PSI - Issue 62

Andrea De Flaviis et al. / Procedia Structural Integrity 62 (2024) 871–878 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3. Results and validation of GBT model GBT results are here reported, both in the static and dynamic case, taking into account the minimum number of deformation modes to have good results, through a convergence analysis. Analytical results are compared to the experimental ones, both in statics, with loading tests, and in dynamics, with OMA to validate the GBT model. 3.1. Static case In GBT, static results have been found by applying a piecewise constant load, equivalent to that of some stationary trucks on the bridge deck, acting along three parallel columns A, B, and C (each one made of three vehicles) one by one; only the GBT-FEM approach has been used. Displacements at midspan ( z = L/2 ) are reported in Fig. 4 (a), for the GBT model with distributed restraints and all conventional modes, together with those of loading tests on the same viaduct and with those of a simply supported Euler-Bernoulli beam model. Displacements given by GBT are quite close to those measured during loading tests. Moreover, if displacements of points located at the top of webs of the cross section (see Fig. 2) are plotted, (Fig. 4 (b)), the GBT model with distributed restraints (with or without extensional modes) behaves better than the model with punctual restraints with only conventional modes and absolutely better of the latter with conventional and extensional modes (not shown). Finally, in Fig. 4 (c) a convergence analysis is represented, when only the first column of loads (A) is applied, with reference to the points 2, 5 and the central point of the cross-section between them (see Fig. 2). After the first 5 deformation modes (plus the first 4 rigid ones) the response becomes asymptotic. Something can be said about the mechanical non linear behavior of the viaduct which can be seen in the red curves of Fig. 4 (a-b) when load is increased: it cannot be taken into consideration with this GBT model with linear elastic behavior but it requires further studies.

Fig. 4. (a) M-v at (b1+b2/2; L/2) fully loaded; (b) M-v at (b1+b2; L/2) fully loaded; (c) convergence analysis with one column of loads.

3.2. Dynamic case In GBT, dynamic results have been found with both approaches explained in Section 2.2, referred to as GBT bvp4c and GBT-FEM, with only distributed restraints in the first case and with distributed and punctual restraints in the second case. Results, reported in the following, are related to GBT-FEM with distributed restraints and all conventional modes, which are in complete agreement with GBT-bvp4c with same restraints and deformation modes; however, a big difference between them is in the computational time, much lower in GBT-FEM, and in the need to make initial assumptions in GBT-bvp4c which implies some knowledge of the expected results. On the other hand, for punctual restraints, results in terms of frequencies and MAC are reported and they differ most from those obtained with OMA: this phenomenon can be explained by the presence of stiffeners at the end sections, made of r.c. plates, so that, even if restraints are punctual, the end sections are somehow prevented to move and the real restraint is close to a distributed one. GBT results have been compared to those obtained through Operational Modal Analysis (natural frequencies,