Deformation of Euler-Bernoulli Beams by Means of Modified Green's Function: Application of Fredholm Alternative Theorem

Abstract

This article deals with the determination of the static displacement function of an Euler-Bernoulli beam with two guided supports. To this end, the Green's function method is employed and exact solution is obtained. The Green's function of the problem is constructed, using pertinent boundary conditions of the problem. Nevertheless, the problem does not admit a Green's function due to a mathematical contradiction. In order to eliminate the trouble, the Fredholm Alternative Theorem is utilized and the Green's function is modified. In this case, application of this theorem adds a particular term to the Green's function which gives rise to an arbitrary constant in the Green's function. Moreover, it is shown that the problem may have no solution or an infinite number of solutions. Besides, the necessary condition for having any solution is investigated. This requirement, which states a significant rule in the mechanics of solids, is the static equilibrium of vertical forces acting on the beam. Some examples are presented and results are thoroughly discussed.

Keywords:

• Euler-Bernoulli beam
• Fredholm alternative theorem
• Guided supports
• Modified Green's function
• Static displacement

Acknowledgments

^#Communicated by Janos Logo.

Notes

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lmbd.