Keywords: masonry constructions, non-linear elasticity, dynamic analysis, finite element method.

In order to perform dynamic analyses of real masonry structures many researchers model the masonry as a linear elastic material and apply the mode superposition method to integrate the equations of motion [1,2,3]. Alternatively, the dynamic behaviour of very simple structures has been investigated using an elastic-plastic constitutive model imbedded into a direct integration method [4,5,6].

In this paper masonry is modelled as a non-linear elastic material with zero tensile strength and infinite compressive strength [7,8,9]. The resulting constitutive equation, known as the equation of masonry-like or no-tension material, is able to account for some of masonry's peculiarities, in particular, its inability to withstand large tensile stresses. Assumptions underlying the model are that the infinitesimal strain is the sum of an elastic part and a fracture part, and that the stress, negative semi-definite, depends linearly and isotropically on the former and is orthogonal to the latter, which is positive semi-definite. Thus, the stress is a non-linear function of the infinitesimal strain. In order to study the structural behaviour of existing masonry structures, both the static and dynamic problem can be solved via the finite-element method [9,10]. The equation of masonry-like materials has been implemented into the finite-element code NOSA, developed entirely at ISTI-CNR with the purpose of studying the behaviour of masonry solids and modelling restoration and reinforcement operations on constructions of particular architectural importance. The code has been successfully applied to the analysis of some buildings of historical interest: the S. Nicolò Motherhouse in Noto, the Medici Arsenal in Pisa, the Baptistery of the Volterra Cathedral, the Church of S. Pietro in Vinculis in Pisa, the Buti bell tower and the dome of the Church of S. Maria Maddalena in Morano Calabro.

As far as numerical solution of the dynamic problem is concerned, the equations of motion must be integrated directly. In fact, due to the non-linearity of the adopted constitutive equation, the mode-superposition method is meaningless. We instead apply the Newmark method implemented in NOSA [10] to perform the time integration of the system of ordinary differential equations obtained by discretising the structure into finite elements.

In this paper, the numerical method presented in [10] is applied to age-old masonry constructions, in particular to the dome of a church located in the Calabria Region, Italy. We present the numerical method implemented in the NOSA code for the dynamic analysis of masonry structures, with particular focus on vaults and domes. Subsequently, we study the damaged dome and tambour of the Church of Santa Maria Maddalena in Morano Calabro. The structure has been discretised with shell elements and analyses performed using the NOSA code. Initially, the structure is at rest, in static equilibrium under the action of its own weight. Subsequently, the base of the tambour, which is clamped, is subjected to a cyclic horizontal acceleration acting over a given time interval, and then the construction is left to oscillate freely. The results obtained considering the masonry to be a non-linear elastic material are compared with the results of an analogous dynamic analysis carried out on the same structure, while however assuming linear elastic material behaviour. This comparison has highlighted the profound differences between the results obtained using the two assumptions.

1

E. Juhásová, M. Hurák, Z. Zembaty, "Assessment of seismic resistance of masonry structures including boundary conditions", Soil Dynamics and Earthquake Engineering 22, 1193-1197, 2002. doi:10.1016/S0267-7261(02)00147-1

2

E. Mele, A. De Luca, A. Giordano, "Modelling and analysis of a basilica under earthquake loading", Journal of Cultural Heritage, 4, 355-367, 2003. doi:10.1016/j.culher.2003.03.002

3

S. Ivorra, F.J. Pallarés, "Dynamic investigation on a masonry bell tower", Engineering Structures 28, 660-667, 2006. doi:10.1016/j.engstruct.2005.09.019

4

R. Cerioni, R. Brighenti, G. Donida, "Use of incompatible displacement modes in a finite element model to analyze the dynamic behaviour of reinforced masonry panels", Computers & Structures, 57, 47-57, 1995. doi:10.1016/0045-7949(94)00590-Y

5

E. Papa, A. Nappi, "Numerical modelling of masonry: A material model accounting for damage effects and plastic strains", Appl. Math. Modelling, 21, 319-335, 1997. doi:10.1016/S0307-904X(97)00011-5

6

C. Wu, H. Hao, Y. Lu, "Dynamic response and damage analysis of masonry structures and masonry infilled RC frames to blast ground motion", Engineering Structures, 27, 323-333, 2005. doi:10.1016/j.engstruct.2004.10.004

7

J. Heyman, The masonry Arch. J. Wiley & Sons, 1982.

8

G. Del Piero, "Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials", Meccanica, 24, 150-162, 1989. doi:10.1007/BF01559418

9

M. Lucchesi, C. Padovani, A. Pagni, "A numerical method for solving equilibrium problems of masonry-like solids", Meccanica, 29, 175-193, 1994. doi:10.1007/BF01007500

10

S. Degl'Innocenti, C. Padovani, G. Pasquinelli, "Numerical methods for the dynamic analysis of masonry structures", Structural Engineering and Mechanics, 22(1), 107-130, 2006.

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