Kantorovich-Vlasov Method for Simply Supported Rectangular Plates under Uniformly Distributed Transverse Loads

C.U, Nwoji; Mama, Benjamin; Onah, Hyginus; Ike, Charles Chinwuba

doi:10.24001/ijcmes.3.2.1

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…The elastic buckling problem considered in the study is given for biaxial compressive forces by Equation (9) or Equation ( 10) when the biaxial compressive forces are equal for both the x and y directions (Nx = Ny). The domain PDE for uniaxial buckling in the x direction is given by Equation (11).…”

Section: Discussionmentioning

confidence: 99%

“…In [27], the author presented two-dimensional Fourier cosine series technique to solve bending problems of rectangular thin plates supported by Winkler one-parameter foundations with the plate domain subject to transverse loading. Kantorovich-Vlasov method was used for the thin plate flexure problems by authors in [9], [28]- [31].…”

Section: Review Of Solution Methods For Plate Problemsmentioning

confidence: 99%

“…In [9] the authors have studied the use of Kantorovich-Vlasov method for solving bending problems of thin plate with Dirichlet conditions under uniformly distributed transverse loading. In [28], the author presented a mixture of Kantorovich, Euler-Lagrange and Galerkin's techniques and used it for solving bending problems of rectangular thin plates.…”

Section: Review Of Solution Methods For Plate Problemsmentioning

confidence: 99%

“…Plates are applied extensively in the varied fields of engineering and used in civil, mechanical, aeronautical, naval, spacecraft and structural components. This has resulted in the extensive studies done by many previous scholars on the subject matter [1][2][3][4][5][6][7][8][9][10]. Plates are subject to loads that produce dynamic ASTESJ ISSN: 2415-6698 This work considers rectangular thin plates made with isotropic, homogeneous, linear elastic materials.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Integral Transform Method for Bending and Buckling Analysis of Rectangular Thin Plate with Two Opposite Edges Simply Supported and Other Edges Clamped

Ike¹,

Onyia²,

Rowland-Lato³

2021

Adv. sci. technol. eng. syst. j.

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This paper presents the generalized integral transform method for solving flexural and elastic stability problems of rectangular thin plates clamped along /2 yb = and simply supported along remaining boundaries (x = 0, x = a) (CSCS plate). The considered plate is homogeneous, isotropic and carrying uniformly distributed transversely applied loading causing bending. Also studied, is a plate subject to (i) biaxial (ii) uniaxial uniform compressive load. The method uses the eigenfunctions of vibrating thin beams of equivalent span and support conditions in constructing the basis functions for the plate deflection and the integral kernel function. The transform is applied to the governing domain equation, converting the problem to integral equations for both cases of bending and elastic buckling. The integral equation reduces to algebraic problems for the bending problem, and algebraic eigenvalue problem for the elastic buckling problem. The deflections are obtained as double infinite series with rapidly convergent properties. Bending moments expressions are double series with infinite terms which are rapidly convergent. Maximum deflections and bending moments values occur at the plate centre in agreement with symmetry. The present results gave double series solutions with good convergent properties in closed form for bending problems. The resulting bending solutions were exact. Solving the resulting eigenvalue equation gave closed analytical equation for the buckling loads. Buckling loads are computed for the cases of biaxial and uniaxial uniform compression of square thin plates using one term approximations. The buckling load obtained for one term approximation of the eigenfunction gave results that are 12.23% greater than the exact solution. The use of more terms in the eigenfunction expansion could give more acceptable results for the eigenvalue problem of buckling of CSCS plates.

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Section: Discussionmentioning

confidence: 99%

Section: Review Of Solution Methods For Plate Problemsmentioning

confidence: 99%

Section: Review Of Solution Methods For Plate Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized Integral Transform Method for Bending and Buckling Analysis of Rectangular Thin Plate with Two Opposite Edges Simply Supported and Other Edges Clamped

Ike¹,

Onyia²,

Rowland-Lato³

2021

Adv. sci. technol. eng. syst. j.

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Review of the Methods of Transition from Partial to Ordinary Differential Equations: From Macro- to Nano-structural Dynamics

Awrejcewicz

Кrysko

Kalutsky

et al. 2021

Arch Computat Methods Eng

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This review/research paper deals with the reduction of nonlinear partial differential equations governing the dynamic behavior of structural mechanical members with emphasis put on theoretical aspects of the applied methods and signal processing. Owing to the rapid development of technology, materials science and in particular micro/nano mechanical systems, there is a need not only to revise approaches to mathematical modeling of structural nonlinear vibrations, but also to choose/propose novel (extended) theoretically based methods and hence, motivating development of numerical algorithms, to get the authentic, reliable, validated and accurate solutions to complex mathematical models derived (nonlinear PDEs). The review introduces the reader to traditional approaches with a broad spectrum of the Fourier-type methods, Galerkin-type methods, Kantorovich–Vlasov methods, variational methods, variational iteration methods, as well as the methods of Vaindiner and Agranovskii–Baglai–Smirnov. While some of them are well known and applied by computational and engineering-oriented community, attention is paid to important (from our point of view) but not widely known and used classical approaches. In addition, the considerations are supported by the most popular and frequently employed algorithms and direct numerical schemes based on the finite element method (FEM) and finite difference method (FDM) to validate results obtained. In spite of a general aspect of the review paper, the traditional theoretical methods mentioned so far are quantified and compared with respect to applications to the novel branch of mechanics, i.e. vibrational behavior of nanostructures, which includes results of our own research presented throughout the paper. Namely, considerable effort has been devoted to investigate dynamic features of the Germain–Lagrange nanoplate (including physical nonlinearity and inhomogeneity of materials). Modified Germain–Lagrange equations are obtained using Kirchhoff’s hypothesis and relations based on the modified couple stress theory as well as Hamilton’s principle. A comparative analysis is carried out to identify the most effective methods for solving equations of mathematical physics taking as an example the modified Germain–Lagrange equation for a nanoplate. In numerical experiments with reducing the problem of PDEs to ODEs based on Fourier’s ideas (separation of variables), the Bubnov–Galerkin method of static problems and Faedo–Galerkin method of dynamic problems are employed and quantified. An exact solution governing the behavior of nanoplates served to quantify the efficiency of various reduction methods, including the Bubnov–Galerkin method, Kantorovich–Vlasov method, variational iterations and Vaindiner’s method (the last three methods include theorems regarding their numerical convergence). The numerical solutions have been compared with the solutions obtained by various combinations of the mentioned methods and with solutions obtained by FDM of the second order of accuracy and FEM for triangular and quadrangular finite elements. The studied methods of reduction to ordinary differential equations show high accuracy and feasibility to solve numerous problems of mathematical physics and mechanical systems with emphasis put on signal processing.

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Computing static behavior of flexible rectangular von Kármán plates in fast and reliable way

Awrejcewicz

Kalutsky²,

Кrysko³

2022

International Journal of Non-Linear Mechanics

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Kantorovich-Vlasov Method for Simply Supported Rectangular Plates under Uniformly Distributed Transverse Loads

Abstract: Abstract-

Cited by 4 publications

References 11 publications

Generalized Integral Transform Method for Bending and Buckling Analysis of Rectangular Thin Plate with Two Opposite Edges Simply Supported and Other Edges Clamped

Generalized Integral Transform Method for Bending and Buckling Analysis of Rectangular Thin Plate with Two Opposite Edges Simply Supported and Other Edges Clamped

Review of the Methods of Transition from Partial to Ordinary Differential Equations: From Macro- to Nano-structural Dynamics

Computing static behavior of flexible rectangular von Kármán plates in fast and reliable way

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