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

Abstract

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.

Some analytic and finite element solutions of the graphene Euler beam

“…In this work, we are interested in providing some novel analytic solutions and finite element solutions of loaded graphene Euler beams. The method of solutions in this paper is similar to the earlier work on solutions of Euler-Bernoulli power-law beams by the authors [16]. It is known that the stress-strain equation for a large class of graphene can be modelled by the following quadratic equation:…”

“…Downloaded by [University of Sherbrooke] at 04:08 12 April 2015 Based on finite element approximation (56) of I(v) and its gradient function (62), the nodal value vector v can be computed by numerical nonlinear optimization techniques [3,13,16], e.g. conjugate gradient method, for the optimization problem (51).…”

Buckling and post-buckling of graphene tubes

“…The constant c is determined along with the solutions of (14) and (15) subject to the conditions in (16) and x(0) = y(0) = 0.…”

“…and(16) for every λ > 0, which we name as the basic solution. The bifurcation points and bifurcation branches off this basic solution for each N ≥ 2 are determined in this section.…”

A predictive thermal dynamic model for parameter generation in the laser assisted direct write process

“…numerical methods may suit certain types of problems better than others, due to the nature of the physics involved, geometry, requirement of accuracy, efficiency of computation or other factors[29,[201][202][203][204].…”