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# Natural Boundary Element Method for Bending Problem of Infinite Plate with a Circular Opening under the Boundary Loads

Shuncai Li1,2,3, Shichuang Zhuo4, Qiang Zhang5
School of Mechanical and Electrical Engineering, Jiangsu Normal University, Xuzhou 221116, China
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou Jiangsu 221008, China
Corresponding author. Tel: +86-13615115336; +86-516-83899037; fax: +86-516-83500251; E-mail address: zsclsc@263.net
School of Physics and Electronics, Jiangsu Normal University, Xuzhou 221116, China
Department of Aeronautics Four Stations, the Air Force Service College, Xuzhou 221000, China

Computer Modeling in Engineering & Sciences 2012, 84(3), 283-296. https://doi.org/10.3970/cmes.2012.084.283

### Abstract

Based on the complex functions theory in elastic mechanics, the bending deflection formula expressed by the complex Fourier series is derived for the infinite plate with a circular opening at first, then the boundary conditions of the circular opening are expanded in Fourier Series, and the unknown coefficients of the Fourier series are determined by comparing coefficients method. By means of the convolution of the complex Fourier series and some basic formulas in the generalized functions theory, the natural boundary integral formula or the analytical deflection formulas expressed by the boundary displacement or loads are developed for the infinite plates with a circular opening under the three common boundary conditions of the circular opening. These analytical formulas can be directly used to solve the bending problems of the infinite plates with a circular opening under the conditions of the clapped edge, simply supported edge and free edge. At last, some examples of using these analytical formulas indicate that under simple boundary conditions we can easily obtain the analytical solutions for the bending problem of the infinite plate with a circular opening, while for the bending problems with some complicated boundary conditions we can get their numerical solutions by these developed formulas.

### Keywords

infinite plate with a circular opening, bending deflection, complex Fourier series, generalized functions, natural boundary element method

Li, S., Zhuo, S., Zhang, Q. (2012). Natural Boundary Element Method for Bending Problem of Infinite Plate with a Circular Opening under the Boundary Loads. CMES-Computer Modeling in Engineering & Sciences, 84(3), 283–296.

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