concrete buildings. Constr Build Mater 23(1):40–53
Pasticier L, Amadio C, Fragiacomo M (2008) Non-linear seismic analysis and vulnerability evaluation of a masonry building by means of the sap2000 v. 10 code. Earthq Eng Struct Dyn 37(3):467–485
Grande E, Imbimbo M, Sacco E (2011) A beam finite element for nonlinear analysis of masonry elements with or without fiber-reinforced plastic (frp) reinforcements. Int J Archit Herit 5(6):693–716
Addessi D, Mastrandrea A, Sacco E (2014) An equilibrated macro-element for nonlinear analysis of masonry structures. Eng Struct 70:82–93
Addessi D, Liberatore D, Masiani R (2015) Force-based beam finite element (fe) for the pushover analysis of masonry buildings. Int J Archit Herit 9(3):231–243
Liberatore D, Addessi D (2015) Strength domains and return algorithm for the lumped plasticity equivalent frame model of masonry structures. Eng Struct 91:167–181
Lagomarsino S, Penna A, Galasco A, Cattari S (2012) TREMURI program: Seismic analyses of 3D masonry buildings. Release 2.0, University of Genoa (mailto: tremuri@gmail.com)
Cattari S, Lagomarsino S (2013) Masonry structures
Cattari S, Camilletti D, Lagomarsino S, Bracchi S, Rota M, Penna A (2018) Masonry italian code-conforming buildings. Part 2: nonlinear modelling and time-history analysis. J Earthq Eng 22(sup2):2010–2040
Raka E, Spacone E, Sepe V, Camata G (2015) Advanced frame element for seismic analysis of masonry structures: Model formulation and validation. Earthq Eng Struct Dyn 44(14):2489–2506
Chen S-Y, Moon F, Yi T (2008) A macroelement for the nonlinear analysis of in-plane unreinforced masonry piers. Eng Struct 30(8):2242–2252
Gambarotta L, Lagomarsino S (1996) On the dynamic response of masonry panels. In: Proceedings of the national conference on masonry mechanics between theory and practice, Messina, 18–20 Sept 1996, pp 451–462
Brencich A, Lagomarsino S (1998) A macroelement dynamic model for masonry shear walls. In: Computer methods in structural masonry, pp 67–75
Penna A, Lagomarsino S, Galasco A (2014) A nonlinear macroelement model for the seismic analysis of masonry buildings. Earthq Eng Struct Dyn 43(2):159–179
Caliò I, Marletta M, Pantò B (2012) A new discrete element model for the evaluation of the seismic behaviour of unreinforced masonry buildings. Eng Struct 40:327–338
Caliò I, Pantò B (2014) A macro-element modelling approach of infilled frame structures. Comput Struct 143:91–107
Chácara C, Cannizzaro F, Pantò B, Caliò I, Lourenço PB (2018) Assessment of the dynamic response of unreinforced masonry structures using a macroelement modeling approach. Earthq Eng Struct Dyn. https://doi.org/10.1002/eqe.3091
Rinaldin G, Amadio C, Macorini L (2016) A macro-model with nonlinear springs for seismic analysis of urm buildings. Earthq Eng Struct Dyn 45(14):2261–2281
Mobarake AA, Khanmohammadi M, Mirghaderi S (2017) A new discrete macro-element in an analytical platform for seismic assessment of unreinforced masonry buildings. Eng Struct 152:381–396
Xu H, Gentilini C, Yu Z, Wu H, Zhao S (2018) A unified model for the seismic analysis of brick masonry structures. Constr Build Mater 184:733–751
O’Dwyer D (1999) Funicular analysis of masonry vaults. Comput Struct 73(1–5):187–197
Andreu A, Gil L, Roca P (2007) Computational analysis of masonry structures with a funicular model. J Eng Mech 133(4):473–480
Block P, Ciblac T, Ochsendorf J (2006) Real-time limit analysis of vaulted masonry buildings. Comput Struct 84(29–30):1841–1852
Block P, Ochsendorf J (2007) Thrust network analysis: a new methodology for three-dimensional equilibrium. J Int Assoc Shell Spat Struct 48(3):167–173
Block P, Lachauer L (2014) Three-dimensional (3d) equilibrium analysis of gothic masonry vaults. Int J Archit Herit 8(3):312–335
Block P, Lachauer L (2014) Three-dimensional funicular analysis of masonry vaults. Mech Res Commun 56:53–60
Fantin M, Ciblac T (2016) Extension of thrust network analysis with joints consideration and new equilibrium states. Int J Space Struct 31(2–4):190–202
Fraternali F (2010) A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mech Res Commun 37(2):198–204
Angelillo M, Babilio E, Fortunato A (2013) Singular stress fields for masonry-like vaults. Contin Mech Thermodyn 25(2–4):423–441
Angelillo M (2015) Static analysis of a guastavino helical stair as a layered masonry shell. Compos Struct 119:298–304
Fraddosio A, Lepore N, Piccioni MD (2019) Lower bound limit analysis of masonry vaults under general load conditions. In: Structural Analysis of Historical Constructions. Springer, pp 1090–1098
Marmo F, Masi D, Rosati L (2018) Thrust network analysis of masonry helical staircases. Int J Archit Herit 12:1–21
D’Altri AM, Castellazzi G, de Miranda S, Tralli A (2017) Seismic-induced damage in historical masonry vaults: a case-study in the 2012 Emilia earthquake-stricken area. J Build Eng 13:224–243
Giuffrè A (1991) Letture sulla meccanica delle murature storiche. Kappa
Ordinanza del Presidente del Consiglio dei Ministri (OPCM). Norme tecniche per il progetto, la valutazione e l’adeguamento sismico degli edifici (2005)
Circolare 2009. Circolare n. 617 del 02/02/2009. Istruzioni per l’applicazione delle nuove Norme Tecniche per le Costruzioni di cui al D.M. del 14/01/2008
NTC2008, Norme Tecniche per le Costruzioni, D.M. 14/01/2008
Milani G (2015) Upper bound sequential linear programming mesh adaptation scheme for collapse analysis of masonry vaults. Adv Eng Softw 79:91–110
Chiozzi A, Milani G, Grillanda N, Tralli A (2018) A fast and general upper-bound limit analysis approach for out-of-plane loaded masonry walls. Meccanica 53(7):1875–1898
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D’Altri, A.M., Sarhosis, V., Milani, G. et al. Modeling Strategies for the Computational Analysis of Unreinforced Masonry Structures: Review and Classification. Arch Computat Methods Eng 27, 1153–1185 (2020). https://doi.org/10.1007/s11831-019-09351-x
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DOI: https://doi.org/10.1007/s11831-019-09351-x