This paper deals with free oscillations of a masonry panel modelled as a unilateral (no-tension) material a la Heyman. The aim of this analysis is to evaluate the role of elasticity in the geometrically nonlinear dynamic response of structures made of such material. Two unilateral models are investigated: in the first one the material is assumed as rigid, while in the second the material is no-tension and elastic in compression. The responses of a simple panel subjected to fixed vertical compression and different initial lateral perturbations are compared, to estimate the role of elasticity for realistic values of Young’s modulus. It is shown that the time histories are periodic or quasi-periodic for rigid or elastic panels. In both cases, the oscillation periods are markedly dependent on the magnitude of the initial (rotation) data. The periodicity of elastic panels and its dependence on initial magnitude is detectable even in the purely elastic phase and is due to the physical non-linearity induced by the unilateral behaviour of the material. The comparison between rigid and elastic time histories shows the main differences between rigid and elastic cases and allows to quantify the energy distance between ‘rocking-like’ or ‘harmonic type’ models.
NTC. Norme Tecniche per le Costruzioni. D. Min. Infrastrutture e Trasporti 17 gennaio 2018 [in Italian], Gazzetta Ufficiale Serie Generale 42, Italy.
2.
Eurocode 8. Design of structures for earthquake resistance - part 1: general rules, seismic actions and rules for buildings. Brussels, Belgium: CEN (European Committee for Standardisation, 2005.
3.
ASCE/SEI 7-22. American Society of Civil Engineers. Minimum design loads and associated criteria for buildings and other structures. American Society of Civil Engineers, 2022.
4.
HousnerGW. The behavior of inverted pendulum structures during earthquakes. Bull Seismol Soc Am1963; 53(2): 403–417.
SpanosPKohAS. Rocking of rigid blocks due to harmonic shaking. J Eng Mech1984; 110(11): 1627–1642.
7.
IshiyamaY. Motion of rigid bodies and criteria for overturning by earthquake excitations. Earthq Eng Struct Dyn1982; 10: 635–650.
8.
OppenheimIJ. The masonry arch as a four-link mechanism under base motion. Earthq Eng Struct Dyn1992; 21(11): 1005–1017.
9.
ShentonHW. Criteria for initiation of slide, rock, and slide-rock rigid-body modes. J Eng Mech1996; 122(7): 690–693.
10.
MakrisN. A half-century of rocking isolation. Earthq Struct2014; 7(6): 1187–1221.
11.
DeJongMJ. Amplification of rocking due to horizontal ground motion. Earthq Spectra2012; 28(4): 1405–1421.
12.
De LorenzisLDeJongMOchsendorfJ. Failure of masonry arches under impulse base motion. Earthq Eng Struct Dyn2007; 36: 2119–2136.
13.
OlivieriCCockingSFabbrocinoF, et alSeismic capacity of purely compressed shells based on Airy stress function. Contin Mech Thermodyn2025; 37(2): 21.
14.
CrespinoEAdriaenssensSFraddosioA, et alA multi-objective optimization approach for novel shell/frame systems under seismic load. Structures2024; 65: 106625.
15.
CastellanoAEliaIFraddosioA, et alA new experimental approach for small-scale dynamic tests on masonry arches aimed at seismic assessment. Int J Maso Res Innov2022; 7(1–2): 89–112.
16.
NodargiNABisegnaP. A variational event-driven approach for the dynamic analysis of multi-block historical masonry structures under ground excitation. Eng Struct2024; 306: 117804.
17.
RamosLFMarquesLLourençoPB, et alMonitoring historical masonry structures with operational modal analysis: two case studies. Mech Syst Signal Process2010; 24(5): 1291–1305.
18.
AtamturkturSBornnLHermezF. Vibration characteristics of vaulted masonry monuments undergoing differential support settlement. Eng Struct2011; 33: 2472–2484.
19.
ChiorinoMACeravoloRSpadaforA, et alDynamic characterization of complex masonry structures: the Sanctuary of Vicoforte. Int J Archit Herit2011; 5(3), 296–314.
20.
GiuffrèA. Restauro e sicurezza in zona sismica. La cattedrale di Sant’Angelo dei Lombardi. Palladio, nuova serie1988; I(1): 1–26.
21.
AddessiDMarfiaSSaccoE, et alModeling approaches for masonry structures. Open Civil Eng J2014; 8(1): 288–300.
22.
D’AltriAMSarhosisVMilaniG, et alModeling strategies for the computational analysis of unreinforced masonry structures: review and classification. Arch Comput Methods Eng2020; 27(4): 1153–1185.
