, Volume 53, Issue 7, pp 1571–1589 | Cite as

Discrete element analysis of stone cantilever stairs

  • Balázs Rigó
  • Katalin Bagi
New Trends in Mechanics of Masonry


Stone cantilever staircases are present in case of both new constructions and reconstructions. The aim of the present paper is to understand the mechanical behaviour of these staircases with the help of discrete element simulations, and to compare the calculated behaviour to the estimations given by the existing manual calculation methods. First a literature review is presented on the statical calculation of cantilevered staircases: manual calculation methods suggested in the 1990s for straight and spiral staircases are introduced, focusing on Heyman’s theory and its improved counterparts. Then the discrete element method is used as a tool to perform virtual experiments, in order to evaluate the mechanical behaviour of the straight and spiral staircases for selfweight, live loads and support movement. The results obtained (internal forces, stresses, deflections) are then compared with the manual calculation results. The most important conclusions are: (1) the term “cantilever stair” is misleading: significant torsion moments occur in the treads, while the bending moments are much smaller than in a free cantilever; (2) the type of the connection between wall and treads (i.e. the end of the tread is simply supported by the wall against translation and torsion, or it is also partly clamped) has a fundamental influence on the internal forces and stress distributions; (3) for simply supported treads the existing manual methods are conservative for straight stairs, but for spiral stairs they dangerously underestimate the torsional moments.


Masonry Structural mechanics 3DEC Spiral and straight stair Helicoidal stair 



This research was inspired by the discussions with Professor Santiago Huerta, Universidad Politecnica de Madrid; his generous help in exploring the literature of the subject is gratefully acknowledged. The authors express their gratitude to Itasca Consulting Group for providing the 3DEC code under the frame of the IEP program. The presented investigations were supported by the Hungarian National Research Fund under Grant No. 100770.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Angelillo M (2015) Static analysis of a Gustavino helical stair as a layered masonry shell. Comput Struct 119:298–304CrossRefGoogle Scholar
  2. 2.
    Angelillo M (2016) The equilibrium of helical stairs made of monolithic steps. Int J Archit Herit 10(6):675–687CrossRefGoogle Scholar
  3. 3.
    Bagi K (2014) When Heyman’s Safe Theorem of rigid block systems fails: non-Heymanian collapse modes of masonry structures. Int J Solids Struct 51:2696–2705CrossRefGoogle Scholar
  4. 4.
    Brik JE (1896) Zur Verwerthung der Ergebnisse aus dem Berichte des Stiegenstufen – Ausschusses. Zeitschrift der oesterr. Ingenieur- und Architekte-Vereines, XLVIII. Jg., Nr. 22. (in German)Google Scholar
  5. 5.
    Brik JE (1898) II. Bericht der Stiegenstufen – Ausschusses. Zeitschrift der oesterr. Ingenieur- und Architekte-Vereines, L. Jg., Nr. 31–32. (in German)Google Scholar
  6. 6.
    Cundall PA (1971) A computer model for simulating progressive large scale movements in blocky rock systems. In: Proceedings of the symposium of the international society of rock mechanics, Vol 1, paper no II-8. Academic Press, Nancy, FranceGoogle Scholar
  7. 7.
    Cundall PA, Hart DH (1992) Numerical modelling of discontinua. J Eng Comput 9:101–113CrossRefGoogle Scholar
  8. 8.
    Godio M, Stefanou I, Sab K (2017) Effects of the dilatancy of joints and of the size of the building blocks on the mechanical behaviour of masonry structures. Meccanica (submitted)Google Scholar
  9. 9.
    Heyman J (1966) The stone skeleton. Int J Solids Struct 2:249–279CrossRefGoogle Scholar
  10. 10.
    Heyman J (1995) The mechanics of masonry stairs. In: Brebbia CA and Leftheris B (eds.): Structural studies of historical buildings. vol 2. Dynamics, repairs and restoration, southampton: computational mechanics Publ., pp. 259–265Google Scholar
  11. 11.
    Kooharian A (1952) Limit Analysis of Voussoir (Segmental) and Concrete Arches. J Am Concr Inst 24(4):317–328Google Scholar
  12. 12.
    Little P, Hough M, Mullarkey E (2009) Stone cantilever stairs—inspection and analysis of cantilever stairs. Struct Eng 87(8):26–33Google Scholar
  13. 13.
    Huerta S (2008) The analysis of masonry architecture: a historical approach. Archit Sci Rev 51(4):297–328CrossRefGoogle Scholar
  14. 14.
    Lemos JV (2007) Discrete element modelling of masonry structures. Int J Archit Herit 1:190–213CrossRefGoogle Scholar
  15. 15.
    Maunder EAW (2005) Staircases as cantilevers or arches?—a question for limit analysis. In: Modena C, Lourenco PB, Roca C (eds) Structural analysis of historical constructions, 1st edn. Taylor and Francis, London, pp 569–576Google Scholar
  16. 16.
    O’Sullivan C (2011) Particulate discrete element modelling: a geomechanics perspective. Spon Press, Taylor and Francis, LondonGoogle Scholar
  17. 17.
    Price S, Rogers H (2005) Stone cantilevered staircases. Struct Eng 83:29–36Google Scholar
  18. 18.
    Sarhosis V, Bagi K, Lemos JV, Milani G (2016) Computational modeling of masonry structures using the discrete element method. IGI Global, HersheyCrossRefGoogle Scholar
  19. 19.
    Simon J, Bagi K (2016) Discrete element analysis of the minimum thickness of oval masonry domes. Int J Archit Herit 10(4):457–475CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Budapest University of Technology and EconomicsBudapestHungary
  2. 2.Department of Structural MechanicsBudapest University of Technology and EconomicsBudapestHungary

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