【作者】 谢扬波；

【导师】 陈山林；

【作者基本信息】 重庆大学 ， 工程力学， 2009， 硕士

【摘要】 变截面细长压杆由于其节省材料且具有良好的受力特性,故在起重机械、桥梁结构、飞机结构中被大量使用,但是与之相关的稳定性分析理论极不完善,这主要是工程中变截面失稳问题所求得的微分方程不是常系数,求解时往往要遇到数学上的困难。常用变截面压杆稳定计算的方法有:静力法、能量法、半解析求解法、有限单元法等。文中基于小挠度理论简要分析了变截面压杆失稳问题,展开了对变截面压杆构件失稳承载力的初步研究。本文主要的工作是:(1)分析了两种常见的变截面情况,沿杆轴线线性变化的变截面压杆在两种不同支承条件(两端铰支、一端固定一端弹性支承)的失稳及沿杆轴线阶梯形变截面压杆(两端铰支)失稳问题进行了理论分析及数值求解。文中采用静力法,根据临界状态平衡的二重性,建立了平衡微分方程,将二阶偏微分方程化为贝塞尔函数进行理论分析,采用Matlab图形法及C++编程求得几种常见截面形式临界荷载。(2)采用能量法分析了变截面压杆失稳。工程中大部分变截面压杆失稳问题的求解在数学上存在困难或过于复杂,一般采用近似方法求解。求解变截面压杆的稳定问题中对于简单的构件采用能量法较为实用且具有较好的精度,但对变截面段数较多的构件所涉及的工作量较大,文中采用能量法分析了两端铰支变截面压杆失稳问题,并与前节静力分析计算结果进行了对比,结果表明所得的临界荷载与静力分析求得的临界荷载吻合较好。(3)简要分析了变截面压杆另一种数值计算方法:有限单元法。有限单元法具有对复杂几何构形的适应性,对于各种物理问题的可应用性,建立于严格理论基础上的可靠性,适合计算机实现的高效性等。随着有限元单元法通过计算机的实现,其具有效率高,更适合工程结构的分析,在文中采用了三维退化梁单元几何非线性有限元方程,对几何非线性方程进行适当改造,得到了变截面压杆的刚度矩阵,很适合于工程结构的总体分析。(4)采用ANSYS有限元分析软件进行特征值屈曲分析。文中分析了两端铰支的变截面压杆失稳构件的屈曲变形模态和稳定承载力,并将所得的结果与静力理论分析值进行了比较,验正了静力分析结果。更多还原

【Abstract】 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.更多还原