圓筒薄殼受力穩定分析研究

A Study on Stability Analysis for Loaded Cylindrical Shells

曾煜群(Yuh-Chyun Tzeng)

指導教授 ： 陳清泉

國立臺灣大學/工學院/土木工程學研究所/博士(2008年)

https://doi.org/10.6342/NTU.2008.00657

摘要

本文旨在求解圓筒及圓弧薄殼受力作用下之挫屈應力，以小變形彈性理論以及一組適宜座標系統由力平衡原理求得一組包含各彎矩、各面內力與變位w之聯立平衡方程式。再由應變與變位、應力與應變之關係，彎矩與曲率之關係，面內力與變位之關係等，求得一組包含三個未知變位函數u、v、w之三個偏微分聯立方程式組。最後再定義一新的運算子 及簡化合併運算過程，將三個未知變位函數u、v、w之三個偏微分聯立方程式組，簡化合併為以變位w為函數之八階偏微分控制方程式，且以此八階偏微分控制方程式求解圓筒及圓弧薄殼受力作用下之挫屈應力。
在實例分析上，本文分別探討不同邊界條件之圓筒薄殼受軸向應力作用下之挫屈行為，圓筒薄殼受扭力作用下之挫屈行為及圓弧薄殼受軸向應力作用下之挫屈行為。其結果並與傳統薄殼理論及SAP2000有限元素分析程式所得之結果相比較探討而研究之。此外，另探討圓筒薄殼軸向受動力載重作用下之動力特性行為。
經探討研究得知，本文所推導而得之八階偏微分控制方程式可順利求得與傳統薄殼理論中，圓筒及圓弧薄殼受力作用下相同之挫屈載重。而藉由SAP2000有限元素分析程式，可進一步求得圓筒及圓弧薄殼在不同厚度比及不同長徑比受力作用下之挫屈模態。所得結果可作為圓筒薄殼承擔能力之研判及圓筒薄殼相關設計之參考。

關鍵字

圓筒薄殼 ； 挫屈應力 ； 挫屈模態 ； 厚度比 ； 長徑比 ； 八階偏微分方程式

並列摘要

A general formulation is presented for the buckling of a cylindrical shell subjected to external loads. In this study, a small shell element is defined using a convenient coordinate system. The governing equation in terms of the radial deflection is derived for the element by adopting a new operator. The 8th-order partial differential equation derived can be applied for cylindrical shells with various boundary conditions.
For illustration, the cylindrical shells subjected to axial compressive forces and torsions are studied by using the 8th-order partial differential equation to obtained the critical stresses. The critical stresses obtained for the buckling of cylindrical shells are compared with those by the finite element program SAP2000 and the classical theory. Moreover, free vibration analysis and dynamic behaviors of cylindrical shells with simply supported ends are also presented in this study.
As results, the critical stresses obtained by using the 8th-order partial differential equation are the same with those by the classical theory, while they are similar to those by the finite element program SAP2000. Good agreement has been obtained for most of the comparative cases.

並列關鍵字

cylindrical shell ； critical stress ； buckling mode ； thickness ratio ； slenderness ratio ； 8th order.

參考文獻

3.S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, 2nd edn. (McGraw-Hill, New York, 1961).

8.Lloyd H. Donnell, Thick general shells under general loading, Journal of Applied Mechanics, 56 (2) (1989) 391-394.

11.R. Szilard, Theory and Analysis of Plates: Classical and Numerical Method (Prentice-Hall, Englewood Cliffs, N.J., 1974).

12.S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd edn. (McGraw-Hill, New York, 1970).

13.B. Venkatraman and S. A. Patel, Structural Mechanics with Introductions to Elasticity and Plasticity (McGraw-Hill, New York, 1970).