Seite - 789 - in Book of Full Papers - Symposium Hydro Engineering

Figure 1: Analytical and numerical results for the ratio between the natural frequencies of Timoshenko and Euler theories, for a simply supported constant cross-section beam. The analytical results, in general, presents a greater impact of the effects of Timoshenko, denoting that the numerical model implemented produces a more rigid model. However, such behavior is expected from numerical models if compared to analytical results. In addition, a validation was also made through the comparison with a finite element model developed in Ansys software. In this case, a cantilever beam, with constant section and typical properties, was modeled both in the finite element software and using the FDM algorithm. It is possible to observe an acceptable approximation between the results, validating the implemented algorithm through the comparison with Ansys, which besides being a consecrated software, uses a stronger formulation (finite elements) than the finite differences. It is important to note that, as the h / L ratio is increased, making the beam more rigid and the Timoshenko effects more important, the FDM implemented formulation differs more and consistently in all modes from the formulation of finite elements. The results via FDM present visibly more rigid models, with higher natural frequencies, which evidences the difficulty of the implemented modeling to correctly capture the real behavior of the beam. However, due to the consistency of this increase over all modes, the algorithm is considered as valid for a parametric and qualitative investigation of the effects of Timoshenko on clamped free beams with variable cross-section 789