**1. Introduction**

114 Advances in Computational Stability Analysis

Article ID: 42072.

851-857.

1262.

Coskun, S.B. & Atay, M.T. (2007). Analysis of convective straight and radial fins with temperature- dependent thermal conductivity using variational iteration method with comparison with respect to finite element analysis. *Mathematical Problems in Engineering*,

Coskun, S.B. & Atay, M.T. (2008). Fin efficiency analysis of convective straight fins with temperature dependent thermal conductivity using variational iteration method.

Coskun, S.B. & Atay, M.T. (2009). Determination of critical buckling load for elastic columns of constant and variable cross-sections using variational iteration method. *Computers* 

Ganji, D.D. & Sadighi, A. (2007). Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. *Journal of* 

Galambos, T.V. (1998). *Guide to Stability Design Criteria for Metal Structures* (fifth edition),

He, J.H. (1999). Variational iteration method - a kind of nonlinear analytical technique: some examples. *International Journal of Non Linear Mechanics*, Vol.34, No.4, pp. 699-708. He, J.H.; Wu, G.C. & Austin, F. (2010). The variational iteration method which should be

Li, Q.S. (2001). Buckling of multi-step non-uniform beams with elastically restrained boundary conditions. *Journal of Constructional Steel Research*, Vol.57, pp. 753–777. Miansari, M.; Ganji, D.D. & Miansari M. (2008). Application of He's variational iteration method to nonlinear heat transfer equations. *Physics Letters A*, Vol. 372, pp. 779-785. Okay, F.; Atay, M.T. & Coskun S.B. (2010). 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 Science Numerical Simulation*, Vol.11, No.10, pp.

Ozturk, B. (2009). Free vibration analysis of beam on elastic foundation by variational iteration method. *International Journal of Non Linear Mechanics*, Vol.10, No.10, pp. 1255-

Pinarbasi, S. (2011). Lateral torsional buckling of rectangular beams using variational

Salmon, C.G.; Johnson, E.J. & Malhas, F.A. (2009*). Steel Structures, Design and Behavior* (fifth

Sweilan, N.H. & Khader, M.M. (2007). Variational iteration method for one dimensional nonlinear thermoelasticity. *Chaos Solitons & Fractals,* Vol.32, No.1, pp. 145-149. Timoshenko, S.P. & Gere, J.M. (1961). *Theory of Elastic Stability* (second edition), McGraw-

iteration method. *Scientific Research and Essays*, Vol.6, No.6, pp. 1445-1457.

edition), Pearson, Prentice Hall, ISBN-10: 0-13-188556-1, New Jersey.

Hill Book Company, ISBN- 0-07-085821-7, New York.

*Applied Thermal Engineering*, Vol.28, No.17-18, pp. 2345-2352.

*and Mathematics with Applications*, Vol.58, pp. 2260-2266.

*Computational and Applied Mathematics*, Vol.207, pp. 24-34.

John Wiley & Sons, Inc., ISBN 0-471-12742-6, NewYork.

followed. *Nonlinear Science Letters A*, Vol.1, No.1, pp. 1-30.

In most of the real world engineering applications, stability analysis of compressed members is very crucial. There have been many researches dedicated to the buckling behavior of axially compressed members. On the other hand, obtaining analytical solutions for the buckling behavior of columns with variable cross-section subjected to complicated load configurations are almost impossible in most of the cases. Some of the works related to obtaining analytical or analytical approximate solutions for the column buckling problem are provided below.

