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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/24230

Title: 應用剛體運動和力平衡探討樑的挫曲理論
Study on the buckling of beam using the force equilibrium and rigid body motion
Authors: 郭世榮;葉為忠;紀志昌
Contributors: NTOU:Department of Harbor and River Engineering
國立臺灣海洋大學:河海工程學系
Keywords: 幾何非線性;剛體運動;增量力平衡
the geometrically nonlinear behaviors;rigid body motion;the incremental force equilibrium
Date: 2011
Issue Date: 2011-10-20T08:10:46Z
Publisher: 國立海洋大學河海工程學系,國科會研究計畫報告
Abstract:   本文主要是提出一種新的方法,藉由剛體運動及增量力平衡的基本力學法則,推導樑構件的幾何非線性虛應變能。本文的方法,首先是探討變形後 狀態的剛體運動及增量力平衡的基本力學法則,由此建立在滿足此二項力學法則時,樑桿件幾何勁度矩陣的條件方程式。其中幾何勁度矩陣滿足增量力平衡的條件方程式,在已知的文獻中不曾被提出,本文是第一個建立此方程式。接著由此二組條件方程式,可求得樑元素近似的幾何非線性虛應變能。最後將任一樑桿件切成多個微小的樑元素,且將樑元素節點自由度以樑桿件的位移函數表示,並藉由黎曼積分原理,求得樑桿件的幾何非線性虛應變能。本文首先是推導直樑在平面變形的幾何非線性虛應變能,藉此說明本文的理論與方法。接著藉由本文的方法,建立直樑在空間的變形的幾何非線性虛應變能。最後本文提出二種方法推導圓形曲樑的幾何非線性虛應變能,其中第一種方法同推導直樑桿件的步驟一樣,首先建立圓形曲樑元素幾何勁度矩陣,在滿足變形後 狀態的剛體運動及增量力平衡的條件方程式,由此推導圓形曲樑的幾何非線性虛應變能。而第二種方法主要是先建立圓形曲樑在曲線座標方向位移與元素直線座標之間的轉換關係,藉由元素長度趨近於無窮小的條件下,求得直樑位移導數與曲樑位移及其導數之間的對應關係,最後將此關係式代入本文已求得的直樑幾何非線性虛應變能,由此建立圓形曲樑的幾何非線性虛應變能。由第二種方法可知,只要建立任意形狀曲樑的曲線座標位移及位移導數與直樑位移導數的關係,即可求得其幾何非線性虛應變能,所以本文提出求圓形曲樑幾何非線性虛應變能的第二種方法,應可推廣至求取任意形狀曲樑桿件的幾何非線性虛應變能。
A novel method that deduces the virtual strain energy for the geometrically nonlinear behaviors of a beam, using the incremental force equilibrium and rigid body motion is proposed. First, we discuss the incremental force equilibrium and rigid body motion in the deformed configuration. The incremental force equation describing the constraint of equilibrium the geometric stiffness matrix has never been mentioned previously in literature. Using these two constraint equations, the virtual strain energy of the beam element can be approximated. A continuous beam can be cut into a lot of ting elements, then the virtual strain energy for this arbitrary beam can be obtained using Riemann integral theorem. In this paper, we deduce the virtual strain energy of 2-D straight solid beams first, which guide readers to understand state the theory and method in this paper. Using this method, we can set up the virtual strain energy of 3-D straight solid beams. Finally, two methods are proposed to deduce the virtual strain energy of 3-D circular curve beams. The first method is as similar to the method we used for deducing the virtual strain energy of 3-D straight solid beams. We use the constraint equation of the geometric stiffness matrix that satisfies the incremental force equilibrium and rigid body motion to deduce the nonlinear virtual strain energy for the geometrically nonlinear behaviors of 3-D circular curve beams. The second method mains establish the conversion relation between displacements described in the curvilinear coordinate and the local element coordinate of curved beam first. Taking the length of the ting beam element to infinitesimal, we can obtain the relation about the straight solid beams and the circular curve beams. Finally we can obtain the virtual strain energy of 3-D circular curve beams by substituting the relation into the nonlinear virtual strain energy of 3-D straight solid beams. The second method provides a novel view to establish the virtual strain energy for an arbitrarily shaped beam. Once the relationship between the local element coordinate and the global curvilinear coordinate can be expressed, it is expected the virtual strain energy for any curvilinear beam can be obtained from that of a straight solid beam and such a transformation relation.
Relation: 計畫編號:NSC94-2211-E-019-010
URI: http://ntour.ntou.edu.tw/handle/987654321/24230
Appears in Collections:[河海工程學系] 期刊論文

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