, Volume 53, Issue 7, pp 1629–1643 | Cite as

Effects of the dilatancy of joints and of the size of the building blocks on the mechanical behavior of masonry structures

  • Michele Godio
  • Ioannis Stefanou
  • Karam Sab
New Trends in Mechanics of Masonry


The effect of the dilatancy of masonry interfaces and of the size of the building blocks on the strength of masonry structures is quantified herein. The study focuses mainly on out-of-plane loadings, which can appear due to various factors such as wind, earthquakes or explosions. The analysis is performed using the Discrete Element Method (DEM), which allows to access directly various micro-mechanical parameters, such as the joints dilatancy angle and the size of the building blocks. Detailed DEM numerical models of existing experimental configurations are presented. The numerical results are first compared and validated towards the experimental observations and then they are used to derive qualitative and quantitative conclusions regarding the effects of joints dilatancy and blocks size. It is shown that dilatancy plays an important role on the overall strength of masonry even under low confinement. The size of the blocks is also an important parameter that needs to be considered in the modeling of masonry structures.


Dilatancy Scale effect Masonry Discrete Element Method (DEM) Experimental tests Limit analysis 



The authors would like to acknowledge the valuable help of Dr Jose Lemos for the technical support he provided in using 3DEC.

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.


  1. 1.
    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. doi: 10.1016/j.euromechsol.2010.03.001.
  2. 2.
    Azevedo J, Sincraian G, Lemos JV (2000) Seismic behavior of blocky masonry structures. doi: 10.1193/1.1586116
  3. 3.
    Bacigalupo A, Gambarotta L (2012) Computational two-scale homogenization of periodic masonry: characteristic lengths and dispersive waves. Comput Methods Appl Mech Eng 213–216:16–28. doi: 10.1016/j.cma.2011.11.020.
  4. 4.
    Baggio C, Trovalusci P (1998) Limit analysis for no-tension and frictional three-dimensional discrete systems. Mech Struct Mach 26(3):287–304. doi: 10.1080/08905459708945496 CrossRefGoogle Scholar
  5. 5.
    Baraldi D, Cecchi A, Tralli A (2015) Continuous and discrete models for masonry like material: a critical comparative study. Eur J Mech A/Solids 50:39–58. doi: 10.1016/j.euromechsol.2014.10.007. MathSciNetCrossRefGoogle Scholar
  6. 6.
    Besdo D (1985) Inelastic behavior of plane frictionless block-systems described as cosserat media. Arch Mech 37(6):603–619zbMATHGoogle Scholar
  7. 7.
    Bui TT, Limam A (2012) Masonry walls under membrane or bending loading cases : experiments and discrete element analysis. In: Proceedings of the eleventh international conference on computational structures technologyGoogle Scholar
  8. 8.
    Çakt E, Saygl Ö, Lemos JV, Oliveira CS (2016) Discrete element modeling of a scaled masonry structure and its validation. Eng Struct 126:224–236. doi: 10.1016/j.engstruct.2016.07.044.
  9. 9.
    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.  10.1061/(ASCE)0733-9399(2005)131:2(185) CrossRefGoogle Scholar
  10. 10.
    Cecchi A, Milani G, Tralli A (2007) A ReissnerMindlin limit analysis model for out-of-plane loaded running bond masonry walls. Int J Solids Struct 44(5):1438–1460. doi: 10.1016/j.ijsolstr.2006.06.033.
  11. 11.
    Cecchi A, Sab K (2004) A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int J Solids Struct 41(9–10):2259–2276. doi: 10.1016/j.ijsolstr.2003.12.020.
  12. 12.
    Colas AS, Morel JC, Garnier D (2008) Yield design of dry-stone masonry retaining structures-Comparisons with analytical, numerical, and experimental data. Int J Numer Anal Methods Geomech 32(14):1817–1832. doi: 10.1002/nag.697.
  13. 13.
    D’Ayala D, Speranza E (2003) Definition of collapse mechanisms and seismic vulnerability of historic masonry buildings. Earthq Spectra 19(3):479–509. doi: 10.1193/1.1599896 CrossRefGoogle Scholar
  14. 14.
    de Bellis M, Addessi D (2011) A cosserat based multi-scale model for masonry structures. Int J Multiscale Comput Eng 9(5):543–563CrossRefGoogle Scholar
