Page - 788 - in Book of Full Papers - Symposium Hydro Engineering

3. ANALYTICAL PARAMETRIC ANALYSIS OF SHEAR DEFORMATION AND ROTATIONAL INERTIA OF CROSS-SECTION EFFECTS AND DEVELOPMENT OF A FINITE DIFFERENCE METHOD ALGORITHM From Eq. [13], making 𝜏 = 1 one obtains the classical natural frequencies equation of Euler’s beam, which considers only the flexural effect: 𝜔 𝑚 𝑓 = ( 𝑚 ∙𝜋 𝐿 ) 2 ∙ √ 𝐸 𝐼 ?̅? ∙𝐿 4 [14] The division of Eq. [13] by Eq. [14] defines the parametric ratio between these two frequencies: 𝜔 𝑚 𝑓 +𝑠 +𝑟 𝜔 𝑚 𝑓 = √ 1 1+𝑟 2∙𝛽 2+𝑟 2∙𝛽 2∙𝛾 ′ [15] To perform the parametric analysis, consider a beam with cross-section height equal to ℎ and Poisson ration 𝜈 . Substituting the definitions of Eq. [11] and [12] in Eq. [15], and performing some arithmetic manipulations, one obtains: 𝜔 𝑚 𝑓 +𝑠 +𝑟 𝜔 𝑚 𝑓 = √ 1 1+ (𝑚 ∙𝜋 )2 12 ∙( ℎ 𝐿 ) 2 + (𝑚 ∙𝜋 )2 5 ∙( ℎ 𝐿 ) 2 ∙(1+𝜈 ) [16] The ratio ℎ ⁄ 𝐿 is a typical measurement of the importance of considering shear effects in beams. The higher this ratio more the beam behavior approaches to deep beam rather than slender beam. Due to the complexity of the analytical approach for the dynamic problem of clamped beams, with variable section, based on the theory of Timoshenko, the numerical approach of the problem is justified. In the present work, it was used the Finite Differences Method (FDM), implemented using the Python programming language, to discretize the general dynamic equation of the Timoshenko beam in free vibration, presented in Eq [6]. The FDM is a method to solve differential equations that is based on the approximation of derivatives by finite differences. The approximation formula is derived from the Taylor series of the derived function. In Eq [6], the differential operators of fourth and second order can be replaced by finite expressions of FDM. After a mesh convergence study, it was chosen to use a 30-node model in all simulations made in this work. The first validation of the implemented algorithm was done comparing the analytical results for a typical simply supported beam with constant cross-section, given by the parametric equation presented in Eq. [13], with the numerical results. Considering a typical beam, it is obtained the parametric curves that associate the variation of the h / L ratio, measure of beam depth, to the relation between frequency obtained by Euler's theory and Timoshenko's theory. The results are presented in Figure 1. It is observed a good conformity between the analytical curves and the numerical ones, obtained by the algorithm implemented. 788