Enriched finite elements and local rescaling for vibrations of axially inhomogeneous Timoshenko beams

Abstract

This work presents a new enriched finite element method dedicated to the vibrations of axially inhomogeneous Timoshenko beams. This method relies on the "half-hat" partition of unity and on an enrichment by solutions of the Timoshenko system corresponding to simple beams with a homogeneous or an exponentially-varying geometry. Moreover, the efficiency of the enrichment is considerably increased by introducing a new formulation based on a local rescaling of the Timoshenko problem, that accounts for the inhomogeneity of the beam. Validations using analytical solutions and comparisons with the classical high-order polynomial FEM, conduced for several inhomogeneous beams, show the efficiency of this approach in the time-harmonic domain. In particular low error levels are obtained over ranges of frequencies varying from a factor of one to thirty using fixed coarse meshes. Possible extensions to the research of natural frequencies of beams and to simulations of transient wave propagation are highlighted.

Improved FEM Natural Frequency Calculation for Structural Frames by Local Correction Procedure

“…Higher-order finite elements [31][32][33][34] provide better accuracy but are more complex to implement, have to solve larger systems of equations and lead to worse conditioned matrices. In contrast, our method works well with the standard finite element method and could work with higher-order finite elements as well to improve their accuracy.…”

“…Higher-order Euler-Bernoulli elements [31] and the Timoshenko element [32][33][34] provide improved accuracy due to the better representation of displacements. This leads to a reduction in the number of elements required to calculate natural frequencies.…”