  15. 15.
    DeJong MJ, Vibert C (2012) Seismic response of stone masonry spires: computational and experimental modeling. Eng Struct 40:566–574. doi: 10.1016/j.engstruct.2012.03.001.
  16. 16.
    Dimitrakopoulos EG, DeJong MJ (2012) Revisiting the rocking block: closed-form solutions and similarity laws. Proc R Soc A Math Phys Eng Sci 468(2144):2294–2318. doi: 10.1098/rspa.2012.0026 ADSCrossRefGoogle Scholar
  17. 17.
    Dimitri R, De Lorenzis L, Zavarise G (2011) Numerical study on the dynamic behavior of masonry columns and arches on buttresses with the discrete element method. Eng Struct 33(12):3172–3188. doi: 10.1016/j.engstruct.2011.08.018 CrossRefGoogle Scholar
  18. 18.
    Gazzola E, Drysdale R (1986) A component failure criterion for blockwork in flexure. In: Structures congress. New Orleans, LouisianaGoogle Scholar
  19. 19.
    Gazzola E, Drysdale RRG, Essawy AA (1985) Bending of concrete masonry walls at different angles to the bed joints. In: Third North American masonry conference. Arlington, TexasGoogle Scholar
  20. 20.
    Germain P (1973) The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J Appl Math 25(3):556–575. doi: 10.1137/0125053. CrossRefzbMATHGoogle Scholar
  21. 21.
    Godio M, Stefanou I, Sab K, Sulem J (2015) Dynamic finite element formulation for Cosserat elastic plates. Int J Numer Methods Eng 101(13):992–1018. doi: 10.1002/nme.4833 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Godio M, Stefanou I, Sab K, Sulem J (2014) Cosserat elastoplastic finite elements for masonry structures. Key Eng Mater 624:131–138. doi: 10.4028/
  23. 23.
    Housner G (1963) The behavior of inverted pendulum structures during earthquakes. Bull Seismol Soc Am 53(2):403–417Google Scholar
  24. 24.
    Itasca consulting group: 3DEC 5.0 (2013)Google Scholar
  25. 25.
    Kourkoulis SK, Ganniari-Papageorgiou E (2010) Experimental study of the size- and shape-effects of natural building stones. Constr Build Mater 24(5):803–810. doi: 10.1016/j.conbuildmat.2009.10.027.
  26. 26.
    Lemos JV (2007) Numerical issues in the representation of masonry structural dynamics with discrete elements. Compdyn 13–16 June.
  27. 27.
    Lemos JV (2007) Discrete element modeling of masonry structures. Int J Archit Herit 1(2):190–213. doi: 10.1080/15583050601176868 CrossRefGoogle Scholar
  28. 28.
    Lengyel G, Bagi K (2015) Numerical analysis of the mechanical role of the ribs in groin vaults. Comput Struct 158:42–60. doi: 10.1016/j.compstruc.2015.05.032.
  29. 29.
    Lourenço PB (2000) Anisotropic softening model for masonry plates and shells. J Struct Eng 126(9):1008–1016CrossRefGoogle Scholar
  30. 30.
    Lourenço PB, Ramos LSF (2004) Characterization of cyclic behavior of dry masonry joints. J Struct Eng 130(5):779–786. doi: 10.1061/(ASCE)0733-9445(2004)130:5(779) CrossRefGoogle Scholar
  31. 31.
    Makris N, Konstantinidis D (2003) The rocking spectrum and the limitations of practical design methodologies. Earthq Eng Struct Dyn 32(2):265–289. doi: 10.1002/eqe.223 CrossRefGoogle Scholar
  32. 32.
    Makris N, Vassiliou MF (2013) Planar rocking response and stability analysis of an array of free-standing columns capped with a freely supported rigid beam. Earthq Eng Struct Dyn 42(3):431–449. doi: 10.1002/eqe.2222 CrossRefGoogle Scholar
  33. 33.
    Masiani R, Rizzi N, Trovalusci P (1995) Masonry as structured continuum. Meccanica 30(6):673–683. doi: 10.1007/BF00986573 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Milani G, Lourenço PB, Tralli A (2006) Homogenization approach for the limit analysis of out-of-plane loaded masonry walls. J Struct Eng 132(October):1650–1663. doi: 10.1061/(ASCE)0733-9445(2006)132:10(1650) CrossRefGoogle Scholar
  35. 35.
    Milani G (2009) Homogenized limit analysis of FRP-reinforced masonry walls out-of-plane loaded. Comput Mech 43(5):617–639. doi: 10.1007/s00466-008-0334-7 CrossRefGoogle Scholar
  36. 36.
    Milani G, Lourenço PB, Tralli A (2009) Homogenized rigid-plastic model for masonry walls subjected to impact. Int J Solids Struct 46(22–23):4133–4149. doi: 10.1016/j.ijsolstr.2009.08.007 CrossRefzbMATHGoogle Scholar
