A Study on the Efficient Derivation of Geometrically Nonlinear Governing Equations for a Plate

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簡歷

基本資料

Project title

Code/計畫編號

NSC92-2211-E019-015

Translated Name/計畫中文名

平板幾何非線性控制方程式簡易推導方法的研究

Project Coordinator/計畫主持人

Shyh-Rong Kuo

Funding Organization/主管機關

National Science and Technology Council

Co-Investigator(s)/共同執行人

葉為忠

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=918801

Year

2003

Start date/計畫起

01-08-2003

Expected Completion/計畫迄

01-07-2004

Co-Investigator(s)

Wei-Chung Yeih

Bugetid/研究經費

409千元

ResearchField/研究領域

土木水利工程

"本研究主要是提出一種新的簡易方法，用來建立平板的幾何非線性控制方程式。傳統推導平板的幾何非線性增量虛功方程式，基本上是由連體力學的大變形理論出發，推導過程中必須合理處理某些物理意義不明確的非線性虛應變能，及完整考慮變形後 2 C 狀態的虛功量，才能建立正確的幾何非線性控制方程式，一般而言這樣的推導過程極為繁雜且不易理解，因此文獻上大都簡化推導，只探討平板面內作用力的幾何非線性控制方程式及幾何勁度矩陣，文獻上若有考慮平板面外彎矩、剪力作用力的幾何非線性行為，則都無法通過剛體運動及力平衡的基本要求。本計畫首先是對平板元素的節點位移做泰勒級數展開，由此建立節點自由度與剛體平移、剛體旋轉及常應變等低階運動模態的關係式。接著利用剛體運動法則及增量力平衡式建立平板元素在此三種變形模態所對應的節點增量力，由此關係式求得微小平板元素的幾何非線性增量虛應變能。最後將平板結構以微小平板元素組合，並利用微積分的黎曼和（Riemann sum）原理，求得板的幾何非線性增量虛應變能積分式。此簡易推導方法可避開傳統推導幾何非線性方程式的繁雜過程，且建立的控制方程式能自動滿足 2 C 狀態剛體運動及增量力平衡。" "In this research proposal, a kind of new simple approach is proposed to construct a geometrically nonlinear stiffness matrix of a simple plate element. For the traditional derivation of the geometrically nonlinear incremental virtual work equations of a plate element, it is based on the large deformation of continua mechanics in the beginning, and then by considering reasonable treatments of some nonlinear virtual strain energy in the deriving process and by carefully addressing the incremental virtual work in the C 2 state after the deformation, the geometrically nonlinear governing equations can be correctly yielded. Due to such a complicate nature, the geometrical stiffness matrices for the plate element in the previous published literatures sometimes were obtained under some simplifications or add assumptions such that most of them even could not satisfy the rigid body motion test. In this research, we will first develop the geometrical stiffness matrix of a plate element, which satisfies the tests of six rigid body motions and force equilibrium of C 2 state after the deformation; and then by using the expansion of the displacement Taylor’s series and the simple matrix operation with the Riemann Sum principle in the calculus, the geometrically nonlinear strain energy can be yielded. Finally, the geometrically nonlinear governing equations for a plate element can be formulated through the variation principle. The proposed approach can not only avoid the tedious derivation process but also automatically satisfy the tests of rigid body motions and force equilibrium. Besides, the current approach will be used to validate the accuracy of the geometric stiffness matrix of a plate element by recasting the previous research."

Keyword(s)

幾何非線性
平板元素幾何勁度矩陣
變形狀態的剛體運動法則
變形狀態增量力平衡
geometric nonlinear formulation
geometric stiffness matrix of plate element
rigid body rule in the deformation state
incremental force equilibrium condition in the deformation state

DSpace CRIS

A Study on the Efficient Derivation of Geometrically Nonlinear Governing Equations for a Plate

基本資料

Description