组合载荷作用下Ludwick型材料悬臂柱后屈曲载荷优化算法

Optimization Algorithm of Post-Buckling Load for Cantilever Columns with Ludwick Type Material under a Combined Load

目的 针对组合载荷作用下的Ludwick型材料悬臂柱,提出不同端部角度下屈曲载荷的优化算法。方法 由Ludwick本构关系推导弯矩-曲率之间的表达式,以曲率半径表示子段在局部坐标系中的位置;根据局部坐标系与整体坐标系之间的变换式,得出整体坐标系下子段两端点的坐标关系,并通过迭代实现悬臂柱两端点坐标之间的联系;以无量纲屈曲载荷为设计变量,固支端转角形成目标函数,建立用于求解屈曲载荷的优化算法,并对比两种不同载荷作用下的无量纲屈曲载荷以及水平位移和垂直位移。结果 随着端部角度的增大,悬臂梁自由端处水平位移减小,垂直位移增大;两种不同载荷作用下无量纲屈曲载荷均表现为:当端部转角从10°增加到40°时,无量纲屈曲载荷减小;当端部转角从40°增加到90°时,无量纲屈曲载荷增大。结论 相较于数值方法,该优化算法的优势在于计算不同端部角度下屈曲载荷、水平位移及垂直位移时更加直接快速;硬化指数是决定后屈曲平衡路径是否稳定的关键因素,笔者取硬化指数为1/0.9;单一载荷或组合载荷作用下Ludwick型材料悬壁柱的后屈曲平衡路径均为不稳定的。

The purpose of this paper is to propose the optimization algorithm of buckling load under different end angles for Ludwick cantilever columns by a combined load.The expression between bending moment and curvature is derived from Ludwick constitutive relationship,and the position of the sub segment in the local coordinate system is expressed by the radius of curvature.According to the transformation between the local coordinate system and the global coordinate system,the coordinate relationship between the two endpoints of the sub segment in the global coordinate system is obtained,and the association between the two endpoints coordinates of the cantilever column is realized through iteration.The dimensionless buckling load as the design variable and the fixed end angle as the objective function,the optimization algorithm for solving the buckling load is established,and the dimensionless buckling load,horizontal displacement and vertical displacement under two different loads are compared.The results show that with the increase of the end rotation angle,the horizontal displacement at the free end of the cantilever beam decreases and the vertical displacement increases.The non-dimensional buckling loads decrease when the corner angle increases from 10° to 40°,but increase when the corner angle increases from 40° to 90°.Compared with the numerical method,this optimization algorithm is more direct and fast in calculating the buckling load,horizontal displacement and vertical displacement at different end angles.It is concluded that the hardening index is a key factor to determine whether the post-buckling equilibrium path is stable.The hardening index in this paper is 1/0.9,the post-buckling equilibrium path is unstable for Ludwick material cantilever columns subjected to a single or a combined load.