Problem Of Rectangular Plate Research Articles

Bridging plate theories and elasticity solutions

In this work, we present an exact 3D plate solution in the conventional form of 2D plate theories without invoking any of the assumptions inherent to 2D plate formulations. We start by formulating a rectangular plate problem by employing Saint Venant’s principle so that edge effects do not appear in the plate. Then the exact general 3D elasticity solution to the formulated interior problem is examined. By expressing the solution in terms of mid-surface variables, exact 2D equations are obtained for the rectangular interior plate. It is found that the 2D presentation includes the Kirchhoff, Mindlin and Levinson plate theories and their general solutions as special cases. The key feature of the formulated interior plate problem is that the interior stresses of the plate act as surface tractions on the lateral plate edges and contribute to the total potential energy of the plate. We carry out a variational interior formulation of the Levinson plate theory and take into account, as a novel contribution, the virtual work due to the interior stresses along the plate edges. Remarkably, this way the resulting equilibrium equations become the same as in the case of a vectorial formulation. A gap in the conventional energy-based derivations of 2D engineering plate theories founded on interior kinematics is that the edge work due to the interior stresses is not properly accounted for. This leads to artificial edge effects through higher-order stress resultants. Finally, a variety of numerical examples are presented using the 3D elasticity solution.

New method for solving the bending problem of rectangular plates with mixed boundary conditions

A new method is used to solve the rectangular plate bending problem with mixed boundary conditions. The method overcomes the complicated derivation of the classical solution by Fourth-order differential problem into integrating question. Under uniform loading rectangular plate bending problem with one side fixed the opposite side half simply supported half fixed the other two sides free rectangular plate, one side simply supported the opposite side half simply supported half fixed the other two sides free rectangular plate is systematically solved. According to the actual boundary conditions of the rectangular plate, the corresponding characteristic equation can easily be set up. It is presented deflection curve equation and the numerical calculation. By compared the results of the equation to the finite element program, we are able to demonstrate the correctness of the method. So the method not only has certain theoretical value, but also can be directly applied to engineering practice.

The stability analysis of a rectangular plate supported at points of the corners

The stability problem of a rectangular plate supported at points of corners under uniform pressures in its median plane is studied by using the reciprocal theorem. At first, according to the differential equation of transversal buckling displacement function for a thin rectangular plate, dual trigonometric series, and the reciprocal theorem, the equation of the deflection surface for a rectangular plate with four simple supported edges under uniform pressures in its median plane and a transversal concentrated unit load is established. Secondly, using the reciprocal theorem, the buckling equation for a rectangular plate supported at points of corners is derived. The unknowns here can be determined by the boundary conditions. The exact solution of critical load is given. Finally, as an example, a square plate has been calculated. The results show that the method is accurate and effective to solve the stability problem of rectangular plates. It can serve as technical reference for engineering designs.

Accurate buckling analysis of thin rectangular plates under locally distributed compressive edge stresses

The buckling analysis of thin rectangular plates under locally distributed compressive edge stresses is a challenging problem if the point discrete methods are to be used. To obtain accurate buckling stress, one of the important factors is that the in-plane stress distributions within the plate prior to buckling should be accurate enough. Although it is possible to get analytical solutions for the in-plane stress distributions, but the expressions are very complicated since a stress-diffusion phenomenon exists. The differential quadrature method (DQM), being a point discrete method, has been successfully used in a variety of fields including the buckling analysis of thin rectangular plates under nonlinearly distributed edge compressions. However, it is rare to employ the DQM directly to solve problems of rectangular plates under locally distributed or point loads. To solve the challenging problem by using the DQM, novel formulations are presented in this paper. The locally distributed stress is first work-equivalently to point loads at all inner grid points on the loaded edge, then the normal stress boundary condition is numerically integrated before being discretized in terms of the differential quadrature. In this way accurate in-plane stress distributions can be obtained by the DQM without any difficulties. Buckling analysis of rectangular plates under either uniaxial or biaxial locally distributed compressive stresses is successfully performed. The accuracy of the DQ data is validated by comparing them with existing analytical solutions and finite element data. It is demonstrated that the compactness and computational efficiency of the DQM are retained. Accurate buckling loads are presented for rectangular plates with nine combinations of boundary conditions, various aspect ratios and load ratios. Some new results are also provided for references.

A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis

Finite element analysis of functionally graded plates based on a general third-order shear deformation plate theory with a modified couple stress effect and the von Karman nonlinearity is carried out to bring out the effects of couple stress, geometric nonlinearity and power-law variation of the material composition through the plate thickness on the bending deflections of plates. The theory requires no shear correction factors. The principle of virtual displacements is utilized to develop a nonlinear finite element model. The finite element model requires C1 continuity of all dependent variables. The microstructural effects are captured using a length scale parameter via the modified couple stress theory. The variation of two-constituent material is assumed through the thickness direction according to a power-law distribution. Numerical results are presented for static bending problems of rectangular plates with various boundary conditions to bring out the parametric effects of the power-law index and length scale parameter on the load–deflection characteristics of plates with various boundary conditions.

