Uniformly loaded rectangular plate supported at the corners

Lee, S.L.; Ballesteros, Pío

doi:10.1016/0020-7403(60)90005-9

Cited by 36 publications

(13 citation statements)

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“…The trial functions satisfied the geometric boundary conditions at the outset while the natural boundary conditions were enforced in an integral sense along the boundary. Subsequently, Pan (1961) proved that the results of Lee and Ballesteros (1960) were in reasonable agreement with those of Galerkin (1953). Shanmugam and his associates used the polynomial deflection function to solve the cornersupported plates with rhombic (Shanmugam et al, 1988) and triangular shapes (Shanmugam et al, 1989).…”

Section: Introductionsupporting

confidence: 57%

“…In order to further substantiate the accuracy of symplectic approach, a comparison of results from various sources including Galerkin (1953), Lee and Ballesteros (1960), the Ritz method and modified Ritz method (Wang et al 2002), finite element solutions (Wang et al 2002) and the symplectic solutions is presented in Table 3. In the present finite element simulation using ABAQUS, the plate thickness-towidth ratio is 0.01, the Poisson's ratio is n ¼ 0.3, and a Without absolute agreement, one may argue that the accuracy of symplectic solutions is not justified.…”

Section: Resultsmentioning

confidence: 99%

“…The approach trespasses an area of practical engineering interest beyond the ability of the classical methods such as Timoshenko's method, Navier method, Levy method, and the approximate polynomial method of Lee and Ballesteros (1960). The variational principle is formulated in the symplectic geometry space and the zero bending moment and shear force boundary conditions at the free edges are satisfied.…”

Section: Discussionmentioning

confidence: 99%

“…One of the earliest comprehensive analyses was presented by Lee and Ballesteros (1960), who adopted a semi-inverse approach by using a predetermined trial function to approximate the plate deflection similar to that of Timoshenko and Woinowsky-Krieger (1959) while the unknown constants were determined by the boundary conditions. The trial functions satisfied the geometric boundary conditions at the outset while the natural boundary conditions were enforced in an integral sense along the boundary.…”

Section: Introductionmentioning

confidence: 99%

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Benchmark symplectic solutions for bending of corner-supported rectangular thin plates

Lim

Yao

Cui

2008

The IES Journal Part A: Civil & Structural Engineering

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This work presents the first ever analytical solutions for bending of a rectangular, thin plate supported only at its four corners. This breakthrough analysis employs a new symplectic elasticity approach that extends beyond the limitation of the classical plate bending methods such as Timoshenko's method, Navier method, Levy method and the polynomial approximation analysis of Lee and Ballesteros (Int J Mech Sci 1960;2:206). The classical methods are, in fact, special cases of this symplectic approach in the real eigenvalue regime for wavenumber with at least one pair of opposite sides of plates simply supported. For plate problems that do not fall into this category, the classical methods fail to yield any analytical solutions, but the symplectic approach does because in these cases the plate bending problems enter the complex eigenvalue regime for wavenumber. Another distinctive feature of this new approach is its necessity to pose an eigenvalue problem even for plate bending. In short, this innovative approach establishes the relationship between eigenvalue problem and bending. The novelty of this approach lies in the use of the Hamiltonian principle in a symplectic geometry space to derive a Hamiltonian system and a full state vector. The free boundaries with corner supports are dealt with using the variational principle. Analytical bending modes are then derived by expansion of eigenfunctions. The solutions are compared with other known (approximate) results and numerical finite element solutions but some of the results are not in agreement. Because the analytical bending moment and shear force solutions thus derived fulfill all natural and geometric boundary conditions, it leaves ample room for authentication of the benchmarks in the future. In addition, the twisting moment at the corners satisfies the condition for static bending equilibrium, in which the finite element solutions fail.

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Section: Introductionsupporting

confidence: 57%

Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Benchmark symplectic solutions for bending of corner-supported rectangular thin plates

Lim

Yao

Cui

2008

The IES Journal Part A: Civil & Structural Engineering

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“…This number of legs was chosen to minimize the base plate deflection due to its weight. The following formula gives the maximum deflection at the centre of a rectangular plate simply supported at its four corners [95]: where P is the mean pressure at the point for which the pressure coefficient is being calculated. Details of the pressure measurement process and the location of P static measurements are explained in section 4.5.…”

Section: Support Legsmentioning

confidence: 99%

Experimental and Numerical Study of Flow Structures Associated With Low Aspect Ratio Elliptical Cavities

Khadivi¹,

Savory

2013

Journal of Fluids Engineering

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The flow regimes associated with a 2:1 aspect ratio, elliptical planform cavity in a turbulent flat plate boundary layer have been systematically examined for various depth/width ratios (0.1 to 1.0) and yaw angles (0° to 90°), using a combination of wind tunnel experiments (involving Particle Image Velocimetry and helium bubble visualization) and Computational Fluid Dynamics (CFD) simulations (employing threedimensional steady calculations with the Reynolds Stress model turbulence closure).Satisfactory agreement has been found between the results using the two methods, indicating that the steady numerical simulations can be a cost-effective tool to predict the mean flow features.The flow has been found to be highly dependent on yaw angle and cavity depth. For each of the three broad flow categories specified according to yaw angle, which include the symmetric flow regime (yaw angle = 0°), the straight vortex regime (yaw angle = 90°) and the asymmetric flow regime (15° ≤ yaw angle ≤ 60°) different regimes are found to exist, depending on cavity depth. For each combination of yaw angle and depth, the flow has been analyzed through investigation of shear layer parameters, the three-dimensional vortex structure, pressure distribution and drag, wake flow, and vortex oscillations.While the elliptical cavity flows have been found to have some similarities with those of nominally two-dimensional and rectangular cavities, the three-dimensional effects due to the low aspect ratio and curvature of the walls give rise to features exclusive to low aspect ratio elliptical cavities, including formation of cellular structures at intermediate depths and unique vortex structures within and downstream of the cavity.The three-dimensional flow structure of the flow is most pronounced in the asymmetric regimes with large yaw angles (45° and 60°). The dominant feature in this regime is the formation of a trailing vortex that is associated with high drag and flow oscillations within the cavity.iv Keywords

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A stabilized one‐point integrated quadrilateral Reissner–Mindlin plate element

Gruttmann

Wagner

2004

Numerical Meth Engineering

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A new quadrilateral Reissner–Mindlin plate element with 12 element degrees of freedom is presented. For linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition can be integrated analytically. This leads to a part obtained by one‐point integration and a stabilization matrix. The element possesses a correct rank, does not show shear locking and is applicable for the evaluation of displacements and stress resultants within the whole range of thin and thick plates. The bending patch test is fulfilled and the computed numerical examples show that the convergence behaviour is better than comparable quadrilateral assumed strain elements. Copyright © 2004 John Wiley & Sons, Ltd.

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Uniformly loaded rectangular plate supported at the corners

Cited by 36 publications

References 1 publication

Benchmark symplectic solutions for bending of corner-supported rectangular thin plates

Benchmark symplectic solutions for bending of corner-supported rectangular thin plates

Experimental and Numerical Study of Flow Structures Associated With Low Aspect Ratio Elliptical Cavities

A stabilized one‐point integrated quadrilateral Reissner–Mindlin plate element

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