Timoshenko梁单元的有限元屈曲分析程序解

FEM Program Solutions of Buckling for Timoshenko Beam Element

为提高欧拉梁理论在梁类结构屈曲失稳载荷求解的适用性,提出了一种基于Timoshenko梁单元的数值求解方法。首先根据最小势能原理推导出了梁单元的弹性刚度矩阵与几何刚度矩阵,建立了有限元屈曲失稳求解方程,并采用Matlab软件对其进行数值求解程序的开发。通过将数值解与欧拉公式解进行对比分析验证表明,当梁柔度系数较大时,两者之间较小的相对误差验证了本文有限元程序解的精确性;当梁柔度系数较小时,两者之间相对误差较大。同时,从梁单元有限元理论角度,给出了两者相对误差产生的理论原因。最后,以欧拉公式解相对程序解的误差小于5%为基准,给出了不同边界条件下对应柔度系数的推荐值。

In order to improve the applicability of Euler beam theory in solving the critical buckling load of beam structures,a numerical solution method based on the Timoshenko beam element is proposed.Firstly,the element elastic stiffness matrix and element geometric stiffness matrix of the beam element were derived according to the principle of minimum potential energy,the finite element buckling instability equation was established,and the numerical solution program was developed based on Matlab.By comparing the numerical solution with the Euler formula solution,the results show that when the compliance coefficient of the beam is large,the smaller relative error between the numerical solution and the Euler formula solution matches the beam theory,which verifies the accuracy of the numerical solution in this paper;however,the results show that when the compliance coefficient of the beam is small,the relative error is large.Meanwhile,the theoretical reasons for the relative errors are given by using the finite element theory of the beam element.Finally,taking the error of Euler formula solution relative to program solution less than 5%as the benchmark,the recommended value of the corresponding compliance coefficient under different boundary conditions is given.