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

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.


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.
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 Coşkun investigated the elastic stability of a homogenous and nonhomogenous Euler beam [33][34][35][36][37][38][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. Eryılmaz and Atay investigated the buckling loads of Euler column with a continuous elastic restraint by using HAM [43].

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

Column under End Load Dependent on Direction
Consider a fixed-free, uniform homogeneous column of flexural rigidity EI, length which is subjected to a load that is dependent on the deflection and slope of the free end of the buckled column as shown in Figure 1 [44]. The governing buckling equation is given by [45] subject to the boundary conditions: where 1 and 2 are nondimensional parameters defined in Figure 1. The general solution of (1) is Substituting the general solution into the aforementioned boundary conditions, the stability criteria take the following form [45]: The stability criteria for the columns in Figure 1 are given in Table 1.

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: where is a nonlinear operator, denotes the independent variable, and ( ) is an unknown function. Liao [2] constructs the so-called zero order deformation equation as follows: where ∈ [0, 1] is the embedding parameter, ℏ is a nonzero auxiliary linear parameter, ( ) is nonzero auxiliary function, 0 ( ) is the initial guess of ( ), is an auxiliary linear operator, and ( ; ) is an unknown function. As increases from 0 to 1, ( ; ) varies from the initial guess 0 ( ) to the exact solution ( ). By expanding ( ; ) in a Taylor's series with respect to , one has International Scholarly Research Notices If the initial guess, auxiliary linear operator, auxiliary parameter, and auxiliary function are properly chosen the series (8) converges at = 1; then we have Let us define the vector Differentiating equation (6) -times with respect to and then setting = 0 and finally dividing by !, Liao has the so-called th order deformation equation: where 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 ( ) = 1. For any given operator , the term [ ⃗ −1 ( )] can be easily expressed by (12). So we can obtain 1 ( ), 2 ( ), . . . by means of solving the linear high order deformation equation (11). The th order approximation of ( ) is given by The approximate solution consists of ℏ, 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 ℏ. To obtain valid region of ℏ we first plot the so-called ℏ-curves of ( , ℏ). The valid region of ℏ 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 ( ), where ( ) 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].

HAM Formulation of the Problem
To solve (1) Using the embedding parameter ∈ [0, 1], we construct a family of equations: The high order deformation equation is as follows: where By using (17) and (18), choosing ( ) = 1, the high order deformation equation (17) yields the equation Starting with an initial approximation 0 ( ), we successively obtain ( ), = 1, 2, 3, . . ., by (19). The solution is of the form Since the governing equation (1) is a fourth order differential equation we choose the initial approximation as 0 ( ) = 3 + 2 + + a polynomial of third degree with four unknown coefficients , , , . Then we obtained ( ), = 1, 2, 3, . . ., by using the th order deformation equation (19) as follows: By substituting (22) into the boundary conditions, we obtained four homogeneous equations. By representing the coefficient matrix of these equations with [ ( , , 1 , 2 , ℏ)] we get the following equation: where , , , and are the unknown constants of initial approximation 0 ( ) and denotes the transpose of the matrix. For nontrivial solution the determinant of the coefficient matrix [ ( , , 1 , 2 , ℏ)] must vanish. Thus the problem takes the following form: The smallest positive real root of (24) is the critical buckling load. We defined the function ( , 0 , 1 , , ℏ) as follows: and then we pilot the ℏ-curves of the ( , , 1 , 2 , ℏ) in order to find convergence region of the ℏ.
The ℏ curves of ( , , 1 , 2 , ℏ) and ( , , 1 , 2 , ℏ) are given in Figure 2. The valid region of ℏ is the region which corresponds to the line segments nearly parallel to the horizontal axis. The valid region of ℏ is about −1.5 < ℏ < −0.4.
Finally we obtained the critical buckling loads from (24) for ℏ = −0.99. We compared the exact solutions given by Wang et al. [45] and HAM solutions in Tables 2 and 3.   Table 3: Comparison of exact and HAM solutions of critical buckling loads for the column in Figure 1(b) with 1 = − , 2 = 0, and = ∞.

Conclusions
In this work, a reliable algorithm based on the HAM to solve the critical buckling load of Euler column with elastic end International Scholarly Research Notices 5 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 ℏ. In general, by means of the so-called ℏ-curve, it is straightforward to choose a proper value of ℏ which ensures that the series solution is convergent. Figure 2 shows the ℏ-curves obtained from the th order HAM approximation solutions. In Tables 2 and  3 the critical buckling loads for various values of 0 , 1 , 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.