Bending Of Rectangular Plates Research Articles

Ritz Variational Method for the Analysis of Thin Rectangular Plate Bending Problems with Adjacent Edges Clamped and Simply Supported Using the Super position of Trigonometric Series and Polynomial Basis Functions

Ritz Variational Method for the Analysis of Thin Rectangular Plate Bending Problems with Adjacent Edges Clamped and Simply Supported Using the Super position of Trigonometric Series and Polynomial Basis Functions

A deviatoric couple stress Mindlin plate model and its degeneration

Based on the classical couple stress theory, the deviatoric couple stress theory (DCST), where both couple stress and curvature tensors are traceless, is developed. The deviatoric couple stress Mindlin plate is established based on a new hypothesis of the vanishing curvature of the transverse normal, and the governing equations and associated boundary conditions of the simplified size-dependent Mindlin plate are respectively derived by the principle of virtual work. In the absence of couple stresses, a simplified Mindlin plate model between the classical Kirchhoff plate model and classical Mindlin plate model is derived. To illustrate the simplified DCST-based Mindlin plate model, the static bending and free vibration of rectangular plates are studied. The dependence of the length scale parameter and curvature ratio on the deflection, rotation and natural frequency is analyzed. The numerical results show that the consideration of couple stresses and curvatures hardens the stiffness of the microstructure, resulting in a decrease in its deflection and rotation compared with that of the macroscopic structure under the same conditions, while the natural frequency increases, which is consistent with the size effect observed in experiments.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

Analysis of Berger nonlinear elastic static plate bending of rectangular plates

This paper deals with a nonlinear static plate model based on Berger theory, which is a specific case of a generalization of the Woinowsky-Krieger mathematical model of beam bending. It is considered a plate bending with forces acting in the middle plane of the plate and a contact problem with an elastic foundation, where the normal compliance condition is employed. A variational equation of the problem and a functional of the total potential energy corresponding to the variational equation are derived. Under additional assumptions on the data (e.g., clamped plate), the existence and uniqueness of the solution are proved. A numerical solution is based on the Galerkin method and Courant approximation. The theory is illustrated by a numerical example.

Static Analysis of thin Rectangular Plate Resting on Elastic Foundation Using Modified Vlasov Model

A four-nodded rectangular plate bending element based on Kirchhoff theory resting on elastic foundation using modified Vlasov model. For evaluate the two soil parameter for Vlasov foundation the modulus of elasticity and Poisson ratio of the soil is assumed constant from top surface to the top of bottom rigid base. All the deformation stiffness matrix of plate and subsoil are evaluated using finite element method. A Matlab code is developed and convergence study is carried out and then some realistic cases of plate under static load on elastic foundations are solved to determine the static response. The results, thus obtained, are compared, with the available results obtained by other researchers. It behaves extremely well for thin plates and convergence rate is high.

Bending of variable thickness rectangular thin plates resting on a double-parameter foundation: integral transform solution

PurposeThe purpose of this study is to propose a generalized integral transform technique (GITT) to investigate the bending behavior of rectangular thin plates with linearly varying thickness resting on a double-parameter foundation.Design/methodology/approachThe bending of plates with linearly varying thickness resting on a double-parameter foundation is analyzed by using the GITT for six combinations of clamped, simply-supported and free boundary conditions under linearly varying loads. The governing equation of plate bending is integral transformed in the uniform-thickness direction, resulting in a linear system of ordinary differential equations in the varying thickness direction that is solved by a fourth-order finite difference method. Parametric studies are performed to investigate the effects of boundary conditions, foundation coefficients and geometric parameters of variable thickness plates on the bending behavior.FindingsThe proposed hybrid analytical-numerical solution is validated against a fourth-order finite difference solution of the original partial differential equation, as well as available results in the literature for some particular cases. The results show that the foundation coefficients and the aspect ratio b/a (width in the y direction to height of plate in the x direction) have significant effects on the deflection of rectangular plates.Originality/valueThe present GITT method can be applied for bending problems of rectangular thin plates with arbitrary thickness variation along one direction under different combinations of loading and boundary conditions.

