臺灣博碩士論文加值系統

本研究係探討一預應力下簡易支撐不對稱疊層圓型板件由於均佈面壓所致之大撓曲問題，其方法係以馮卡門大撓曲理論為基礎，且藉由無因次合併技巧，以推導得出一側向斜率與徑向力合之非線性聯立藕合掌控方程組。求解過程則係先忽略其非線性項來考量一線性問題，以得出一側向斜率之修正貝索方程，再利用修正貝索函數之相關定義，並配合邊界條件來求得其解析解。另一方面，其非線性問題則運用有限差分法配合迭代技巧，且以先前所得線性解為初始值以尋求其數值解。
所提出之方法係針對兩層不對稱疊層板件在各式物理參數，包含不同預拉伸力與側向面壓大小、不同疊層模數比與厚度比條件下作探討。針對擬單層簡支板件在極低預拉伸力下所得之結果與古典板理論單層板之結果吻合，說明本文所提方法之正確性。針對典型兩層板件所得結果也發現，簡易支撐板件並無夾固板件之邊區效應，雖然預拉伸力會如夾固板件情況一樣延伸其線性行為之範圍，其線性化容許的側負荷則相對較低。此外，在低模數比、高預拉伸力下，線性解析解有其準確性，但在高模數比之下，則需仰賴非線性數值解以求其更準確之結果。此外也發現，預拉伸力影響為最大，其次是面壓，而模數比之影響，則僅在低預拉伸力與面壓之作用下，才會顯現。而在探討板件之各式幾何回應中，預拉伸力與面壓之作用有類似之趨勢，並且在低預拉伸力下，板件受面壓之影響較為直接，其中又以本文所探討之簡易支撐更為明顯。從討論之結果中也得出，板件之邊界條件對其板中心的影響範圍較小，易言之，板件中心之薄膜行為最早出現。

Abstract
The problem of large deflection of a pre-stressed simply supported un-symmetric layered circular plate due to lateral pressure is studied. Nonlinear governing equations were derived base on von-Karman plate theory for large deflection, which are further formulated in terms of lateral slope and radial force resultant via a non-dimensional and merging scheme. The reduced linear problem is considered, first, by neglecting the nonlinear terms, yielding a modified Bessel equation for the lateral slope. The linear analytical solution is obtained following the definitions of modified Bessel functions and the boundary conditions of the problem. For the nonlinear problem, on the other hand, numerical solutions are sought using a finite difference method incorporated with an iteration scheme, by taking the previously obtained linear solution as an initial gues
The developed approaches were implemented mainly for 2-layered unsymmetric plate with various physical parameters including different magnitudes of pretensions, lateral pressures, and ratios in modulus and thickness between the layers. For a nearly monolithic plate, the results for a very low pretension correlate well with those obtained following CPT (classical plate theory) method and thus the present approach is checked. For typical two-layered plates, the obtained results show that, there is no edge effect like a clamped plate. In addition, although pretension may extend the range of linear behavior as in the case of a clamped plate, the allowable lateral load for linearity is comparatively lower. For a low modulus ratio between the layers of the plate with a high pretension, linear analytical solution may provide reasonable results. For the case of a high modulus ratio, however, the problem would require the presented nonlinear numerical approach to obtain more accurate results.
Keywords : Initial Tension, Un-symmetric Layered Plate, Von-Karman Large Deflection, Modified Bessel Function, Finite Difference Method.

中文摘要 i
Abstract ii
致謝 iii
目錄 iv
圖表目錄 vi
符號說明 vii
第一章 前言 1
1-1研究動機與目的: 1
1-2文獻回顧 5
1-3本文架構 8
第二章 物理問題與基礎理論描述 9
2-1問題描述 9
2-2不對稱疊層圓型板大撓曲掌控方程 9
I. 不對稱圓型板件基礎掌控方程: 9
II. 馮卡門大撓曲理論: 11
2-3預應力下不對稱疊層大撓曲非線性掌控方程之推導 12
I. 增量型式: 12
II. 無因次化: 12
第三章 解析與數值方法推導 14
3-1線性小撓曲問題 14
I. 斜率解析解: 14
II. 撓度解析解: 15
III. 曲率解析解: 15
IV. 徑向應力解析解: 16
3-2非線性大撓曲問題 17
I. 差分法之應用 17
II. 非線性數值解收斂技巧 21
III. 非線性數值解之應力解 21
第四章數值結果與討論 22
4-1大撓曲問題之線性解結果 23
4-1.1擬單層線性解與古典板理論解之比較 23
4-1.2線性解與典型單層夾固式板件之比較 24
4-1.3典型矽質疊層板件之各式幾何回應之線性結果 27
4-1.4不同模數比與厚度比下各式幾何回應結果 28
4-2大撓曲問題之非線性數值結果 28
4-2.1擬單層與大模數比板件之線性解析解與非線性數值解比較 29
4-2.2含不同模數比下之最大線性面壓與拉伸參數關係圖 30
4-2.3不同預拉伸力下中心撓度對無因次面壓關係之結果 31
4-2.4不同預拉伸力下中心曲率對無因次面壓關係之結果 32
4-2.5不同模數比下非線性中心力合與面壓關係之結果 32
4-2.6不同面壓下之全域式斜率結果 34
4-2.7不同面壓下之全域式撓度結果 35
4-2.8不同面壓下之全域式曲率結果 36
4-2.9不同面壓下之全域式無因次徑向力合(Sr/S0)結果 37
4-2.10不同面壓下之全域式環向力合(Sθ/S0)結果 38
4-2.11不同預拉伸力之全域式徑向應力(Σr/P)解結果 39
第五章結論 40
參考文獻 42
圖表整彙 47

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