A semi-analytical approach for stress concentration of cantilever beams with circular holes under bending

Abstract

In the paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve bending problems of a circular beam with circular holes. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. After moving the null-field point to the boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The present method is treated as a “semi-analytical” since analytical expressions as much as possible before numerical implementation. Finally, an example, including four holes, is given to demonstrate the validity of the proposed method. The present formulation can be extended to handle beam problems with arbitrary number and various positions of circular holes.本文利用退化核及傅立業級數展開搭配零場積分方程求解圓形斷面梁含圓型孔洞的彎曲問題。藉由分離核函數的表示式，可免於計算邊界積分中計算主值的困擾。文中採用自適性觀察座標系統來充分掌握分離核函數的特性。透過零場積分方程將零場點推向邊界，滿足邊界條件後可以得到線性代數方程式，其中未知的傅立葉係數可輕易地求得。本法可稱之為“半解析＂法，其主要誤差來源為所截取的傅立葉項數。最後，以一個包含四個圓孔洞的例子來驗證此方法的正確性，且探討應力集中發生之位置。藉由此方法可進行任意個數及不同位置之孔洞的彎矩分析。