Free vibration analysis of stepped rectangular plates resting on non-homogeneous elastic foundations

Abstract A Half Boundary Method (HBM) is proposed for analyzing the free vibration problem of rectangular plates with stepped thickness resting on non-homogeneous elastic foundations. The unknown quantities of the method exist only on half of the boundary. The non-homogeneous elastic foundation discussed here consists of two-segment elastic Winkler foundation. The fundamental differential equations are established for the bending problem of the plate on elastic foundations. The Green function, which is obtained by transforming these differential equations into integral equations and using numerical integration, is used to establish the characteristic equation of the free vibration. The effects of the modulus of the foundation, the stepped thickness and aspect ratio on the frequency parameters are considered. By comparing the present numerical results with those previously published, the efficiency and accuracy of the present method are investigated.

Extension of Rayleigh–Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates

The Rayleigh–Ritz (R–R) method is extended to eigenvalue problems of rectangular plates with discontinuous boundary conditions (DBC). Coordinate functions are defined as sums of products of orthogonal polynomials and consist of two parts, each satisfying the BC in its respective region. These parts are matched by minimizing the mean square error of functions and their x-derivatives at the interface between regions. Matching defines a positive definite 2N^2 × 2N^2 matrix Q whose eigenvectors form the orthogonal coordinate functions. The corresponding eigenvalues measure the matching error of the two parts at the interface. When applying the R–R method, the total error is the sum of the matching error and that arising from the finite number of coordinate functions. Although most of the coordinate functions correspond to the zero eigenvalue, these do not suffice and additional functions corresponding to small but finite eigenvalues must be included. In three examples with discontinuous BC of the clamped, simply supported and free kind, the calculated frequencies match closely those from a finely discretized finite element method.

The Collapse Load of Plates by Hierarchical C<sup>0</sup>-Plate Element

Although a structural analysis of plates based on the linear elastic theory yields good results for deformations and stresses produced by working loads, it fails to assess the real load-carrying capacities of plates on the verge of yielding. In the case of a limit analysis of plates, the yield line theory is widely used on the basis of the upper bound theorem and theoretically it overestimates the strength of plates. That is why the p-version of the finite element method has been proposed for determining the accurate limit load of plates causing collapse. In this method, the hierarchical Co -plate element for bending of elastic-plastic plates accounting for transverse shear effects has been formulated, and is based on the incremental theory of plasticity and the Reissner-Mindlin plate theory. The numerical results are presented for a variety of rectangular plate problems and are compared to the results obtained by the h-version software ADINA, as well as with the available analytical solutions in literature.

Simple iterative method for solving problems for plates with partial internal supports

We consider problems for rectangular plates on one and on two partial internal supports with two opposite edges simply supported and two other edges clamped. The problems are modeled by the biharmonic equation with strongly mixed boundary conditions in the sense that there is a change of types of boundary conditions on a side of the rectangle. These boundary value problems lead to sequences of problems for the Poisson equation with strongly mixed boundary conditions. Finally, in solving these problems a version of the domain decomposition method is applied. The convergence of the obtained approximate solution is proved. Several numerical experiments show the fast convergence of the method. The efficiency of the proposed method is clear from the comparison of it with the dual series equations method used by other authors in the case of a plate with one internal support.

Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions

Recently, the present authors proposed a simple mixed Ritz-differential quadrature (DQ) methodology for free and forced vibration, and buckling analysis of rectangular plates. In this technique, the Ritz method is first used to discretize the spatial partial derivatives with respect to a coordinate direction of the plate. The DQ method is then employed to analogize the resulting system of ordinary or partial differential equations. The mixed method was shown to work well for vibration and buckling problems of rectangular plates with simple boundary conditions. But, due to the use of conventional Ritz method in one coordinate direction of the plate, the geometric boundary conditions of the problem can only be satisfied in that direction. Therefore, the conventional mixed Ritz-DQ methodology may encounter difficulties when dealing with rectangular plates involving adjacent free edges and skew plates. To overcome this difficulty, this paper presents a modified mixed Ritz-DQ formulation in which all the natural boundary conditions are exactly implemented. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of thick rectangular and skew plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of thick rectangular plates involving adjacent free edges and skew plates using a small number Ritz terms and DQ sampling points.

Numerical Solution of Bending Problems for Rectangular Plates

The stress–strain state of thick rectangular plates is determined using a numerical-analytic approach based on the three-dimensional theory of elasticity. The problem is made one-dimensional using the method of spline collocation in two coordinate directions. The boundary-value problem for a system of ordinary differential equations of high order is solved with the stable numerical method of discrete orthogonalization. The stress–strain state of plates with different geometrical parameters and hinged or clamped edges is analyzed

Some steady-state thermoelastic problems of rectangular plates

Two steady-state thermoelastic problems of rectangular plates are studied in the paper. The first problem is the inverse steady-state thermoelastic problem of determining the temperature, displacement and stress functions of a thin rectangular plate. On the two horizontal edges and one vertical edge the temperature is maintained at zero. On a vertical line the temperature is prescribed which serves as the interior condition. At the end a discussion on the inverse thermoelastic problem is included. The second problem is the direct steady-state thermoelastic problem of determining temperature, displacement and stress functions in a thin rectangular plate with the stated boundary conditions, which are special cases of third kind boundary conditions.

