A Study on the Efficient Derivation of Buckling Theory for a Circular Curve Beam

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Project title

Code/計畫編號

NSC98-2221-E019-018

Translated Name/計畫中文名

圓形曲梁挫屈理論之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

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=1915807

Year

2009

Start date/計畫起

01-08-2009

Expected Completion/計畫迄

01-07-2010

Bugetid/研究經費

439千元

ResearchField/研究領域

土木水利工程

"本研究主要是提出一個新的簡易方法，利用剛體運動及增量力平衡此二項基本的力學原理，推導圓形曲梁的幾何非線性虛應變能。傳統文獻中，以虛功法推導曲梁桿件幾何非線性虛應變能之過程較為複雜，有些會無法通過剛體運動及增量力平衡的基本要求。本計劃首先藉由剛體運動法則，建立圓形曲梁在剛體增量位移條件下，曲梁節點力及節點彎矩所作的增量虛功，並藉著增量虛功恆等式，求得在剛體增量位移條件下的曲梁桿件幾何非線性虛應變能，此虛應變能可用來對曲梁挫屈理論做剛體運動法則的檢驗。接著利用曲梁桿件節點力在前階段變形已知的1C 狀態及現階段變形未知的2C 狀態之力平衡關係式，化簡曲梁桿件在剛體虛位移條件下，節點力及節點彎矩所作的增量虛功，並藉由增量虛功恆等式，求得剛體虛位移條件下，曲梁桿件幾何非線性虛應變能，此虛應變能可用來對曲梁挫屈理論進行增量力平衡的驗證。最後由上述利用剛體運動及增量力平衡方程式，分別求得曲梁桿件在剛體位移及剛體虛位移條件下的二組幾何非線性虛應變能積分式，並藉由將微分運算子分成剛體運動微分運算子及高次微分運算子的方式，證明只要比對此二組幾何非線性虛應變能在同一作用力相互對應的虛應變能積分項，即可求得以任意增量位移及虛位移為表示式的曲梁幾何非線性虛應變能。本計劃所提出的方法，僅需進行簡單的積分運算及虛應變能比對，物理意義明確，且可避免傳統繁雜且不易理解的推導過程，求得的圓形曲梁幾何非線性虛應變能，能自動滿足剛體運動法則及力平衡此二項基本力學條件。" "The rigid body motion rule and incremental force equilibrium condition are adopted to derive the geometrically nonlinear strain energy of the circular curve beam. In the literature, virtual work method for deriving the geometrically nonlinear strain energy is more complex and can not conform to the rigid body motion rule and incremental force equilibrium condition. It is proposed deriving the incremental virtual work be done by beam node forces and moments when the circular curve beam incremental rigid body displacement in this proposal. This method can test curve beam buckling theory that satisfy the rigid body motion rule. Using curve beam member node forces in former 1C and present 2C of force equilibrium equality, to obtain curve beam member node forces and moments of incremental virtual work at virtual rigid body displacement condition. Furthermore, to obtain given geometrically nonlinear strain energy of the curve beam member by using incremental virtual work equality. This virtual strain energy can be tested curve beam buckling theory whether it satisfy force equilibrium condition. At last we use rigid body motion rule and incremental force equilibrium condition to find two geometrically nonlinear virtual strain energy integral equations at the rigid body displacement and virtual rigid body displacement of the beam. Dividing the differential operation equation into rigid body motion differential operation and high order differential operation to proof comparing the two integral equations of geometrically nonlinear virtual strain energy at the same force can find virtual strain energy for geometrically nonlinear of beam with any displacement and virtual displacement. In this study only requires simple integration and comparison between strain energy obtained from two rules, and it can avoid complex derivation. In addition, the geometrically nonlinear strain energy obtained from this method can satisfy the rigid body motion rule and incremental force equilibrium condition."

Keyword(s)

曲梁
幾何非線性虛應變能
剛體運動法則
變形狀態力平衡方程式
剛體虛位移
curve beam
virtual strain energy for the geometric nonlinear
rigid body motionrule
force equilibrium in the deformation state
virtual rigid body displacement

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A Study on the Efficient Derivation of Buckling Theory for a Circular Curve Beam

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