Thin Plate Problems Research Articles

Bending analysis of thin plates with variable stiffness resting on elastic foundation via a two-network strategy physics-informed neural network method

A physics-informed neural network method based on a two-network strategy is introduced to address the bending problem of thin plates with variable stiffness resting on an elastic foundation. This problem is governed by a fourth-order partial differential equation (PDE), and the use of a one-network strategy to solve the PDE directly may lead to convergence issues due to singular points. Following the principles of Kirchhoff plate theory, the governing PDE is equivalently transformed into four second-order PDEs. A two-network strategy is employed for solution. We present numerical examples under various load conditions, plate geometries, foundation types, thicknesses, and material properties. The obtained results are validated against finite element method (FEM) solutions and literature. Discussions on alternative network strategies were conducted, and this study reveals that the presented two-network strategy have proven to be effective and robust. It also facilitates easier imposition of boundary conditions.

Theoretical study on multi-physics coupling problems of functionally-graded lead-free piezoelectric rectangular thin plates

This paper carried out a theoretical research on multi-physics coupling problems of functionally-graded lead-free piezoelectric rectangular thin plates. Firstly, the perturbation theory was used to study the bending problem of functionally-graded lead-free piezoelectric rectangular thin plates, and the nonlinear governing equations for this bending problem were established. Then, the three piezoelectric coefficients reflecting the piezoelectric property were taken as the perturbation parameters, and the nonlinear governing equations were transformed into multiple sets of linear equations by the multi-parameter perturbation method. Finally, the solutions of these linear equations were solved by the semi-inverse solving method. Based on the finite element simulations, the effectiveness of the perturbation solutions obtained here was verified. The research results showed that it was reasonable to choose three piezoelectric coefficients as perturbation parameters, which was accord with the basic idea of perturbation theory. The analysis results indicated that there was a cross transfer relationship between external loads and perturbation solutions of every order, which could be used to simplify the perturbation expansion. Moreover, the deformation of the functionally-graded lead-free piezoelectric thin plate was less than that of the functionally-graded thin plate, that is, piezoelectric properties in lead-free piezoelectric materials had a piezoelectric enhancement effect.

Symplectic superposition method for vibration problems of orthotropic rectangular thin plates with four corners point-supported

Symplectic superposition method for vibration problems of orthotropic rectangular thin plates with four corners point-supported

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.

Comparison of free vibration behaviors for simply supported and clamped T-shaped thin plate resting on Winkler elastic foundation

The analysis of free vibration problems for thin plates is essential for the design of various structural systems. However, it is difficult to find analytical solutions due to the complexity of mathematical computing. Based on the symplectic superposition method, the T-shaped thin plate on the Winkler elastic foundation is divided into four sub-plates and are solved by using the symplectic eigen expansion method, and the modes and frequencies are studied. The method begins directly with the fundamental equations and undergoes a rigorous mathematical derivation without assuming the form of the solution beforehand. This approach helps circumvent the drawbacks associated with traditional semi-inverse solution methods. In addition, the theoretical calculation model and finite element analysis model of T-shaped thin plates on elastic foundation are established by using Mathematic software and ABAQUS software in present paper. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Results show that the boundary condition, foundation stiffness and aspect ratio have great influences on vibration frequency and mode shape for T-shaped structures.

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes.

Multi-Scale-Matching neural networks for thin plate bending problem

Physics-informed neural networks are a useful machine learning method for solving differential equations, but encounter challenges in effectively learning thin boundary layers within singular perturbation problems. To resolve this issue, multi-scale-matching neural networks are proposed to solve the singular perturbation problems. Inspired by matched asymptotic expansions, the solution is decomposed into inner solutions for small scales and outer solutions for large scales, corresponding to boundary layers and outer regions, respectively. Moreover, to conform neural networks, we introduce exponential stretched variables in the boundary layers to avoid semi-infinite region problems. Numerical results for the thin plate problem validate the proposed method.

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.