Vibration of an initially stressed rectangular plate due to an accelerated traveling mass

This paper presents a combined application of the Ritz method, the Differential Quadrature (DQ) method, and the Integral Quadrature (IQ) method to vibration problem of rectangular plates subjected to accelerated traveling masses. In this study, the Ritz method with beam eigenfunctions is first used to discretize the spatial partial derivatives with respect to a co-ordinate direction of the plate. The DQ and IQ methods are then employed to analogize the resultant system of partial differential equations. The resulting system of ordinary differential equations is then solved by using the Newmark time integration scheme. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those of the existing literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQM sampling points. Finally, the effects of following parameters having something to do with the title problem are investigated: moving load speed and acceleration, and transverse inertia of the moving load. Numerical results show that all the above-mentioned parameters have significant effects on the transient response of such structures under traveling dynamic loads.

Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates

This paper presents a combined application of the finite element method (FEM) and the differential quadrature method (DQM) to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.

A mixed method for free and forced vibration of rectangular plates

This paper presents a very first combined application of Ritz method and differential quadrature (DQ) method to vibration problem of rectangular plates. In this study, the spatial partial derivatives with respect to a coordinate direction are first discretized using the Ritz method. The resulting system of partial differential equations and the related boundary conditions are then discretized in strong form using the DQ method. The mixed method combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The results are obtained for various types of boundary conditions. Comparisons are made with existing analytical and numerical solutions in the literature. Numerical results prove that the present method is very suitable for the problem considered due to its simplicity, efficiency, and high accuracy.

Thermo-microstretch elastic bodies and plane waves

In the present work, vibration problems of rectangular plates are considered for the determination of upper bounds to the unknown microstretch material properties. The frequencies are obtained by extending the Ritz method to this case. The analysis shows that some additional frequencies characterizing the microstretch effects appear among the classical frequencies. Furthermore, by the increasing values of the microstretch constants, the additional frequencies disappear and only the classical frequencies remain in the spectrum. Considering this phenomenon, an optimization problem is established for the identification of the upper bounds of microstretch elastic constants. In the second part of the work, thermal effects are considered and several theories are discussed. Finally, propagation of the plane waves is investigated.

Exact solution for buckling of FCFC stepped rectangular plates

This paper investigates the buckling problem of rectangular plates with stepped thickness. In the present work two opposite and parallel edges are clamped, while two others are free. The supposed deflection function satisfies the clamped boundary conditions along the related edges. A rectangular plate with a single step, with only two subplates, is considered. Our theory is also applicable to multistepped plates. Furthermore, the assumed function of deflection must also satisfy the governing differential equation. Having satisfied that condition, an ordinary nonlinear differential equation is obtained. In order to get an ordinary differential equation, as it was in the Levy theory, we had to apply the Galerkin–Bubnov theorem of identity. After getting the linear differential equation, eight boundary conditions needed to be specified. Only two of them were constant (Levy case), but six boundary conditions were functions of x. The Galerkin–Bubnov theorem had to be applied to these boundary conditions in order to obtain the corresponding system of algebraic equations, thus solving for the critical force N x . The proposed procedure could be considered as an extension of the Levy buckling theory.

On Feasibility of Variable Separation Method Based on Hamiltonian System for a Class of Plate Bending Equations

The eigenfunction system of infinite-dimensional Hamiltonian operators appearing in the bending problem of rectangular plate with two opposites simply supported is studied. At first, the completeness of the extended eigenfunction system in the sense of Cauchy's principal value is proved. Then the incompleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator of this kind of plate bending equation is proved. At last the general solution of the infinite dimensional Hamiltonian system is equivalent to the solution function system series expansion, so it gives to theoretical basis of the methods of separation of variables based on Hamiltonian system for this kind of equations.

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...

DSC ANALYSIS FOR BUCKLING AND VIBRATION OF RECTANGULAR PLATES WITH ELASTICALLY RESTRAINED EDGES AND LINEARLY VARYING IN-PLANE LOADING

This paper presents the discrete singular convolution (DSC) method for solving buckling and vibration problems of rectangular plates with all edges transversely supported and restrained by uniform elastic rotational springs. The opposite plate edges are subjected to a linearly varying uni-axial in-plane loading. The rationale for using DSC method stems from its numerical stability and flexible implementation for structural analysis. To verify the present approach, convergence and comparison studies for rectangular plates with different combinations of elastically restrained and classical edges are carried out. Accurate buckling and vibration solutions of plates having two opposite edges elastically restrained and the other two sides clamped, or all edges elastically restrained are presented.