圓淺殼於邊緣受均勻張力之變形與反折斷式挫曲

Abstract

本文研究一圓淺殼受均勻邊緣張力後之變形與穩定分析，分理論分析與實驗兩部分。由von Karman’s板殼模型可推導出運動方程式，並用Galerkin’s method將方程式離散化求解。淺殼所受之邊緣張力與初始形狀雖均假設為軸對稱形式，在分析變形時亦考慮非軸對稱變形模態。對一初始形狀為軸對稱形式的淺殼來說，受均勻邊緣張力作用後其平衡位置可分為軸對稱與非軸對稱解。軸對稱的平衡位置有穩定與不穩定解，但是非軸對稱的平衡位置均為不穩定，因此在分析準靜態問題的時候，不需要考慮非軸對稱的假設模態。若淺殼一開始是在不受內應力的平衡位置，當邊緣張力增加時淺殼只會被拉平。但是，當淺殼一開始處於受內應力的平衡位置時，當邊緣張力增加至臨界值時，淺殼將會突然跳回另一平衡位置，此現象稱為反折斷式挫曲。實驗部分是以不同初始高度的軸對稱淺殼來印證理論預測。大致上來說，當淺殼初始高度 ，實驗量測出的變形與臨界挫曲值與理論較吻合。

In this paper we study the deformation and stability of a shallow shell under uniform edge tension, both theoretically and experimentally. von Karman’s plate model is adopted to formulate the equations of motion. Although the initial shape of the shallow shell is assumed to be axisymmetrical, the possibility of unsymmetrical deformation is also examined. For an axisymmetrical shell, the equilibrium positions can be classified into axisymmetrical and unsymmetrical solutions. While there may exist both stable and unstable axisymmetrical solutions, all the unsymmetrical solutions are unstable. Since the unsymmetrical solutions will not affect the stability of the axisymmetrical solutions, it is concluded that for quasi-static analysis, there is no need to include unsymmetrical assumed modes in the calculation. If the shell is initially in the unstrained configuration, it will only be flattened smoothly when the edge tension is applied. No snap-through buckling is possible in this case. On the other hand, if the shell is initially in the strained position, it will be snapped back to the stable position on the other side of the base plane when the edge tension reaches a critical value. Experiment is conducted on several free copper shells of different initial heights to verify the theoretical predictions. Generally speaking, for the range of initial height H<6 the experimental measurements of the deformation and the reverse snapping load agree well with theoretical predictions.