Bending Of Rectangular Plates Research Articles

Closure on discussion of ‘Benchmark symplectic solutions for bending of corner-supported rectangular thin plates’ by M. Batista

Closure on discussion of ‘Benchmark symplectic solutions for bending of corner-supported rectangular thin plates’ by M. Batista

Discussion of ‘Benchmark symplectic solutions for bending of corner-supported rectangular thin plates’ by C.W. Lim, W.A. Yao and S. Cui

Discussion of ‘Benchmark symplectic solutions for bending of corner-supported rectangular thin plates’ by C.W. Lim, W.A. Yao and S. Cui

Symplectic system based analytical solution for bending of rectangular plates on Winkler foundation

Based on an appropriate definition of symplectic inner product, we establish a symplectic space formed by generalised displacements and their dual variables for the bending of rectangular plates on Winkler foundation. By using the Hellinger–Reissner variational principle, dual equations in the symplectic space are obtained and its operator matrix proved to be a Hamilton operator matrix. So the original plate bending problem is converted into the symplectic system. By using separation of variables and symplectic eigenfunction expansion, a novel analytical method is presented for solving the aforementioned plate bending problem. The symplectic eigenvalue problems for rectangular plates with two opposite sides simply supported and the other two opposite sides clamped are discussed. The transcendental equation of its eigenvalues and symplectic eigenvectors is derived in analytical forms. Analytical solutions of two examples are presented by using this method. The solution for a simply supported plate under un...

Closed form solutions for element matrices of 4-node rectangular plate element using IFM

This paper presents closed form solutions for equilibrium and flexibility matrices of the Mindlin- Reissner theory based 4-node rectangular plate bending element using Integrated Force Method (IFM). Use of closed form solutions of equilibrium and flexibility matrices reduce the computational time significantly and more suitable for the plate bending problems with square/rectangular boundaries. Large number of standard square/rectangular plate bending benchmark problems have been analyzed for central deflections and moments using the presented closed form solutions. Results are compared with those of similar displacement based plate bending elements available in the literature. The results are also compared with the exact solutions.

Three-dimensional elasticity solution of simple-supported rectangular plate on point supports, line supports and elastic foundation

This paper studies the bending of simple-supported rectangular plate on point supports, line supports and elastic foundation. On the basis of three-dimensional elasticity theory, the exact expressions of the displacement functions, which satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived. The reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the plate. The unknown coefficients are then determined by the boundary conditions on the upper and lower surfaces of the plate. Comparing the numerical results obtained from the proposed method to those obtained from Kirchhoff plate theory, Mindlin plate theory and those obtained from the commercial finite element software ANSYS, the high accuracy of the present method has been demonstrated.

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated.

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

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 a...

Efficient finite element analysis by graph-theoretical force method; triangular and rectangular plate bending elements

The calculation of null basis for equilibrium matrix is the main part of finite elements analysis via force method. For an optimal analysis, the selected null basis matrices should be sparse and banded corresponding to sparse, banded and well-conditioned flexibility matrices. There are many algorithms for the formation of null bases among which the algebraic methods benefit from the generality. However, the efficiency of these methods is highly dependent on the size of problems, and their computational times are very high for such problems. In this paper, a graph-theoretical method is presented for the formation of sparse, banded and highly accurate null basis matrices for finite element models with triangular and rectangular plate bending elements. These bases are generated much faster than those obtained by the algebraic methods. The efficiency of the present method is illustrated through some examples.

Exact analytical solutions for bending of rectangular plates with a partial internal line support

This paper studies the behavior of uniformly loaded rectangular thin plates with a partial internal line support. The highlight of the problem is that the analytical formulation explicitly considers the moment singularities that occur at the tips of partial internal line supports. The proper finite Hankel transform is used to transform a pair of dual-series equations obtained from the mixed conditions along the partial internal line support to a single Fredholm integral equation. Numerical results concerning deflection, bending moment, resultant forces, and bending-stress intensity factors are given for a square plate. Some results are also compared with the case of a square plate without partial internal line support.

On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported

This paper presents a bridging research between a modeling methodology in quantum mechanics/relativity and elasticity. Using the symplectic method commonly applied in quantum mechanics and relativity, a new symplectic elasticity approach is developed for deriving exact analytical solutions to some basic problems in solid mechanics and elasticity which have long been bottlenecks in the history of elasticity. In specific, it is applied to bending of rectangular thin plates where exact solutions are hitherto unavailable. It employs the Hamiltonian principle with Legendre’s transformation. Analytical bending solutions could be obtained by eigenvalue analysis and expansion of eigenfunctions. Here, bending analysis requires the solving of an eigenvalue equation unlike in classical mechanics where eigenvalue analysis is only required in vibration and buckling problems. Furthermore, unlike the semi-inverse approaches in classical plate analysis employed by Timoshenko and others such as Navier’s solution, Levy’s solution, Rayleigh–Ritz method, etc. where a trial deflection function is pre-determined, this new symplectic plate analysis is completely rational without any guess functions and yet it renders exact solutions beyond the scope of applicability of the semi-inverse approaches. In short, the symplectic plate analysis developed in this paper presents a breakthrough in analytical mechanics in which an area previously unaccountable by Timoshenko’s plate theory and the likes has been trespassed. Here, examples for plates with selected boundary conditions are solved and the exact solutions discussed. Comparison with the classical solutions shows excellent agreement. As the derivation of this new approach is fundamental, further research can be conducted not only on other types of boundary conditions, but also for thick plates as well as vibration, buckling, wave propagation, etc.

Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method

The main objective of this study is to give a numerical solution of three-dimensional analysis of thick rectangular plates. The analysis uses discrete singular convolution (DSC) method. Free vibration, bending and buckling of rectangular plates have been studied in this paper. Regularized Shannon's delta (RSD) kernel is selected as singular convolution to illustrate the present algorithm. In the proposed approach, the derivatives in both the governing equations and the boundary conditions are discretized by the method of DSC. The obtanied results are compared with those of other numerical methods. It is found that the convergence of the DSC approach is very good and the results agree well with those obtained by other researchers.

Elasticoplastic bending of rectangular plates reinforced with fibers of constant cross section

Elasticoplastic bending of rectangular plates reinforced with fibers of constant cross section

Bending of rectangular plates carried by vacuum cups

In order to carry thin plates, vacuum cups are frequently used. When the over-hang is large, the deflections and stresses of the plate have considerably large values. In this paper, the rectangular plate hung by circular vacuum cups is treated. The analysis is carried out for the plate, which is subjected to line loads and radial bending moments at the inner circular boundary and free at the outer rectangular boundary. In addition to these boundary conditions, the plate is subjected to different distributed loads on inner and outer domains. First, the general solutions for the deflections on each domain are obtained by using infinite series, which are expressed by the polar coordinate system. The several undetermined constants in these equations are decreased by the conditions of continuity at the inner boundaries. Satisfying the boundary conditions at the finite points on the outer edges of the plate, the deflections and stresses of the plate and the contact pressures between the plate and the vacuum cup are calculated. Typical results are presented in dimensionless graphical form for different parameters and vacuum cup edge conditions.

Analysis and Applications of Extended Kantorovich–Krylov Method

In our prior work, the two-dimensional bending and in-plane mode shape functions of isotropic rectangular plates were solved based on the extended Kantorovich–Krylov method. These plate modes were then applied to sandwich plate analysis using the assumed modes method. Numerical results has shown these two-dimensional plate modes improved our sandwich plate analysis. However, the rigorous mathematical convergence proof of the extended Kantorovich–Krylov method is lacking. In this article, we provide a rigorous mathematical convergence proof of the extended Kantorovich–Krylov method using the example of rectangular plate bending vibration, in which the governing equation is a biharmonic equation. The predictions of natural frequency and mode shape functions based on the extended Kantorovich–Krylov method were calculated and the results were numerically validated by other analyses. A similar convergence proof can be applied to other types of partial differential equations (PDEs) that govern vibration problems in engineering applications. Based on these results, the extended Kantorovich–Krylov method was proven to be a powerful tooi for the boundary value problems of partial differential equations in the structural vibrations.

An exact solution for the bending of thin rectangular plates with uniform, linear, and quadratic thickness variations

In this paper, the analysis of the titled problem is based on classical thin-plate theory, and its numerical solution is carried out by using the small parameter method and Lévy-type approach. The thin rectangular plate considered herein is simply supported on two opposite edges. The boundary conditions at the other two edges may be quite general, and between these two edges the plate may have varying thickness. Closed-form solutions have been developed for the static response of isotropic rectangular plates with non-uniform thickness variation and subjected to arbitrary loading. The accuracy of the present model is demonstrated via problems for which the exact solutions and numerical results are available, and results are also compared with those obtained by using the finite-difference method.

Discrete Fourier-series method in problems of bending of variable-thickness rectangular plates

The approach to the solution of the boundary-value problems of bending of elastic rectangular plates of variable thickness is presented. It is proposed to introduce into the resolving system of partial differential equations additional functions which enables the variables to be formally separated and the problem to be reduced to a unidimensional one by representing all the functions as a Fourier series in a single coordinate. In this case the problem can be solved by the stable numerical method of discrete orthogonalization. To calculate the additional functions, Fourier series of discretely assigned functions with allowance for variations in the plate thickness are used. The boundary-value problems for rectangular plates of variable thickness were solved assuming that their weight is unchanged.

Thin Plates and Shells: Theory, Analysis, and Applications

Thin Plates and Shells: Theory, Analysis, and Applications

Three-dimensional static solutions of rectangular plates by variant differential quadrature method

An endeavor to exploit three-dimensional elasticity solutions for bending and buckling of rectangular plates via the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods is performed. Unlike other works, the priority of this paper is to examine the computational characteristics of the two methods; therefore, we focus our studies only on the simply supported and clamped rectangular plates. To start with, we first outline the basic equations and boundary conditions describing the bending and buckling of rectangular plates followed by normalizing and discretizing them according to the DQ and HDQ algorithms. The resulting algebraic equation systems are then solved to obtain the solutions. Based on these solutions, the computational characteristics of the DQ and HDQ methods are investigated in terms of their numerical performances. It is found that the DQ method displays obvious superior convergence characteristics over the HDQ method for the three-dimensional static analysis of rectangular plates.

Refined Global Approximation Theory of Multilayered Plates and Shells

A refined global approximation theory of thin multilayered anisotropic shells is developed. The effects of the laminated material response, transverse shear, and transverse normal strains are included. The material of each layer of the shell is assumed to be linearly elastic, anisotropic, homogeneous, or fiber reinforced. As unknown functions the tangential displacements of the face surfaces and transverse displacements of the face or middle surfaces of the shell are chosen. It is an important feature of the proposed theory. This fact simplifies, for example, an analysis of the contact problems and allows to elaborate universal numerical algorithms. An exact solution for the problem of the bending of homogeneous isotropic rectangular plates subjected to a sinu- soidal load has been found. Numerical solutions for the problem of the bending of multilayered composite plates and cylindrical shells have been also obtained. The influence of anisotropy, and transverse shear and transverse normal deformation response of the stress state of the shell is examined.

Bending of rectangular plate with each edges arbitrary a point supported under a concentrated load

The reciprocal theorem was applied to solve the bending of the rectangular plates with each edges arbitrary a point supported under a concentrated load, the exact solutions and computation example are given.