A general moderately thick rectangular plate model is proposed and its analytical solutions are obtained by using the symplectic elasticity approach (SEA). First, the equilibrium equations of the model are transformed into a Hamiltonian dual equation and the eigenvalues and eigenvectors of the corresponding Hamiltonian operators are calculated. Furthermore, the symplectic orthogonality and the completeness of eigenvectors are proved, and the analytical solutions of the problem are presented based on boundary conditions. The feasibility of the proposed framework and the effectiveness of the SEA are verified by numerical examples of the bending problems of moderately thick rectangular plates on the different elastic foundations and the free vibration problem of moderately thick rectangular plates.
Citation: Jianan Qiao, Guolin Hou, Jincun Liu. Analytical solutions for the model of moderately thick plates by symplectic elasticity approach[J]. AIMS Mathematics, 2023, 8(9): 20731-20754. doi: 10.3934/math.20231057
A general moderately thick rectangular plate model is proposed and its analytical solutions are obtained by using the symplectic elasticity approach (SEA). First, the equilibrium equations of the model are transformed into a Hamiltonian dual equation and the eigenvalues and eigenvectors of the corresponding Hamiltonian operators are calculated. Furthermore, the symplectic orthogonality and the completeness of eigenvectors are proved, and the analytical solutions of the problem are presented based on boundary conditions. The feasibility of the proposed framework and the effectiveness of the SEA are verified by numerical examples of the bending problems of moderately thick rectangular plates on the different elastic foundations and the free vibration problem of moderately thick rectangular plates.
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Jianan Qiao, Guolin Hou, Jincun Liu. Analytical solutions for the model of moderately thick plates by symplectic elasticity approach[J]. AIMS Mathematics, 2023, 8(9): 20731-20754. doi: 10.3934/math.20231057
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