Analysis and design of elastic plastic structures subjected to dynamic loads

In the last decades, the concept of “optimization” has reached considerable value in many different fields of scientific research and, in particular, it has assumed great importance in the field of structural mechanics. The present study describes and shows the scientific path followed in the three years of doctoral studies. The state of the art concerning the optimization of elastic plastic structures subjected to quasi-static loads was already well established at the beginning of the Ph.D. course. Actually, it was already faced the study of structures subjected to quasi-static cyclic loads able to ensure different structural behaviors in relation to different intensity levels of the applied loads. In such a scientific context and with the aim to refer to cases of great practical interest, e.g. civil and industrial buildings affected by catastrophic events such as earthquakes and strong wind loads, the research effort has been addressed to reach an optimal design problem formulation for elastic perfectly-plastic structures subjected to static and dynamic loads, rigorously taking into account the dynamic and random nature of the loads and the non-linear behavior of the material. Solving a structural optimization design problem means the reaching, among all the feasible designs, of the one which minimizes or maximizes a suitable chosen quantity according to given constraints usually representing some required behaviors for the structure. The formulation of the relevant search problem requires the choice of the structural model together with the definition of some parameter used as design variables, the choice of an objective function, and the imposition of suitable admissibility conditions. In particular, the minimum volume of resistant structure has been chosen as objective function for a structure constituted by beam elements, in the hypothesis of small strains and displacements. The cross sections of the beams have been completely defined by means of a geometric parameter (thickness) used in the optimization problem as design variable. Furthermore the admissibility conditions of the problem have been used to impose different structural behaviors. As known, a structure made of elastic plastic material subject to loads, which are variable in a quasi-static and/or in a dynamic manner, can exhibit five different structural behaviors depending on the relevant load intensity: the purely elastic behavior in which the structure does not suffer any plastic strain and the stress does not reach the yielding condition at no point; the elastic shakedown behavior in which the structure responds in an elastic manner after a first initial phase where there is the production of some limited plastic deformations; the plastic shakedown or low cycle fatigue behavior in which one can assist in the cycle to the production of plastic deformation of opposite sign that can lead to fatigue failure; the ratchetting or incremental collapse behavior for which there is the production of increasing plastic deformations along the load path; finally, the instantaneous collapse behavior for which the structure is immediately transformed into a mechanism. Some practical considerations permit to say that a civil or industrial building made of elastic perfectly plastic material must behave in an elastic manner for permanent loads, must shake down for a combination of permanent and exceptional loads with low intensity, in such a way to exploit all the ductility resource that the structure possesses beyond the elastic limit, and should not instantaneously collapse for a combination of permanent and exceptional loads with high intensity. In particular, the adaptation behavior and the instantaneous collapse are studied by means of the shakedown theory and limit analysis respectively. This choice is particularly fruitful since both are first-order theories which do not require the solution of incremental elastic plastic analysis problems. It is useful to remember that the shakedown theory is based on a dual couple of theorems. These theorem, commonly known as Melan and Koiter theorems, permit to establish if the shakedown occurs in the structure subjected to quasi-static loads just looking at the elastic response. Even the limit analysis is founded on a dual couple of well-known theorems which allow to identify the origin of a local or global mechanism of plastic zones (instantaneous collapse) for a structure subject to a monotonically increasing quasi-static load. Referring to the classical shakedown theory and to international standards, the optimal design problem has been first formulated by modeling the seismic load through the response spectrum and calculating the purely elastic response through the modal combination. This response from a theoretical point of view has been idealized as the response to a quasi-static and perfectly cyclic load. Obviously, in order to simultaneously impose the shakedown and the collapse behavior, two different seismic responses, related to different probabilities of exceedance during the life of the structure, have been considered. This approach has been used for the optimal design formulation of structures subjected to wind and seismic loads. In order to take into account some dangerous phenomena for the optimal structures such as the bucking and the P-Delta effects, various specializations have been proposed. Furthermore, the optimization of existing buildings equipped with seismic isolation system has been studied. In order to more rigorously formulate the optimal design problem of structures subjected to seismic loads, a further scientific development has been reached making reference to the so-called “dynamic shakedown theory” which allows a better formulation of the optimal design problem of structures subject to dynamic actions. This theory was born with the pioneering work of Ceradini. Then it has experienced an enormous theoretical development in the eighties and nineties, and in the present thesis, it is used in the optimal design problem in order to take into account the real dynamic nature of seismic and wind loads. The Ceradini's theorem is an extension in the dynamic range of the Melan's one. However, there are some theoretical differences. The former investigates the conditions under which a structure made of elastic plastic material subjected to a single dynamic and infinite load history, eventually shakes down (“restricted shakedown”), while the latter considers the same structure but subjected to an unknown quasi-static load within a given load domain. These difference can be overcome by the “unrestricted shakedown theory” in which the dynamic loads are conceived to appertain to a particular excitation domain so that the structure is exposed to an infinite and unknown sequence of potentially active excitations like in the classical theory. Following the unrestricted dynamic shakedown approach a complete optimal design formulation has been given. The bibliographic study conducted on “dynamic shakedown” during the last part of the Ph.D. course have led to a new and original interpretation of the dynamic shakedown analysis problem in a probabilistic key. This modeling has paved the way to an additional and unexpected scientific improvement of the optimal design with the dynamic shakedown criterion in which the loads are modeled in all their stochastic nature. Finally, consolidated linearization techniques have been used for the solution of formulated problems with continuous variables while a new heuristic algorithms, suitably modified and adapted for multi-constrained problems, have been used for the solution of the formulated design problems with both continuous and discrete variables.