On Critical Buckling Loads of Columns under End Load Dependent on Direction

Abstract

Most of the phenomena of various fields of applied sciences are nonlinear problems. Recently, various types of analytical approximate solution techniques were introduced and successfully applied to the nonlinear differential equations. One of the aforementioned techniques is the Homotopy analysis method (HAM). In this study, we applied HAM to find critical buckling load of a column under end load dependent on direction. We obtained the critical buckling loads and compared them with the exact analytic solutions in the literature.

Musa Basbük 1 and Aytekin Eryilmaz 1 and M. Tarik Atay 2

Academic Editor:George Fikioris

1, Department of Mathematics, Nevsehir Haci Bektas Veli University, 50300 Nevsehir, Turkey
2, Department of Mechanical Engineering, Abdullah Gül University, 38039 Kayseri, Turkey

Received 30 April 2014; Revised 18 September 2014; Accepted 29 September 2014; 29 October 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Nonlinear differential equations arise in a wide range of scientific studies from physics to biology, from engineering to economics. However it is not possible to find an exact analytical solution for the nonlinear equations every time. Analytical approximate solution techniques such as perturbation and nonperturbative techniques have been used to solve these nonlinear equations in recent years. These techniques have been widely applied in many fields of engineering and science. Neither perturbation techniques nor nonperturbation techniques ensure the convergence of solution series and adjust or control the convergence region and rate of approximation series.

On the other hand an analytic approach, the homotopy analysis method (HAM) which is proposed by Liao, provides a convenient way to adjust and control the convergence region and the rate of approximation series by the auxiliary parameter [figure omitted; refer to PDF] and auxiliary function [figure omitted; refer to PDF] [1, 2]. HAM has been applied successfully to obtain the series solution of various types of linear and nonlinear differential equations such as the viscous flows of non-Newtonian fluids [3-13], the KdV-type equations [14-16], nanoboundary layer flows [17], nonlinear heat transfer [18, 19], finance problems [20, 21], Riemann problems related to nonlinear shallow water equations [22], projectile motion [23], Glauert-jet flow [24], nonlinear water waves [25], ground water flows [26], Burgers-Huxley equation [27], time-dependent Emden-Fowler type equations [28], differential difference equation [29], difference equation [30], Laplace equation with Dirichlet and Neumann boundary conditions [31], and thermal-hydraulic networks [32].

One of the fields that nonlinear differential equations arise is the stability analysis of columns in mechanical engineering. Many researchers applied analytical approximate solution techniques to the stability analysis of various types of columns with different end conditions. Atay and Coskun investigated the elastic stability of a homogenous and nonhomogenous Euler beam [33-39]. Pinarbasi investigated the buckling analysis of nonuniform columns with elastic end restraints [40]. Huang and Luo determined critical buckling loads of beams with arbitrarily axial inhomogeneity [41]. Recently, Yuan and Wang [42] solved the postbuckling differential equations of extensible beam-columns with six different cases. Eryilmaz and Atay investigated the buckling loads of Euler column with a continuous elastic restraint by using HAM [43].

In this study we apply HAM to find the critical buckling load of a column under end load dependent on direction.

2. Column under End Load Dependent on Direction

Consider a fixed-free, uniform homogeneous column of flexural rigidity EI, length [figure omitted; refer to PDF] which is subjected to a load [figure omitted; refer to PDF] that is dependent on the deflection and slope of the free end of the buckled column as shown in Figure 1 [44].

Buckling of various types of columns [45].

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The governing buckling equation is given by [45] [figure omitted; refer to PDF] subject to the boundary conditions: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are nondimensional parameters defined in Figure 1. The general solution of (1) is [figure omitted; refer to PDF] Substituting the general solution into the aforementioned boundary conditions, the stability criteria take the following form [45]: [figure omitted; refer to PDF] The stability criteria for the columns in Figure 1 are given in Table 1.

Table 1: Stability criteria for the various columns given in Figure 1.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Stability criteria
[figure omitted; refer to PDF]	0	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0	[figure omitted; refer to PDF]

3. Basic Idea of Homotopy Analysis Method (HAM)

Liao introduced the homotopy analysis method (HAM) in [1, 2]. To demonstrate the homotopy analysis method, let us consider the following differential equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a nonlinear operator, [figure omitted; refer to PDF] denotes the independent variable, and [figure omitted; refer to PDF] is an unknown function. Liao [2] constructs the so-called zero order deformation equation as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the embedding parameter, [figure omitted; refer to PDF] is a nonzero auxiliary linear parameter, [figure omitted; refer to PDF] is nonzero auxiliary function, [figure omitted; refer to PDF] is the initial guess of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is an auxiliary linear operator, and [figure omitted; refer to PDF] is an unknown function. As [figure omitted; refer to PDF] increases from 0 to 1, [figure omitted; refer to PDF] varies from the initial guess [figure omitted; refer to PDF] to the exact solution [figure omitted; refer to PDF] . By expanding [figure omitted; refer to PDF] in a Taylor's series with respect to [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF] If the initial guess, auxiliary linear operator, auxiliary parameter, and auxiliary function are properly chosen the series (8) converges at [figure omitted; refer to PDF] ; then we have [figure omitted; refer to PDF] Let us define the vector [figure omitted; refer to PDF] Differentiating equation (6) [figure omitted; refer to PDF] -times with respect to [figure omitted; refer to PDF] and then setting [figure omitted; refer to PDF] and finally dividing by [figure omitted; refer to PDF] , Liao has the so-called [figure omitted; refer to PDF] th order deformation equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] In order to obey both of the rule of solution expression and the rule of the coefficient ergodicity [2], the corresponding auxiliary function is determined by [figure omitted; refer to PDF] . For any given operator [figure omitted; refer to PDF] , the term [figure omitted; refer to PDF] can be easily expressed by (12). So we can obtain [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] by means of solving the linear high order deformation equation (11). The [figure omitted; refer to PDF] th order approximation of [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] The approximate solution consists of [figure omitted; refer to PDF] , which is a cornerstone of the HAM in determining convergence of series solution rapidly. We may adjust and control the convergence region and rate of the solution series (14) by means of the auxiliary parameter [figure omitted; refer to PDF] . To obtain valid region of [figure omitted; refer to PDF] we first plot the so-called [figure omitted; refer to PDF] -curves of [figure omitted; refer to PDF] . The valid region of [figure omitted; refer to PDF] is the interval, which corresponds to the line segments nearly parallel to the horizontal axis.

