# The Picard’s iteration method for determining the critical buckling load of Euler elastic columns

• 1 University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria

## Abstract

The Picard’s successive iteration method is applied to determine the Euler critical force for a slender column. The initial approximation of the buckling mode of the column is a polynomial function that satisfies the boundary conditions. A numerical example is solved and the obtained result is compared with that obtained by the Euler formula. An assessment of the accuracy of the solution is made at each iteration step. In the paper is shown that the Picard’s iteration method could also be used for obtaining the closed form exact solution of the buckling load. The investigated column is hinged at its upper end and supported by a Q- apparatus at the other. It is loaded with a compressive force at the upper end.

## References

1. M.T.Atay, M. T. , Determination of critical buckling loads for variable stiffness Euler columns using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, Volume 10, Issue 2, 2009.
2. A. Eryılmaz, M.T. Atay, S.B. Coşkun, M. Başbük, Buckling of Euler columns with a continuous elastic restraint via homotopy analysis method. Journal of Applied Mathematics, Volume 2013, Special Issue ,2013.
3. C. C. Ike, E. U. Ikwueze, I.O. Ofondu, Picard’s successive iteration method for the elastic buckling analysis of Euler columns with pinned ends, Saudi Journal of Civil Engineering Vol.2, Issue 2, 2018.
4. K. Mladenov, J. Klecherov, S. Lilkova-Markova, V. Rizov, Strength of Materials, ABC Technics, 2012.
5. F. Okay, M.T. Atay, S.B. Coşkun, Determination of buckling loads and mode shapes of a heavy vertical column under its own weight using the variational iteration method. International Journal of Nonlinear Sciences and Numerical Simulation, Volume 11, Issue 10, 2010.
6. K. Tisdell, On Picard’s iteration method to solve differential equations and a pedagogical space for otherness, International Journal of Mathematical Education, Vol.50, Issue 5, 2019.