Kantorovich-Euler Lagrange-Galerkin’s method for bending analysis of thin plates

Ike, Charles Chinwuba

doi:10.4314/njt.v36i2.5

Cited by 15 publications

(19 citation statements)

References 6 publications

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“…Numerical methods are methods that aim to obtain approximate solutions to the governing boundary value problem of plates. They include: finite difference methods [10], finite element methods, boundary element methods, variational Ritz methods [11], variational Galerkin method [12,13], collocation methods, Bubnov-Galerkin method, Kantorovich method [14,15], Vlasov method [14,15].…”

Section: Methods Of Solving Plate Problemsmentioning

confidence: 99%

Kantorovich-Vlasov Method for the Flexural Analysis of Kirchhoff Plates with Opposite Edges Clamped, and Simply Supported (CSCS Plates)

H.N¹,

B.O²,

-³

et al. 2017

IJET

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The Kantorovich-Vlasov method was used, in this study, for the flexural analysis of rectangular Kirchhoff plates with opposite edges (x = 0, and x = a) simply supported and the other opposite edges (y = 0, and y = b) clamped (CSCS plates). The plate was subjected to a linear distribution of load over the entire plate domain. Vlasov method was used in finding the coordinate function in the x-direction, and Kantorovich method was used to consider the displacement function for the plate. The total potential energy functional, and the corresponding Euler-Lagrange differential equations were then obtained for the plate problem. This was solved subject to the boundary conditions in the y direction to obtain the displacement function which minimized the total potential energy functional. Bending moment distributions were obtained using the bending moment-displacement equations. The solutions obtained for deflection and bending moment distributions were found to be rapidly convergent single series. Deflections and bending moment computed at the center of the plate were also rapidly convergent series. The solutions obtained for deflections and bending moments (M xx and M yy) were exactly identical with solutions presented by Timoshenko and Woinowsky-Krieger who used the method of superposition.

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Section: Methods Of Solving Plate Problemsmentioning

confidence: 99%

Kantorovich-Vlasov Method for the Flexural Analysis of Kirchhoff Plates with Opposite Edges Clamped, and Simply Supported (CSCS Plates)

H.N¹,

B.O²,

-³

et al. 2017

IJET

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“…Nwoji et al [26] applied the Galerkin-Vlasov method to the flexural analysis of rectangular Kirchhoff plates with clamped and simply supported edges for the case of uniformly distributed transverse loads. Ike [27] used the Kantorovich-Euler-Lagrange-Galerkin's method to solve the flexural problem of Kirchhoff plates with clamped and simply supported edges. Other researchers who have worked on the plate problem include Mama et al [28], Ezeh et al [29] and Aginam et al [30].…”

Section: Review Of Plate Theoriesmentioning

confidence: 99%

Ritz variational method for bending of rectangular kirchhoff plate under transverse hydrostatic load distribution

Nwoji¹,

Onah²,

Mama³

et al. 2018

MMEP

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In this study, the Ritz variational method was used to analyze and solve the bending problem of simply supported rectangular Kirchhoff plate subject to transverse hydrostatic load distribution over the entire plate domain. The deflection function was chosen based on double series of infinite terms as coordinate function that satisfy the geometric and force boundary conditions and unknown generalized displacement parameters. Upon substitution into the total potential energy functional for homogeneous, isotropic Kirchhoff plates, and evaluation of the integrals, the total potential energy functional was obtained in terms of the unknown generalized displacement parameters. The principle of minimization of the total potential energy was then applied to determine the unknown displacement parameters. Moment curvature relations were used to find the bending moments. It was found that the deflection functions and the bending moment functions obtained for the plate domain, and the values at the plate center were exactly identical as the solutions obtained by Timoshenko and Woinowsky-Krieger using the Navier series method.

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“…Kantorovich method has been used in the literature to solve thin plate flexure problems. Ike [18] used the Kantorovich method to solve the thin plate flexure problem with fixed edge at = 0 and simple supports at = 0, = , = for the case of uniform load. Nwoji et al [17] used the Kantorovich-Vlasov method to solve the flexural problem of simply supported Kirchhoff-Love plates under uniformly distributed load on the entire plate domain.…”

Section: Advantages Of the Kantorovich Variational Methodsmentioning

confidence: 99%

“…The classical methods used include: the Navier's double trigonometric series methods [13], the symplectic elasticity methods [14], the Levy's single trigonometric series methods [13], the methods of eigen function expansions, the method of integral transformations [15]. The approximate (numerical) methods used include variational method [16][17][18][19] weighted residual method, Finite Difference method [20] finite element method, finite grid methods, Improved finite difference methods.…”

mentioning

confidence: 99%

Kantorovich variational method for the flexural analysis of CSCS Kirchhoff-Love plates

Ike¹,

Mama²

2018

Math. models, eng.

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Abstract. The Kantorovich variational method was used in this study to solve the flexural problem of Kirchhoff-Love plates with two opposite edges = ± 2 ⁄ clamped and the other two edges = ± 2 ⁄ simply supported, for the case of uniformly distributed transverse load over the entire plate domain. The plate considered was assumed homogeneous, and isotropic. The total potential energy functional for the Kirchhoff-Love plate was found as the sum of the potential energy of the applied distributed load and the strain energy functional of the plate. The symmetrical nature of the plate and the load was used to choose the deflection function as a single series of infinite terms involving the cosine function of the y coordinate variable with the unknown function being dependent on the coordinate variable. Integration with respect to the coordinate variable simplified the total potential energy functional to be dependent on the unknown function of , and its derivatives. Euler-Lagrange differential equations were used to find the differential equations of equilibrium of the plate as a system of homogeneous fourth order ordinary differential equations (ODE) in the unknown function. The system of fourth order ODE was solved using the method of differential D operators, or the method of trial functions to obtain the general solution as the linear superposition of the homogeneous and particular solutions. The demands of symmetry of unknown function about the point = 0, and the boundary conditions at the clamped edges were used to obtain all the four integration constants; thus, completely determining the deflection. Bending moment distributions were determined using the bending moment deflection relations. The deflection was determined at the centre of the plate. The bending moment values were also determined at the centre of the plate, and at the middle of the clamped edges. Expressions obtained for the deflection at the centre, and bending moment values were found to be single series of infinite terms with highly converging properties. The solutions obtained converged to the exact solutions with the use of only four terms of the series.

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Kantorovich-Euler Lagrange-Galerkin’s method for bending analysis of thin plates

Cited by 15 publications

References 6 publications

Kantorovich-Vlasov Method for the Flexural Analysis of Kirchhoff Plates with Opposite Edges Clamped, and Simply Supported (CSCS Plates)

Kantorovich-Vlasov Method for the Flexural Analysis of Kirchhoff Plates with Opposite Edges Clamped, and Simply Supported (CSCS Plates)

Ritz variational method for bending of rectangular kirchhoff plate under transverse hydrostatic load distribution

Kantorovich variational method for the flexural analysis of CSCS Kirchhoff-Love plates

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