具軸力變形影響之塑性極限面

現今土木的設計潮流以極限設計為主軸，然而在分析方面，主要仍是利用彈性分析作為理論根據，故發展一套與極限設計能夠相互配合的極限分析法是很重要的。塑性之重要特徵為路徑相依，即構架行為與載重是路徑相關的。對於一個材料施加載重，則當內力達到降伏點時會形成一個「初始降伏面」，後續繼續加載，則會依不同路徑而產生無窮多個「接續降伏面」。最後，當載重大到產生靜態崩坍時，各路徑之崩坍點會圍出一個唯一的「塑性極限面」。此種分析方式不必經過繁複冗長的歷時分析，並且可以清楚知道不同路徑最後產生的崩塌模態會是如何。計算過程中，為了簡化分析過程，通常只考量彎矩變形的影響。然而事實上，許多構件都有承受不小的軸力，無論是壓或拉力，故在分析上，「具軸力變形影響之塑性極限面」是更能夠反應現實情況的。本文主要在探討將軸力變形納入考量之塑性極限面的計算方法，本來若是不利用矩陣運算，則納入軸力會使問題非常複雜。但若是利用線彈性完全塑性之組成模式，配合線性代數的運算，以及正齊一次性之不等式，則可以用較為簡潔的方式計算得出崩塌模態及塑性極限面。而此方法主要是解決構架具軸力彎矩的情況，若要應用於桁架或梁亦可套用，反而會使問題更加單純，不失為一個可靠的方法。

關鍵字

極限分析；軸力變形；構架；彈塑性；軸彎

並列摘要

Nowadays, the design trend of civil engineering is based on the limit design.However, in terms of analysis, it is still mainly based on the use of elastic analysis as a theoretical basis. Therefore, it is very important to develop a limit analysis method that can cooperate with the limit design. The important feature of plasticity is path dependence, ie, the behavior of the frame and the loads are path-dependent. When loads are applied to a material, an "initial yield surface" is formed when the internal force reaches the yield point. Further, subsequent loading will result in an infinite number of "subsequent yield surfaces" which are depended on the paths. Finally, when the loads are big enough to reach the static collapse, the collapse point of each path will construct an unique "plastic limit surface." This kind of analysis does not have to go through a lengthy and time-consuming analysis, and it is clear how different paths eventually result in different collapse modalities. During the process of calculation, in order to simplify the whole analysis process, it is usually considered the influence of the bending deformation only. However, in fact, many components are borne with axial forces which are not small, whether they are pressure or tension. Therefore, in analysis, "plastic limit surface with the effect of axial deformation" is more capable to reflect the reality. This article mainly discusses the calculation method of plastic limit surface that takes axial deformation into consideration. If we do not use matrix operations, incorporating axial forces will make the problem very complicated. However, by using linearly elastic-perfectly plastic model, the linear algebra operation, as well as the inequalities of positive homogeneity of degree one, the collapse modalities and the plastic limit surface can be calculated in a more concise manner. This method is mainly used to solve the situation that the structure has axial forces and bending moments. If it is applied to trusses or beams, it will make the problem more simple. So, we can say that it is also a reliable and useful method.