GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 2.3 439 can be efficiently optimized assuming the thicknesses of the FRP jackets as the major design variables, and solving the optimization problem by using the principle of virtual work and the Taylor series approximation. The cited work shows the advantages of the FRP; however, this approach presents some drawbacks. Firstly, in order to have tractable analytical expressions for the objective function, simplified assumptions are to be considered, e.g. bilinear moment- rotation curve for the plastic hinges, without proper consideration of strength degradation in the constitutive relationship. Secondly, even though improvements in strength, ductility and collapse mechanism are observed, they are not made explicit in the optimization analysis, formulated simply as to minimize the FRP weight while satisfying interstorey drift code prescriptions. In this paper, based on the work developed in (Chisari and Bedon, 2016). it is shown that remarkable improvement in the formulation of the problem may be achieved if a more general approach is utilised. In particular, multi-objective optimization by means of Genetic Algorithms (GA) is used for the design of FRP jackets. To show the feasibility of the proposed approach, a reference case study is investigated. The goals of the multi-objective optimization analysis are then given by i) maximization of the RC frame ductility and ii) minimization of the volume (hence the cost) of FRP jackets, while satisfying the current provisions of the seismic design standards in use for concrete structures. Modelling FRP-reinforced RC members . In this work, the typical RC frame is modelled in the form of nonlinear force-based BeamColumn elements within the finite element (FE) software Opensees. The constitutive law at element level is evaluated according to the fibre approach, consisting of integrating the stress-strain relationship of the materials into the cross- section. According to this strategy, both plain concrete and steel reinforcement are modelled separately at the cross-section level. Conversely, the FRP jacket is represented by its effect on the mechanic characteristics of the plain concrete. It is well known that given a RC member, transverse reinforcements in general, such as steel stirrups, or FRP jackets, produce a confinement action which opposes the expansion of the concrete core, thus causing a state of triaxial stress inside the element. Such stress state is beneficial to the overall behaviour of the element, as it increases both strength and ductility. In the current study, the modelling approach proposed in (D������ ’����� Amato et al., 2012) for the calculation of the confining pressure acting in the section core of a FRP reinforced concrete member was taken into account. The advantage of the assumed approach is that the material model is able to estimate the increment of strength and ductility due to the assigned FRP confinement for a given RC member. In addition, the latter model accounts also for the tensile strength of plain concrete, and was specifically developed to evaluate the cyclic non-linear response of RC structures with degraded linear unloading/ reloading stiffness. The computed confined stress������� ������������� ��� ���� �������� �� ��� –������ ������������� ��� ���� �������� �� ��� strain relationships are then utilized in the non-linear structural analysis of the given system under the assigned design loads. The material model for the steel reinforcement was the Giuffr����������������� ������ è-Menegotto-Pinto model. Optimal design by means of GeneticAlgorithms . Optimal design in structural engineering consists of finding the best structure according to N pre-defined objectives f i to minimize, M equality constraints g j ( x )=0 and P inequality constraints h k ( x )<0. Any structure is parameterized by a design variable vector x , which collects the factors which can be varied in the search for the optimum. The space of the values that can be assumed by the variables x is called ������� “������ design space�� �� ��� ���� �� ���������� ������������� ������� ��� ������� ��������� �� �������� �� ��� ” �� ��� ���� �� ���������� ������������� ������� ��� ������� ��������� �� �������� �� ��� . In the case of structural optimization, usually the primary objective to minimize is the cost of the structure while the constraints reproduce the structural requirements prescribed by the codes. A practitioner could also wish to design the seismic retrofit maximizing its effect (ductility increment, decrease of actions/displacements/accelerations on the structure). These requirements naturally lead to a multi-objective optimization problem, where more than one conflicting objectives are to be optimized. In the context of multi-objective optimization, the concept of Pareto optimality replaces the usual notion of optimality. In a minimization problem with N objectives, a solution x 1 is said to dominate a solution x 2 if and only if and . A solution is referred to as Pareto optimal if it is