Bending of Plates with Complex Shape Made from Materials that Differently Resist to Tension and Compression

A new numerical-analytical method for solving physically nonlinear bending problems of thin plates with complex shape made from materials that differently resist to tension and compression is developed. The uninterrupted parameter continuation method is used to formulate and linearize the problem of physically nonlinear bending. For the linearized problem, a functional in the Lagrange form, given on the kinematically possible displacement rates, is constructed. The main unknown problems (displacements, strains, stresses) were found from the solution of the initial problem, which was solved by the Runge-Kutta-Merson method with automatic step selection, by the parameter related to the load. The initial conditions are found from the solution of the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method in combination with the R-function method, which allows to submit an approximate solution in the form of a formula – a solution structure that exactly satisfies the boundary conditions and is invariant with respect to the shape of the domain where the approximate solution is sought. The test problem for the nonlinear elastic bending of a square hinged plate is solved. Satisfactory agreement with the three-dimensional solution is obtained. The bending problem of the plate of complex shape with combined fixation conditions is solved. The influence of the geometric shape and fixation conditions on the stress-strain state is studied. It is shown that failure to take into account the different behavior of the material under tensile and compression can lead to significant errors in the calculations of the stress-strain state parameters.

The MLS based numerical manifold method for bending analysis of thin plates on elastic foundations

Compared with the finite element method, H2-regularity in the Galerkin based approximation to the Kirchhoff thin plate model can be easily realized using either the moving least squares (MLS) or the generalized moving least squares (GMLS), which take the Lagrange form and the Hermite form, respectively. Coupling (G)MLS with the numerical manifold method (NMM) can greatly improve numerical properties of NMM in the treatment of plates of complicated shape, thereby denoted by MLS-NMM and GMLS-NMM. In the (G)MLS-NMM, the mathematical cover is composed of simply connected and partially overlapped mathematical patches that are the influence domains of (G)MLS nodes. GMLS-NMM appears to better fit to the Kirchhoff plate because it is equipped with rotation angle degrees of freedom. Through numerical tests and theoretical analysis in solving problems of thin plates on elastic foundations, however, this study shows that MLS-NMM is much more advantageous over GMLS-NMM from the aspects of both accuracy and memory usage.

On simplified deformation gradient theory of modified gradient elastic Kirchhoff–Love plate

A concise modified gradient elastic Kirchhoff–Love plate model with two length-scale parameters is proposed based on simplified deformation gradient theory. A sixth-order basic differential equation and boundary conditions applicable to arbitrary shapes are obtained by applying the principle of minimum potential energy and combining with the generalized strain energy related to classical strain, strain gradient and rotation gradient. By introducing couple stress, the classical bending stiffness corresponding to the fourth-order terms and force-related boundary conditions in the typical gradient elastic Kirchhoff–Love plate model are modified, and the bending deformation of thin plates can be described more flexibly. Under certain conditions, the modified gradient elastic Kirchhoff–Love plate model can be reduced to typical gradient elastic thin plate model, couple stress thin plate model and classical Kirchhoff–Love plate model. The bending boundary value problems of gradient elastic thin plates are further studied, and the specific modified classical boundary conditions and non-classical higher-order boundary condition acceptable to the sixth-order basic equation in Cartesian coordinates are derived. The analytical and numerical bending solutions to gradient Navier-type and Levy-type thin plates with various boundary conditions, including simply supported, clamped and free boundaries are presented, and the size-dependent bending stiffness that is jointly determined by geometric dimensions and length-scale parameters is defined to describe the size effect of thin plates comprehensively.

