A Unified Meshfree Path to Arbitrary Order Hermite Finite Elements for Euler-Bernoulli Beams

The fourth-order governing equation of Euler-Bernoulli beams necessitates the employment of C1 Hermite shape functions in finite element analysis. However, unlike the C0 finite element shape functions that can be easily formulated as Lagrangian polynomials in a unified manner, the Hermite shape functions of Euler-Bernoulli beam elements are usually constructed case by case. In this work, a simple and unified meshfree path is proposed to develop arbitrary order Hermite shape functions of Euler-Bernoulli beam elements. This approach is realized by proving that the Hermite reproducing kernel meshfree shape functions degenerate to the interpolatory Hermite finite element shape functions under a proper choice of support sizes. The proposed approach provides an explicit formalism for Hermite finite element shape functions, which enables an easy construction of arbitrary order Hermite beam elements. Accordingly, in addition to the conventional cubic Hermite beam elements, explicit shape functions, mass and stiffness matrices for quintic and septic Hermite beam elements are presented. Meanwhile, the frequency errors and general higher order mass matrix formulation for these Hermite beam elements are elaborated as well. These theoretical results are consistently demonstrated by numerical examples with respect to both static and free vibration analysis.

Automated summary

This paper proposes a simple and unified method for developing arbitrary order Hermite shape functions of Euler-Bernoulli beam elements. The method utilizes the fact that the Hermite reproducing kernel meshfree shape functions degenerate to the interpolatory Hermite finite element shape functions under a proper choice of support sizes. Using this method, cubic, quintic, and septic Hermite beam elements are constructed, and the shape functions, mass and stiffness matrices, frequency errors, and higher order mass matrix formulation for these elements are presented.