Derivation of Structural Geometric Stiffness Matrices of Spatially Curved Beam Elements Based on Rigid Body Rule and Force Equilibrium

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基本資料

Project title

Code/計畫編號

NSC101-2221-E019-053

Translated Name/計畫中文名

剛體運動法則及力平衡在空間曲梁元素幾何勁度矩陣推導之應用

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

姚忠達

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=2640526

Year

2012

Start date/計畫起

01-08-2012

Expected Completion/計畫迄

01-07-2013

Bugetid/研究經費

575千元

ResearchField/研究領域

海洋科學
土木水利工程

"本計畫主要是提出一種簡易方法，藉由剛體運動法則及增量力平衡方程式，推導空間變曲率曲梁的元素幾何勁度矩陣。傳統有限元素法是由曲梁桿件的控制方程式，經變分法推導出曲梁虛應變能，再藉由形狀函數求得曲梁元素勁度矩陣，此推導過程，必需面對二個困難，(1)求得滿足剛體運動法則及增量力平衡的曲梁幾何非線性虛應能，(2)選取適當的形狀函數，合理模擬曲梁桿件「軸向-撓曲-扭轉」偶合變形的力學行為，以避免曲梁元素勁度矩陣產生鎖住的現象。由於上述二個問題不易克服，因此文獻中大都以直梁元素勁度矩陣近似模擬曲梁桿件的幾何形狀並藉此探討曲梁結構的幾何非線性力學行為。本研究主要是提出一種新的簡易方法，藉由滿足剛體運動法則及增量力平衡此二項基本力學法則，建立曲梁桿件的狀態方程式及傳接矩陣，並由傳接矩陣直接求得曲梁元素的線性勁度矩陣及幾何勁度矩陣。本計畫的內容可分成三個部份，第一部分是說明變曲率梁元素勁度矩陣、元素傳接矩陣及其狀態矩陣之間的關連性。第二部份是建立此三個矩陣滿足剛體運動法則與增量力平衡的條件方程式，並藉此二個基本力學法則之條件方程式求得曲梁桿件的狀態方程式及其挫屈理論。本研究的第三部份是藉由曲梁桿件狀態矩陣，經積分及矩陣運算，求得曲梁桿件的傳接矩陣及曲梁元素線性勁度矩陣、幾何勁度矩陣。本計劃提出的方法，只須進行簡易的積分及矩陣運算，即可求空間變曲率曲梁的元素幾何勁度矩陣，除了避開傳統繁雜又困難的曲梁幾何非線性虛應變能推導，且不需應用形狀函數求取元素勁度矩陣，因此求得的曲梁元素幾何勁度矩陣能自動滿足六個剛體運動及六個增量力平衡檢測，且能避開傳統有限元素法所要面對的鎖住現象。本計畫求得的曲梁元素幾何勁度矩陣，因已考量自然變形效應的內在幾何勁度矩陣，所以應用此幾何勁度矩陣進行曲梁桿件幾何非線性行為數值分析時，將可提高其精度及效率。" "Conventionally, researchers or scientists usually need to select a set of suitable shape functions in formulating structural matrices of curved beam elements. In addition, the geometric stiffness matrix derived for the curved beam element has to satisfy the rigid body test due to equilibrium considerations. These two procedures are not easy to attain because of locking problems. Thus most researchers adopted straight beam elements to simulate curved beam structures for nonlinear structural analyses in existing literature. This proposal intends to develop an alternative approach that can satisfy both conditions of the rigid body test and incremental force equilibrium through the transfer matrices and state equations of a curved beam. The entire project is divided into three parts. The first part is aimed at establishing the relationship of curved beam structural stiffness matrix, the beam element transfer matrix, and state transfer matrix of curved beam. Based on the state transfer matrix method developed in the first part, the second research work is focused on the theoretical development of transformation relation among the three structural matrices that can satisfy both conditions of the rigid body test and incremental force equilibrium. In this stage, the transformation matrix has taken into account the second order effect induced by external forces due to buckling. Finally, the structural matrices including beam transfer matrix, elastic and geometric stiffness matrices of a spatially curved-beam element are derived through the curved-beam state matrix that has been obtained from the second part. The proposed approach has two features: concise physical meanings and simple matrix manipulation in deriving structural stiffness matrices of a spatially curved bar, from which laborious mathematical derivation in dealing with nonlinear strains, potential energy and suitable selection of shape functions can be avoided. Besides, both the qualification of rigid body test and the conditions of force equilibrium are always satisfied in the process of theoretical development. It is emphasized that the locking problem of curved beam elements using conventional variational principle based on displacement shape selections will not occur using the present curved-beam element. For this, a number of examples of buckling and large deformations of curved beam structures will be demonstrated in this study."

Keyword(s)

空間曲梁
傳接矩陣
狀態矩陣
剛體運動法則
增量力平衡
外在幾何勁度矩陣
內在幾何勁度矩陣
geometric stiffness matrix
beam theory
curved beam element
force equilibrium
rigid body rule
transfer matrix

DSpace CRIS

Derivation of Structural Geometric Stiffness Matrices of Spatially Curved Beam Elements Based on Rigid Body Rule and Force Equilibrium

基本資料

Description