Free Vibration and Stability of Axially Functionally Graded Tapered Euler-Bernoulli Beams

TLDR

The authors propose a beam element that leverages shape functions from homogeneous uniform beam elements for axially functionally graded tapered Euler‑Bernoulli beams. Finite‑element analysis is employed to evaluate structural matrices of axially functionally graded tapered Euler‑Bernoulli beams, incorporating variations in cross‑sectional dimensions and material properties. The element is shown to handle arbitrary mass density and modulus distributions with varying cross‑sectional area, and its performance is validated through stability, free longitudinal, and free transverse vibration analyses of double‑tapered beams with different boundary conditions, demonstrating convergence.

Abstract

Structural analysis of axially functionally graded tapered Euler-Bernoulli beams is studied using finiteelement method. A beam element is proposed which takes advantage of the shape functions of homogeneous uniform beam elements. The effects of varying cross-sectional dimensions and mechanical properties of the functionally graded material are included in the evaluation of structural matrices. This method could be used for beam elements with any distributions of mass density and modulus of elasticity with arbitrarily varying cross-sectional area. Assuming polynomial distributions of modulus of elasticity and mass density, the competency of the element is examined in stability analysis, free longitudinal vibration and free transverse vibration of double tapered beams with different boundary conditions and the convergence rate of the element is then investigated.