Analytic and finite element solutions of the power-law Euler–Bernoulli beams

• Published in: Finite elements in analysis and design Vol. 52; pp. 31 - 40
• Main Authors: Wei, Dongming, Liu, Yu
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.05.2012; Elsevier
• Subjects: Approximation; Convergence; Error estimate; Euler-Bernoulli beams; Exact sciences and technology; Exact solutions; Finite element method; Finite element solution; Fundamental areas of phenomenology (including applications); Hermite elements; Hollomon's equation; Inelasticity (thermoplasticity, viscoplasticity...); Mathematical analysis; Mathematical models; Mathematics; Matlab; Methods of scientific computing (including symbolic computation, algebraic computation); Nonlinear Euler–Bernoulli beam; Nonlinearity; Numerical analysis. Scientific computation; Physics; Power-law; Ritz–Galerkin method; Sciences and techniques of general use; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Work hardening material; Work hardening material; Error estimate; Hollomon's equation; Finite element solution; Power-law; Ritz–Galerkin method; Nonlinear Euler–Bernoulli beam; Convergence; Hermite elements; Conjugate gradient method; Work hardening; Lagrangian; Error estimation; Hermite interpolation; Bernoulli Euler model; Modeling; Beam(mechanics); Hooke law; Finite element method; Nonlinear Euler-Bernoulli beam; Ritz-Galerkin method; Cubics; Ritz method; Non linear effect; Power law; Strain hardening
• ISSN: 0168-874X; 1872-6925
• DOI: 10.1016/j.finel.2011.12.007

In this paper, we use Hermite cubic finite elements to approximate the solutions of a nonlinear Euler–Bernoulli beam equation. The equation is derived from Hollomon's generalized Hooke's law for work hardening materials with the assumptions of the Euler–Bernoulli beam theory. The Ritz–Galerkin finite element procedure is used to form a finite dimensional nonlinear program problem, and a nonlinear conjugate gradient scheme is implemented to find the minimizer of the Lagrangian. Convergence of the finite element approximations is analyzed and some error estimates are presented. A Matlab finite element code is developed to provide numerical solutions to the beam equation. Some analytic solutions are derived to validate the numerical solutions. To our knowledge, the numerical solutions as well as the analytic solutions are not available in the literature. ► Hermite cubic finite elements to approximate the solutions of a nonlinear Euler–Bernoulli beam equation. ► Convergence of the finite element approximations is analyzed and some error estimates are presented. ► Analytic solutions are derived to validate the numerical solutions for some special cases.