A New Method to Formulate Buckling Equations of Thin-Walled I-Beam with Variable Curvatures and Warping Deformations Based on Rigid Body Rule and Force Equilibrium

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Project title

Code/計畫編號

NSC102-2221-E019-042

Translated Name/計畫中文名

剛體運動及力平衡在含翹曲效應變曲率工型梁挫屈理論之研究

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=3090313

Year

2013

Start date/計畫起

01-08-2013

Expected Completion/計畫迄

31-07-2014

Bugetid/研究經費

575千元

ResearchField/研究領域

土木水利工程

"本研究主要是提出一種新的簡易方法，藉由剛體運動法則、增量力平衡此二項基本力學法則及應用斷面分解法，建立含翹曲效應之工型變曲率曲梁桿件的挫屈狀態方程式。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，引進線性虛應變能及非線性虛應變能，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此二項基本力學要求。本計劃工作主要可分成二個部分(1)提出一個簡易方法，首先由材料力學的梁理論，建立曲梁增量力(彎矩)與增量應變(曲率)之梁斷面組成律，接著應用剛體運動及增量力平衡，直接求得含分佈載重作用之實心斷面變曲率曲梁挫屈狀態方程式。(2)首先將工型薄壁曲梁的斷面，分解成三個薄壁矩形之子斷面曲梁，考量各子桿件在斷面交界處曳引力的影響，應用第一部分的成果，建立各子斷面曲梁含分佈載重作用之挫屈狀態方程式，接著藉由工型斷面與各子斷面的形心位移連續條件及斷面力平衡關係式，推導工型變曲率曲梁的挫屈狀態方程式及其挫屈理論。本研究提出的方法，除了可省去繁雜且不易理解的數學推導，且本計畫的推導過程，具有明顯的物理意義，並能自動滿足剛體運動法則及增量力平衡此二項基本力學條件。因此也降低了工型變曲率曲梁挫屈理論可能出現的錯誤，並可直觀瞭解直梁與曲梁挫屈理論之相關性。" "This proposal intends to establish an alternative approach to formulate the buckling equations of an thin-walled I-beam with variable curvatures and warping deformation based on the rigid body rule and incremental force equilibrium criterion. Generally, using conventional virtual work method to derive buckling equations of a thin-walled beam, researchers usually have to deal with six nonlinear strains with high order terms and the equilibrium consideration of induced moments caused by cross sectional stress results under rotations. To simplify these laborious procedures, a curved I-beam is decomposed into three narrow curved beam components in conjunction with geometrical hypothesis of rigid cross section so that one can adopt the buckling equations of a 3D Bernoulli-Euler beam to combine the buckling equations of each component of the curved beams. The entire project is divided into two parts. The first part is aimed at establishing the relationship of incremental stress resultant and incremental strain of curved solid beam based on the Bernoulli-Euler beam theory so that we can derive the buckling equations of the curved beam using the concept of rigid body rule and force equilibrium. Then the I-beam element is decomposed into three narrow beams, from which one can obtain the buckling equations of each beam component using the bucking equations derived in the first part. Finally, the total buckling equations of the I-beam are combined by the continuity of the centroid of the three narrow beams with the rigid section hypothesis and the force equilibrium conditions for the combined cross sectional forces. Since the proposed approach has satisfied both the rigid body rule and incremental force equilibrium conditions in the process of formulation, it not only gives concrete physical meanings of the I-beam theory but also prevents possible mistakes or errors from formulating the buckling equations of an I-beam with variable curvatures and warping deformations. Moreover, from the present formulation, one can observe the buckling relations between straight beam and curved beam."

Keyword(s)

工型曲梁
挫屈狀態方程式
剛體運動法則
增量力平衡
翹曲效應
beam theory of state equations
thin-walled I-beam
force equilibrium
rigid body rule
warping

DSpace CRIS

A New Method to Formulate Buckling Equations of Thin-Walled I-Beam with Variable Curvatures and Warping Deformations Based on Rigid Body Rule and Force Equilibrium

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