Version 1
: Received: 10 May 2021 / Approved: 11 May 2021 / Online: 11 May 2021 (15:39:17 CEST)
Version 2
: Received: 29 September 2021 / Approved: 30 September 2021 / Online: 30 September 2021 (16:24:48 CEST)
How to cite:
Fogang, V. Timoshenko Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints2021, 2021050252. https://doi.org/10.20944/preprints202105.0252.v2
Fogang, V. Timoshenko Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints 2021, 2021050252. https://doi.org/10.20944/preprints202105.0252.v2
Fogang, V. Timoshenko Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints2021, 2021050252. https://doi.org/10.20944/preprints202105.0252.v2
APA Style
Fogang, V. (2021). Timoshenko Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints. https://doi.org/10.20944/preprints202105.0252.v2
Chicago/Turabian Style
Fogang, V. 2021 "Timoshenko Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method" Preprints. https://doi.org/10.20944/preprints202105.0252.v2
Abstract
This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. The model developed in this paper consisted of formulating differential equations with finite differences and introducing additional points at the beam’s ends and at positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends. Moreover, grid points with variable spacing were considered, the grid being uniform within beam segments. First-order, second-order, and vibration analyses of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis, vibration 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.
Keywords
Timoshenko beam; finite difference method; additional points; element stiffness matrix; tapered beam; second-order analysis; vibration analysis; direct time integration method
Subject
Engineering, Civil Engineering
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter: Valentin Fogang
Commenter's Conflict of Interests: Author
Graphical abstract
Introduction
Suppression of Section 2.1.2.5
Change of beam structure in Section 3.2.1