New analytic buckling solutions of side-cracked rectangular thin plates by the symplectic superposition method

Hu, Zhe; Zheng, Xinran; An, Dongqi; Zhou, Chao; Yang, Yushi; Li, Rui

doi:10.1016/j.ijmecsci.2020.106051

Cited by 34 publications

(3 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equating the characteristic equation of Eq. ( 8 ) to zero gives the eigensolution of 27 : where and are expressed as with , , , , , and . Substituting Eq.…”

Section: Methodsmentioning

confidence: 99%

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

et al. 2023

Sci Rep

Self Cite

View full text Add to dashboard Cite

This work reports new analytic free in-plane vibration solutions for orthotropic non-Lévy-type rectangular plates, i.e., those without two opposite edges simply supported, by the symplectic superposition method (SSM), which has never been applied to in-plane elasticity problems in any existing works. Such analytic solutions are not accessible through conventional analytic methods as seeking analytic solutions that meet both the governing partial differential equations and various non-Lévy-type boundary conditions is an acknowledged challenge in mechanical analysis of plates. The clamped and free plates are considered as two most representative cases of non-Lévy-type plates. The SSM is implemented in the Hamiltonian system-based symplectic space, where the separation of variables and the symplectic eigen expansion prove to be well-grounded. These two mathematical treatments are adopted to first gain the analytic solutions of two elementary problems. The final analytic free in-plane vibration solutions are obtained by superposition of the two elementary problems. Comprehensive new natural frequencies and vibration modes are studied and validated by reference solutions from the finite element method or other approaches. The rigorous solution procedure, fast convergence, and highly accurate results render the present framework capable of serving as benchmarks for future comparison and applicable to analytic investigation of more plate problems.

show abstract

“…Equating the characteristic equation of Eq. ( 8 ) to zero gives the eigensolution of 27 : where and are expressed as with , , , , , and . Substituting Eq.…”

Section: Methodsmentioning

confidence: 99%

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

et al. 2023

Sci Rep

Self Cite

View full text Add to dashboard Cite

show abstract

“…Current analytical solutions for the buckling phenomenon in rectangular plates are primarily restricted to scenarios featuring uncomplicated boundary conditions with two opposite sides simply supported, commonly referred to as Lévy type plates. As for rectangular plates with non-opposite side simple supported, most research has relied on similar/numerical methods [3][4][5][6][7][8][9][10][11][12]. For rectangular plates under corner support conditions, finite difference method [13], differential quadrature method [14,15], discrete singular convolution method [16,17], meshless method [18], generalized Galerkin method [19,20], etc., have been used to obtain the similar/numerical solution of the buckling problem of such plates.…”

Section: Introductionmentioning

confidence: 99%

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

Yang,

Xu,

Chu

et al. 2024

Mathematics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Rahbar employed a semi-analytical method [43] to study the forced vibration responses of plates with clamped and simply supported edges. Li invented a new analytical approach [44][45][46], which is the combination of the symplectic elastic method and the superposition method, to handle non-Lévy-type thin/thick plate problems. Chen achieved efficient benchmark random vibration solutions for thin plates by using the discrete analytical method [47], in which the edge conditions involved a clamped edge, simply supported edge, and free edge.…”

Section: Introductionmentioning

confidence: 99%

New Natural Frequency Studies of Orthotropic Plates by Adopting a Two-Dimensional Modified Fourier Series Method

Wu,

Li,

et al. 2024

Buildings

View full text Add to dashboard Cite

The free vibration behavior of orthotropic thin plates, which are clamped at three edges and free at one edge, is a matter of great concern in the engineering field. Various numerical/approximate approaches have been proposed for the present problem; however, lack precise analytic benchmark solutions are lacking in the literature. In the present study, we propose a modified two-dimensional Fourier series method to effectively handle free vibration problems of plates under various edge conditions. In the given solution, the adopted trial function automatically satisfies several boundary conditions. After imposing Stoke’s transformation in the trial function and letting it satisfy the remaining boundary conditions, we can change the present plate problem into calculating several systems of linear algebra equations which are easily handled. The present method can be regarded as an easily implemented, rational, and rigorous approach, as it can exactly satisfy both the governing equation and the associated edge conditions. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through the utilization of different types of Fourier series, and it can be extended for the further dynamic/static analysis of plates under different shear deformation theories. Finally, all the novel analytical solutions are confirmed to be sufficiently accurate since they match well with the FEM results. The new analytic solution obtained may serve as a benchmark for validating other numerical and approximate methods.

show abstract

New analytic buckling solutions of side-cracked rectangular thin plates by the symplectic superposition method

Cited by 34 publications

References 59 publications

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

New Natural Frequency Studies of Orthotropic Plates by Adopting a Two-Dimensional Modified Fourier Series Method

Contact Info

Product

Resources

About