Optimal projecting in the task of bending of rectangular orthotropic piecewise-homogeneous plate

Optimal projecting in the task of bending of rectangular orthotropic piecewise-homogeneous plate

Elastic diffusion vibrations of an isotropic Kirchhoff-Love plate under an unsteady distributed transverse load

We investigated an unsteady elastic diffusion vibration of a simply supported rectangular isotropic Kirchhoff-Love plate. The plate is under the action of a distributed transverse load. A model that describes coupled elastic diffusion processes in a multicomponent continuum is used for the mathematical problem formulation. The model is taking into account the diffusion fluxes relaxation. The transverse vibration equations of a rectangular isotropic Kirchhoff-Love plate with diffusion were obtained from the model using the d'Alembert variational principle. The initial-boundary value problem of a freely supported isotropic rectangular plate bending is formulated on the basis of the obtained equations. The plate is under the action of elastic diffusion perturbations distributed over the surface. The problem solution of an unsteady elastic diffusion plate vibration is sought in an integral form. The surface Green's functions are the kernels of the integral representations. To find the Green's functions, we used the Laplace transform in time and the expansion into double trigonometric Fourier series in spatial coordinates. Green's functions in the image domain are represented in the form of rational functions and depend on the Laplace transform parameter. The transition to the original domain is done analytically through residues and tables of operational calculus. The surface Green's function analytical expressions are obtained. As a calculation example, we considered a freely supported elastodiffusive plate under the action of suddenly applied unsteady bending moments distributed over the plate surface. By using a three-component continuum, a numerical study of interactions between unsteady mechanical and diffusion fieldsis done for an isotropic plate. The influence of relaxation effects on the kinetics of mass transfer is investigated. The solution is presented in the analytical form, as well as in the graphs of the displacement fields and concentration increments on time and coordinates. At the end of the publication, the main conclusions are given about the fields coupling effect and the relaxation of diffusion fluxes on the stress-strain state and mass transfer in the plate.

New Straightforward Benchmark Solutions for Bending and Free Vibration of Clamped Anisotropic Rectangular Thin Plates

Abstract This study presents new straightforward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates by a double finite integral transform method. Being different from the previous studies that took pure trigonometric functions as the integral kernels, the exponential functions are adopted, and the unknowns to be determined are constituted after the integral transform, which overcomes the difficulty in solving the governing higher-order partial differential equations with odd derivatives with respect to both the in-plane coordinate variables, thus goes beyond the limit of conventional finite integral transforms that are only applicable to isotropic or orthotropic plates. The present study provides an easy-to-implement approach for similar complex problems, extending the scope of finite integral transforms with applications to plate problems. The validity of the method and accuracy of the new solutions that can serve as benchmarks are well confirmed by satisfactory comparison with the numerical solutions.

Bending of rectangular orthotropic plates with rotationally restrained and free edges: Generalized integral transform solutions

Generalized integral transform solutions for bending problem of orthotropic rectangular thin plates with rotationally restrained and free edges are presented in this work, including four edges rotationally restrained (RRRR), three-edges rotationally restrained and one-edge free (RRRF), and two opposite edges rotationally restrained and the other two edges free (RFRF). The dimensionless governing equation of orthotropic rectangular plate bending with RRRR, RRRF or RFRF boundary conditions are integral transformed in each spatial direction by adopting Euler–Bernoulli beam eigenfunctions with appropriate boundary conditions (RR, RF or FF). The eigenfunctions for beams with RR and RF boundary conditions in numerically stable exponential function form are presented for the first time, as well as the respective characteristic equations for the determination of the eigenvalues. Free edges of the plate are integral transformed and then incorporated into the transformed governing equations by integration by parts. Transversal deflection of the orthotropic rectangular thin plate under arbitrary transversal load distribution can be readily obtained by solving a system of linear algebraic equations. Numerical results are presented for uniform load, hydrostatic pressure, and concentrated load, with the rotational fixity parameter varying between the limiting cases of simply-supported and clamped edges. Through the comparison with available analytical solutions and finite element solutions by Abaqus, the proposed solutions show the rapid convergence and high accuracy. The effects of the rotational restraints on the deflection of orthotropic rectangular plates with the three combinations of boundary conditions are also investigated.

