Euler-Bernoulli梁的高阶二次摄动解及收敛性讨论

首次用解析的方式给出了Euler-Bernoulli梁后屈曲与非线性弯曲问题的高阶二次摄动解答.假定梁的中线不可伸长,用精确曲率公式与能量变分原理导出了非线性Euler-Bernoulli梁的模型.通过与精确解或高阶摄动解的比较,讨论了二次摄动解答的收敛性及适用域.得到主要结论如下:低阶摄动解适用于描述梁的初始后屈曲阶段及初始非线性弯曲阶段;更高阶次的摄动解适用于描述梁的深度后屈曲以及深度非线性弯曲.从这个意义上去说,该文不仅仅指出某些文献上的部分结果不精确是由于摄动解答超出了其特定的适用域,并且还进一步发展与完善了二次摄动法.

High-Order Analytical Solutions and Convergence Discussions of the 2-Step Perturbation Method for Euler-Bernoulli Beams

High-order analytical solutions of the 2-step perturbation method were first obtained for post-buckling and nonlinear bending of Euler-Bernoulli beams. The nonlinear model with centerline inextensibility was derived with the exact curvature expression according to the energy variational principle. Based on the comparison with the exact solutions or high-order perturbation solutions, the asymptotic property and the suitable range of 2-step perturbation solutions were also discussed. The results show that, the lower-order perturbation solutions are suitable for the initial post-buckling stage and the initial nonlinear bending stage, and the higher-order perturbation solutions are necessary for the late post-buckling stage and the highly nonlinear bending stage. Therefore, the reason why some previous perturbation solutions are inaccurate lies in the offside beyond suitable ranges, and the 2-step perturbation method is developed and improved herein.