The calculation method of Plastic collapse

在塑性力學中，極限分析是很重要的主題。塑性的重要特徵就是「路徑對路徑」，位移與載重的關係是「路徑相關的」。首先彈性達到降伏時，會出現一個「初始降伏面」，接著就進去塑性的階段，隨著載重的路徑不同，會有對應的「接續降伏面」，接續降伏面是無窮多的，也就是不唯一，但在最終將可以建立出在載重空間下唯一的一個「崩塌面」，也就是說它是與路徑無關的，意思用個比較貼切的成語，就是所謂的「殊途同歸」吧！ 這部分前人做過很多的探討，包括利用最佳化問題的Karush-Kuhn-Tucker（KKT）條件分析，與設定目標的不同，發展出單目標最佳化，與多目標最佳化等，但最大的缺點就是式子最後都會過於的複雜。本文主要是在研究載重空間下崩塌面的計算方法，利用片段線性模型來模擬線彈性材料，然後建構出我們的降伏式，接著利用本文的計算方法，跳過接續降伏面，便可直接連結到崩塌面，節省掉很多繁雜的計算。這個計算方法不論是桁架、梁或者是剛架，都可以使用。在剛架的部分，我們還有去考慮至軸力彎矩互制的影響，使得結果更準確。In plasticity, limit analysis is a very important topic.The important characteristics of plasticity is ""path to path ,"" and the relationship of load and displacement is ""path-related."" First, the behavior is elastic until yield occurs, when there will be an ""initial yield surface"" in the state space. Then, in the plastic stage, the yield surface is evolving in the state space and the so-called “subsequent yield surfaces"" are infinite in number, depending upon the load paths. But it will eventually be able to build in the load space a unique ""collapse surface,"" which means that it is path-independent. It is the same collapse surface to which infinitely many different load paths together with different evolving subsequent yield surfaces will eventually reach! Our predecessors, engineers and scholars, have done a lot of discussions, including the use of optimization problem, Karush-Kuhn-Tucker (KKT).conditions, and the setting of different goals, the development of single-objective optimization, and multi-objective optimization, etc. but the biggest drawback is the final formula are too complicated. In view of these, the present thesis focuses on the study of calculating the collapse surface on the load space for trusses and frames. The method developed herein can calculate the collapse surface directly on the basis of the compatibility and equilibrium equations of the structure and also on the algebraic part of the constitutive relations, skipping all subsequent yield surfaces and thereby saving out a lot of complicated calculations. This calculation method may be used for trusses, beams, or frames. In the part of frames, we attempt to consider the axial force and moment interaction in order that the result is more accurate