Buckling of Nonprismatic Column on Varying Elastic Foundation with Arbitrary Boundary Conditions

Citation: Ahmad A. Ghadban, Ahmed H. Al-Rahmani, Hayder A. Rasheed, and Mohammed T. Albahttiti, “Buckling of Nonprismatic Column on Varying Elastic Foundation with Arbitrary Boundary Conditions,” Mathematical Problems in Engineering, vol. 2017, Article ID 5976098, 14 pages, 2017. doi:10.1155/2017/5976098Buckling of nonprismatic single columns with arbitrary boundary conditions resting on a nonuniform elastic foundation may be considered as the most generalized treatment of the subject. The buckling differential equation for such columns is extremely difficult to solve analytically. Thus, the authors propose a numerical approach by discretizing the column into a finite number of segments. Each segment has constants  (modulus of elasticity),  (moment of inertia), and  (subgrade stiffness). Next, an exact analytical solution is derived for each prismatic segment resting on uniform elastic foundation. These segments are then assembled in a matrix from which the critical buckling load is obtained. The derived formulation accounts for different end boundary conditions. Validation is performed by benchmarking the present results against analytical solutions found in the literature, showing excellent agreement. After validation, more examples are solved to illustrate the power and flexibility of the proposed method. Overall, the proposed method provides reasonable results, and the examples solved demonstrate the versatility of the developed approach and some of its many possible applications

On Critical Buckling Loads Of Euler Columns With Elastic End Restraints

I n recent years, a great number of analytical approximate solution techniques have been introduced to find a solution to the nonlinear problems that arised in applied sciences. One of these methods is the homotopy analysis method HAM . HAM has been successfully applied to various kinds of nonlinear differential equations. In this paper, HAM is applied to find buckling loads of Euler columns with elastic end restraints. The critical buckling loads obtained by using HAM are compared with the exact analytic solutions in the literature. Perfect match of the results veries that HAM can be used as an efficient, powerfull and accurate tool for buckling analysis of Euler columns with elastic end restraint

Analysis of tilt‐buckling of euler columns with varying flexural stiffness using homotopy perturbation method

In this paper, the Homotopy Perturbation Method (HPM), is introduced for elastic stability analysis of tilt‐buckled columns with variable flexural stiffness. Buckling loads and corresponding mode shapes are determined considering different types of variations in flexural stiffness of columns. The proposed approach is an efficient technique for the elastic stability analysis of specified problems. First published online: 09 Jun 201

Investigation of nonuniform rod elements stability by direct integration method

The calculation for the equilibrium stability of a rod system with variable stiffness by the direct integration method is proposed. The method is based on the exact solutions of the corresponding differential equations. Using this method, the rod stability problem with arbitrary continuous variable flexural stiffness is solved. The formulas for the parameters of the frame structure stability are expressed in the analytical form and the method of their numerical implementation is provided. The frame structures with binomial distribution of stiffness are considered

Buckling Analyses of Axially Functionally Graded Nonuniform Columns with Elastic Restraint Using a Localized Differential Quadrature Method

A localized differential quadrature method (LDQM) is introduced for buckling analysis of axially functionally graded nonuniform columns with elastic restraints. Weighting coefficients of differential quadrature discretization are obtained making use of neighboring points in forward and backward type schemes for the reference grids near the beginning and end boundaries of the physical domain, respectively, and central type scheme for the reference grids inside the physical domain. Boundary conditions are directly implemented into weighting coefficient matrices, and there is no need to use fictitious points near the boundaries. Compatibility equations are not required because the governing differential equation is discretized only once for each reference grid using neighboring points and variation of flexural rigidity is taken to be continuous in the axial direction. A large case of columns having different variations of cross-sectional profile and modulus of elasticity in the axial direction are considered. The results for nondimensional critical buckling loads are compared to the analytical and numerical results available in the literature. Some new results are also given. Comparison of the results shows the potential of the LDQM for solving such generalized eigenvalue problems governed by fourth-order variable coefficient differential equations with high accuracy and less computational effort