GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 2.3 441 (D1) with horizontal loads increasing proportionally to the building height, and the second (D2) with horizontal loads proportional to the seismic masses. The total mass of the RC frame was estimated to be equal to 285.86t, as given by the concrete density ������ ���� γ=2300 kg/m 3 plus the distributed loads applied on the beams. Finally, P-� ������� ���� ��� ����� ���� �������� Δ effects were not taken into account. As design variable of the optimization problem, the thickness of the FRP wraps applied to the columns was considered. In particular, different thicknesses were assumed for external and internal columns and for each floor. The problem consisted thus of 6 design variables (3 floors times 2 column typologies). The FRP thicknesses were allowed to vary between 0 (no reinforcement) to 2 mm, with 0.001 mm increments. Once the trial values for the design variables were set for a generic individual of any population, a FE model of the reinforced structure was created and a push-over analysis performed. From the results of the pushover analysis the following quantities were then extracted: a) capacity curve of the RC frame, i.e. in the form of base shear-top displacement; b) inter-storey drift at each imposed load increment; c) maximum steel strain in each RC member, at each imposed load increment. From this amount of information, several key values were thus evaluated, including: (i) the first yielding point ( F y , u y ), i.e. the time step in which the first yielding in steel; (ii) the peak load point ( F max , u max ); (iii) the ultimate state point ( F u , u u ), defined as the point after the peak load point where either a 20% drop of the peak base shear occurred or where the analysis did not converge anymore due to achievement of the ultimate strain in the concrete (and thus rotation capacity of the section). Finally, ductility: , behaviour factor: , DLS design base shear: , and DLS inter-storey drift ratio: were calculated. The objectives of the multi-objective optimization analysis were set as 1) the maximization of ��μ, and 2) the minimization of the volume of FRP, with the reference constraint that d DLS ≤ 0.005. For each FE model, two different optimization analyses were carried out depending on the horizontal load distributions, i.e. (D1) a first-mode proportional distribution and (D2) a mass- proportional distribution over the frame height. A preliminary analysis shows that under these loading conditions the bare frame undertakes a maximum drift ratio d DLS = 0.0136 > 0.005, hence it does not comply to the DLS prescription. A purposely designed FRP reinforcement is thus needed. The Pareto Fronts obtained for both loading conditions are displayed in Fig. 2. It is clear that while for low levels of FRP volume the relationship between this and ductility is almost linear, the increase of ductility degrades as the cost increases. In other words, after a certain point, the advantages in terms of ductility are negligible, compared to the cost increase. A more detailed analysis of the obtained results shows how the � ������ �� ����������� μ ������ �� ����������� values of individuals in the Pareto Fronts are related to the thickness of FRP wraps, for the external and internal columns respectively. In Fig. 3 it can be seen that there is strongcorrelationbetween ductility and FRP thickness at the first floor for the internal columns, as the former increases almost linearly with Fig. 2 – Pareto fronts of the optimization analyses, as obtained for the D1 and D2 loading conditions respectively.