- 學位論文
受平面拘束及集中力作用之彈性板條的變形與穩定性分析
Deformation and Stability of an Elastica under a Point Force and Constrained by a Flat Surface
摘要
本論文共分為兩個部份，第一部份研究彈性板條(elastica)受一集中力從一端半靜態地(quasi-statially)移動到另一端的變形及穩定性，且彈性板條受一剛性平面及兩端銷接(pin)拘束。根據彈性板條和剛性平面的接觸狀態我們可以發現三種平衡變形：未接觸、一點接觸及一邊線接觸。利用振動法可以決定所求變形的穩定性，為了考慮振動時彈性板條與剛性平面間的滑動，我們將以Eulerian的描述方式來替代Largrangian的描述式。結果發現所有的一點接觸皆為不穩定的平衡位置，而一邊線接觸會經由兩種不同途徑產生不穩定現象，一種為二次挫曲(secondary buckling)，另一種為極限點(limit-point bifurcation)造成的不穩定。發生二次挫曲時，線接觸的長度與軸力會滿足Euler挫曲理論的銷接-夾持樑；而因極限點造成的不穩定，軸力不會超過Euler挫曲臨界負載。同時我們利用實驗驗證理論的預測。 第二部份則利用小變形分析，且將剛性平面改為彈性基底，研究一挫曲樑(buckled beam)受銷接拘束及集中力作用的變形與穩定性分析，且挫曲樑的側向變形受一單向抗壓(tentionless)彈性基底拘束，且彈性基底在變形前為平面。從靜態分析可以發現此系統有五種變形：(1)未接觸，(2)完全接觸，(3)一邊接觸，(4)中間接觸，和(5)兩邊接觸變形。在某些特定的幾何條件下會有多個平衡位置，為了預測集中力從挫曲樑-彈性基底系統的一端移動至另一端的現象，我們必須對這些平衡位置進行穩定性分析，穩定性分析的方法為，使用振動法求出系統的自然頻率，並將接觸區域在振動時的微小變動列入考慮，分析結果發現以上的五種變形，變形(1)、(2)、(3)和(4)在某些負載條件下可能為穩定的平衡位置；而當彈性基底趨近於剛性平面時，只有變形(1)和(3)兩種為穩定的平衡位置。
並列摘要
This thesis is divided into two parts. In the first part we study the deformation and stability of a pinned elastica under a point force moving quasi-statically from one end to the other. The elastica is constrained by a rigid plane wall containing the two ends. Three types of equilibrium configurations can be found; they are non-contact, one-point contact, and one-line contact on the side. A vibration method is adopted to determine the stability of the calculated deformations. In order to take into account the variation of the contact region between the elastica and the plane wall during vibration, an Eulerian version of the governing equations is adopted. It is found that all the point-contact deformations are unstable. On the other hand, there are two different mechanisms a line-contact deformation becomes unstable; one through a secondary buckling and the other through a limit-point bifurcation. In the secondary buckling, the length of the line-contact segment and the axial force satisfy the Euler buckling criteria for a pinned-clamped column. On the other hand, when a line-contact deformation becomes unstable via a limit-point bifurcation, the axial force does not exceed the Euler buckling load. The theoretical predictions are confirmed by experimental observations. In the second part of the thesis, we adopt a small deformation analysis and replace the rigid surface with an elastic foundation. We study the deformation and stability of a pinned buckled beam under a point force. The buckled beam is constrained by a tensionless elastic foundation, which is flat before deformation. From static analysis, we found a total of five different deformation patterns; they are (1) non-contact, (2) full contact; (3) one-sided contact; (4) isolated contact in the middle, and (5) two-sided contact. For a specified set of parameters, there may coexist multiple equilibria. In order to predict the response of the buckled beam-foundation system as the point force moves from one end to the other, we have to determine the stability of these equilibrium configurations. In order to achieve this, a vibration method is adopted to calculate the natural frequencies of the system, taking into account the slight variation of the contact range between the buckled beam and the tensionless foundation during vibration. It is concluded that among all five deformation patterns, deformations (1), (2), (3), and (4) may become stable for certain loading parameters. In the extreme case when the foundation is rigid, on the other hand, only two types of solutions are stable; i.e., deformations (1) and (3).
參考文獻
延伸閱讀
- 林勇志（2007）。兩端角度固定之彈性板條的變形與穩定性分析〔碩士論文，國立臺灣大學〕。華藝線上圖書館。https://doi.org/10.6342/NTU.2007.02103
- 李佳彥（2014）。能量法分析受拘束彈性板條之穩定性〔碩士論文，國立臺灣大學〕。華藝線上圖書館。https://doi.org/10.6342/NTU.2014.00803
- 林佑承（2012）。置於剛性基底上之彈性板條受重力影響之振動分析〔碩士論文，國立臺灣大學〕。華藝線上圖書館。https://doi.org/10.6342/NTU.2012.01196
- 李佩君（2008）。彈性支撐壓電複合層板之非線性振動分析〔碩士論文，淡江大學〕。華藝線上圖書館。https://doi.org/10.6846/TKU.2008.00286
- 林家豪（2012）。Transient Response of an Elastic Half Space Subjected to Rotating Surface Forces〔碩士論文，國立臺灣大學〕。華藝線上圖書館。https://doi.org/10.6342/NTU.2012.03213