Study On The Buckling Of Non-uniform Cross-section Slender Compressive Bar With Centrally Load

The members of non-uniform cross-section are commonly used as column in the design of various structures such as building frames, cranes, aircraft manufacturing, bridge structure, masts etc for its favorable capacity of carrying compressive loading. But the theory of elastic stability for arbitrary variable cross is not perfect. Since the cross-sectional properties of the column vary along its axis, the coefficients of the governing differential equation are variable and the solutions of differential equations are difficult in practical projects. Historically, solutions of elastic stability have developed many methods, such as theoretical analysis method; power series solutions; semi-analytic method; the finite element method etc. The main purpose of this paper is to investigate buckling of the slender bars which have involved the main methods of non-uniform cross-section in the theory of small deflection.In this paper, there are four aspects I have been done. Firstly, I introduce the buckling of the two kinds of common variable cross-section along the axis of the compressive rods, linear (the two ends hinged, the one end fixed and another end elastic bearing) and stepped (the two ends hinged) with the regular cross-sectional shapes considered. In theoretical analysis method, the differential equations are obtained by dual property of critical conditions and solved by Bessel function. Secondly, the other method of solving equations is introduced. Power series solutions have two methods which are Timoshenko power solution and Riz power solution. The differential equations in the practical project are implicated to solve in most condition and the approximate methods are applied in common condition. The power series solution is useful, but the workloads are very huge, especially when the numbers of stage of variable cross-section are increased. As an illustration, the non-uniform cross-section of the two ends hinged constrained is analyzed. Thirdly, the other method of analysis which is the finite element method is explained in brief. The finite element method based on rigorous mathematic theory is extensive adapted to the complicated geometric configurations. It is also high efficiency, especially development of the computer technology. In this paper, the finite element equation of geometric non-linearity is obtained. It is modified to obtain the non-uniform cross-section stiffness matrices and is the very useful in project analysis.In the end, the common software of ANSYS is introduced to solve the eigenvalue problem of variable cross-section. According to the needing, the two ends hinged is analyzed by ANSYS for verification purposes only and both analytical and numerical results correlate with reasonable accuracy. The results are based on the assumption that material failure occurs for small lateral displacements.