Equivalent Formulations of Euler–Bernoulli Beam Theory for a Simple Gradient Elasticity Law
Publication: Journal of Engineering Mechanics
Volume 149, Issue 1
Abstract
Existing Euler-Bernoulli beam theories in classical elastostatics suffer from the inconsistency that either the elasticity law or the equilibrium equations are not satisfied in local form. It has recently been shown that by assuming elastic anisotropy subject to internal constraints, it is possible to make the theory consistent. This has been proved to be true also for a simple gradient elasticity law. Usually, bending of beams is viewed as a one-dimensional problem. We consider in this paper two known one-dimensional formulations for Euler-Bernoulli beam and gradient elastic material behavior. The two formulations seem to be different, as the free energy functional of the one includes the cross-sectional area of the beam, whereas the other does not. The aim is, by using consistent Euler-Bernoulli beam theory, to derive the two one-dimensional formulations as special cases of a three-dimensional simple gradient elasticity model and to show that these are equivalent to each other.
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Acknowledgments
The authors gratefully acknowledge the Deutsche Forschungsgemeinschaft for partial support of this work under Grant TS 29/13-1.
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© 2022 American Society of Civil Engineers.
History
Received: Mar 29, 2022
Accepted: Jul 20, 2022
Published online: Oct 26, 2022
Published in print: Jan 1, 2023
Discussion open until: Mar 26, 2023
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