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PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping and C.A. Mota Soares
A Numerical Method for the Stability Problem of Masonry Elements
B. Pintucchi and N. Zani
Department of Construction, University of Firenze, Italy
B. Pintucchi, N. Zani, "A Numerical Method for the Stability Problem of Masonry Elements", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 192, 2004. doi:10.4203/ccp.79.192
Keywords: buckling, no-tension material, masonry columns, masonry arches.
The importance of stability problems for masonry columns loaded with an eccentric axial load at the tip has been clarified since the earliest studies, dating back to the 1940s . When load eccentricity induces lateral deflection with a consequent increase in bending moment, due to the non-linear behaviour of the material, which is characterized by a low tensile strength, cracking of sections may occur; such a mechanism induces an unfavorable reduction of structure bending stiffness, implying an amplification of second-order effects. Moreover, the problem of buckling has been recognised as very important for simple masonry columns but, more important, it often turn out very significant also with regard to more complex structures. Namely, walls of masonry buildings often subjected to eccentric vertical loads due to the bending moment induced by the supported deck or by the horizontal loads occurring during earthquake and wind excitations. As widely assumed, bearing walls can be studied by means of a column model. Nevertheless, in such cases structures may become more complex in geometry, being variable in thickness along the height, as well as in the constraint conditions offered by the floor slab. Analogously, for masonry arches analytical solutions of the stability problems are not available, although these structures are often characterized by low thickness and high values of the axial loads, that may induce buckling phenomena.
However, analysis of a large part of applications of real interest need generally to resort to numerical models, because of the difficulties to obtain analytical solutions for general load conditions and refined masonry mechanical properties assumptions [2,3]. For these reasons, a numerical procedure has been developed to solve the transversal equilibrium equation for beams with reference to the current configuration and taking into account effects of axial force.
In this paper, masonry is assumed to be a no-tension material with limited compressive strength, an hypothesis embedded into the constitutive equation used. It is a constitutive equation that expresses generalized stress characteristics (normal force and bending moment) as a function of the strain ones (the extensional strain and curvature of the beam's longitudinal axis) for rectangular cross-section beams under the Euler-Bernoulli hypothesis .
The numerical procedure provides for the structure's discretization with finite conforming elements. Moreover, from the explicit evaluation of the derivatives of generalized stresses with respect to the generalized strains, the tangent stiffness matrix has been evaluated and used in applying with the Newton-Raphson method for the solution of the non-linear algebraic system obtained from the discretization.
The problem is a particularly complex one because of the interaction between the geometric and constitutive non-linearities, as reveals the asymmetry of the so-called geometric stiffness matrix.
The FEM method proposed allows us to perform analysis of masonry structures loaded and constrained in any way, such as columns, arches etc. The reliability of the numerical methods is evaluated by a comparison of results with some available explicit solutions, presented in previous works  where the same constitutive relation has been used to determine the critical load of a column subjected to an eccentric axial load.
Therefore, the method is applied to the study of masonry arches made of no-tension material with limited compressive strength. We considered a series of circular arches, with different values of the springing angle, subjected to different load conditions and for different values of the material compressive resistance. In order to evidence the influence of buckling phenomena from a qualitative point of view, the collapse load obtained by taking into account the second-order geometrical effects have been compared with that obtained negleting the axial load effects. Results show that the buckling effects can be very relevant in the case of very rib arches, when the slenderness can be very small, in the case of particular load condition (uniform per unit span), and for large value of material's compressive strength.
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