الگوریتم تکراری موثر برای تحلیل محدود ساختارهای خمشی خرپایی Effective iterative algorithm for the Limit Analysis of truss-frame structures by a kinematic approach

1. Introduction Limit Analysis (LA) constitutes by today a well-established and consolidated discipline, for evaluating consistent bounds on the collapse (limit) loads acting on engineering structures characterized by a mechanical behavior that may be idealized as perfectly plastic, subjected to increasing live loads. LA may be considered as a milestone in the more recent history of structural mechanics and provides a rather powerful tool for structural design and assessment purposes in a wide variety of engineering situations. It has acquired its rational formulation thanks to the contribution by Drucker et al. [1], who formulated and demonstrated the fundamental theorems of LA for a continuum. More recent consolidated contributions, such as those by Massonet and Save [2], Kaliszky [3], Lubliner [4] and Jirasek and Bazant [5], have further made the theory and methods of LA rather fundamental in the applications of mechanics of solids and structures, becoming by now classical references on the topic (see also contributions in Spiliopoulos and Weichert [6] and Foreword by G. Maier in it). LA is a methodology characterized by intrinsic peculiarities that may be regarded as bringing considerable advantages in the engineering practice. These include the fact that it determines safety coefficients with respect to the ultimate limit state of a structure and the kinematics of a collapse mechanism, providing essential and very expressive results to structural designers. For this reason, it appears as particularly suitable for comparative evaluations among different design options, in order to easily guide engineers toward selecting the most efficient and feasible structural choices. LA provides theorems for the determination of the ultimate load of a perfectly plastic solid or a structure. The lower bound theorem, or equivalently the static (‘‘safe”) principle, states that if, for a given set of loads, an equilibrium stress field (satisfying also the stress boundary conditions) can be found, which nowhere violates the yield condition, the body will not collapse. The upper bound theorem, or equivalently the kinematic principle, states that, for a given collapse mechanism whose kinematics is compatible, the ratio between the internal and external rates of dissipation will be higher than or equal to that actually found at collapse, thus providing an upper bound on the ultimate load. It is a well-known fact that, in both cases, the plastic behavior intrinsically renders the problem to be non-linear. Essentially, two different approaches are usually adopted to deal with this issue: (i) the use of a polyhedral approximation to the yield surface; (ii) the implementation of direct non-linear programming algorithms.