Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Bibliography
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.
Angelillo M., Cardamone L., Fortunato A., A new numerical model for masonry structures. J Mech. Mater. Stuct. 5, 583-615 (2010).
Bathe K. J., Wilson E. L., Numerical methods in finite element analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.
Bennati S., Barsotti R.: Optimum radii of circular masonry arches. Proc. Third Int. Arch Bridges Confer., Paris (2001).
Brenner S. C., Scott L. R. The mathematical theory of finite element method, Second edition, Springer 2002.
Ciarlet P. G., The finite element method for elliptic problems. North-Holland Publishing Company, 1978.
Cottle R. W., Pang J. S., Stone R. E., The linear complementarity problem. Academic Press Inc. 1992.
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.
Dacorogna B.: Introduction to the Calculus of Variations. Imperial College Press, (2004).
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).
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).
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.
Gurtin M. E., An introduction to continuum mechanics. Academic Press, Boston, 1981.
Lucchesi M., Padovani C., Pagni A., A numerical method for solving equilibrium problems of masonry-like solids. Meccanica, vol. 29, pp. 175-193, 1994.
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.
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).
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.
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.
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).
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).
Lucchesi M., Silhavy M., Zani N.: Equilibrium problems and limit analysis of masonry beams. J. Elasticity, 106, 165-188, (2012).
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).
http://www.nosaitaca.it/en/
Oden J. T., Carey G. F., Finite elements-Special problems in solid mechanics, Vol. 5. Prenctice-Hall, Inc., Englewood Cliffs, New Jersey, 1984.
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).
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).
Romano G., Sacco E., Sulla proprietà di coassialità del tensore di fessurazione. Atti Ist. Scienza delle Costruzioni, Facoltà di Ingegneria, Napoli, n. 351, 1984.
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.
http://www.salome-platform.org
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).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 CISM, Udine
About this chapter
Cite this chapter
Lucchesi, M. (2014). A numerical method for solving BVP of masonry-like solids. In: Angelillo, M. (eds) Mechanics of Masonry Structures. CISM International Centre for Mechanical Sciences, vol 551. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1774-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-7091-1774-3_3
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-1773-6
Online ISBN: 978-3-7091-1774-3
eBook Packages: EngineeringEngineering (R0)