Theorem 1 (Convergence Theorem [2]).

As long as the series (9) converges to [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is governed by the high order deformation equation (11) under the definitions (12) and (13), it must be the exact solution of (1) subject to the boundary conditions (2).

For the proof see [2].

4. HAM Formulation of the Problem

To solve (1) by means of homotopy analysis method, we define the nonlinear operator [figure omitted; refer to PDF] and the auxiliary linear operator [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Using the embedding parameter [figure omitted; refer to PDF] , we construct a family of equations: [figure omitted; refer to PDF] The high order deformation equation is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By using (17) and (18), choosing [figure omitted; refer to PDF] , the high order deformation equation (17) yields the equation [figure omitted; refer to PDF] Starting with an initial approximation [figure omitted; refer to PDF] , we successively obtain [figure omitted; refer to PDF] , by (19). The solution is of the form [figure omitted; refer to PDF] Since the governing equation (1) is a fourth order differential equation we choose the initial approximation as [figure omitted; refer to PDF] a polynomial of third degree with four unknown coefficients [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then we obtained [figure omitted; refer to PDF] , by using the [figure omitted; refer to PDF] th order deformation equation (19) as follows: [figure omitted; refer to PDF] Ten iterations are conducted and we get [figure omitted; refer to PDF] By substituting (22) into the boundary conditions, we obtained four homogeneous equations. By representing the coefficient matrix of these equations with [figure omitted; refer to PDF] we get the following equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the unknown constants of initial approximation [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denotes the transpose of the matrix. For nontrivial solution the determinant of the coefficient matrix [figure omitted; refer to PDF] must vanish. Thus the problem takes the following form: [figure omitted; refer to PDF] The smallest positive real root of (24) is the critical buckling load. We defined the function [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] and then we pilot the [figure omitted; refer to PDF] -curves of the [figure omitted; refer to PDF] in order to find convergence region of the [figure omitted; refer to PDF] .

The [figure omitted; refer to PDF] curves of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are given in Figure 2. The valid region of [figure omitted; refer to PDF] is the region which corresponds to the line segments nearly parallel to the horizontal axis. The valid region of [figure omitted; refer to PDF] is about [figure omitted; refer to PDF] .

Figure 2: The [figure omitted; refer to PDF] curves of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Finally we obtained the critical buckling loads from (24) for [figure omitted; refer to PDF] . We compared the exact solutions given by Wang et al. [45] and HAM solutions in Tables 2 and 3.

Table 2: Comparison of exact and HAM solutions of critical buckling loads for the column in Figure 1(a) with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	Critical load [figure omitted; refer to PDF]
Exact solution [45]	HAM solution
0.1	2.86277	2.86277
0.2	2.65366	2.65366
0.3	2.49840	2.49840
0.4	2.38064	2.38064
0.5	2.28893	2.28893
0.6	2.21571	2.21571
0.7	2.15598	2.15598
0.8	2.10638	2.10638
0.9	2.06453	2.06453
1	2.02876	2.02876

Table 3: Comparison of exact and HAM solutions of critical buckling loads for the column in Figure 1(b) with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	Critical load [figure omitted; refer to PDF]
Exact solution [45]	HAM solution
0.1	1.428870	1.428870
0.2	1.313840	1.313840
0.3	1.219950	1.219950
0.4	1.142230	1.142230
0.5	1.076870	1.076870
0.6	1.021110	1.021110
0.7	0.972911	0.972911
0.8	0.930757	0.930757
0.9	0.893519	0.893519
1	0.860334	0.860334

5. Conclusions

In this work, a reliable algorithm based on the HAM to solve the critical buckling load of Euler column with elastic end restraints is presented. Two cases are given to illustrate the validity and accuracy of this procedure. The series solutions of (1) by HAM contain the auxiliary parameter [figure omitted; refer to PDF] . In general, by means of the so-called [figure omitted; refer to PDF] -curve, it is straightforward to choose a proper value of [figure omitted; refer to PDF] which ensures that the series solution is convergent. Figure 2 shows the [figure omitted; refer to PDF] -curves obtained from the [figure omitted; refer to PDF] th order HAM approximation solutions. In Tables 2 and 3 the critical buckling loads for various values of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] obtained by HAM are tabulated. The HAM solutions and the exact solutions in [45] are compared. As a result HAM is an efficient, powerful and accurate tool for buckling loads of columns.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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