### 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.

Original language | English |
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Pages (from-to) | 31-40 |

Number of pages | 10 |

Journal | Finite Elements in Analysis and Design |

Volume | 52 |

DOIs | |

Publication status | Published - May 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Convergence
- Error estimate
- Finite element solution
- Hermite elements
- Hollomon's equation
- Nonlinear Euler-Bernoulli beam
- Power-law
- Ritz-Galerkin method
- Work hardening material

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Engineering(all)
- Computer Graphics and Computer-Aided Design

### Cite this

**Analytic and finite element solutions of the power-law Euler-Bernoulli beams.** / Wei, Dongming; Liu, Yu.

Research output: Contribution to journal › Article

*Finite Elements in Analysis and Design*, vol. 52, pp. 31-40. https://doi.org/10.1016/j.finel.2011.12.007

}

TY - JOUR

T1 - Analytic and finite element solutions of the power-law Euler-Bernoulli beams

AU - Wei, Dongming

AU - Liu, Yu

PY - 2012/5

Y1 - 2012/5

N2 - 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.

AB - 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.

KW - Convergence

KW - Error estimate

KW - Finite element solution

KW - Hermite elements

KW - Hollomon's equation

KW - Nonlinear Euler-Bernoulli beam

KW - Power-law

KW - Ritz-Galerkin method

KW - Work hardening material

UR - http://www.scopus.com/inward/record.url?scp=84855778519&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855778519&partnerID=8YFLogxK

U2 - 10.1016/j.finel.2011.12.007

DO - 10.1016/j.finel.2011.12.007

M3 - Article

VL - 52

SP - 31

EP - 40

JO - Finite Elements in Analysis and Design

JF - Finite Elements in Analysis and Design

SN - 0168-874X

ER -