New analytic buckling solutions of moderately thick clamped rectangular plates by a straightforward finite integral transform method

Ullah, Salamat; Wang, Haiyang; Zheng, Xinran; Zhang, Jinghui; Zhong, Yi; Li, Rui

doi:10.1007/s00419-019-01549-6

Cited by 19 publications

(13 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The boundary conditions for SSCF plates are given by Equations (5 -10). Using the Galerkin-Vlasov methodology, the shape function in the x coordinate direction Fm(x) is chosen as the eigenfunctions of a vibrating Euler-Bernoulli beam of equivalent span and support conditions, as Equation (11). The deflection function used is thus given as the product of unknown function in the ycoordinate Gn(y) and Fm(x) as Equation (12).…”

Section: Discussionmentioning

confidence: 99%

“…Since the scholarly works of Navier and Saint Venant, several other studies have contributed significantly to the present day knowledge of buckling. Some of these works are presented by: Ullah et al [11,12,13,14], Timoshenko [2], Timoshenko and Gere [3], Yu [8], Xiang et al [15], and Abodi [16]. Oguaghamba [17] and Oguaghamba et al [18] investigated the buckling and post-buckling loads of rectangular plates assumed to be thin and with material properties that are isotropic and homogeneous.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Galerkin-Vlasov Variational Method for the Elastic Buckling Analysis of SSCF and SSSS Rectangular Plates

Onyia¹,

Rowland-Lato²,

Ike³

2020

International Journal of Engineering Research and Technology

View full text Add to dashboard Cite

This paper presents the Galerkin-Vlasov variational method for the elastic buckling analysis of SSCF and SSSS rectangular plates. The thin plate problems studied are: (i) simply supported along two opposite sides x = 0, and x = a, clamped along the third side y = 0, and free along the fourth side y = b; (ii) simply supported along the four sides x = 0, x = a, y = 0 and y = b. In each case the edges x = 0 and x = a are subjected to uniform compressive load. Mathematically, the considered stability problem is a Boundary Value Problem (BVP) expressed by a domain fourth order partial differential equation (PDE) whose general solution should satisfy all the boundary conditions determined by the edge support conditions. By the Galerkin-Vlasov method, the unknown deflection shape function is chosen as the product of the eigenfunctions of a vibrating thin beam of identical span in the x direction and an unknown function of y(Gn(y)). The Galerkin-Vlasov variational integral equation is simplified using the Leibnitz rule, integration by parts and the orthogonality properties of the eigenfunctions of simply supported thin beams to a system of fourth order ordinary differential equations (ODEs). The general solution of the system of ODEs is obtained using trial function methods in terms of hyperbolic and trigonometric functions. The imposition of boundary conditions is used in each of the two cases to find the characteristic buckling equation. The buckling equation is obtained in each case as a transcendental equation, which is solved to obtain the eigenvalues from which the buckling loads are determined. The results obtained in each case for the buckling equation are identical to previous results obtained by other scholars who used classical methods and energy minimization methods. The results obtained for the buckling loads are in agreement with previously obtained solutions in the literature. The results obtained in each presented case in this study are found to be exact because exact shape functions were used in the x direction and the general solution was obtained for the domain PDE at every point in the plate domain. In addition, the solution obtained was made to satisfy all the boundary conditions at all the edges of the plate.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Galerkin-Vlasov Variational Method for the Elastic Buckling Analysis of SSCF and SSSS Rectangular Plates

Onyia¹,

Rowland-Lato²,

Ike³

2020

International Journal of Engineering Research and Technology

View full text Add to dashboard Cite

show abstract

“…Relieving the virtual displacements, δu and δv, in equation (8) via the integration by parts and then applying the calculus of variations yields…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

“…Zhang et al [5,6] presented a finite integral transform solution to determine the vibration behavior of rectangular orthotropic thin plates with different boundary conditions. Ullah et al [7,8] extended the integral transform method to study the buckling behavior of rectangular plates. Of course, there are many other excellent achievements, and there is no detailed discussion here.…”

Section: Introductionmentioning

confidence: 99%

Free In‐Plane Vibrations of Orthotropic Rectangular Plates by Using an Accurate Solution

