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Keywords:Masonry Church of the Nativity Narthex FE model Non-linear dynamic analysis Vault a b s t r a c tThe Church of the Nativity in Bethlehem has a narthex in the front that is as long as the façade of the Church and about six meters wide. Currently, the narthex is covered by five cross vaults, three of which in a dangerous state of decay, and it is internally divided by three walls perpendicular to the façade, which appears to be strongly rotated outwards with a maximum horizontal top displacement of about 40 cm. Inside the central cross vault, the narthex has been heavily damaged and propped since the thirties of the last century. Numerous attempts have been made over the time to identify the causes of such damage. Some archival researches, in-situ inspections of the subsoil and detailed laser scanner surveys, which were carried out during the recent restoration works in the Church and in the narthex, allowed for gaining a deeper insight into the construction features of the cross vaults and for putting forward some hypotheses about the possible causes of damage. This paper provides a scientific validation of these hypotheses by means of finite element numerical simulations, which try to reproduce the seismic response of the Church and the deformation process of a three-dimensional simplified model of the narthex from an assumed initial configuration up to an ultimate state of damage, comparable with the current one. Such models are discretized by means of tetrahedron elements obeying a damage plasticity law that exhibits a softening behavior in both tension and compression. The numerical simulations carried out provide some results that fit reasonably with the actual deformed configuration of the narthex and can be considered as a useful tool for further insights.

The paper concisely reviews the available numerical methods to analyze masonry vaults up to collapse, putting in evidence pros and limitations of each approach. To be reliable, any procedure adopted should take into account the distinctive aspects of masonry mechanical behavior, namely the scarce (or zero) tensile strength, the good compressive resistance and the observed formation of failure mechanisms constituted by rigid macro-blocks in mutual roto-translation. Classic no-tension material models disregard the little but not null tensile strength and make the hypothesis of (1) infinitely elastic behavior in compression and (2) isotropy, giving thus the possibility to deal with either semi-analytical approaches (especially for arches) or robust numerical procedures. More advanced but rather complex models are nowadays able to deal also with anisotropy induced by texture, little tensile strength and softening in tension, as well as finite strength in compression. Traditionally, limit analysis has proven to be the most effective for a fast and reliable evaluation of the load carrying capacity of vaulted masonry structures. Classic lower and upper bound theorems recall respectively the concepts of equilibrium and formation of failure mechanisms with rigid elements. The so-called Thrust Network Method moves its steps from lower bound theorem, whereas FE limit analysis approaches with rigid elements take inspiration from the upper bound point of view. An alternative to limit analysis is represented by traditional FEM combined with either elasto-plastic or damaging models with softening, usually adapted to masonry from other materials, which are capable of providing a large set of output numerical information but that still remain very demanding

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In this paper the numerical solution of the elastic frictionless contact problem is obtained by means of boundary discretization techniques. Variational formulations in terms of boundary tractions are given in presence of both bilateral and unilateral constraints. The discretization of the boundary functional is examined from the point of view of the theory of approximation and it is proved that the coerciveness (but not the symmetry) of the continuum problem is preserved when standard B.E.Ms are employed. As a consequence, the contact problem can be cast as a L.C.P. having, as coefficient matrix, a generally non symmetric P matrix. A simple, but meaningful example is discussed in some detail