, Volume 53, Issue 7, pp 1793–1802 | Cite as

Limit analysis of masonry structures with free discontinuities

  • A. Fortunato
  • F. Fabbrocino
  • M. Angelillo
  • F. Fraternali
New Trends in Mechanics of Masonry


We formulate a novel procedure for the limit analysis of two-dimensional masonry structures subject to arbitrary loading conditions. The proposed approach works in the framework of free discontinuity methods, on examining collapse mechanisms that exhibit free crack opening discontinuities. The load bearing capacity and the collapse mechanism of the structure are obtained through a fully variational approach, by minimizing a kinetic functional that admits the collapse crack pattern as a variable. Numerical examples illustrate the practical application of the proposed procedure to the limit analysis of a variety of masonry walls and arches subject to foundation settlements, vertical and horizontal forces.


Masonry structures Load bearing capacity Limit analysis Kinematic collapse multiplier Free discontinuities 



The authors wish to thank Antonino Iannuzzo from the Department of Structures for Engineering and Architecture of the University of Naples ‘Federico II” for his helpful assistance with the numerical results presented in Sect. 6.1.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest to report.


  1. 1.
    Heyman J (1966) The stone skeleton. Int J Solids Struct 2:249–279CrossRefGoogle Scholar
  2. 2.
    Como M, Grimaldi A (1985) A unilateral model for the limit analysis of masonry walls. In: Del Piero G, Maceri F (eds) Unilateral problems in structural analysis. CISM courses and lectures. Springer, Berlin, pp 25–45CrossRefGoogle Scholar
  3. 3.
    Di Pasquale S (1984) Statica dei solidi murari. Atti Dipartimento Costruzioni. University of Firenze, Florence, ItalyGoogle Scholar
  4. 4.
    Romano G, Romano M (1979) Sulla soluzione di problemi strutturali in presenza di legami costitutivi unilaterali. Rendiconti Accademia Nazionale dei Lincei 67:104–113MATHGoogle Scholar
  5. 5.
    Baratta A, Toscano R (1982) Stati tensionali in pannelli di materiale non resistente a trazione. In: Proceedings VI AIMETA Congress, GenovaGoogle Scholar
  6. 6.
    Giaquinta M, Giusti E (1985) Researches on the equilibrium of masonry structures. Arch Ration Mech Anal 88:359–392MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Del Piero G (1989) Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials. Meccanica 24:150–162MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Como M (1992) Equilibrium and collapse of masonry bodies. Meccanica 27(3):185–194CrossRefMATHGoogle Scholar
  9. 9.
    Angelillo M (ed) (2014) Mechanics of masonry structures. Series: CISM International Centre for Mechanical Sciences, vol 551. Spriger-verlag, WienGoogle Scholar
  10. 10.
    Milani G (2012) New trends in the numerical analysis of masonry structures. Open Civ Eng J 6:119–120CrossRefGoogle Scholar
  11. 11.
    Tralli A, Alessandri C, Milani G (2014) Computational methods for masonry vaults: a review of recent results. Open Civ Eng J 8:272–287CrossRefGoogle Scholar
  12. 12.
    Gesualdo A, Monaco M (2015) Constitutive behaviour of quasi-brittle materials with anisotropic friction. Lat Am J Solids Struct 12(4):695–710CrossRefGoogle Scholar
  13. 13.
    Addessi D, Sacco E (2016) Enriched plane state formulation for nonlinear homogenization of in-plane masonry wall. Meccanica 51(11):2891–2907MathSciNetCrossRefGoogle Scholar
  14. 14.
    Addessi D, Sacco E (2014) A kinematic enriched plane state formulation for the analysis of masonry panels. Eur J Mech A Solids 44:188–200CrossRefGoogle Scholar
  15. 15.
    Del Piero G (1998) Limit analysis and no-tension materials. Int J Plast 14:259–271CrossRefMATHGoogle Scholar
  16. 16.
    Fraldi M, Nunziante L, Gesualdo A, Guarracino F (2010) On the bounding of multipliers for combined loading. Proc R Soc A Math Phys Eng Sci 466(2114):493–514ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Block P, Ochsendorf J (2007) Thrust network analysis: a new methodology for three-dimensional equilibrium. J Int Assoc Shell Spat Struct 48(3):167–173Google Scholar
  18. 18.
    Fraternali F (2010) A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mech Res Commun 37:198–204CrossRefMATHGoogle Scholar
  19. 19.
    Angelillo M, Fortunato A, Lippiello M, Montanino A (2014) Singular stress fields for masonry walls: Derand was right. Meccanica 49(5):1243–1262MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Fraternali F (2007) Free discontinuity finite element models in two-dimensions for in-plane crack problems. Theor Appl Fract Mech 47:274–282CrossRefGoogle Scholar
  21. 21.
    Hawksbee S, Smith C, Gilbert M (2013) Application of discontinuity layout optimization to three-dimensional plasticity problems. Proc R Soc A Math Phys Eng Sci 469:20130009ADSCrossRefGoogle Scholar
  22. 22.
    Chiozzi A, Milani G, Tralli A (2016) Fast kinematic limit analysis of FRP reinforced masonry vaults through a new genetic algorithm Nurbs-based approach. In: Proceedings of the ECCOMAS congress 2016, Crete, Greece, 05–10 JuneGoogle Scholar
  23. 23.
    Chiozzi A, Milani G, Grillanda N, Tralli A (2016) An adaptive procedure for the limit analysis of FRP reinforced masonry vaults and applications. Am J Eng Appl Sci 9(3):735–745CrossRefGoogle Scholar
  24. 24.
    Ascione L, Feo L, Fraternali F, Fraternali F, Marini A, El Sayed T, Della Cioppa A (2011) On the structural shape optimization via variational methods and evolutionary algorithms. Mech Adv Mater Struct 18:225–243CrossRefGoogle Scholar
  25. 25.
    European Committee for Standardization (2014) Eurocode 8: design of structures for earthquake resistance. Part 1: general rules, seismic actions and rules for buildings. EN 1998-1:2004, Brussels, BelgiumGoogle Scholar
  26. 26.
    Angelillo M (2015) Static analysis of a Guastavino helical stair as a layered masonry shell. Compos Struct 119:298–304CrossRefGoogle Scholar
  27. 27.
    Lang AF, Benzoni G (2014) Modeling of nonlinear behavior of confined masonry using discrete elements. In: NCEE 2014—10th U.S. national conference on earthquake engineering: frontiers of earthquake engineeringGoogle Scholar
  28. 28.
    Carpentieri G, Modano M, Fabbrocino F, Feo L, Fraternali F (2017) On a tensegrity approach to the minimal mass reinforcement of masonry structures with arbitrary shape. Meccanica 52(7):1561–1576MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Civil EngineeringUniversity of SalernoFiscianoItaly
  2. 2.Department of EngineeringPegaso UniversityNaplesItaly

Personalised recommendations