Cao¹,

Zhong

Shao

et al. 2019

Mathematical Problems in Engineering

View full text Add to dashboard Cite

Many numerical methods have been developed for in-plane vibration of orthotropic rectangular plates with various boundary conditions; however, the exact results for such structures with elastic boundary conditions are very scarce. Therefore, the object of this paper is to present an accurate solution for free in-plane vibration of orthotropic rectangular plates with various boundary conditions by the method of reverberation ray matrix (MRRM) and improved golden section search (IGSS) algorithm. The boundary condition studied in this paper is defined as that a set of opposite edges is with one kind of simply supported boundary conditions, while the other set is with any kind of classical and general elastic boundary conditions or their combination. Its accuracy, reliability, and efficiency are verified by some numerical examples where the results are compared with other exact solutions in the published literature and the FEA results based on the ABAQUS software. Finally, some new accurate results for free in-plane vibration of orthotropic rectangular plates with elastic boundary conditions are examined and further can be treated as the reference data for other approximate methods or accurate solutions.

show abstract

“…Considerable research works have been presented on the stability of plates of various shapes, types and boundary conditions. Some significant contributions to the research on plate stability are reported by: Gambhir [2], Bulson [3], Chajes [4], Timoshenko and Gere [5], Shi [8], Shi and Bezine [9], Ullah et al [10,11,12,13], Wang et al [14], Abodi [15], Yu [16], Abolghasemi et al [17], Xiang et al [18] and Bouazza et al [19] Contemporary research work on the plate stability problems have used various numerical methods such as the differential quadrature method (DQM), discrete singular convolution (DSC) method, harmonic differential quadrature method, ordinary finite difference method (FDM), meshfree method, generalized Galerkin method, finite strip method, B-spline finite strip method, exact finite strip method, hp-cloud method, modified Ishlinskii's solution method, meshless analog equation method, finite element method (FEM), extended Kantorovich method (EKM) and pb2-Ritz method. Very recent research work on the subject of plate stability using various numerical and analytical techniques have been reported by Lopatin and Morozov [20], Ghannadpour et al [21], Jafari and Azhari [22], Zureick [23], Seifi et al [24], Li et al [25], Wang et al [26], Mandal and Mishra [27], Shama [28], and Yao and Fujikubo [29].…”

Section: Introductionmentioning

confidence: 99%

Elastic Buckling Analysis of SSCF and SSSS Rectangular Thin Plates using the Single Finite Fourier Sine Integral Transform Method

Onyia¹,

Rowland-Lato²,

Ike³

2020

International Journal of Engineering Research and Technology

View full text Add to dashboard Cite

The analysis of the stability problem of plate subjected to inplane compressive load is important due to the relatively poor capacity of plates in resisting compressive forces compared to tensile forces. It is also significant due to the nonlinear, sudden nature of buckling failures. This study presents the elastic buckling analysis of SSCF and SSSS rectangular thin plates using the single finite Fourier sine integral transform method. The considered plate problems are (i) rectangular thin plate simply supported on two opposite edges, clamped on one edge and free on the fourth edge; (ii) rectangular thin plate simply supported on all edges. The plates are subject to uniaxial uniform compressive loads on the two simply supported edges. The governing domain equation is a fourth order partial differential equation (PDE). The problem solved is a boundary value problem (BVP) since the domain PDE is subject to the boundary conditions at the four edges. The single finite sine transform adopted automatically satisfies the Dirichlet boundary conditions along the simply supported edges. The transform converts the BVP to an integral equation, which simplifies upon use of the linearity properties and integration by parts to a system of homogeneous ordinary differential equations (ODEs) in terms of the transform of the unknown buckling deflection. The general solution of the system of ODEs is obtained using trial function methods. Enforcement of boundary conditions along the y = 0, and y = b edges (for the SSCF and SSSS plates considered) results in a system of four sets of homogeneous equations in terms of the integration constants. The characteristic buckling equation in each case considered is found for nontrivial solutions as a transcendental equation, whose roots are used to obtain the buckling loads for various values of the aspect ratio and for any buckling modes. In each considered case, the obtained buckling equation is exact and identical with exact expressions previously obtained in the literature using other solution methods. The buckling loads obtained by the present method are validated by the observed agreement with results obtained by previous researchers who used other methods.

show abstract

New analytic buckling solutions of moderately thick clamped rectangular plates by a straightforward finite integral transform method

Cited by 19 publications

References 25 publications

Galerkin-Vlasov Variational Method for the Elastic Buckling Analysis of SSCF and SSSS Rectangular Plates

Galerkin-Vlasov Variational Method for the Elastic Buckling Analysis of SSCF and SSSS Rectangular Plates

Free In‐Plane Vibrations of Orthotropic Rectangular Plates by Using an Accurate Solution

Elastic Buckling Analysis of SSCF and SSSS Rectangular Thin Plates using the Single Finite Fourier Sine Integral Transform Method

Contact Info

Product

Resources

About