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Analyzing Free-Free Beams by Green's Functions and Fredholm Alternative Theorem

Published online by Cambridge University Press:  10 July 2017

M. Rezaiee-Pajand ,
A. Aftabi Sani  and
S. M. Hozhabrossadati
M. Rezaiee-Pajand*
Affiliation:
Department of Civil EngineeringFerdowsi University of MashhadMashhad, Iran
A. Aftabi Sani
Affiliation:
Department of Civil EngineeringFerdowsi University of MashhadMashhad, Iran
S. M. Hozhabrossadati
Affiliation:
Department of Civil EngineeringFerdowsi University of MashhadMashhad, Iran
*
*Corresponding author ([email protected])
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Abstract

This article deals with the analysis of free-free beams by an analytical method. The well-known Green's function method is employed, and exact solution for the problem is obtained. As a second problem, the simply supported-free beam with rotational rigid body motion is analyzed. It is initially shown that ordinary Green's functions cannot be constructed due to a mathematical contradiction. To remedy this limitation, the Fredholm Alternative Theorem is utilized. This theorem eliminates the contradiction and enables analysts to obtain modified Green's functions. The fundamental existence conditions are derived and thoroughly investigated from the structural point of view. Finally, the deflection functions of these beams are found using modified Green's functions.

Keywords

Modified Green's functionsFredholm Alternative TheoremDeflection functionsSolvability conditions
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 1 , February 2019 , pp. 27 - 39
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2019 

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