在塑性力學的領域中，極限分析是一個很重要的主題。塑性反應是路徑相關的，但極限分析的最大好處是，對於塑性結構可以在不給定路徑的情況下，直接快速地求得崩塌載重。在過去，最常見的作法即是結合數學規劃法，列出最佳化問題來求解彈塑性結構的崩塌載重。

而最近，有人透過片段線性多降伏面模型描述彈塑性材料的組成律，並且透過降伏式與崩塌平衡機構的關係，發展了一套方法去求解結構物的崩塌面，並且建立出其在外力空間的安全載重區域。但是對於軟化材料，如何正確地去描述組成律，還有軟化結構物的安全載重空間分析，有過於保守的問題需要改進和研究。

在本文裡面，我們使用元件的並聯描述軟化桿件的組成律，並且使其中一個並聯元件具有耗散非正的特性。然後我們將軟化桿件視為一個封閉系統仍然會遵守熱力學定律。透過這樣的觀念，我們可以透過不同的機械模型像是雙線性模型或是完全彈塑性模型來描述軟化桿件的組成律。

對於軟化結構物的安全載重空間分析，我們認為使用軟化桿件的殘餘應力強度去計算崩塌面是過於保守的，而且透過結構設計，整體結構物會發生互制的行為。基於上面兩點的考量，本文加入了相對應的位移空間，當無限制變形(塑流平台)發生，對應到的即是載重空間的崩塌面去重新定義軟化結構物的崩塌載重。最後去找到正確的崩塌機構並且擴大了軟化結構物可使用的安全載重空間。

In the eld of theory of plasticity, limit analysis is an important topic. Plastic responses
are known to be path-dependent. The greatest advantage of limit analysis is that we can obtain plastic collapse loads directly without prescribing loading paths. In the past, the most common approach is applying mathematical programming to calculating the collapse loads of elastoplastic structures.
Recently, some people use the piecewise linear model to describe the constitutive law of elastoplastic materials, and develop a method via the relationship of evolving yielding surface and collapse mechanism in equilibrium to calculate the collapse surfaces of structures and
to construct the safe domain in load space. But for softening materials, the di cuties of how to describe the constitutive law correctly and the analysis of safe load domain being too conservative are two major problems which need to be improved and studied.
In this thesis, we use a parallel connection of components to describe the constitutive law of a softening member of truss. We make one of the components with the characterisation of negative dissipation, and regard the softening member as a closed system, which still follows the
laws of thermodynamics. With this concept, we can formulate the constitutive law with various mechanical models such as bilinear model of ealstoplasticity or model of perfect elastoplasticity.
For the analysis of safe load domain for a softening structure, we deem using the residual strengthes of softening members to calculate the collapse surface to be too conservative, and there will exist structural interactions between structural members in states of stresses higher than residual strengthes. With these two viewpoints, we use the corresponding displacement space to rede ne the collapse load of a softening structure when unrestricted deformation (plastic ow plateau) takes place, and we nd correct collapse modes and thus expand the safe load domain for softening structures.

口試委員會審定書 i
誌謝 ii
摘要 iii
ABSTRACT iv
目錄 v
表目錄 viii
圖目錄 ix
Chapter 1 導論 1
1.1 研究動機與目的 1
1.2 文獻回顧 2
1.3 研究內容與本文架構 3
1.4 符號說明與假設 5
1.4.1 符號說明 5
1.4.2 假設 6
Chapter 2 軟化彈塑姓組成模式 7
2.1 彈塑性組成模式 7
2.1.1 片段線性多降伏平面模式 7
2.1.2 元件並聯模式 9
2.2 軟化桿件儲存耗散的新解 12
2.2.1 桿件模式的熱力學定律 12
2.2.2 耗散非負的問題 13
2.2.3 建立軟化桿件的能量模型 14
2.3 數值運算例一、軟化彈塑性模型組成律關係 16
2.3.1 數值運算例一之一、片段雙線性彈塑模式 16
2.3.2 數值運算例一之二、完全彈塑性模式 17
2.4 軟化結構層次模式 17
2.5 小節 20
Chapter 軟化桁架的崩塌面與安全載重空間 21
3.1 崩塌載重 21
3.1.1 傳統極限分析的崩塌載重 21
3.1.2 崩塌載重的定義探討 21
3.1.3 軟化結構的崩塌載重 21
3.2 軟化桁架崩塌面之求解 24
3.2.1 降伏面與崩塌面的關係 24
3.2.2 定義桁架崩塌面模式 25
3.2.3 桿件前處理 26
3.2.4 模態向量與崩塌模態式的定義 28
3.2.5 模態向量的條件式與求解 29
3.3 數值運算例二、軟化桁架崩塌面求解 27
3.3.1 數值運算例二之一、二桿桁架 32
3.3.2 數值運算例二之二、三桿桁架 33
3.3.3 數值運算例二之三、橋型桁架 36
3.4 軟化桿件的安全載重空間 37
3.3.1 軟化結構物安全載重空間的考量一 37
3.3.2 軟化結構物安全載重空間的考量二 38
3.3.3 重新定義軟化結構物的崩塌面與安全載重空間 39
3.5 數值運算例三、軟化桁架安全載重空間探討 40
3.5.1 數值運算例三之一、五桿桁架 40
3.5.2 數值運算例三之二、九桿桁架 42
3.6 小節 45
Chapter 4 結論與未來展望 46
4.1 結論 46
4.2 未來展望 47
參考文獻 49
附錄A、元件並聯模式推導 90
附錄B、桿件能量模式 94

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