A New Approach to Develop the Geometric Nonlinear Theory of Thin Shells Based on the Rigid Body Motion Rule and Force Equilibrium

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簡歷

基本資料

Project title

Code/計畫編號

MOST103-2221-E019-010

Translated Name/計畫中文名

薄殼幾何非線性理論之新推導法－基於力平衡與剛體運動法則

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=8360430

Year

2014

Start date/計畫起

01-08-2014

Expected Completion/計畫迄

31-07-2015

Bugetid/研究經費

645千元

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，利用剛體運動法則及力平衡此二項基本的力學原理，推導薄殼結構的幾何非線性虛應變能。傳統文獻中，以虛功法推導薄殼結構幾何非線性虛應變能之過程極為複雜且不易理解，並且又無法通過剛體運動法則及力平衡的基本要求。本研究首先應用剛體運動法則，建立在剛體增量位移條件下，薄殼結構邊界力及邊界彎矩在變形後狀態所作的增量虛功，並藉由增量虛功恆等式求得在剛體增量位移條件下薄殼的幾何非線性虛應變能，此虛應變能可用來檢測薄殼挫屈理論是否通過剛體運動法則。接著利用薄殼結構邊界力(彎矩)在變形前及變形後狀態的力平衡關係式，求得在剛體虛位移的條件下，薄殼結構邊界力所作的增量虛功，並藉由增量虛功恆等式，求得在剛體虛位移條件下，薄殼結構幾何非線性虛應變能，此虛應變能可驗證薄殼挫屈理論是否滿足力平衡的要求。最後將上述分別在剛體增量位移及剛體虛位移狀況下建立的兩條虛功式，應用史托克定理(Stokes’ theorem)，將沿薄殼中曲面邊界之曲線積分的虛功表示式，轉換成在薄殼中曲面之曲面積分的虛應變能表示式，藉此求得變分型式對稱的虛應變能曲面積分式，並由比對上述二組虛應變能互相對應的變分型式，求得以任意增量位移及虛位移表示的薄殼幾何非線性虛應變能曲面積分式。本研究所提出的方法，僅需進行簡單的積分運算及虛應變能比對，故可省去傳統繁雜且不易理解的數學推導過程；由於本文所建議的幾何非線性虛應變能，具有明顯的物理意義，並能自動滿足剛體運動法則及力平衡此二項基本力學條件，因此，也降低了在推導過程可能出現的錯誤產生。 This study is to develop a new approach to derive the geometrically nonlinear strain energy of a thin shell element based on the rigid body motion rule and the incremental force equilibrium condition. Conventionally, the virtual work method is usually used to derive the geometrically nonlinear strain energy; however, it is a sophisticated process because the whole derivation is complicate and not easy to understand, especially for nonlinear higher order terms. In addition, some of the results derived in literatures cannot pass the rigid body motion test and satisfy the incremental force equilibrium conditions. In this research, the applicant proposes a new approach to derive the incremental virtual work done by boundary forces and boundary moments in deformed state when an incremental rigid body displacement is superimposed using the rigid body motion rule. Using the equilibrium conditions for the incremental virtual work, the geometrically nonlinear strain energy of the thin shell element can be obtained by a set of given incremental rigid body displacements. This strain energy derived by following this procedure has obeyed the rigid body motion rule. Further, the incremental virtual work done by the boundary forces and moments for the shell element can be obtained by considering the force equilibrium conditions for the shells at the initial C1 state and the deformed C2 state by superimposing a virtual rigid body displacement. Similarly, the geometrically nonlinear strain energy for the thin shell element can be obtained by superimposing a virtual rigid body displacement by using the virtual work equilibrium conditions. The strain energy obtained by this procedure should satisfy the incremental force equilibrium rule. By the union of the geometrically nonlinear strain energies derived from the above-mentioned two procedures, one can find the complete geometrically nonlinear strain energy based on the arbitrary nature of incremental and virtual displacement. The present procedure proposed in this study only requires simple integration and comparison between the strain energies obtained from the two rules, which can avoid cumbersome derivation. In addition, the geometrically nonlinear strain energy obtained from this method can satisfy both the rigid body motion rule and the incremental force equilibrium conditions.

Keyword(s)

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
薄殼結構
geometric nonlinear strain energy of thin shell
rigid body motion rule
force equilibrium in the deformed state
virtual rigid body displacement

DSpace CRIS

A New Approach to Develop the Geometric Nonlinear Theory of Thin Shells Based on the Rigid Body Motion Rule and Force Equilibrium

基本資料

Description