硬軟化桁架崩塌面分析

Abstract

傳統的極限分析有許多的限制,其中在載重形式上會給定單一比例載重,等於將多維的載重空間特化為一維來處理而且限制要單調加載,不能卸載或反覆循環;另外在組成律方面則必須限於剛塑性或完全彈塑性。然而在真實情況中,作用在結構上的載重往往沒有比例關係,為各自獨立的載重,且硬化、軟化的材料在結構物中也比比皆是。因此,在本文中會放鬆以上限制,以處理更接近真實狀況的問題。 極限分析最大的好處為,可以在不需給定路徑的情況下,直接快速地求得崩塌載重。在極限分析的領域中,最常見的作法是結合數學規劃法,列出最佳化問題來求解彈塑性結構的崩塌載重。在高維載重空間中,崩塌載重即是以崩塌面的型式存在,本文認為崩塌面不單單只是最佳化後的數值結果,而是在結構喪失靜不定性後具有跟降伏面對應的模式,意即崩塌面是載重空間的多面體,它的每一個面代表一種崩塌模式, 不但是一種崩塌機構,而且也同時滿足平衡條件與廣義應力允許條件,因此可以由參與該機構的各個桿件的降伏面合成出來。本文即利用此種載重空間中降伏面與崩塌面的關係,定義具有數學、物理意義的機構向量條件式並給出證明,藉由搜索崩塌機構的方式直接求解崩塌面的模式,建立出結構在載重空間中的安全區域,並由硬化、 軟化桁架的數值實例來說明本文的方法。

There are some severe restrictions in the traditional limit analysis. For the load type, the load space is one-dimensional and monotonic by imposing the restriction of uni-directional proportional loading. On the other hand, only can the constitutive law be perfectly elastic, either rigid-perfectly plastic or elastic-perfectly plastic. However, the load space is usually high dimensional in true situation, and the hardening/softening behavior prevails in almost all engineering structures. To deal with these problems, it is important to loose the above restrictions. The greatest advantage of limit analysis is that it can obtain collapse loads directly without giving loading paths. In the field of limit analysis, the most common approach is applying mathematical programming to calculating the collapse loads of elastoplastic structures, and the problem becomes an optimization problem of maximizing the collapse load. Collapse loads form a collapse surface in high dimensional load space; we deem a collapse surface to be not merely numerical results of optimization, but an equivalent model of yield surfaces once the structure in equilibrium loses its static indeterminancy and forms mechanisms. That is, we observe that each piece of a collapse surface represents a collapse mode of the structure and is corresponding to the yield surfaces of those structural members that form a collapse mechanism.Therefore, by using the relationship between the collapse surface and the yield surface in load space, we define the conditions of mechanism vectors which have mathematical and physical meaning. After searching each mechanism, the model of collapse surface can be constructed, and then we can construct the safety region in load space. Finally, some examples for truss structures with hardening and softening are given to verify this method.