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  • 學位論文

軟化彈塑性桁架的崩塌載重

The Collapse Loads of Softening Elastoplastic Trusses

指導教授 : 洪宏基

摘要


傳統的極限分析有許多侷限, 其中最重要的就是組成行為必須是完全塑性, 然而, 軟化行為在土木結構物中相當常見, 因此放鬆此項限制便有其必要性. 本文先從一般的材料模型, 推廣到廣義線性軟化模型, 再特化到軟化桁架模型, 並得到其所代表的最佳化問題, 另外也重新定義崩塌載重為結構的外功增量最大, 得到求取崩塌載重的平衡約束數學規 劃(mathematical programs with equilibrium constraints, MPEC) 問題, 發現整個求取崩塌載重的過程, 包括廣義模型, 實際上是一個非線性最佳化問題, 而原來的平衡約束, 其實是最佳化問題的KKT(Karush-Kuhn-Tucker) 條件. 在「單調加載」的前提下, 最佳化結果有實用意義; 也試圖解開「單調加載」的外力形式限制, 文中將軟硬化行為分為「等向」及「走動」, 探討其循環圈, 並假設等向軟硬化會飽和, 猜測如果對外力乘子α 給與下限, 就能得到工程上有意義的結果, 即「在崩塌載重之下的外力, 結構物一定安全」. 對於此最佳化方法提出一個簡易算法, 對桁架內功增量分段加以處理, 便能先解開二套互補三元, 將原最佳化問題所需擾動的變數減少至只有節點位移, 並以現有商業最佳化軟體MATLAB 求解, 後以五個例子搭配增量法來驗證此理論, 結果顯示此理論方法在中小型結構上, 可快速求得軟化桁架崩塌載重, 且對於區域不穩定也能夠處理.

並列摘要


In the conventional limit analysis, only can perfect plasticity be dealt with. However, softening structures exist all over the world and they should be considered. Therefore, it is important to loose this restriction. At first, a general material model is introduced. From it, we derive a generalized linearly softening model which can be specialized to be a softening truss model. The equivalent optimization problem is also carried out here. Then, the problem of calculating the collapse load in the form of MPEC(mathmatical programming with equilibrium constraints) is developed by redefining the collapse load as maximum external work of the structure. We find out that the whole process of calculation, including the generalized model, is a nonlinear programming problem. The original equilibrium constraints are actually the KKT conditions of the nonlinear programming problem. To enlarge the variety of the load history, not just“monotonic load”, we divide the hardening/softening rule into “isotropic” and “kinematic” parts. By assuming the isotropic part has a saturated limit, we conjecture if giving a lower limit of the force multiplier, the result of optimization will make sense in engineering: “Subjected to a varying load under the collapse load but above the lower limit, structures will be safe.” Also, a simple algorithm is given: By classifying the internal work of trusses at first, we can uncouple two complementary trios. Therefore, the variables which need perturbation can be reduced and only nodal displacements remain. Then, we call the Optimization Toolbox of MATLAB to solve this unconstrained nonlinear programming problem. Moreover, five examples with an incremental method are given to verify this theory and method. The results show that the theory and algorithm are practicable for small and medium structures. They can also deal with local instability.

參考文獻


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被引用紀錄


林冠宇(2014)。軟化桁架結構組成律、崩塌面與安全載重空間之探討〔碩士論文,國立臺灣大學〕。華藝線上圖書館。https://doi.org/10.6342%2fNTU.2014.02043

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