# An Analytical Solution for Large Displacements of End-Loaded Beams

## Abstract

The paper presents a closed-form analytical solution for the finite displacement analysis of cantilever beams end-loaded, having an unvarying section. Extension to supported beams loaded at the middle point is obvious. The solution accounts of slender beams, so that shear deformation effect is neglected. Material elasticity holds. Concentrated end loads are considered within the solution. Several former approaches to this classical problem can be found in the scientific literature. Typical methods provides an integral form, solved by means of incomplete elliptic functions. Alternatively, a massive number of numerical integrations techniques have been proposed, such as “Non linear shooting method”, “Automatic Taylor expansion technique (ATET)”, “Runge-Kutta method”, etc.

In the present paper, the solution is gained through a parametric representation of the average line of the bending beam, after an expansion based on Gaussian hypergeometric functions. One of the main advantage is that an end-to-end analysis is possible, thus avoiding the integration throughout the full length of the beam. The method considers straight lines as well as circular shaped lines; the way to shift between the two cases is suggested within the paper. The reference system to get an easier solution of the integrals is taken so that it is aligned with the resulting load applied. Graphic presentations of some examples compare the analytic solutions here provided with numerical solutions, finite element method results and an experimental evidence.

## Keywords

Large displacements beams Hypergeometric series Analytic solution## References

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