Equilibrium approach in the derivation of differential equations for homogeneous isotropic mindlin plates

Ike, Charles Chinwuba

doi:10.4314/njt.v36i2.4

Cited by 10 publications

(14 citation statements)

References 7 publications

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“…The unknown displacement parameters of the deflection function are evaluated by solving the matrix algebraic equation. 4. A one unknown parameter choice of the deflection function that satisfied the geometric and force boundary conditions yielded seasonally accurate prediction of the maximum deflection at the center with a relative error of 1.9 %.…”

Section: Discussionmentioning

confidence: 92%

“…The mathematical problems of plate analysis belongs to the three dimensional theory of elasticity governed by the simultaneous satisfaction of the requirements of the material stress-strain laws, the kinematic (geometric) relations between strain and displacements, the differential equations of equilibrium and the loading and restraint boundary conditions [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

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Systematic Presentation of Ritz Variational Method for the Flexural Analysis of Simply Supported Rectangular Kirchhoff–Love Plates

Ike¹

2018

JES

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In this work, the Ritz variational method for solving the flexural problem of Kirchhoff-Love plates under transverse distributed load has been presented systematically in matrix form. An illustrative application of the matrix presentation was done for simply supported rectangular Kirchhoff-Love plate under uniformly distributed load. The application used a one term Ritz approximating displacement (coordinate, or basis) function. A one term Ritz approximate solutions obtained for center displacement of square plates showed a difference of 1.9 % from the exact solution for displacement. Solution obtained for the bending moment at the center showed a difference of 7.9 % from the exact solution for bending moment. The one term Ritz approximation for the maximum shear force showed a difference of-10.7 % from the exact solution. The results obtained for a one term Ritz approximation of the displacement shape function was reasonably close for practical purposes.

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Section: Discussionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Systematic Presentation of Ritz Variational Method for the Flexural Analysis of Simply Supported Rectangular Kirchhoff–Love Plates

Ike¹

2018

JES

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“…Also, the shear force and the tangential displacement will be zero. The differential equation of equilibrium of axisymmetric circular Kirchhoff-Love plate will simplify to [30,31]…”

Section: Introductionmentioning

confidence: 99%

Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates

Ike¹

2018

Math. models, eng.

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In this work, the problem of first order shear deformable solid circular plate under transverse load was solved mathematically. The problem considered was assumed axisymmetric. The plate and loading were considered axisymmetric. The problem was defined as a boundary value problem of a system of differential equations of equilibrium in terms of the stress resultants and the stress-resultants-displacement relations. The set of equations were considered simultaneously to express them in variable separable form. The mathematical technique of separation of variables was then used to obtain solutions for the unknown generalised displacements. Specific problems of clamped edge plates and simply supported edge plates under uniformly distributed load and point load at the centre were considered and solved using the same technique of separation of variables. The mathematical expressions obtained showed that in all cases, the deflection was expressible in terms of flexural and shear components. The maximum deflection was found to occur at the plate centre as is expected from the symmetrical nature of the problem. The shear component of the transverse deflection was found to significantly increase with significant increase in the ratio of the plate thickness to the radius (ℎ ⁄).

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“…The classical Kirchhoff plate theory has been found to give satisfactory results for thin plates, and despite the obvious mathematical simplicity of the governing equations, the theory has the following limitations [12] (i) the accuracy of the Kirchhoff plate theory decreases with increase in plate thickness, with rapidly increasing or localized forces and in problems of stress concentration around openings in the plate. (ii) the Kirchhoff plate theory assumes a displacement field for which the strain field is derived.…”

Section: Y mentioning

confidence: 99%

Theory of Elasticity Formulation of the Mindlin Plate Equations

Nwoji¹,

C.U.²,

H.N³

et al. 2017

IJET

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Abstract-In this work, the mathematical theory of elasticity has been used to formulate and derive from fundamental principles, the first order shear deformation theory originally presented by Mindlin using variational calculus. A relaxation of the Kirchhoff's normality hypothesis was used to account for the effect of the transverse shear strains in the behaviour of the plate. This made the resulting theory appropriate for use for moderately thick plates. A simultaneous use of the strain-displacement relations for small-deformation elasticity, stress-strain laws and the stress differential equations of equilibrium was used to obtain the differential equations of static flexure for Mindlin plates in terms of three unknown generalised displacements. The equations were found to be coupled in the unknown displacements; but reducible to the Kirchhoff plate equations when the removed Kirchhoff normality hypothesis was introduced. This showed the Kirchhoff plate theory to be a specialization of the Mindlin plate theory. The theory of elasticity foundations of the Mindlin and Kirchhoff plate theories are thus highlighted.

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Equilibrium approach in the derivation of differential equations for homogeneous isotropic mindlin plates

Cited by 10 publications

References 7 publications

Systematic Presentation of Ritz Variational Method for the Flexural Analysis of Simply Supported Rectangular Kirchhoff–Love Plates

Systematic Presentation of Ritz Variational Method for the Flexural Analysis of Simply Supported Rectangular Kirchhoff–Love Plates

Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates

Theory of Elasticity Formulation of the Mindlin Plate Equations

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