  37. 37.
    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. doi: 10.1016/j.ijsolstr.2005.02.010.
  38. 38.
    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. doi:10.1016/j.ijsolstr.2005.02.011.
  39. 39.
    Papadopoulos, C., Basanou, E., Vardoulakis, I., Boulon, M., Armand, G.: Mechanical behaviour of Dionysos marble smooth joints under cyclic loading: II constitutive modelling. In: Proceedings of the international conference on mechanics of jointed and faulted rock. Vienna, Austria (1998)Google Scholar
  40. 40.
    Papantonopoulos C, Psycharis IN, Papastamatiou DY, Lemos JV, Mouzakis HP (2002) Numerical prediction of the earthquake response of classical columns using the distinct element method. Earthq Eng Struct Dyn 31(9):1699–1717. doi: 10.1002/eqe.185 CrossRefGoogle Scholar
  41. 41.
    Petry S, Beyer K (2014) Scaling unreinforced masonry for reduced-scale seismic testing. Bull Earthq Eng 1–25. doi: 10.1007/s10518-014-9605-1
  42. 42.
    Restrepo-Vélez L, Magenes G, Griffith M (2014) Dry stone masonry walls in bending—Part I: static tests. Int J Archit Herit 8(1):1–28. doi: 10.1080/15583058.2012.663059 CrossRefGoogle Scholar
  43. 43.
    Rondelet J (1834) Traité théorique et pratique de l’art de bâtir. Chez l’auteur, ParisGoogle Scholar
  44. 44.
    Salençon J (1983) Calcul à la Rupture et Analyse Limite. Presses de l’Ecole Nationale des Ponts et Chaussées, ParisGoogle Scholar
  45. 45.
    Salerno G, de Felice G (2009) Continuum modeling of periodic brickwork. Int J Solids Struct 46(5):1251–1267. doi: 10.1016/j.ijsolstr.2008.10.034.
  46. 46.
    Simon J, Bagi K (2014) Discrete element analysis of the minimum thickness of oval masonry domes. Int J Archit Herit. doi: 10.1080/15583058.2014.996921
  47. 47.
    Stefanou, I., Fragiadakis, M., Psycharis, I.N.: Seismic reliability assessment of classical columns subjected to near source ground motions. In: Seismic assessment, behavior and retrofit of heritage buildings and monuments, chap. 3, Springer, pp. 61–82 (2015). doi:  10.1007/978-3-319-16130-3_3
  48. 48.
    Stefanou I, Sulem J, Vardoulakis I (2008) Three-dimensional Cosserat homogenization of masonry structures: elasticity. Acta Geotech 3(1):71–83. doi: 10.1007/s11440-007-0051-y CrossRefGoogle Scholar
  49. 49.
    Stefanou I, Psycharis I, Georgopoulos IO (2011) Dynamic response of reinforced masonry columns in classical monuments. Constr Build Mater 25(12):4325–4337. doi: 10.1016/j.conbuildmat.2010.12.042 CrossRefGoogle Scholar
  50. 50.
    Stefanou I, Vardoulakis I, Mavraganis A (2011) Dynamic motion of a conical frustum over a rough horizontal plane. Int J Non-Linear Mech 46:114–124. doi: 10.1016/j.ijnonlinmec.2010.07.008 CrossRefGoogle Scholar
  51. 51.
    Stefanou I, Sab K, Heck JV (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. doi: 10.1016/j.ijsolstr.2014.10.007.
  52. 52.
    Stefanou I, Sulem J, Vardoulakis I (2010) Homogenization of interlocking masonry structures using a generalized differential expansion technique. Int J Solids Struct 47(11–12):1522–1536. doi: 10.1016/j.ijsolstr.2010.02.011.
  53. 53.
    Trovalusci P, Pau A (2013) Derivation of microstructured continua from lattice systems via principle of virtual works: the case of masonry-like materials as micropolar, second gradient and classical continua. Acta Mech. doi: 10.1007/s00707-013-0936-9
  54. 54.
    van der Pluijm R (1999) Out-of-plane bending of masonry behaviour and strength. Eindhoven: Technische Universiteit Eindhoven. doi: 10.6100/IR528212
  55. 55.
    Vannucci P, Masi F, Stefanou I (2017) A study on the simulation of blast actions on a monument structure. (Working paper or preprint)

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.EPFL ENAC IIC EESD, Earthquake Engineering and Structural Dynamics Laboratory (EESD), School of Architecture, Civil and Environmental Engineering (ENAC)Ecole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland
  2. 2.NAVIER, UMR 8205, École des Ponts, IFSTTAR, CNRSUniversité Paris-EstChamps-sur-MarneFrance

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