Skip to main content

Singular stress fields for masonry-like vaults


In the present paper, we apply the theorems of limit analysis to vaults modeled as masonry-like materials, that is, unilateral continuous bodies. On allowing for singular stresses, we consider statically admissible stress field concentrated on surfaces lying inside the masonry. Such structures are unilateral membranes, whose geometry is described a la Monge, and the equilibrium of them, under vertical loads, is formulated in the Pucher form. The problem is reduced to a singular partial differential equation of the second order where the shape f and the stress function F appear symmetrically. The unilateral restrictions require that the membrane surface lies in between the extrados and intrados surfaces of the vault and that the stress function be concave. Such a constraint is, in general, not satisfied on a given shape for given loads: in such a case, the shape has to be modified to fit the constraint. In a sense, the unilateral assumption renders the membrane an underdetermined structure that must adapt its shape in order to satisfy the unilateral restrictions. A number of simple examples are presented to illustrate how the method works.

This is a preview of subscription content, access via your institution.


  • Huerta, S.: Mechanics of masonry vaults: The equilibrium approach. In: Lorenco, Roca (eds.), Historical Constructions, pp. 47–70 (2001)

  • Heyman J.: The stone skeleton. Int. J. Solids Struct. 2, 249 (1966)

    Article  Google Scholar 

  • Di Pasquale, S.: Statica dei solidi murari, Atti Dipartimento Costruzioni, Università di Firenze (1984)

  • Del Piero G.: Constitutive equations for linear elastic masonry-like materials and compatibility of the external loads. Meccanica 24, 150–162 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  • Angelillo, M.: Constitutive relations for no-tension materials. Meccanica 28(2), (1993)

  • Angelillo M., Rosso F.: On statically admissible stress fields for a plane masonry-like structure. Quarterly of Applied Mathematics 53(4), 731–751 (1995)

    MathSciNet  MATH  Google Scholar 

  • Angelillo M., Cardamone L., Fortunato A.: A numerical model for masonry-like structures. JoMMS 5(4), 583–615 (2010)

    Article  Google Scholar 

  • Del Piero G.: Limit analysis and no-tension materials. Int. J. Plast. 14(1–3), 259–271 (1998)

    MATH  Article  Google Scholar 

  • Moseley H.: On a new principle in statics, called the principle of least pressure. Philos. Mag. 3, 285–288 (1833)

    Google Scholar 

  • Wolfe W.S.: Graphical Analysis: An Handbook on Graphic Statics. McGraw-Hill, New York (1921)

    Google Scholar 

  • Bloch P., Ochsendorf J.: Thrust network analysis: A new methodology for three-dimensional equilibrium. J. Int. Assoc. Shell and Spatial Structures 48(3), 167–173 (2007)

    Google Scholar 

  • Fraternali F., Angelillo M., Fortunato A.: A lumped stress method for plane elastic problems and the discrete-continuum approximation. Int. Jou. Solids Struct. 39, 6211–6240 (2002)

    MATH  Article  Google Scholar 

  • Angelillo, M., Fortunato, A.: Equilibrium of masonry vaults, Novel approaches in Civil Engineering. In: Maceri, Frémond (eds.) Lecture notes in Applied and Computational Mechanics, 16, pp. 105–109. Springer, Berlin (2002)

  • Baratta A., Corbi I., Corbi O.: Arches, walls and vaults. In: Fodde, E. (ed.) Structural Analysis of Historic Constructions: Preserving, CRC Press, Boca Raton (2008)

    Google Scholar 

  • Baratta A., Corbi O.: On the equilibrium and admissibility coupling in NT vaults of general shape. Int. J. Solids Struct. 47(17), 2276–2284 (2010)

    MATH  Article  Google Scholar 

  • Lucchesi M., Silhavy M., Zani N.: Integration of measures and admissible stress fields for masonry bodies. J. Mech. Mater. Stuct. 3, 675–696 (2008)

    Article  Google Scholar 

  • Babilio, E., Fortunato, A., Angelillo, M.: Singular stress fields and the equilibrium of masonry walls and arches. In: Proceedings IX AIMETA Congress, Bologna (2011)

  • Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, Oxford (2000)

    MATH  Google Scholar 

  • Témam, R.: Problémes mathématiques in plasticité, Gauthier Villars (1983)

  • Pucher, A.: Uber der spannungzustand in gekrummten flachen, Beton u Eisen (1939). (2002)

  • Babilio, E., Fortunato, A., Lippiello, M.: A stress approach to the equilibrium of masonry domes: a case study. In: De Luca, A., Spinelli, P. (eds.) Proceedings of the workshop “WonderMasonry 2007”: Edifici in muratura: Progetto e Riabilitazione. Polistampa (2008)

  • Como, M.: Statica delle Costruzioni Storiche in muratura, Aracne (2010)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Maurizio Angelillo.

Additional information

Communicated by Francesco dell'Isola and Samuel Forest.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Angelillo, M., Babilio, E. & Fortunato, A. Singular stress fields for masonry-like vaults. Continuum Mech. Thermodyn. 25, 423–441 (2013).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Masonry equilibrium
  • Vaults
  • Unilateral materials