考慮大旋轉之非線性梁與構架動力反應之理論與計算

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Code/計畫編號

NSC98-2923-E002-005-MY3

Translated Name/計畫中文名

考慮大旋轉之非線性梁與構架動力反應之理論與計算

Funding Organization/主管機關

National Science and Technology Center for Disaster Reduction

Co-Investigator(s)/共同執行人

楊永斌(計畫主持人)
彭瑞麟

Department/Unit

Department of Civil Engineering,NTU

Website

https://www.grb.gov.tw/search/planDetail?id=1925324

Year

2009

Start date/計畫起

01-08-2009

Expected Completion/計畫迄

31-07-2010

Co-Investigator(s)

Shyh-Rong Kuo

Bugetid/研究經費

1343千元

ResearchField/研究領域

土木水利工程

在過去二十年裡，台灣和奧地利的研究群均各自以有限元素法推導出古典的梁理論，而且也各自在著名的國際期刊發表了一系列的論文，台灣的研究群以台灣大學的楊永斌教授為主，奧地利的研究群則以Johannes Kepler University of Linz 的Hans Irschik 教授為主，然而，這兩個研究群所採用的方法並不相同，兩個方法各自有其優點，也可望能夠繼續加以改善，以形成一個新的共同的方法。台灣的研究群主要是採用更新式拉格蘭治法，來推導梁的理論，比較著重於各階段分析中剛體運動法則(Yang and Chiou 1987)的運用，在每個增量步驟中，區分每個元素的剛體旋轉和自然變形，並且各以不同程序加以處理。所採用的節點力矩係定義為段面之合應力，而相對的旋轉自由度則定義為位移之導數，採用此組物理量的優點是，它們可以直接被工程師採用，但缺點則是不具能量對應性(conjugate)(Reissner 1972)，在引進節點於變形位置之平衡條件後，該法被延伸至三維構架的非線性分析。相較之下，奧地利研究群則是以連體力學的虛功法，來推導梁的大位移─有限變形理論，從2nd Piola-Kirchhoff stresses 和格林應變開始著手，此一連體力學作法具有能量對應的優點，因係以應力和應變來定義組合律，所以可模擬更先進的材料。而Reissner (1972)採用廣義應力和廣義應變的結構力學作法，則存在不易定義組合律的問題，特別是針對三維梁而言。奧地利研究群後來又引進了絕對節點座標推導法，此法更方便組合律的考慮，不論是線彈性、超彈性、或彈塑性均可輕易加以模擬。絕對節點座標推導法的缺點，是需要大量的自由度和較高的計算成本。然而，它也曾被成功使用於梁在剛體運動下大變形的動力分析和模擬。在本計畫中，台奧兩地的研究方法將被整合，以發展出新的梁元素，用來析梁和空間構架含大旋轉角的動力問題，特別是針對元素慣性力和起始力在大剛體旋轉下的物理現象。雖然絕對節點座標推導法，可模擬先進的材料組合律和動力效應，該法卻有低階收斂效率不高的問題。因此，我們將採用台灣研究群的結果，也就是區分剛體位移和自然變形的作法，來改善奧方所建立的有限元素，以發展一具高階收斂性的元素，在此，奧方過去所用的跟隨旋轉(corotational)推導法(Gerstmayr and Schöberl, 2006)，可望與台方的更新式拉格蘭治推導法結合成一個新的體系。為確認研究的成果，我們將採用一些既存的的測試例，也會引進一些新的測試例，在可能的情況下，也打算進行一些簡單的實驗，以印證理論分析結果的正確性。為達成以上共同的研究目標，在合作計畫進行中，我們還需要互訪並舉辦研討會。 Classical beam theories have been studied independently by the research groups using the finite element method in Taiwan and Austria in the past two decades, each resulting in a mainstream of research papers published in reputed technical journals. The former was mainly led by Prof. Y. B. Yang of National Taiwan University and the latter by Prof. H. Irschik of the Johannes Kepler University Linz, Austria. However, different approaches were adopted by the two groups. Both approaches have their own merits, but can be still improved to form a new and possibly better joint approach. The Taiwan side has formulated their beam theory based on the updated Lagrangian formulation, with specific emphasis on application of the rigid body rule (Yang and Chiou 1987) in each stage of the analysis. In each incremental step of the analysis, division is made between the rigid rotation and natural deformation of each finite element, with procedures developed for dealing with each component. The nodal moments adopted for each element are the moments defined as stress resultants and the corresponding degrees of freedom are the rotations defined as the derivatives of transverse displacement. Such a pair of nodal quantities can be conveniently used by engineers working on design practice, but have the drawback of not being energetically conjugate (Reissner 1972). With the equilibrium of each joint defined in the deformed configuration, this approach has been extended to analysis of three-dimensional frames. In contrast, the Austrian side has put much emphasis on the large displacement – finite deformation of beams based on the continuum mechanics virtual work principle, i.e., in terms of the 2nd Piola-Kirchhoff stress and Green strain. Such a continuum mechanics approach allows us to consider technically advanced material laws, which are based on generally nonlinear constitutive relations between stress and strain. The approach by Reissner (1972) based on structural mechanics is considered to release some conceptual difficulties of defining the constitutive laws for three-dimensional beams. Among others, an absolute nodal coordinate formulation has been presented by the Austrian side, which allows inclusion of linearly elastic, hyperelastic or elasto-plastic materials in an easy way. At the present stage, the absolute nodal coordinate formulation suffers from a larger number of degrees of freedom and high computational costs. The formulation has been developed and tested for the analysis and simulation of dynamical problems of beams undergoing large rigid body motion and large deformations. In this project, both the Taiwan and Austrian approaches will be merged to obtain new beam finite elements for the dynamic analysis of beams and space frames involving large rotations. Of particular interest is the consideration of the inertial forces and initial forces of each beam element undergoing large rigid rotations. While the absolute nodal coordinate formulation up to now allows the inclusion of advanced material laws and dynamical effects, it is not efficient and shows low order convergence. Thus, the results of the Taiwan group will help to improve existing elements of the Austrian group and to develop efficient and high order convergent elements using e.g. the division into rigid body rotation and natural deformation. Existing numerical techniques based on corotational approaches of the Austrian group (Gerstmayr and Schöberl, 2006) will lead to synergies with the updated Lagrangian formulation of the Taiwan group. As a verification of the outcome of the joint project, existing benchmark problems will be evaluated and new benchmark problems will be established, along with some experimental testing of simple problems. As a main part, the joint project will include exchange visits and joint workshops.

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考慮大旋轉之非線性梁與構架動力反應之理論與計算

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