Development of Numerical Method with High Accuracy using Incremental Perturbation Method for Predicting Critical Behaviors of Spatial Frames

在结构设计中，尽可能准确地预测结构的最终状态是必不可少的。最确定的终态是结构在各种外扰作用下的力学临界状态。本研究的目的在于发展一套能模拟空间钢框架各种临界行为的高精度数值分析方法。在分析方法的推导过程中，对三维有限转动进行了细致的处理，并采用了增量摄动法来保证临界行为的模拟精度。本文的主要研究成果如下：（1）本文改进的梁柱有限元法充分模拟了钢节点木结构单层网壳的临界受力性能。在足尺试验中观察到的节点转动屈曲几乎是完美的，通过这种方法。在数值试验中首次发现了节点转动屈曲的二次分叉现象。(2)本文提出了一种数值方法，它能够很好地预测具有弯扭的弹性空间框架的临界性能。翘曲函数用于估计单元的弯曲扭转。用该方法分析了弹性梁的侧向屈曲行为。(3)为了在结构动力数值分析中计算合适的时间长度，提出了一种有效的截断误差控制策略。该策略估计相对截断误差的奇次幂项和偶次幂项的摄动展开系列，并被证明是有效的和准确的线性单自由度系统的自由振动响应。

In structural design, it is indispensable to predict the final state of the structure as accurately as possible. The most definite final state is a mechanical critical state of structure under various external disturbances. The aim in this research is to develope the high-precise numerical analysis method that is able to simulate various critical behaviors of spatial steel frames. In the derivation of the analysis method, 3-dimensional finite rotation is very carefully dealt and the incremental perturbation method is adopted to ensure the accuracy of simulation of the critical behaviors. The results in this research are summarized as follows :(1) A critical behavior of timber single-layr lattice dome with steel joint-members is simulated sufficiently by the beam-column FEM method refined in this research. The rotational buckling of joints observed in a full-scale experiment is almost perfectly by this method. The second bifurcation of rotational buckling of joints is shown at first in this numerical experiment.(2) A numerical method is proposed which is able to predict with good accuracy the critical behavior of elastic spatial frames with flexural torsion. The warping function is used for estimating the flexural torsion in elements. The lateral buckling behaviors of elastic beams are analyzed by the proposed method.(3) An effective strategy for truncation error control is proposed in order to compute an appropriate value of the time length in a numerical dynamic analysis of structure. The strategy estimates relative truncation errors both in odd power terms and in even power terms of perturbation expansion series, and is demonstrated to be efficient and accurate in a free vibration response of a linear single-degree-of-freedom system.