On Timoshenko’s Beams Coefficient of Sensibility to Shear Effect

Abstract

Classical beams theory usually neglect shear contribution to deformation. Timoshenko´s Beam Theory (TBT) corrects this negligence. This work is a step in the research to check accuracy of TBT in piezoactuators and in nano beams, particularly in Carbon nanotubes (CNT). Before immerse into the nanoscale problem, the superior limit of applicability of TBT must be investigated. This work introduces a proposition to the range of validation of TBT. A sensibility coefficient related to free vibratory beams is proposed, in terms of a ratio of deflected geometry to aspect ratio of the beams. Finite difference and Galerkin finite element were used to formulate the computational model. This approach was checked with experimental results to propose a top validation of TBT. Convergence in solution process was verified with precedent works. Results are related to symmetric dynamic bending of beams considering shear deformability and rotatory inertia, at small change of configuration. Timoshenko’s beam models show a displacement field greater than those obtained with Euler-Bernoulli Theory (EBT). The magnitude of the difference is of 6 to 34 % greater in TBT in relation to EBT, for tree kinds of boundary conditions, for the same beam geometry and load pattern.