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.
Finite Elements in Analysis & Design
Dongming Wei, Yu Liu, Analytic and finite element solutions of the power-law Euler–Bernoulli beams, Finite Elements in Analysis and Design, Volume 52, May 2012, Pages 31-40, ISSN 0168-874X, 10.1016/j.finel.2011.12.007. (http://www.sciencedirect.com/science/article/pii/S0168874X1100237X)