Issue 51

M. Pepe et alii, Frattura ed Integrità Strutturale, 51 (2020) 504-516; DOI: 10.3221/IGF-ESIS.51.38 507 where Eqn. (1.1) 1 represents the kinematic compatibility for the whole system of interfaces and blocks in which B is the compatibility matrix as in [42], but also in nonlinear static analysis by [43], Eqn. (1.1) 2 defines the equilibrium for the whole structure, Eqn. (1.1) 3 describes the generalized yield domain of the system where N is the block-diagonal gradient matrix, Eqn. (1.1) 4 represents the flow rule which express the vector ε as a linear combination with non-negative coefficients λ , called inelastic multiplier and M is the block-diagonal matrix of the modes of failures. Eqn. (1.1) 5 is the complementarity condition which defines the plastic behavior of contact surface. Moreover, the collapse mechanism must be characterized by a non-negative work of the live loads, defined by Eqn. (1.1) 6 . Within the framework of the holonomic perfect plasticity, the same relations govern the problem of a non-standard rigid-plastic discrete materials. Resorting this formal analogy, the collapse load for a masonry structure, under the hypothesis of proportional load with the factor 0   , can be determined. After some algebra the authors obtained the following non-linear and non-convex programming problem (NLNCP)                   C 1 2 T T T 0 1 0 L 2 1 2 T T 0 1 L T T T 0 L 2 1 2 min subjected to 0 0 1 0 0                          AM M λ A N f f N AN σ λ A M f λ f f N AN σ (1.2) with the unknowns  , 2 σ , λ and the bounds 0  λ , where 2 σ are the undetermined unknown of the system which represent the statically undetermined term of the generalised contact stress [30] In order to deal with the NLNCP, authors [31] developed a specific code, called ALMA ( A nalisi L imite M urature A ttritive), which used a two-step procedure to solve the problem: in the first step a linear programming problem (LP), obtained by replacing friction with dilatancy, is solved; in the second step, the NLNCP solution is approached using, as initial guess for the unknowns of the problem, the solution of the first step. However, the problem of Limit Analysis of structures with frictional interfaces (non-standard LA) is numerically very difficult to be solved. The solution could not exist and when it exists it could be a local minimum instead of the global one [44]. On the other hand, due to the presence of non-associative flow rules, the Drucker stability postulate no longer holds and the solution in terms of contact actions and collapse load factor loses its uniqueness. Moreover, bi-dimensional or three-dimensional real structures, characterized by many degrees of freedom, increase the computational complexity of the problem. From Limit Analysis theory it is well known that if normality rule holds, i.e. the vector of inelastic strain results normal to the yield surface, the static and kinematic theorems of limit analysis could be formulated in a linear programming context, resulting in two dual problems, which lead to a unique solution. In particular, following the approach of [31], results of this work refer to a linear programming optimization problem related to the kinematic approach of Limit Analysis, which provides the collapse multiplier and the corresponding mechanism. Friction is considered in term of dilatancy. The kinematic problem is defined as (1.3) with the bounds on the unknowns 0  λ . To overcome some computational limits of the original code ALMA, mainly related to the number of blocks and interfaces involved into analysis, a new version of the code, ALMA 2.0, was implemented using MATLAB  for linear optimization and a Python  interface for pre- and post-processing operations. F EM / DEM A NALYSIS he limit analysis model is compared with the models adopted in [40,41], here regarded as a benchmark. In the referred works, two micro-mechanical models have been proposed for the in-plane failure analysis of masonry walls: a discrete element method (DEM) and a combined finite/discrete element method (FEM/DEM). Both T         0 1 0 1 2 0 1 min 1 0 T T C T T L subjected to        λ A N f AN N λ 0 λ A N f