A Levy solution for bending, buckling, and vibration of Mindlin micro plates with a modified couple stress theory

Based on the modified couple stress and first-order shear deformation plate theories, a Levy-type solution is presented for bending, buckling, and vibration analyses of rectangular isotropic micro plates with simple supports at opposite edges and different boundary conditions at the other two ones. The governing equations are derived using the Hamilton's principle, and then solved by a single Fourier series expansion and the state-space method, which its implementation has not been straightforward. The results are verified with the existing ones for a fully simply supported micro plate in the literature. Finally, the effect of geometric parameters and length scale parameter on bending, buckling, and vibration behaviors of micro plates is studied. Since, there are no analytical solutions for bending, buckling loads, and natural frequencies of Mindlin micro plates with different boundary conditions in the literature and the Navier method is the only available analytical solution for the Mindlin and higher order shear deformable micro plates within the modified couple stress theory, the results presented here can be used as a benchmark in future studies. In addition, it is shown that the difference between results of the Kirchhoff and the Mindlin plate models depends not only on the plate thickness but also on the length scale parameter to thickness ratio (\(l/h)\) as well as the boundary supports. This result emphasizes the significance of analytical solutions for shear deformation models of micro plates with different boundary supports.

Linear piezoelectric motor using a hollow rectangular stator

We propose a linear piezoelectric motor using a hollow rectangular stator that can translate a load placed inside it by a direct-drive. This stator structure enables a quick response and high resolution by few components. These advantages are suited for controlling autofocus and zoom mechanisms in imaging devices. In this study, we verify the feasibility of the driving principle that uses two vibration modes in the bending and longitudinal direction of a hollow rectangular plate. We design a hollow linear piezoelectric motor by a finite element method analysis and evaluate the performance of the prototype motor by experiments. In the prototype motor, the stator measures 60 mm in height and width and 4.1 mm in thickness (the length in slider travel direction), and a slider with a side length of 42 mm is inserted into the stator. The performance parameters of the linear motion, such as thrust force and speed, are evaluated experimentally. By optimizing the preload between the stator and the slider, the motor thrust force has improved to over 0.5 N at 70 Vp−p, which is a practical force for moving a certain lens unit.

Technical Theory of Bending of Elastic Rectangular Plate Pivotally Supported on the Perimeter or Pinched Along Two Adjacent Sides

There is a need to improve methods for calculating elements of engineering structures and equipment for strength and stiffness, which directly affects the safety of nuclear power facilities. For the first time the theory of pure bending of rectangular plates pivotally supported at corner points is generalized to the case of their bending both when the plates are hinged around the perimeter and when a rectangular plate is pinched on two adjacent sides. It should be noted that the obtained system of equilibrium equations does not allow satisfying the equilibrium equations exactly, but only in the sense of the average integral value. However, the authors think this method of solving the deflection problem is much more mathematically and physically justified in comparison with the use of Kirchhoff hypotheses, which lead to contradictions when zero shear forces are assumed to be nonzero, only in order to obtain a deliberately equilibrium equation. The proposed approach allows us to evaluate the deflection of the plate in the case when the main vector of forces applied to the plate can be applied to its geometric center. The article indicates the conditions for the distribution of the transverse load under which it can be assumed that the main vector of forces can be applied to the geometric center of the plate. As examples, the problems of deflection of a rectangular plate under its own weight have been solved both when hinged around the perimeter and when pinched on two adjacent sides.

Bending of orthotropic rectangular thin plates with two opposite edges clamped

This work presents integral transform solutions of the bending problem of orthotropic rectangular thin plates with constant thickness, subject to five sets of boundary conditions: (a) fully clamped; (b) three edges clamped and one edge simply supported; (c) three edges clamped and one edge free; (d) two opposite edges clamped, one edge simply supported, and one edge free; and (e) two opposite edges clamped and two edges free. By adopting eigenfunctions of Euler–Bernoulli beams with corresponding boundary conditions for each direction of the plate, the governing fourth-order partial differential equation is integral transformed into a system of linear algebraic equations. Boundary conditions at the free edges are treated exactly by carrying out integral transform in the boundary formulations, which are incorporated in the transformed governing equations by integration by parts. The numerical difficulties with the high-order beam functions are overcome by using modified exponential forms, thus limiting the eigenfunctions to the range between −2 and 2. Analytical integration forms are used for the integrals of the coefficients of the transformed equations, further avoiding numerical difficulties with large high-order eigenvalues. The accuracy and convergence of the solutions are shown through numerical examples in comparison with available solutions in the literature and with finite element solutions obtained by using Abaqus program.