23.
D’AltriAMCannizzaroFPetraccaM, et alNonlinear modelling of the seismic response of masonry structures: calibration strategies. Bull Earthq Eng2022; 20(4): 1999–2043.
24.
FabbrocinoFOlivieriCLucianoR, et alSeismic performance of historic masonry buildings: a comparative analysis of equivalent frame and block-based methods. Alex Eng J2024; 109: 359–375.
25.
MilaniGLourençoPBTralliA. Homogenised limit analysis of masonry walls, part I: failure surfaces. Comput Struct2006; 84(3–4): 166–180.
26.
CecchiASabK. A homogenized Reissner-Mindlin model for orthotropic periodic lates: application to brickwork panels. Int J Solids Struct2007; 44(18–19): 6055–6079.
27.
SaccoE. A nonlinear homogenization procedure for periodic masonry. Eur J Mech A Solids2009; 28(2): 209–222.
28.
DrougkasASarhosisV. Micro-mechanical homogenisation of three-leaf masonry walls under compression. Eng Struct2021; 245: 112890.
29.
BatiSBRanocchiaiGRoveroL. Suitability of micromechanical model for elastic analysis of masonry. J Eng Mech1999; 125(8): 922–929.
30.
FouchalFLebonFTiteuxI. Contribution to the modelling of interfaces in masonry construction. Construct Build Mater2009; 23(6): 2428–2441.
31.
ZucchiniALourençoPB. A micro-mechanical homogenisation model for masonry: application to shear walls. Int J Solids Struct2009; 46(3–4): 871–886.
32.
FouchalFLebonFRaffaML, et alAn interface model including cracks and roughness applied to masonry. Open Civil Eng J2014; 8: 263–271.
33.
CavalagliNCluniFGusellaV. Evaluation of a statistically equivalent periodic unit cell for a quasi-periodic masonry. Int J Solids Struct2013; 50(25–26): 4226–4240.
34.
ScimemiGFGiambancoGSpadaA. The interphase model applied to the analysis of masonry structures. Comput Methods Appl Mech Eng2014; 279: 66–85.
35.
TaliercioA. Closed-form expressions for the macroscopic in-plane elastic and creep coefficients of brick masonry. Int J Solids Struct2014; 51(17): 2949–2963.
36.
RaffaMLLebonFRizzoniR. Derivation of a model of imperfect interface with finite strains and damage by asymptotic techniques: an application to masonry structures. Meccanica2018; 53(7): 1645–1660.
37.
CavalagliNCluniFGusellaV. Failure surface of quasi-periodic masonry by means of statistically equivalent periodic unit cell approach. Meccanica2018; 53(7): 1719–1736.
38.
AddessiDDi RePSaccoE. Micromechanical and multiscale computational modeling for stability analysis of masonry elements. Eng Struct2020; 211: 110428.
39.
WildingBVGodioMBeyerK. The ratio of shear to elastic modulus of in-plane loaded masonry. Mater Struct2020; 53(2): 40.
40.
KoochekiKPietruszczakS. Mesoscale analysis of fracture process in brick masonry structures. Math Mech Solids2025; 30(1): 73–92.
41.
PortioliFCasapullaCCasciniL. An efficient solution procedure for crushing failure in 3D limit analysis of masonry block structures with non-associative frictional joints. Int J Solids Struct2015; 69: 252–266.
42.
HeymanJ. The stone skeleton. Int J Solids Struct1966; 2(2): 249–279.
43.
LivesleyRK. Limit analysis of structures formed from rigid blocks. Int J Numer Methods Eng1978; 12(12): 1853–1871.
44.
RomanoGRomanoM. Sulla soluzione di problemi strutturali in presenza di legami costitutivi unilaterali. Atti della Accademia Nazionale dei Lincei. Rend Lincei Sci Fis Nat1979; 67(1-2): 104–113.
45.
ComoM. Equilibrium and collapse of masonry bodies. Meccanica1992; 27(3): 185–194.
46.
AngelilloMFortunatoAMontaninoA, et alSingular stress fields in masonry structures: Derand was right. Meccanica2014; 49(5): 1243–1262.
47.
GesualdoACennamoCFortunatoA, et alEquilibrium formulation of masonry helical stairs. Meccanica2017; 52(8): 1963–1974.
48.
FortunatoABabilioELippielloM, et alLimit analysis for unilateral masonry-like structures. Open Constr Build Technol J2016; 10(Suppl 2:M12): 346–362.
49.
