GNGTS 2014 - Atti del 33° Convegno Nazionale

GNGTS 2014 S essione 2.3 325 . (4) Concerning the transversal ( X -direction) stiffness of the same devices, due to transverse deformability of the deck in the horizontal plane suggested by the measured deformed shape of EMA Mode 2 (Fig. 1d), a different analytical approach was taken into account. The bridge deck, specifically, was described in the form of a two-span continuous beam having a uniform cross-section along the total deck length and resting on three elastic supports (Fig. 3b). In this hypothesis, the elastic stiffness of the end supports is in fact reasonably given by two seismic isolators only ( K x a = 2 K x ), while the transversal stiffness offered by the mid-span “pier-isolators” support should be calculated by taking into account both the contribution of two isolators and the deformability of the pier along the transverse direction: . (5) While the pier stiffness contribution pier x K was calculated in accordance with Eq.(2), the transverse in-plane free vibration response of the deck (with fundamental frequency ω and modal amplitude u = u(y) ) was estimated by means of a well-defined eigenvalue problem. In doing so, the moment of inertia I of the deck was assumed constant and coincident with the nominal value of its mid-span cross-section (Fig. 1a). Solution of the eigenvalue problem in closed form allowed then to estimate the stiffness parameter K X as the optimal value able to provide full agreement between the fundamental analytical frequency ω and the corresponding experimental frequency EMA f 2 (Tab. 1). Application of the aforementioned methods to the studied bridge resulted in K Y = 172.4 MN/m and K X = 151.2 MN/m for the longitudinal and transversal directions respectively. An average value K X = K Y = 161.8 MN/m was then reasonably assumed in the 3D-REF model. The isolator stiffness in the vertical direction, finally, was kept equal to its nominal value ( K Z = 7631 MN/m). Compared to the 3D-NOM model, refined calculation of the shear stiffnesses K X and K Y resulted in marked improvement of correlation between analytical modal predictions and experimental data (Tab. 2), hence justifying the discussed analytical assumptions and suggesting their possible applications to similar bridges. Maximum discrepancies between analytical and test frequencies generally resulted lower than 7%. Optimal 3D FE-model (3D-OPT). Further improvement of the 3D-REF model was finally obtainedby sensitivitynumerical studies performedbymeans of parametricmodal investigations. The shear stiffness K X = K Y of each isolator, in this sense, was progressively increased – starting from the aforementioned 3D-REF value – so that discrepancy between EMA frequencies and corresponding 3D-OPT predictions could be minimized. The optimized value K X OPT = K Y OPT = 194.2 MN/m was identified as the stiffness value able to provide the best correlation between analytical predictions and test measurements. The obtained discrepancies were in fact generally lower than 3% (Tab. 2). In this case, it is interesting to notice that the optimal K X = K Y was detected to be ≈ 2.2 times the nominal value suggested by producers (Fig. 1b), hence confirming the importance of a proper calibration of FE-model components for structural investigations or monitoring programs. The equivalent 2D FE-model of the bridge. Based on accurate model updating of 3D models, a 2D-version of the 3D-OPT model was successively developed in SAP2000, to increase its computational efficiency but preserving its original accuracy (Tab. 2). The deck was described by means of shell elements belonging to two different horizontal planes, located at the barycentre of the central part of the cross section and of the two lateral portions respectively (Fig. 3c). Discrete Kirchhoff, isotropic, four node shell elements, with 6 DOFs at each node were used. The kinematical continuity conditions between shell element nodes