A finite element solution of Bergan-Wang plate model

Bergan-Wang approach has led to a formulation of the strain energy of a plate bending deflection as function of only the transversal deflection of the plate. In this paper, two rectangular plate bending finite elements are introduced, using new degrees of freedom based on Bergan-Wang approach for analysis of thin, moderately thick plates, in terms of this unique variable. The first element has four nodes with 24 DOF while the second has 36 DOF. These two elements are conforming in case of thin plates. Adopting the usual 3 boundary conditions of Reissner-Mindlin theory, variety of examples have been analysed for thin and moderately thick plate bending problems with plurality of finite element meshes and a variety of thickness to plate length ratios with different boundary conditions on sides. As typical characteristics of Bergan-Wang approach, there is no locking as the thickness decreases and convergence to the classical thin plate solution is achieved. Comparison with Reissner-Mindlin and 3D solutions supports the study.

A Finite Element Formulation for Kirchhoff Plates in Strain-gradient Elasticity

The current contribution is centered on bending of rectangular plates using the finite element method in the strain-gradient elasticity. To this aim, following introducing stresses and strains for a plate based on the Kirchhoff hypothesis, the principle of the virtual work is adopted to derive the weak form. Building upon Hermite polynomials and by deeming convergence requirements, four rectangular elements for the static analysis of strain-gradient plates are presented. To explore the performance of the proposed elements, particularly in small scales, some problems are solved and the results are compared with analytical solutions.

Advanced models for nonlocal magneto-electro-elastic multilayered plates based on Reissner mixed variational theorem

In the present work, nonlocal layer-wise (LW) models for the analysis of magneto-electro-elastic multilayered plates are formulated. An Eringen non-local continuum behavior is assumed for the layers material; in particular, as usual in plate theories, partial in-plane nonlocality is assumed whereas local constitutive behavior is considered in the thickness direction. The proposed plate theories are obtained via the Reissner mixed variational theorem (RMVT), assuming the generalized displacements and generalized out-of-plane stresses as primary variables, and expressing them as through-the-thickness expansions of suitably selected functions, considering the expansion order as a free parameter. In the framework of the Carrera Unified Formulation, this allows the systematic generation of advanced high order plate theories via a layer-based assembly algorithm of the so-called fundamental nuclei associated with the variables expansion terms. The use of the LW approach and RMVT allows for (i) the explicit fulfillment of the transverse generalized stress interface equilibrium, which is crucial for a correct description of the plate fields, (ii) the straightforward analysis of plates with layers exhibiting different characteristic lengths in their nonlocal behavior. A Navier solution for the developed models has been implemented and tested for the static bending and free vibrations analyses of rectangular simply-supported plates. The obtained representative results favorably compare against available three-dimensional analytic results and demonstrate the features of the proposed theories.

Exact solutions of boundary value problems in the theory of plate bending in a half-strip: basics of the theory

Using the boundary value problem on the bending of a thin elastic semi-infinite plate in which the long sides are free, while a self-balanced bending moment and a generalized shearing force are specified at its end as an example, we consider the main steps in constructing exact solutions to the boundary value problems of bending of thin elastic rectangular plates. The solutions are constructed in the form of series in Papkovich–Fadle eigenfunctions. The unknown expansion coefficients are determined in the same way as in classical periodic solutions in trigonometric series and have the same structure, i.e., are expressed via the Fourier integrals of the boundary functions specified at the half-strip end. The systems of functions biorthogonal to the Papkovich–Fadle eigenfunctions constructed here are used in this case. The exact solutions possess properties that are not inherent in any of the known solutions in the theory of plate bending. Some of them are discussed in the paper. The final formulas describing the exact solution of the boundary value problem are simple and can be easily used in engineering practice. The results that we have obtained previously when solving the boundary values problems of the plane theory of elasticity in a rectangular region form the basis for our work.

An analytical Solution for Bending Problem of micro Plate using New Displacement Potential Function

In this paper to include small scale effect, the augmented Love displacement potential functions (DPF) are developed for isotropic micro or Nano scales medium based on couple stress theory. By substituting the new DPF in equilibrium equations, governing equations are simplified to two linear partial differential equations of sixth and fourth order. Then the governing differential equations are solved for simply supported rectangular plate using the separation of variable method with satisfying exact boundary conditions without any simplification assumptions. Displacements, bending and torsional moments of rectangular plate are obtained for different length scale parameters, aspect and Poisson’s ratios. The obtained results are compared with other studies which show excellent agreement between them.