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。最确定的最终状态是结构在各种外部扰动作用下的力学临界状态。本研究的目的是发展能够模拟空间钢框架各种临界行为的高精度数值分析方法。在分析方法的推导中，仔细处理了三维有限旋转，并采用增量摄动法保证了临界行为模拟的准确性。研究结果总结如下：(1)本文改进的梁柱有限元法较好地模拟了带钢节点的木质单层格构穹顶的临界性能。在全尺寸实验中观察到的节点旋转屈曲几乎是完美的。在数值试验中，首次出现了节点旋转屈曲的二次分岔现象。(2)提出了一种能够较准确地预测具有弯曲扭转的弹性空间框架临界性能的数值方法。翘曲函数用于估算构件的弯曲扭转。用该方法对弹性梁的侧向屈曲行为进行了分析。(3)为了在结构数值动力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。该方法对扰动展开级数的奇幂项和偶幂项的相对截断误差进行了估计，并在线性单自由度系统的自由振动响应中证明了该方法的有效性和准确性。

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.