Analysis of Euler-Bernoulli Beam with Piecewise Quadratic Hermite Finite Elements

Abstract

The piecewise quadratic Hermite polynomials are employed in the finite element context to analyze the static and free vibration behaviors of Euler-Bernoulli beam. The desirable C1 continuity is achieved for the piecewise quadratic Hermite element that is required for the numerical solution of the Galerkin weak form of Euler-Bernoulli beam. In contrast to the classical cubic Hermite element, the piecewise quadratic Hermite element has a piecewise constant curvature representation within each element and thus the integration of the stiffness matrix is trivial. Several benchmark problems are shown to demonstrate the convergence properties of the piecewise quadratic Hermite element. The frequency error of the beam free vibration with this quadratic Hermite element is derived as well. Numerical examples consistently verify the analytical convergence rates.