Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 10 May 2021 / Approved: 11 May 2021 / Online: 11 May 2021 (15:39:17 CEST)

How to cite: Fogang, V. Timoshenko Beam Theory:First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints 2021, 2021050252 (doi: 10.20944/preprints202105.0252.v1). Fogang, V. Timoshenko Beam Theory:First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method. Preprints 2021, 2021050252 (doi: 10.20944/preprints202105.0252.v1).

Abstract

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. First-order, second-order, and vibration analyses of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. In addition, tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis, and vibration analysis). Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, considering the damping. The efforts and displacements could be determined at any time.

Subject Areas

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

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