Issue 51

A. Gesualdo et alii, Frattura ed Integrità Strutturale, 51 (2020) 376-385; DOI: 10.3221/IGF-ESIS.51.27 377 such as arches and vaults, columns and plane slabs [2]. Henceforth existing masonry buildings subjected to seismic loads need a correct modeling of these structural elements and their interaction [3]. In particular, especially in the cases in which the out of plane mechanisms can be considered absent [4], the role of the in plane behavior of masonry elements (piers and spandrels) and their interaction with the horizontal structural elements is a key one [5, 6]. Given the importance of the masonry heritage and in view of its rehabilitation, efficient and consistent tools are needed [7], especially constitutive models able to describe the complex patterns of cracks after static and dynamic actions [8, 9, 10]. Several constitutive models have been proposed in the last decades to describe the behaviour of masonry, and among them a preminent role can be assigned to the No-Tension (NT) model. The first approaches to the unilateral model date back to nineteenth century, [11], although in the indeterminate case of the rectangular table with four legs the problem was for the first time posed in an indirect way by Euler at the end of XVIII century [12]. The first rational consideration on the constitutive model were developed in the first decades of XIX century by Signorini [13]. The papers by Heyman [14, 15] considered the low tensile stress of masonry negligible, so that a NT material could be taken into account. He introduced the safe theorem of limit analysis for particular masonry structures, according to which an unreinforced masonry vault will stand if a network of compression forces in the section of the structure and in equilibrium with the applied loads can be found. This solution can be considered according the statements of limit analysis a lower-bound solution [16, 17] and was presented for the first time as an application to the case of “voussoir” (or segmental) arch [18], as the author pointed out. Since that time, and with limited exceptions [19-21] the NT problem has become an almost exclusively Italian question. The first rational assessment was developed since the beginnings of ’80 and mainly thanks to Italian contribute, see for example [22, 23]. The classical Heyman hypotheses of null tensile strength, infinite compressive strength and no-sliding were since then the basis his theory, together with the static theorem of limit analysis, used mainly for the analysis of arches and vaults [24]. The Heyman hypotheses and the limits of their application to masonry structures were successively discussed [25]. This paper presents two NT approaches to model the behaviour of masonry walls subjected to in-plane actions. In both cases a variational strategy is proposed. An extremum problem is in fact solved, in the first case with reference to the complementary energy, in the second one to the potential energy. The first method considers an approximate solution when small strains are involved, with the constitutive hypothesis of unilateral constraints on normal stresses. The solution is a kinematically consistent configuration obtained as a minimum for the complementary energy c E . The numerical problem is solved introducing a curvilinear coordinate system corresponding to the distribution of compression rays [26]. The second approach solves the problem of a 1-D element with variable cross section and symmetric shape defined in the panel domain and corresponding to the compressed area. The solution corresponds to the minimum of total potential energy [27]. The stress maps in case of monotonic increase of shear load is provided [28]. E LASTIC N O -T ENSION MODEL General definitions he mechanics of masonry-like –materials, developed by Giaquinta and Giusti [22] and Del Piero [23]. Fortunato [26] considered the boundary value problem for a masonry-like rectangular panel, traction free on the lateral sides and subjected to zero body forces as well as prescribed rigid body displacements of the top and bottom bases. In particular, the problem is that of a NT body occupying a two dimensional regular region  (Fig. 1), with: Figure 1 : The NT body. T