FortunatoAFabbrocinoFAngelilloM, et alLimit analysis of masonry structures with free discontinuities. Meccanica2018; 53(7): 1793–1802.
50.
FortunatoAFraternaliFAngelilloM. Structural capacity of masonry walls under horizontal loads. Int J Earthq Eng2014, 31(1): 41–51.
51.
IannuzzoADe LucaAFortunatoA, et alFractures detection in masonry constructions under horizontal seismic forces. Int J Earthq Eng2018; 35(3): 87–103.
52.
OchsendorfJ. Collapse of masonry structures. Ph.D. Dissertation, University of Cambridge, Cambridge, UK, 2002.
53.
Huerta FernándezS. Geometry and equilibrium: the gothic theory of structural design. Struct Eng2006; 84(2): 23-28.
54.
Huerta FernándezS. Galileo was wrong: the geometrical design of masonry arches. Nexus Netw J2006; 8(2): 25–52.
CocciaSComoMDi CarloF. Minimum thrust and minimum thickness of hemispherical masonry domes. Acta Mech2016; 227(9): 2415–2425.
57.
NikolićD. Thrust line analysis and the minimum thickness of pointed masonry arches. Acta Mech2017; 228(6): 2219–2236.
58.
MonacoMBergamascoIBettiM. A no-tension analysis for a brick masonry vault with lunette. J Mech Mater Struct2019; 13(5): 703–714.
59.
CusanoCMontaninoAOlivieriC, et alGraphical and analytical quantitative comparison in the domes assessment: the case of San Francesco di Paola. Appl Sci2021; 11(8): 3622.
60.
AmendolaAFortunatoAFraternaliF, et alLimit analysis of strut nets. Math Mech Solids2023; 28(8): 1760–1775.
61.
OlivieriCAdriaenssensSCennamoC. A novel graphical assessment approach for compressed curved structures under vertical loading. Int J Space Struct2023; 38(2): 141–155.
62.
AngelilloMOlivieriCDeJongMJ. A new equilibrium solution for masonry spiral stairs. Eng Struct2021; 238: 112176.
63.
OlivieriCCennamoCCusanoC, et alMasonry spiral stairs: a comparison between analytical and numerical approaches. Appl Sci2022; 12(9): 4274.
64.
BlockP. Thrust network analysis: exploring three-dimensional equilibrium. Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, MA, 2009.
65.
BlockPOchsendorfJ. Thrust network analysis: a new methodology for three-dimensional equilibrium. J Int Assoc Shell Spat Struct2007; 48(3): 167–173.
66.
AvelinoRMIannuzzoAVan MeleT, et alAssessing the safety of vaulted masonry structures using thrust network analysis. Comput Struct2021; 257: 106647.
67.
AngelilloMFortunatoA. Equilibrium of masonry vaults. Lecture Notes in Applied and Computational Mechanics. In: FremondMMaceriF (eds) Novel approaches in civil engineering. vol. 14. Berlin: Springer, 2004, pp.105–111.
De ChiaraECennamoCGesualdoA, et alAutomatic generation of statically admissible stress fields in masonry vaults. J Mech Mater Struct2019; 14(5): 719–737.
70.
GesualdoABrandonisioGDe LucaA, et alLimit analysis of cloister vaults: the case study of Palazzo Caracciolo di Avellino. J Mech Mater Struct2019; 14(5): 739–750.
71.
OlivieriCFortunatoADeJongMJ. A new membrane equilibrium solution for masonry railway bridges: the case study of Marsh Lane Bridge. Int J Maso Res Innov2021; 6(4): 446–471.
72.
MontaninoAOlivieriCZuccaroG, et alFrom stress to shape: equilibrium of cloister and cross vaults. Appl Sci2021; 11(9): 3846.
73.
OlivieriC. FORMERLY-math: constrained form-finding through membrane equilibrium analysis in Mathematica. Softw Impacts2023; 16: 100512.
74.
IannuzzoA. Energy based fracture identification in masonry structures: the case study of the church of “Pietà dei Turchini”. J Mech Mater Struct2019; 5(14): 683–702.
75.
IannuzzoAOlivieriCFortunatoA. Displacement capacity of masonry structures under horizontal actions via PRD method. J Mech Mater Struct2019; 5(14): 703–718.
76.
IannuzzoAAngelilloMDe ChiaraE, et alModelling the cracks produced by settlements in masonry structures. Meccanica2018; 7(53): 1857–1873.
77.
IannuzzoAVan MeleTBlockP. Piecewise rigid displacement (PRD) method: a limit analysis-based approach to detect mechanisms and internal forces through two dual energy criteria. Mech Res Commun2020; 107: 103557.
