GNGTS 2014 - Atti del 33° Convegno Nazionale

324 GNGTS 2014 S essione 2.3 measured on the other span. Mode 7 (6.93 Hz), is reminiscent of the third bending mode of a two-span continuous beam, although the limited number of measurement points on the span opposite to Dogna side made difficult the complete description of the modal shape. Mode 6 (6.89 Hz) and Mode 8 (8.00 Hz) are representative of the fundamental and second torsional modes of the bridge deck. A significant coupling between bending and torsional deformation of the deck was finally detected for the higher four EMAModes 9-12. FE-model calibration. A 3D FE-model was developed with the SAP2000 computer package, based on the nominal geometrical details of the bridge (Fig. 3a). The typical FE-model was carried out using 3D solid FEs to model the deck and the pier. The inertial/stiffening effects of footways were neglected, while the abutments were considered as non-deformable (e.g. rigid foundations) and the pier was assumed to be rigidly connected to ground, hence neglecting possible soil-structure interaction effects on the dynamic behavior of the bridge. Concrete was described as a linear elastic, isotropic material. Based on experiments performed on 150 mm diameter cylinders casted during construction of deck slab and piers, Young’s modulus was set equal to E c = 43.2 GPa, being ν c = 0.15 and ρ c = 2300 kg/m 3 the Poisson’s ratio and density. Each seismic isolator was also described in the form of 3D solid elements, by taking into account the mechanical properties of an equivalent, orthotropic, linear elastic material able to provide – in the form of well-calibrated Young’s and shear moduli E X , E Y , E Z and G Z , G Y , G Z – the desired shear K X , K Y and axial K Z stiffnesses for the devices. Different 3D FE-models partly discussed below (e.g. 3D-NOM, 3D-REF, 3D-OPT) were in fact characterized by the horizontal elastic stiffness of seismic isolators, being the geometrical and mechanical description of the other bridge components identical for all of them. Preliminary 3D FE-model (3D-NOM). In the first case, preliminary FE-modal simulations were developed by taking into account the nominal shear stiffnesses K X and K Y given by the producer of seismic isolators ( K X = K Y = 89.7 MN/m, based on Fig. 1b), with K Z = 7631 MN/m the corresponding nominal stiffness in the vertical direction. Modal analysis results are collected in Tab. 2. As shown, discrepancies up to 28% were found between analytical and experimental predictions, for the first ten vibration modes. Further studies and refined FE-models were consequently taken into account. Refined 3D FE-model (3D-REF). For the 3D-REF model, a rigorous analytical method was taken into account for a proper calibration of seismic devices. The stiffnesses K X and K Y , specifically, were separately estimated based on some experimental frequencies collected in Tab. 1 and simple analytical models able to properly describe the expected behavior of the bridge deck. In the longitudinal (Y) direction, in particular, the deck was rationally assumed to behave as a rigid-body simply supported on the abutments (with 2 K y the stiffness provided by two isolators, on each abutment) and the mid-span pier (with K y C the stiffness contribution given by two seismic devices working in series with the concrete pier), being: and . (1), (2) In Eq.(2), H is the pier height; A and χ are representative of the area and the shear factor of the pier cross-section; I y its moment of inertia respect to the bending axis and G c the shear elastic modulus of concrete. Assuming for the longitudinal mode of the bridge (RB Longitudinal, Tab. 1) a rigid-body motion, the sum K y,TOT of stiffness contributions offered by the abutments and the pier supports should in fact result in: , (3) with M deck the total mass of the deck plate, hence leading, based on Eqs.(1) and (2), to: