傅利葉級數方法於板式構造應力分析

Abstract

本研究以傅利葉級數表示的邊緣函數及角隅函數，處理任意形狀之多邊形平板的邊界值問題。這些問題包含：1.平面雙軸應力問題(Bi-axial stress)。2.平面彈性問題(Plane elasticity)。3.平板撓曲問題(Plate bending)。4.雙軸應力與平板撓曲應力組合的板式構造。5.平面應力與平板撓曲應力組合的板式構造。以平板半平面之雙軸應力、平面彈性、平板撓曲控制方程式的解析解，作為各邊緣及角隅位移影響函數的基本解，利用座標轉換及邊緣積分，得到邊緣對邊緣影響函數解析解（包含邊緣函數及角隅函數）。各邊緣之解析解邊界值經疊加後，與已知邊界值經由傅利葉級數轉換之諧合，以得出解析函數中的未知數。對於同平面的凸形多邊形平板，本方法僅需一個元素，非凸形區域則需切割為數個凸形平面，藉由共同邊界諧合求解。除未知數為平板區域邊緣的震幅外，計算步驟與邊界元素法甚為類似。 相同於邊界積分法，傅利葉級數法不需網格劃分。所使用的元素與自由度較有限元素、有限差分等方法少。此外，傅利葉級數法所形成的矩陣，其數值皆由解析解直接積分得出，較邊界元素法使用數值積分的運算更具效率。而藉由雙軸應力與平板撓曲自由度組合，或平面應力與平板撓曲自由度組合，不同平面平板可經由共同邊界之諧合，以組成工程實務上常使用的板式構造(plated-structure)，此部分亦於本文中完成。

This study presents a novel method based on edge function and corner function approach using the Fourier series for boundary value problems and applies it to polygonal domains. This method is also applied to (1).plane bi-axial stress,(2).plane elasticity problems, (3).plate bending problems,(4).plated structures assembled by bi-axial stress and plate bending,(5).plated structures assembled by plane elasticity and plate bending. In addition, analytical solutions for bi-axial stress, plane elasticity and plate bending for each edge serving as a set of fundamental functions, utilized coordinate transformation and boundary integral, the solution function of each edges are obtained. The problem can be solved by superimposing the solution functions and matching the Fourier harmonics of the prescribed boundary conditions. By this approach, a convex polygon can be solved with one element only; non-convex domain is divided into several convex sub-domains with appropriate continuity conditions at the interface. The process closely resembles the boundary element method, except that the unknowns are the amplitudes of Fourier harmonics. Similar to the boundary integral method, the proposed method does not involve any mesh generation. In addition, the number of elements and the degrees of freedom for the proposed method are significantly smaller than the finite element and finite difference methods. The proposed method also holds an advantage over boundary elements in that the matrices are directly integrated from an analytical solution instead of numerical integration. Combining bi-axial stress and plate bending element, or combining plane elasticity and plate bending element, allows us to readily extend the proposed method to plated structures.