Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 337 In the literature, 2D thrust-line, membrane, and 3D thrust-network [4, 5], along with convex and concave contact models [3, 6, 7], have been developed within the limit analysis framework to find the ultimate load factor (or the minimum thickness) for the feasible models of masonry structures. Referring to basic geometries, 2D thrust-line and membrane were applied to find the limit equilibrium of semi-circular arches [8-16], hemispherical domes [17-20] and vaults of given geometries including skew arches, and pavilion, cross and groin vaults [21-26]. Finding the ultimate load factor, together with the stress state and failure mechanisms for models with basic geometries, is a classical problem also solved through other structural analysis methods such as Finite Element (FE) analysis, including detailed and simplified finite element micro models [27-29], and Discrete Element (DE) analysis [30-32]. On the other hand, further approaches have been proposed to simplify the static and dynamic analysis of masonry structures assumed as assemblages of rigid blocks, such as equivalent macro-elements constituted by a set of trusses to describe masonry vaults [33] and one-sided motion of rocking rigid blocks [34, 35]. However, limit analysis approach is less computationally expensive still highly accurate, compared to other computational methods. According to the limit analysis approach applied to assemblages of rigid blocks, the stability of a block masonry structure is obtained when the internal forces distributed at block interfaces satisfy two physical constraints: the forces normal to the interfaces should be in compression and the forces tangential to the interfaces must be less than the interface sliding resistance, including associative [3, 6] or non-associative [36, 37] isotropic frictional resistance. D’Ayala and Casapulla [1] developed a limit analysis procedure to find the minimum thickness required for a hemispherical masonry dome composed of rigid blocks with finite isotropic friction, to withstand a weight-like load distribution. That work first proved that due to the symmetry of geometry and loading, this configuration belongs to a special class of non-associative friction problems for which unique solutions within standard limit analysis can be found. This solution is the optimized axisymmetric membrane surface that everywhere lies within the thickness of the dome and satisfies equilibrium and frictional constraints. The resistant surface is characterised by meridional and hoop stresses, and it was observed that its profile might not coincide with the mid hemispherical surface for small frictional resistances. This means that the curvature of its generating meridian is not known a–priori and is generally not constant. So, in the cited paper, a meridional thrust-line per unit length of parallels (circle of latitude) was constructed by connecting a set of control points on the blocks, assumed as the optimization variables. The objective function of the optimization was to determine the coordinates of these control points for the unique membrane surface placed in the thinnest structure, satisfying the equilibrium condition and sliding constraints. To develop the sliding constraint, the stress state at the points where interfaces intersect the mentioned thrust-line was calculated. In the first part of this paper, the equilibrium formulations and the sliding constraints presented in the mentioned work [1] are to some extent revisited, since in that work the sliding constraint in the parallel direction was not very accurate. Furthermore, the revised method can present the stress states at block interfaces through the construction of the membrane or discrete network of forces for the lunes of the dome. The second part of this paper addresses the extension of limit analysis to interlocking interfaces with non-isotropic sliding resistance. In the literature, the sliding behaviour of interlocking blocks with different geometries has been taken into the consideration very recently. For example, the in-plane and out-of-plane capacity of masonry walls with blocks having corrugated interfaces have lately been studied through experimental and numerical investigations [38-40]. Particularly interesting is the numerical simulation of brick infill walls with locks along vertical interfaces under out-of-plane loading, developed by means of an innovative discrete macro-modelling strategy [41]. Dyskin et al. [42] and Estrin et al. [43] carried out an experimental test to evaluate the out-of-plane behaviour of osteomorphic blocks, while the out-of-plane capacity of a wall with blocks having cross shaped locks, keeping the blocks together, was demonstrated in [44] through experimental investigations. Furthermore, different structural behaviours of three types of joinery connections with different geometric properties were addressed in [45], experimentally and numerically. Herein, the convex contact model proposed in [2] for 2D interlocking interfaces is adopted and extended to perform the orthotropic sliding behaviour of 3D corrugated hoop interfaces with locks having rectangular cross section (Fig. 1). This behaviour is governed by the Coulomb’s friction law in one direction and by the shear resistance of the locks in the orthogonal direction. A heuristic method using this contact model is then introduced to find the sliding resistance of such interlocking blocks. The geometry of their interfaces is idealized and a relation between the geometric parameters of the locks, i.e. g x b x h (width x length x height), including their orientation and number and the sliding resistance of the interfaces, is defined (Fig. 1). In the following sections, the revisited limit analysis and optimization of the hemispherical dome proposed in the cited work [1] is first presented. The results obtained by this revisited procedure are compared with the previous ones and the extension of this analysis to hemispherical domes composed of interlocking blocks is discussed, in terms of a construction