Method of Solving Geometrically Nonlinear Bending Problems of Thin Shallow Shells of Complex Shape

A new numerical analytical method for solving geometrically nonlinear bending problems of thin shallow shells and plates of complex shape is given in the paper. The problem statement is performed within the framework of the classic geometrically nonlinear formulation. The parameter continuation method was used to linearize the nonlinear bending problem of shallow shells and plates. An increasing parameter t related to the external load, which characterizes the shell loading process, is introduced. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible movement speeds, is constructed. To find the main unknowns of the problem of nonlinear bending of the shell (displacement, deformation, stress), the Cauchy problem was formulated by the parameter t for the system of ordinary differential equations, which was solved by the fourth order Runge-Kutta-Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometric linear deformation. The right-hand sides of the differential equations at fixed values of the parameter t, corresponding to the Runge-Kutta-Merson scheme, were found from the solution of the variational problem for the functional in the Lagrange form. Variational problems were solved using the Ritz method combined with the R-function method, which allows to accurately take into account the geometric information about the boundary value problem and provide an approximate solution in the form of a formula - a solution structure that exactly satisfies all (general structure) or part (partial structure) of boundary conditions. The test problem for the nonlinear bending of a square clamped plate under the action of a uniformly distributed load of different intensity is solved. The results for deflections and stresses obtained using the developed method are compared with the analytical solution and the solution obtained by the finite element method. The problem of bending of a clamped plate of complex shape is solved. The effect of the geometric shape on the stress-strain state is studied

Gradient reproducing kernel based Hermite collocation method (GHCM) for eigenvalue analysis of functionally graded thin plates with in-plane material

A meshfree Hermite collocation method based on gradient reproducing kernel approximations is proposed for the eigenvalue analysis of thin functionally graded plates with in-plane material inhomogeneity. Compared with direct collocation method (DCM), Hermite collocation method (HCM) can improve the accuracy of the solutions, especially on the boundaries. Moreover, HCM results in a determined system for the eigenproblems while DCM leads to an over-determined system where the solutions may not be the real results. Since derivative calculations of the reproducing kernel function are complex and time-consuming, gradient reproducing kernel approximations are introduced for solving the thin plate problems to avoid the high order direct differentiations which are required in the thin plate strong form solutions. Proper weights that should be imposed for the boundary conditions to balance the errors in the solutions are derived for both DCM and HCM. Convergence studies are also derived to evaluate the convergence of these two methods. Comparison studies with analytical solutions indicate the high accuracy and good performance of the proposed method. Numerical simulations demonstrate that material inhomogeneity has considerable effects on the natural frequencies and mode shapes.

Highly Accurate Wavelet Solution for Bending and Free Vibration of Circular Plates Over Extra-Wide Ranges of Deflections

Abstract The wavelet multiresolution interpolation Galerkin method in which both the unknown functions and nonlinear terms are approximated by their respective projections onto the same wavelet space is utilized to implement the spatial discretization of the highly coupled and nonlinear Von Karman equation for thin circular plates with various types of boundary conditions and external loads. Newton’s method and the assumption of a single harmonic response are then used for solving the static bending and free vibration problems, respectively. Highly accurate wavelet solutions for an extremely wide range of deflections are finally obtained by the proposed method. These results for moderately large deflections are in good agreement with existing solutions. Meanwhile, the other results for larger deflections are rarely achieved by using other methods. Comparative studies also demonstrate that the present wavelet method has higher accuracy and lower computational cost than many existing methods for solving geometrically nonlinear problems of thin circular plates. Moreover, the solutions for large deflection problems with concentrated load support the satisfactory capacity for handling singularity of the proposed wavelet method. In addition, a trivial initial guess, such as zero, can always lead to a convergent solution in very few iterations, even when the deflection is as large as over 46 times thickness of plate, showing an excellent convergence and stability of the present wavelet method in solving highly nonlinear problems.

A [formula omitted]-nonconforming virtual element methods for the vibration and buckling problems of thin plates

In this work, we study the C0-nonconforming VEM for the fourth-order eigenvalue problems modeling the vibration and buckling problems of thin plates with clamped boundary conditions on general shaped polygonal domain, possibly even nonconvex domain. By employing the enriching operator, we have derived the convergence analysis in discrete H2 seminorm, and H1, L2 norms for both problems. We use the Babuška–Osborn spectral theory (Babuška and Osborn, 1991), to show that the introduced schemes provide well approximation of the spectrum and prove optimal order of rate of convergence for eigenfunctions and double order of rate of convergence for eigenvalues. Finally, numerical results are presented to show the good performance of the method on different polygonal meshes.

