Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method
The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formid...
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MDPI AG
2024-01-01
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| author | Yushi Yang Dian Xu Jinkui Chu Rui Li |
| author_facet | Yushi Yang Dian Xu Jinkui Chu Rui Li |
| author_sort | Yushi Yang |
| collection | DOAJ |
| description | The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T10:42:43Z |
| publishDate | 2024-01-01 |
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| spelling | doaj.art-d9ff9017b2eb4f6d99f203dce55217292024-01-26T17:31:32ZengMDPI AGMathematics2227-73902024-01-0112224910.3390/math12020249Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition MethodYushi Yang0Dian Xu1Jinkui Chu2Rui Li3Key Laboratory for Micro/Nano Technology and System of Liaoning Province, School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, ChinaState Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, International Research Center for Computational Mechanics, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian 116024, ChinaKey Laboratory for Micro/Nano Technology and System of Liaoning Province, School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, ChinaState Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, International Research Center for Computational Mechanics, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian 116024, ChinaThe buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes.https://www.mdpi.com/2227-7390/12/2/249symplectic superposition methodrectangular thin platebucklingcorner-point supports |
| spellingShingle | Yushi Yang Dian Xu Jinkui Chu Rui Li Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method Mathematics symplectic superposition method rectangular thin plate buckling corner-point supports |
| title | Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method |
| title_full | Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method |
| title_fullStr | Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method |
| title_full_unstemmed | Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method |
| title_short | Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method |
| title_sort | analytic solution for buckling problem of rectangular thin plates supported by four corners with four edges free based on the symplectic superposition method |
| topic | symplectic superposition method rectangular thin plate buckling corner-point supports |
| url | https://www.mdpi.com/2227-7390/12/2/249 |
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