78.
IannuzzoABlockPAngelilloM, et alA continuous energy-based numerical approach to predict fracture mechanisms in masonry structures: CDF method. Comput Struct2021; 257: 106645.
79.
OlivieriCIannuzzoAMontaninoA, et alA continuous stress-based form finding approach for compressed membranes. Int J Maso Res Innov2024; 9(5–6): 585–605.
80.
Di PasqualeS. Statica dei solidi murari: teoria ed esperienze. Internal Report, Dipartimento di Costruzioni, Università di Firenze, 1984, p.27.
81.
CastellanoG.Un modello cinematico per i materiali non resistenti a trazione (in Italian). In: Nel Cinquantenario della Facoltà di Architettura di Napoli: Franco Jossa e la Sua Opera. Napoli, Italy: Giannini Editore, 1988, pp.241–256.
82.
AngelilloMRossoF. On statically admissible stress fields for a plane masonry-like structure. Q Appl Math1995; 53(4): 731–751.
83.
AngelilloM. Constitutive equation for no-tension materials. Meccanica1993; 28(3): 195–202.
84.
AnzellottiG. A class of convex non-coercive functionals and masonry-like materials. Ann Inst Henri Poincaré C Anal Non Linéaire1985; 2(4): 261–307.
85.
GiaquintaMGiustiE. Researches on the equilibrium of masonry structures. Arch Ration Mech Anal1985; 88(4): 359–392.
86.
Del PieroG. Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials. Meccanica1989; 24(3): 150–162.
87.
AngelilloMCardamoneLFortunatoA. A Numerical model for masonry-like structures. J Mech Mater Struct2010; 5(4): 583–615.
88.
FortunatoA. Elastic solutions for masonry-like panels. J Elast2010; 98(1): 87–110.
89.
LucchesiMPadovaniCPasquinelliG, et alNumerical modelling of the dynamic behaviour of masonry structures. In: BulJW (ed.) Computer analysis and design of masonry structures. Stirlingshire, UK: Saxe-Coburg Publications, 2018, pp.55–98.
90.
GirardiMLucchesiMPadovaniC, et alNumerical methods for slender masonry structures: a comparative study. In: Proceedings of the 11th international conference on computational structures technology 2012 - civil-comp proceedings, 99, 4-7 September 2012, Civil-Comp Press, Topping, B.H.V, Dubrovnik, Croatia, p.102644.
91.
AngelilloM. Practical applications of unilateral models to Masonry Equilibrium. In: AngelilloM (ed.) Mechanics of masonry structures. Vienna: Springer, 2014, pp.109–210.
92.
MannWMüllerH. Failure of shear-stressed masonry. an enlarged theory, tests and application to shear walls. Proc Br Ceram Soc1982; 30: 223–235.
93.
GanzHThürlimannB. Versuche an Mauerwerksscheiben unter Normalkraft und Querkraft [in German]. In: Bericht Nr. 7502-4. May 1984, Zürich, CH: Institut für Baustatik und Konstruktion ET.
94.
AtkinsonRHYanGG. Results of a statistical study of masonry deformability. Mason Soc J1990; 9(1): 81–90.
95.
VermeltfoortA ThRaijmakersTh MJJanssenHJM. Shear tests on masonry walls. In: Proceedings of the 6th North American masonry conference (eds. Ahmad A Hamid and Harry G Harris), June 6-9, 1993, The Masonry Society, Philadelphia, PA, 1993, pp.1183–1193.
EN 1052-3. Methods of test for masonry-part 3: determination of initial shear strength. Brussels, Belgium: CEN European Committee for Standardisation, 2003.
98.
RaijmakersTMJVermeltfoortATh. Deformation controlled Meso shear tests on masonry piers. Report B-92-1156, TNO-BOUW/TU. Building and Construction Research: Eindhoven, The Netherlands, 1992.
99.
LourençoPBRotsJG. Multisurface interface model for analysis of masonry structures. J Eng Mech1997; 123(7): 660–668.
100.
MessaliFEspositoRJafariS, et alA multiscale experimental characterisation of Dutch unreinforced masonry buildings. In: Proceedings of the16th European conference on earthquake engineering (ECEE) (eds. Kyriazis Pitilakis), June 18-21, Springer International PublishingThessaloniki, Greece, 2018.
101.
FortunatoAGesualdoAMascoloI, et alP-Bézier energy optimisation for elastic solutions of masonry-like panels. Int J Maso Res Innov2022; 7(1–2): 113–125.
