Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

Timoshenko Beam Theory: Exact Solution for First-Order Analysis, Second-Order Analysis, and Stability

Version 1 : Received: 17 November 2020 / Approved: 17 November 2020 / Online: 17 November 2020 (14:55:23 CET)
Version 2 : Received: 5 April 2021 / Approved: 5 April 2021 / Online: 5 April 2021 (16:06:31 CEST)
Version 3 : Received: 14 July 2021 / Approved: 15 July 2021 / Online: 15 July 2021 (10:30:53 CEST)

How to cite: Fogang, V. Timoshenko Beam Theory: Exact Solution for First-Order Analysis, Second-Order Analysis, and Stability. Preprints 2020, 2020110457 (doi: 10.20944/preprints202011.0457.v3). Fogang, V. Timoshenko Beam Theory: Exact Solution for First-Order Analysis, Second-Order Analysis, and Stability. Preprints 2020, 2020110457 (doi: 10.20944/preprints202011.0457.v3).

Abstract

This paper presents an exact solution to the Timoshenko beam theory (TBT) for first-order analysis, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory (EBBT) neglects shear deformations. Thus, the Euler–Bernoulli beam is a special case of the Timoshenko beam. The moment-curvature relationship is one of the governing equations of the EBBT, and closed-form expressions of efforts and deformations are available in the literature. However, neither an equivalent to the moment-curvature relationship of EBBT nor closed-form expressions of efforts and deformations can be found in the TBT. In this paper, a moment-shear force-curvature relationship, the equivalent in TBT of the moment-curvature relationship of EBBT, was presented. Based on this relationship, first-order and second-order analyses were conducted, and closed-form expressions of efforts and deformations were derived for various load cases. Furthermore, beam stability was analyzed and buckling loads were calculated. Finally, first-order and second-order element stiffness matrices were determined.

Subject Areas

Timoshenko beam; moment shear force curvature relationship; closed-form solutions; non-uniform heating; stability; second-order element stiffness matrix; elastic Winkler foundation

Comments (1)

Comment 1
Received: 15 July 2021
Commenter: Valentin Fogang
Commenter's Conflict of Interests: Author
Comment: Title
Closed-form expressions of efforts and deformations in second-order analysis
Load case Non-uniform heating
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