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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/30699

Title: 建立在力平衡及剛體運動法則之預扭曲梁挫屈理論研究
Buckling Theory of Curved Beams with Initial Pretwist Based on Rigid Body Rule and Force Equilibrium
Authors: 郭世榮;姚忠達
Contributors: NTOU:Department of Harbor and River Engineering
國立臺灣海洋大學:河海工程學系
Keywords: 預扭曲梁;剛體運動法則;增量力平衡;挫屈理論;狀態矩陣;曲梁斷面座標變化率
buckling theory;pretwisted curved beam;force equilibrium;rigid body rule;state matrix;curved-beam coordinate variation
Date: 2011-08
Issue Date: 2012-04-13T01:20:02Z
Publisher: 行政院國家科學委員會
Abstract: 摘要:本研究主要利用一個新的簡易方法,藉由直梁桿件的狀態方程式,推導預扭空間曲梁的挫屈理 論。因為挫屈方程式是建立在變形後的力平衡狀態,文獻中主要是由連體力學的大變形理論出發,引 進線性應變能及非線性應變能,應用變分法求得曲梁桿件的挫屈方程式。此推導過程須完整考慮六項 非線性應變,及合理處理物理意義不明確的非線性應變能,並且正確的分析旋轉變形後彎矩引量所做 的虛功,才能求得完整合理的曲梁挫屈方程式。由於推導過程較為複雜,文獻中有些曲梁桿件挫屈理 論,無法通過剛體運動及增量力平衡的基本要求。 本計畫主要分成二個部分,其中第一部分是說明直梁元素勁度矩陣與元素傳接矩陣及其直梁挫屈 方程式之間的關連性。首先推導直梁元素勁度矩陣與直梁元素傳接矩陣的關係,接著利用元素長度趨 近於零之極限原理,求得直梁桿件的狀態方程式及其對應的挫屈理論。最後與文獻中的直梁挫屈方程 式作比較,由此驗證本研究推導方法的正確性。本研究第二部分是探討直梁元素傳接矩陣與預扭曲梁 元素傳接矩陣及其預扭空間曲梁挫屈理論之間的關係。由有限元素的理論可知,當預扭曲梁桿件元素 切得過多 (元素長度趨近於零),則可採用直梁元素的勁度矩陣,模擬分析預扭曲梁桿件的力學行為。 因元素傳接矩陣是由元素勁度矩陣求得,由此可知當曲梁元素長度趨近於零,則直梁元素的傳接矩 陣,藉由元素節點自由度之沿預扭曲梁斷面座標與直線座標之間的座標轉換運算後,可用來近似預扭 曲梁元素的傳接矩陣。本研究第二部分是應用上述的論點,由直梁元素的傳接矩陣建立預扭曲梁的傳 接矩陣,並推導預扭曲梁桿件斷面座標沿曲梁中性軸長度的一次導數,接著利用極限原理,求得預扭 曲梁桿件的狀態方程式及其挫屈理論。 本計畫提出的方法僅需將直梁狀態矩陣與預扭曲梁斷面座標矩陣沿曲梁中性軸長度的一次導數 相加,即可求得預扭曲梁的狀態矩陣及其挫屈理論。本研究提出的推導過程物理意義明確,且只需進 行簡單的矩陣運算,除了避開傳統繁雜又不易理解的數學推導,所求得的預扭曲梁桿件之挫屈理論, 能同時滿足力平衡及剛體運動法則之『軸-扭-撓曲』變形互偶的挫屈力學行為。
abstract:Using the principle of energy to formulate the buckling equations of a spatial curved bar with initial pretwist based on continuum mechanics, researchers or scientists usually need to deal with laborious derivations of conventional nonlinear strains and the induced moments caused by cross sectional stresses under rotations for satisfying the qualification of rigid body rule and force equilibrium. To simplify the complicated procedure and satisfy both the qualification of rigid body rule and the conditions of force equilibrium in formulation, this study attempts to introduce the concept of beam-transfer matrix using the well-developed straight-beam element into the pretwisted curved beam theory. The entire project is divided into two parts. The first part is aimed at establishing the relationship between the straight-beam element stiffness matrix (including geometric effect) and the beam element transfer matrix to verify the feasibility of using the present formulation to derive the governing equations of conventional straight beam theory. Based on the state transfer matrix method developed previously, the second research work is focused on the theoretical development of transformation relationship between the straight-beam transfer matrix and the pretwisted curved-beam transfer matrix. In this stage, the transformation matrix has taken into account the second order effect induced by external forces due to buckling. Finally, the pretwsited curved-beam equations at buckled state can be derived from the straight-beam equations in conjunction with the approximations of a curved beam transfer matrix with pretwists through successive coordinate transformations. It is emphasized that since a pretwisted curved bar can be treated in the limit as the composition of an infinite number of infinitesimal straight-beam segments with initial pretwist, the equilibrium conditions at connected joints of the composition of straight-beam segments have been established in the buckled configuration. Concise physical meanings and simple matrix manipulation in deriving the governing equations of a pretwisted curved bar established at buckled configurations are the main features of this project. Besides, both the qualification of rigid body test and the conditions of force equilibrium are always satisfied in the process of theoretical development. Finally, the coupling of extension, torsion, and flexure of a pretwisted curved bar subject to external forces will be investigated using the derived equations.
Relation: NSC100-2221-E019-042
NSC100-2221-E019-042
URI: http://ntour.ntou.edu.tw/handle/987654321/30699
Appears in Collections:[河海工程學系] 研究計畫

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