A numerical method for solving BVP of masonry-like solids

  • Massimiliano Lucchesi
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 551)


In this chapter, we shall first explicitly determine the solution to the constitutive equation of isotropic masonry-like materials (i.e., find the stress tensor T and the fracture strain E f corresponding to a prescribed strain tensor E, so as to satisfy relations (2.4)). The problem can be solved by observing that the stress in the isotropic case is coaxial not only with the fracture strain, but with the elastic strain as well. This enables representing all these tensors with respect to the same principal system and then expressing the constitutive equation as a function of their eigenvalues. In this way, (2.4) can be transformed into a linear complementarity problem (7), (27), (28) whose solution is unique because the tensor of the elastic constants has been assumed to be positive definite. As the solution to the constitutive equation depends on the number of principal stresses that vanish, in order to construct the stress function we need to consider a partition of the strain space into four different regions, each of which corresponds to different material behavior. We then calculate the derivative of the stress function with respect to the strain, as this will be used to construct the tangent stiffness matrix when dealing with numerical solution of the equilibrium problem. The derivative of the stress function turns out to be smooth in each region and its jump has no tangential component at the interfaces between the different regions.


Compressive Strength Constitutive Equation Equilibrium Problem Stress Function Fracture Strain 
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  1. [1]
    Alfano G., Rosati L., Valoroso N., A numerical strategy for finite element analysis of no-tension materials. Int. J. Numer. Meth. Engng, vol. 48, pp. 317-350, 2000.Google Scholar
  2. [2]
    Angelillo M., Cardamone L., Fortunato A., A new numerical model for masonry structures. J Mech. Mater. Stuct. 5, 583-615 (2010).Google Scholar
  3. [3]
    Bathe K. J., Wilson E. L., Numerical methods in finite element analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.Google Scholar
  4. [4]
    Bennati S., Barsotti R.: Optimum radii of circular masonry arches. Proc. Third Int. Arch Bridges Confer., Paris (2001).Google Scholar
  5. [5]
    Brenner S. C., Scott L. R. The mathematical theory of finite element method, Second edition, Springer 2002.Google Scholar
  6. [6]
    Ciarlet P. G., The finite element method for elliptic problems. North-Holland Publishing Company, 1978.Google Scholar
  7. [7]
    Cottle R. W., Pang J. S., Stone R. E., The linear complementarity problem. Academic Press Inc. 1992.Google Scholar
  8. [8]
    Cuomo M., Ventura G., A complementary energy formulation of no tension masonry-like solids. Computer Methods in Applied Mechanics and Engineering 189, pp. 313-339, 2000.Google Scholar
  9. [9]
    Dacorogna B.: Introduction to the Calculus of Variations. Imperial College Press, (2004).Google Scholar
  10. [10]
    Degl’Innocenti S., Lucchesi M., Padovani C., Pagni A., Pasquinelli G., Zani N.: Dynamical analysis of masonry pillars. Proc. Third Int. Cong. on Science and Technology for the Safeguard of Cultural Heritage in the Mediterranean Basin, Alcalá de Henares, (2001).Google Scholar
  11. [11]
    Degl’Innocenti, S., Padovani, C., Pagni, A., Pasquinelli, G.: Dynamic analysis of age-old masonry constructions. Proc. Eighth Inter. Conf. on Computational Structures Technology, B.H.V. Topping, G. Montero and R. Montenegro(Editors), Civil-Comp Press, Stirlingshire, Scotland 2006. Las Palmas de Gran Canaria 12-15 Sept., (2006).Google Scholar
  12. [12]
    Guidotti P., Lucchesi M., Pagni A., Pasquinelli G., Application of shell theory to structural problem using the finite element method. CNR-Quaderni de ”La Ricerca Scientifica”, 115, 1986.Google Scholar
  13. [13]
    Gurtin M. E., An introduction to continuum mechanics. Academic Press, Boston, 1981.Google Scholar
  14. [14]
    Lucchesi M., Padovani C., Pagni A., A numerical method for solving equilibrium problems of masonry-like solids. Meccanica, vol. 29, pp. 175-193, 1994.Google Scholar
  15. [15]
    Lucchesi M., Padovani C., Pasquinelli G., Zani N., The maximum modulus eccentricity surface for masonry vaults and limit analysis. Mathematics and Mechanics of Solids, 4, pp. 71-87, 1999.Google Scholar
  16. [16]
    Lucchesi M., Padovani C., Pagni A., Pasquinelli G., Zani N: COMES-NOSA A finite element code for non-linear structural analysis. Report CNUCE-B4-2000-003 (2000).Google Scholar
  17. [17]
    Lucchesi M., Zani N., Some explicit solution to plane equilibrium problem for no-tension bodies. Structural Engineering and Mechanics, vol. 16, pp. 295-316, 2003.Google Scholar
  18. [18]
    Lucchesi M., Padovani C., Pasquinelli G., Zani N., Static analysis of masonry vaults, constitutive model and numerical analysis. Journal of Mechanics of Materials and Structures, 2(2), pp. 211-244, 2007.Google Scholar
  19. [19]
    Lucchesi M., Pintucchi B.: A numerical model for non-linear dynamic analysis of slender masonry structures. Eur. J. Mech. A. Solids, 26, 88-105, (2007).Google Scholar
  20. [20]
    Lucchesi M., Padovani C., Pasquinelli G., Zani N.: Masonry Constructions: Mechanical Models and Numerical Applications. Lectures notes in applied and computational mechanics, Vol. 39, Springer Verlag (2008).Google Scholar
  21. [21]
    Lucchesi M., Silhavy M., Zani N.: Equilibrium problems and limit analysis of masonry beams. J. Elasticity, 106, 165-188, (2012).Google Scholar
  22. [22]
    Lucchesi M., Pintucchi B. Zani N.: MADY, a finite element code for dynamic analysis of slender masonry structures: Mathematical specifications and user manual (In preparation) (2013).Google Scholar
  23. [23] Scholar
  24. [24]
    Oden J. T., Carey G. F., Finite elements-Special problems in solid mechanics, Vol. 5. Prenctice-Hall, Inc., Englewood Cliffs, New Jersey, 1984.Google Scholar
  25. [25]
    Padovani C., Pagni A., Pasquinelli G.: Un codice di calcolo per l’analisi statica e il consolidamento di volte in muratura. Proc. WONDERmasonry - Workshop on design for rehabilitation of masonry structures, 145 - 153, P. Spinelli (ed.), (2006).Google Scholar
  26. [26]
    Pintucchi B, Zani N.: Effects of material and geometric non-linearities on the collapse load of masonry arches. Eur. J. Mech. A. Solids, 28, 45-61, (2009).Google Scholar
  27. [27]
    Romano G., Sacco E., Sulla proprietà di coassialità del tensore di fessurazione. Atti Ist. Scienza delle Costruzioni, Facoltà di Ingegneria, Napoli, n. 351, 1984.Google Scholar
  28. [28]
    Sacco E., Modellazione e calcolo di strutture in materiale non resistente a trazione. Rend. Mat. Acc. Lincei, s. 9, vol. 1, pp. 235-258, 1990.Google Scholar
  29. [29]
    http://www.salome-platform.orgGoogle Scholar
  30. [30]
    Zani N.: A constitutive equation and a closed form solution for notension beams with limited compressive strength. Eur. J. Mech. A. Solids, 23, 467-484, (2004).Google Scholar

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© CISM, Udine 2014

Authors and Affiliations

  • Massimiliano Lucchesi
    • 1
  1. 1.Dipartimento di Ingegneria Civile e AmbientaleUniversity of FlorenceFlorenceItaly

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