FINITE ELEMENT ANALYSIS OF SHEAR-DEFORMATION AND ROTATORY INERTIA FOR BEAM VIBRATION - Vasconcelos, Ana Carolina Azevedo; Costa Azevêdo, Anderson Soares da; Hoefel, Simone dos Santos (Revista Interdisciplinar de Pesquisa em Engenharia, 2017)
Vibration analysis of a beam is an important subject of study in engineering. All real physical structures, when subjected to loads or displacements, behave dynamically. In case of structure with large aspect ratio of height and length the Timoshenko beam theory (TBT) is used, instead of the Euler-Bernoulli theory (EBT), since it takes both shear and rotary inertia into account. Shear effect is extremely large in higher vibration modes due to reduced mode half wave length. In this paper, the full development and analysis of TBT for the transversely vibrating uniform beam are presented for classical boundary condition. Finally, a finite element is developed in terms of dimensionless parameters of rotatory and shear. The stiffness and mass matrices for a two-node beam element with two degree of freedom per node is obtained based upon Hamilton’s principle. Cubic and quadratic Lagrangian polynomials are made interdependent by requiring them to satisfy both of the homogeneous differential equations associated with TBT. Numerical examples are given for some boundary conditions. The results showed that for frequencies above critical frequency, Timoshenko beams presents distinct mode shapes behavior including the presence of double eigenvalues, shear mode or remarkably modes.