结构稳定理论伽辽金法

用伽辽金法推导两端固支杆的临界荷载及其ANSYS命令流主讲人：郑如杰【例2.4】如图所示，用伽辽金法试求两端固支杆的临界荷载解：设构件变形的曲线方程1122yaa432212xlxlx5423322254xlxlxlx和都满足全部的边界条件12(0)y()0yl(0)y()0yl(0)()0iil(0)()0iil即即则伽辽金的方程组为(4)2(y)yLy22222312[242(66)]a[120(2)2(203012)]axxllxlxlxlxl2252261120(y)(0.80.0191)(00)0lLdxallall2262272120(y)(06)(0.57140.006349)0lLdxallall由上式可确定和的比值，从而确定y1a2a则上式中的参数，不全为零的条件是其系数行列式为零，为此，令其系数行列式为零即得稳定方程为展开后解得=01a2a2241.99l由2/FEI得最小根为241.99crEIFl239.48crEIFl与精确解相比，偏差大约为6.3%ANSYS命令流FINI/CLEAR,START/PREP7ET,1,BEAM189!定义单元SECTYPE,1,BEAM,I,,0!定义截面SECOFFSET,CENT!定义横截面偏移SECDATA,0.066,0.066,0.102,0.006,0.006,0.006,0,0,0,0!定义截面尺寸MP,EX,1,70E9!定义弹性模量MP,NUXY,1,0.3!定义泊松比k,1,0,0,0,!定义关键点k,2,0,0,4,k,3,0,0.5,0.5,!定义方向关键点l,1,2,!连接1,2点成线段LATT,1,,1,,3,,1!指定线的单元属性LESIZE,all,,,20,,,,,1!指定所选线上单元数LMESH,all!网格划分DK,1,all!加约束DK,2,UXDK,2,UYDK,2,ROTXDK,2,ROTYDK,2,ROTZF,2,FZ,-1!加力/SOL!静力计算ANTYPE,STATICPSTRES,ON!计入预应力SOLVEFINISH/SOL!线性屈曲分析ANTYPE,BUCKLEBUCOPT,LANB,3,0,0!指定提取屈曲阶数前三阶MXPAND,3,0,0,1,0.001,!将前三阶振型写入结果文件SOLVEFINISH/POST1!进入一般后处理SET,FIRST!读入第一步结果PLDISP,1!显示结构一阶模态图，保留未变形结构轮廓SET,NEXT!读入下一步结果PLDISP,2!显示结构二阶模态图，保留未变形结构轮廓SET,NEXTPLDISP,3!显示结构三阶模态图，保留未变形结构轮廓FINISH模态图一阶模态图临界荷载(FACT)为49809.7N与精确解49937.56N相比，偏小约0.3%二阶模态图临界荷载为101584N三阶模态图临界荷载为197505NTHANKS