GNGTS 2016 - Atti del 35° Convegno Nazionale

440 GNGTS 2016 S essione 2.3 not dominated by any other solution. The set of Pareto optimal solutions, called Pareto Front (PF), represents the general solution of the problem. Once the design space, the objective and the constraints are set, the solution of an optimization problem may be accomplished by using several different approaches. Among many, GAs are rather popular because of their ability to solve different typologies of problems, and were used in this work, by employing the software TOSCA (Chisari, 2015). GAs are a zero- order, population-based meta-heuristic widely used to solve difficult optimization problems. They mimic the optimum search as observed in nature, where living species evolve through recombination of their genetic pool. The algorithm starts with a population of randomly (or quasi-randomly) generated solutions, which are modified during the process by applying some specific GA operators (selection, crossover, mutation, elitism). During the analysis, the population evolves increasing the number of ������ ��������� ��� �������� ��� ������� �� “����� ��������� ��� �������� ��� ������� �� good” solutions and avoiding the regions of the design space characterised by low fitness (i.e. high f i ). In the approach proposed in this work, the single individual evaluation consists of running each FE model represented by the individual solution, extracting output variables (interstorey drift, internal forces, displacements) and evaluating objectives and constraints. Case study . As a reference case study, a 3-storey, 3-bay RC frame already investigated in (Zou et al. , 2007) was taken into account, see Fig. 1a. The reference frame is considered as part of an office building, supposed to be located in an intensity I seismic zone of Italy, in accordance with the Eurocode 8 (EN1998-1: 2004). The design of the retrofitting was performed according to the seismic actions prescribed by this standard. Conversely, the RC details were based on the information reported in the reference (Zou et al. , 2007). A 250���� �� ×600 mm 2 cross-section was considered for all the beams, while 300 mm- and 400 mm-width square cross-sections were respectively considered for the external and internal columns. The details of the reinforcement for the RC frame elements are shown in Fig. 1b. Concrete was assumed to have an unconfined compressive strength equal to f c =21MPa. According to the formulation proposed by the Model Code 2010, the corresponding Young modulus was set equal to E c = 30660 MPa, while the ultimate strain for confined concrete was assumed as ε 2 =0.0356. The steel reinforcement was then characterized byYoung modulus E s =210 GPa, yielding stress f y =300 MPa, strain-hardening ratio 0.01 and ultimate strain equal to 3%. Finally, the FRP reinforcement was considered in the form of an elastic-brittle material, with Young modulus E f =230 GPa and ultimate strain ε f,u =0.00913, corresponding to an ultimate stress σ f,u =2100 MPa. The typical RC frame was modelled in OpenSees (2009). A first loading step accounting for the gravity load was considered before performing the pushover analyses. The gravity loads consisted of the self-weight plus a distributed loads equal to p =50 kN/m on the beams. In the following step, two load distributions for push-over analyses were considered, the first Fig. 1 – Reference geometry for the RC frame object of investigation. Nominal dimensions in meters.