Accurate assessment of natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by adaptive generalized finite element method

Abstract

Purpose – The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an adaptive generalized finite element method (GFEM). The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. Design/methodology/approach – The variational problem of free vibration is formulated and the main aspects of the adaptive GFEM are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform Euler-Bernoulli beams are checked. The application of this technique in a frame is also presented. Findings – The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. The GFEM, which was conceived on the basis of the partition of unity method, allows the inclusion of enrichment functions that contain a priori knowledge about the fundamental solution of the governing differential equation. The proposed enrichment functions are dependent on the geometric and mechanical properties of the element. This approach converges very fast and is able to approximate the frequency related to any vibration mode. Originality/value – The main contribution of the present study consisted in proposing an adaptive GFEM for vibration analysis of Euler-Bernoulli uniform and non-uniform beams and frames. The GFEM results were compared with those obtained by the h and p-versions of FEM and the c-version of the CEM. The adaptive GFEM has shown to be efficient in the vibration analysis of beams and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems.