Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

Version 1 : Received: 23 February 2021 / Approved: 24 February 2021 / Online: 24 February 2021 (17:03:09 CET)
Version 2 : Received: 26 March 2021 / Approved: 29 March 2021 / Online: 29 March 2021 (17:13:17 CEST)
Version 3 : Received: 22 September 2021 / Approved: 23 September 2021 / Online: 23 September 2021 (13:08:38 CEST)

How to cite: Fogang, V. Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints 2021, 2021020559 (doi: 10.20944/preprints202102.0559.v3). Fogang, V. Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints 2021, 2021020559 (doi: 10.20944/preprints202102.0559.v3).

Abstract

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally, the finite difference approximations are derived based on fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection curve; the FOPH is made for the fourth and third derivative of the deflection curve while the SOPH is made for its second and first derivative. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, the FOPH was made for all of the derivatives of the deflection curve, and additional points were introduced at the beam’s ends and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid. Further research will be needed to investigate polynomial refinements (higher-order polynomials such as fifth-order, sixth-order…) of the deflection curve; the polynomial refinements aimed to increase the accuracy, whereby non-centered finite difference approximations at beam’s ends and positions of discontinuity would be used.

Keywords

Euler Bernoulli beam; finite difference method; additional points; element stiffness matrix; tapered beam; first-order analysis; second-order analysis; vibration analysis; direct time integration method

Comments (1)

Comment 1
Received: 23 September 2021
Commenter: Valentin Fogang
Commenter's Conflict of Interests: Author
Comment: Abstract
Graphical abstract
Introduction
W FDM approximation for tapered beams
+ Respond to this comment

We encourage comments and feedback from a broad range of readers. See criteria for comments and our diversity statement.

Leave a public comment
Send a private comment to the author(s)
Views 0
Downloads 0
Comments 1
Metrics 0


×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.