The problems of buckling of columns under variable distributed axial loads were solved in detail by Vaziri and Xie [1] and others. Some analytical closed-form solutions are given by Dinnik [2], Karman and Biot [3], Morley[4], Timoshenko and Gere [5] and others. One of the detailed references related to the structural stability topic is written by Simitses and Hodges [6] with detailed discussions. Iyengar [7] made some analysis on buckling of uniform with several elastic supports. Wang et al. [8] have given exact mathematical solutions for buckling of structural members for various cases of columns, beams, arches, rings, plates and shells. Ermopoulos [9] found the solution for buckling of tapered bars axially compressed by concentrated loads applied at various locations along their axes. Li [10] gave the exact solution for buckling of non-uniform columns under axially concentrated and distributed loading. Lee and Kuo [11] established an analytical procedure to investigate the elastic stability of a column with elastic supports at the ends under uniformly distributed follower forces. Furthermore, Gere and Carter [12] investigated and established the exact analytical solutions for buckling of several special types of tapered columns with simple boundary conditions. Solution of the problem of buckling of elastic columns with step varying thickness is established by Arbabei and Li [13]. Stability problems of a uniform bar with several elastic supports using the moment-

© 2012 Coşkun and Öztürk, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution method were analyzed by Kerekes [14]. The research of Siginer [15] was about the stability of a column whose flexural stiffness has a continuous linear variation along the column. Moreover, the analytical solutions of a multi-step bar with varying cross section were obtained by Li et al. [16-18]. The energy method was used by Sampaio et al. [19] to find the solution for the problem of buckling behavior of inclined beamcolumn. Some of the important researchers who studied the mechanical behavior of beamcolumns are Keller [20], Tadjbakhsh and Keller [21] and Taylor [22]. Later on, analytical approximate techniques were used for the stability analysis of elastic columns. Coşkun and Atay [23] and Atay and Coşkun [24] studied column buckling problems for the columns with variable flexural stiffness and for the columns with continuous elastic restraints by using the variational iteration method which produces analytical approximations. Coşkun [25, 26] used the homotopy perturbation method for buckling of Euler columns on elastic foundations and tilt-buckling of variable stiffness columns. Pnarbaş [27] also analyzed the stability of nonuniform rectangular beams using homotopy perturbation method. These techniques were also used successfully in the vibration analysis of Euler-Bernoulli beams and in the vibration of beams on elastic foundations. [28-29]

Elastic Stability Analysis of Euler Columns Using Analytical Approximate Techniques 117

(2)

*dx* (3)

*dx* (4)

(5)

**Figure 1.** Elastic columns with various end conditions

*Clamped-Sliding Restraint*.

Pin support:

Free end:

Clamped support:

and

In the case of constant flexural rigidity (*i.e. EI* is constant), Eq.(1) becomes

4 2 4 2 <sup>0</sup> *dy dy <sup>P</sup> dx dx EI*

where *EI* is the flexural rigidity of the column, and *P* is the applied load. Both Eqs. (2) and (3) are solved due to end conditions of the column. Some of these conditions are shown in Fig.1. In this figure, letters are used for a simplification to describe the support conditions of the column. The first letter stands for the support at the bottom and the second letter for the top. Hence, CF is *Clamped-Fixed*, PP is *Pinned-Pinned*, C-P is *Clamped-Pinned* and C-S is

The governing equations (1) and (2) are both solved with respect to the problem's end

*<sup>y</sup>* <sup>0</sup> and <sup>0</sup> *dy*

<sup>3</sup> <sup>0</sup> *d y dy <sup>P</sup> dx EI dx*

3

2 <sup>2</sup> <sup>0</sup> *d y*

conditions. The end conditions for the columns shown in Fig.1 are given below:

*y* 0 and

Recently, by the emergence of new and innovative semi analytical approximation methods, research on this subject has gained momentum. Analytical approximate solution techniques are used widely to solve nonlinear ordinary or partial differential equations, integrodifferential equations, delay equations, etc. The main advantage of employing such techniques is that the problems are considered in a more realistic manner, and the solution obtained is a continuous function which is not the case for the solutions obtained by discretized solution techniques.

The methods that will be used throughout this study are, Adomian Decomposition Method (ADM), Variational Iteration Method (VIM) and Homotopy Perturbation Method (HPM). Each technique will be explained first, and then all will be applied to a selected case study related to the topic of the article.
