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

Ike, Charles Chinwuba; Onyia, Michael E.; Rowland-Lato, Eghosa. O.

doi:10.25046/aj060133

Cited by 3 publications

(3 citation statements)

References 37 publications

(68 reference statements)

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“…Ike et al [46] used the Generalized Integral Transform Method (GITM) to solve the stability problem of rectangular thin plate with two opposite clamped edges and the other edges simply supported. Ike [47] used the Variational Ritz-Kantorovich-Euler-Lagrange method to develop solutions to the elastic stability problem of rectangular Kirchhoff plate with clamped boundaries.…”

Section: Review Of Previous Workmentioning

confidence: 99%

Ritz Variational Method for Solving the Elastic Buckling Problems of Thin-Walled Beams with Bisymmetric Cross-Sections

Oguaghamba¹,

Ike²,

Ikwueze³

et al. 2023

MMEP

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The Ritz variational method was used in this study to solve the lateral torsional buckling problem of simply supported thin-walled beam with doublysymmetric cross-section. Two considered cases of loading were uniform bending moments applied at the two ends, and a point load applied vertically at the midspan. The problem was presented in variational form as the problem of minimizing the total potential energy functional, , with respect to the unknown parameters of the generalized displacement modal functions. The total potential energy functional was found to be a function of two unknown displacement buckling functions v(x) and (x) and their derivatives with respect to the longitudinal coordinate axis. Suitable displacement buckling functions that satisfy the Dirichlet boundary conditions at the ends were used as trial functions to obtain the Ritz variational problem as the minimization of  with respect to the generalized buckling modal displacement amplitudes c1n and c2n. The Ritz variational equations were obtained as the minimum conditions for  with respect to c1n and c2n. The equations were solved for the two cases considered and the buckling moments found for the nth buckling mode from solving the resulting system of homogeneous algebraic equations. It was found that the expressions obtained for the buckling moments in each considered case were the exact expressions obtained by other researchers in literature who solved using classical mathematical methods. It was further found that for each considered case the critical buckling moment occurred at the first buckling mode, and the critical buckling moment expressions for each case agreed with exact solutions from the literature. The effectiveness of the Ritz variational method was thus illustrated for stability problems of thin-walled beams with Dirichlet boundary conditions.

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Section: Review Of Previous Workmentioning

confidence: 99%

Ritz Variational Method for Solving the Elastic Buckling Problems of Thin-Walled Beams with Bisymmetric Cross-Sections

Oguaghamba¹,

Ike²,

Ikwueze³

et al. 2023

MMEP

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show abstract

“…The buckling problem has been very well discussed in the context of several theories of plates, such as the Kirchhoff-Love plate theory (KLPT), shear deformation theories, and refined plate theories (RPTs). KLPT neglects shear deformation and thus does not apply to thick plates where shear deformation plays a significant role [1][2][3]. KLPT overestimates the buckling load capacities of thick plates and is thus unsafe for such plates.…”

Section: Introductionmentioning

confidence: 99%

“…They obtained analytical solutions that are exact within the framework of the thin plate buckling theory adopted. Ike et al, [48] used the generalized integral transform method to solve the thin SSCC plate buckling problem and obtained exact solutions. Nwoji et al, [49] obtained exact buckling solutions for simply supported thin plates using DFSTM.…”

Section: Introductionmentioning

confidence: 99%

Single Variable Thick Plate Buckling Problems Using Double Finite Sine Transform Method

Ike

2023

ETJ

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The double finite sine transform method was utilized to obtain exact buckling solutions for thick plate problems • The method simplified the problem to algebraic ones since the sinusoidal function satisfies the boundary conditions • The buckling solutions aligned with exact solutions for thick plates This paper derives buckling solutions for single variable thick plate buckling problems using the double finite sine transform method (DFSTM). The problem governing partial differential equation (GPDE), originally formulated by Shimpi and others, uses a refined plate theory (RPT) and accounts for transverse shear deformations, rendering it applicable to thick plates. The thick plate is simply supported and subjected to (i) uniform uniaxial compressive load in the x direction. (ii) uniform biaxial compressive load in the x and y axes. The DFSTM was applied to the GPDE, and the problem transformed into an algebraic equation, which was simpler in this case due to the Dirichlet boundary conditions satisfied by the sinusoidal kernel of the DFSTM. Analytical buckling solutions were determined for the two considered cases of uniaxial and biaxial compressive loads in terms of the buckling modes, Poisson's ratio (µ), and thickness (h) to least dimension (a) ratio. Critical buckling loads (Pcr) determined at the first buckling modes agreed with previously obtained Navier solutions for Mindlin, Reddy, and Refined plates. Pcr calculated for ratios of h/a equal to 0.01 converged to the solutions obtained using Kirchhoff-Love plate theory (KLPT), illustrating the applicability of the GPDE to thin and thin plate buckling.

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A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

Ike

2023

Math. models, eng.

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A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

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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

Cited by 3 publications

References 37 publications

Ritz Variational Method for Solving the Elastic Buckling Problems of Thin-Walled Beams with Bisymmetric Cross-Sections

Ritz Variational Method for Solving the Elastic Buckling Problems of Thin-Walled Beams with Bisymmetric Cross-Sections

Single Variable Thick Plate Buckling Problems Using Double Finite Sine Transform Method

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

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