A Study of the Geometric Stiffness Matrix for the Plate Element Using the Rigid Body Motion Concept

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Project title

Code/計畫編號

NSC91-2211-E019-011

Translated Name/計畫中文名

剛體運動法則在平板元素幾何勁度矩陣之推導與探討

Project Coordinator/計畫主持人

Shyh-Rong Kuo

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=782099

Year

2002

Start date/計畫起

01-08-2002

Expected Completion/計畫迄

01-07-2003

Bugetid/研究經費

466千元

ResearchField/研究領域

土木水利工程

本文主要是提出一個簡易的方法，推導合理的三角形平板元素幾何勁度矩陣，此矩陣包含三個薄膜力、三個彎矩及二個橫向剪力等作用力的非線性效應，此方法可避開傳統有限元素推導繁雜冗長的非線性虛應變能及數值積分。由連體力學的觀點，力平衡及客觀性的剛體運動法則，是力學分析中，二項最基本的要求。本文首先是建立三角形平板元素幾何勁度矩陣滿足剛體運動法則及力平衡的條件方程式。接著利用此二條件方程式，求得一合理的平板元素幾何勁度矩陣，此勁度矩陣可滿足剛體運動法則及力平衡二項基本要求。由力平衡及剛體運動法則，可證實三角形平板元素幾何勁度矩陣，為一不對稱矩陣。本文推得的元素幾何勁度矩陣之反對稱項，在組合總體勁度後，會相互抵銷，因此仍可求得一對稱的總體勁度矩陣，此對稱矩陣在板殼結構非線性分析，將可提高電腦運算效率。 In this paper, a geometric stiffness matrix for a plate element is derived based on the rigid body rule. The incremental nodal force and the incremental nodal displacement from the rigid body modes are used to derive the geometric stiffness matrix, which needs to satisfy the rigid body rule. It is proved that infinite many choices exist for the geometric stiffness matrix. By means of decomposing a couple into a symmetric couple system and a skew-symmetric one, it can be proved that the local geometric stiffness matrix is not symmetric due to the skew-symmetric couple system while considering a single plate element but the global geometric stiffness matrix becomes symmetric. A symmetric geometric stiffness matrix is beneficial for the nonlinear analysis. This symmetric property yields better results. The combined stiffness matrix, i.e., the sum of the elastic stiffness and geometric stiffness matrix, is used in the predictor stage and the corrector stage. It is shown numerically under some situation that to use the elastic stiffness matrix in the predictor stage may result in non-convergence.

Keyword(s)

幾何非線性
平板挫屈
有限元素法
剛體運動測試
力平衡測試
geometric nonlinear formulation
buckling analysis of thin plate
finite element method
rigid body test in the nonlinear sense

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A Study of the Geometric Stiffness Matrix for the Plate Element Using the Rigid Body Motion Concept

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