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A 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).

The 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.

The 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.

The 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.

The purpose of this paper is to present elastic buckling behaviour of simply supported and clamped thin rectangular isotropic plates having central circular cutouts subjected to uniaxial partial edge compression. Analysis is carried out for four different kinds of partial edge compression and it is extended to study the effect of aspect ratio of plate on buckling load.

A finite element method technique is used in the current work to solve the buckling problem of plate using eight node quadrilateral element and plate kinematics based on first order shear deformation theory. Results obtained from finite element analysis are first validated for isotropic square plates, without cutouts, subjected to uniaxial partial edge compression with some earlier published literature.

From the current work it is concluded that the buckling strength of square plates is highly influenced by partial edge compression, as compared to plate subjected to uniform edge compression; but with increase in aspect ratio, influence of partial edge compression on plate buckling load decreases.

This paper usefully shows how partial edge compression of plates affects the buckling strength of plate having circular cutouts. Generally, simply supported plates subjected uniaxial partial edge compression of Type I and Type III are found to be stronger than plates subjected to partial edge compression Type II and Type IV, respectively.

Among different types of engineering structures, plates play a significant role. Their analysis necessitates numerical modeling with finite elements, such as triangular, quadrangular or sector plate elements, owing to the intricate geometrical shapes and applied loads. The scope of this study is the development of a new rectangular finite element for thin plate bending based on the strain approach using Airy's function. It is called a rectangular plate finite element using Airy function (RPFEUAF) and has four nodes. Each node had three degrees of freedom: one transverse displacement (w) and two normal rotations (x, y).

Equilibrium conditions are used to generate the interpolation functions for the fields of strain, displacements and stresses. The evolution of the Airy function solutions yielded the selection of these polynomial bi-harmonic functions. The variational principle and the analytical integration approach are used to evaluate the basic stiffness matrix.

The numerical findings for thin plates quickly approach the Kirchhoff solution. The results obtained are compared to the analytical solution based on Kirchhoff theory.

The efficiency of the strain based approach using Airy's function is confirmed, and the robustness of the presented element RPFEUAF is demonstrated. Because of this, the current element is more reliable, better suited for computations and especially intriguing for modeling this kind of structure.

The purpose of this paper is to present a simple HP-cloud method as an accurate meshless method for the geometrically nonlinear analysis of thick orthotropic plates of general shape. This method is used to investigate the effects of thickness, geometry of various shapes, boundary conditions and material properties on the large deformation analysis of Mindlin plates.

Nonlinear analysis of plates based on Mindlin theory is presented. The equations are derived by the Von-Karman assumption and total Lagrangian formulations. Newton-Raphson method is applied to achieve linear equations from nonlinear equations. Simple HP-cloud method is used for the construction of the shape functions based on Kronecker-δ properties, so the essential boundary conditions can be enforced directly. Shepard function is utilized for a partition of unity and complete polynomial is used as an enrichment function.

The suitability and efficiency of the simple HP-cloud method for the geometrically nonlinear analysis of thin and moderately thick plates is studied for the first time. Large displacement analysis of various shapes of plates, rectangular, skew, trapezoidal, circular, hexagonal and triangular with different boundary conditions subjected to distributed loading are considered.

This paper shows that the simple HP-cloud method is well suited for the large deformation analysis of Mindlin plates with various geometries, because it uses a set of a few arbitrary nodes placed in a plate of general shape. Moreover the convergence rate of the proposed method is high and the cost of solving equations is low.

A simple model is proposed for the analytical calculation of the eddy current induced by a time‐varying magnetic field in a conductive plate. It is assumed that the current is uniformly distributed over the plate thickness and circulates in rectangular paths. The plate is considered as an equivalent LR circuit and its equivalent inductance and resistance are calculated analytically and expressed as a function of the plate dimensions. The analytical solution of the equation, governing the LR circuit, yields the total current circulating in the plate as a function of time. The model is applied to a well known benchmark problem and the results are in good agreement with those obtained by experiment.

The main objective of this study is to develop a numerical model based on Isogeometric Analysis to study the dynamic behavior of multi-directional functionally graded plates with variable thickness.

A numerical study was conducted on the dynamic behavior of multi-directional functionally graded plates. Rectangular and circular plates with variable thickness are taken into investigation. The third-order shear deformation plate theory of Reddy is used to describe the displacement field, while the equation of motion is developed based on the Hamilton's principle. Isogeometric Analysis approach is employed as a discretization tool to develop the system equation, where NURBS basis functions are used. The famous Newmark method is used to solve time-dependent problems.

The results obtained from this study indicated that the thickness gradation has a more considerable effect than in-plane variation of materials in MFGM plates. Additionally, the influence of the damping factor is observed to affect the vibration amplitude of the plate. The results obtained from this study could be used for future investigations, where the viscous elasticity and other dynamic factors are considered.

Although there have been a number of studies in the literature devoted to analyzing the linear static bending and free vibration of FGM and MFGM plates with variable thickness, the study on dynamic response of FGM and MFGM plate is still limited. Therefore, this study is dedicated to the investigation of the dynamic behavior of multi-directional functionally graded plates.

Investigations which have been carried out up to the present have shown that trajectories of the arc spot over the extinguishing plate drift always towards the plate axis. Results of calculations presented in this paper are different from the results which have been obtained till now. The calculations have been made with the use of computer programs based on the boundary element method. The results of the calculations show that for some areas of plates of specific shapes potential gradient of a field of forces acting against the arc spot may approach zero. When the arc reaches any point of such an area it is stopped and the arc spot trajectory does not end at the plate axis.

TO establish the strength and stiffness of certain types of structure it is necessary to use a large displacement theory, i.e. one in which allowance is made for the redistribution of the loading effects as a consequence of the deformation produced by the loads. The post‐buckling behaviour of panels, cylinders and other types of structure under compressive endloads require such a theory and another important category is constituted in thin plates under normal pressure, the pressure being partly resisted by tensions in the plane of the plate, in the way that membranes resist pressure, and partly by the bending resistance or stiffness of the plate.

A bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view is given. The bibliography at the end of the paper contains 1,726 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1996‐1999.

Distributed control is an effective method for controlling and suppressing excessive vibrations of continuous systems. Optimal distributed control for a plate problem is solved utilizing a maximum principle after the introduction of a quadratic index of performance in terms of displacement, velocity and a control force as well as an adjoint variable. The problem is reduced to solving a system of partial differential equations for the state variable and the adjoint variable subjected to boundary, initial and terminal conditions. A numerical algorithm is presented to solve the optimal distributed control problem in the space‐time domain which reduces the computational effort required to solve the initial‐terminal‐boundary value problem. Results obtained for a simply supported, rectangular, thin plate are also presented.