OPTIMAL SUPPORT POSITIONING OF A RECTANGULAR PLATE IN A STABILITY PROBLEM UNDER TEMPERATURE FIELD EXPOSURE

Introduction . Thin plates are widely used structural elements in modern civil, mechanical, aeronautical, and marine engineering design. Alongside the temperature field, the middle plain of plates is subjected to uniformly distributed normal mechanical forces, with σφ intensity, which raises the stability problem of thin plates. The strength and stability of rectangular plates have been investigated by S.P. Timoshenko [1], S.A. Ambartsumyan [2] and others. In the field of design of thin structure elements, the optimal positioning of plate supports has been studied in the works of V.Ts. Gnuni [3], M.V. Belubekyan [4], and A.V. Eloyan [5, 6]. However, the problem of optimal support positioning has not been sufficiently studied, particularly regarding the thermoelastic stability of thin plates. The simultaneous exposure of mechanical forces and temperature field on a rectangular plate poses the issue of an efficient parameter с that determines the positions of transverse supports along the length of a plate thus ensuring the maximum buckling load. Aim . To calculate the optimal values of a parameter α = c/a in accordance with σ̅, h̅, values for different side ratios λ=a/b of a plate for specified temperature values. Materials and methods . Elastic isotropic plates were used, the maximum buckling load was determined. Results and conclusions . According to the results, the optimal location of plate supports for values λ = a/b–1/2,1 is determined by σ̅ = 3 β depending on the plate temperature. The maximum buckling load is determined when σ̅ = 3 β , h̅ = 0,01, w* = 1,772, α = 0,37, Т = 300 °С.

New Analytical Solution of the Large Deflection Problem of a Circular Thin Plate with Edge Clamped but Sliding Freely Under a Combined Loading

This paper intends to study the large deflection problem of a circular thin plate with an edge clamped but sliding freely under the combined action of uniformly distributed load and centrally concentrated load by the Adomian decomposition method. If the uniformly distributed load is not decomposed, the parameters obtained by the Adomian decomposition method are the same as the perturbation solution studied by scholars before. If the uniformly distributed load is decomposed into a finite series, a new analysis solution can be obtained through the Adomian decomposition method. From the error comparisons between the new solution in this paper and the perturbation solution, one can see that the result in this paper is better. Finally, the internal stress and deflection are discussed through the graphics based on the new analysis method.

An energy-conserving finite element method for nonlinear fourth-order wave equations

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.

Free Vibration Analysis of Rectangular Thin Plates Resting on Nonhomogeneous Elastic Foundations by Using Matched Interface and Boundary Algorithms and their Multi-Domain Formulations

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations.

New Analytical Free Vibration Solutions of Thin Plates Using the Fourier Series Method

This article aims at analytically solving the free vibration problem of rectangular thin plates with one corner free and its opposite two adjacent edges rotationally-restrained, which is difficult to handle by conventional semi-inverse approaches such as the Levy solution and Naiver solution, etc. Based on the classical Fourier series theory, this work presents a first endeavor to treat the two-dimensional half-sinusoidal Fourier series, which is quite similar to the Navier’s form solution, as the solution form of plate deflection. By utilizing the orthogonality of the present trial function and the Stoke’s transformation technique, the present solution procedure converts the complicated plate problem into solving sets of linear algebra equations, which heavily decreases the difficulties. Therefore, the present approach enables one to solve the title problem in a unified, simple and straightforward way, which is very easily implemented by researchers. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through utilizing different types of Fourier series and it can be extended for further dynamic/static analysis of plates under different shear deformation theories. Moreover, without any extra derivation processes, new, precise analytical free vibration solutions for plates under three non-Levy-type boundary conditions are also obtained by choosing different rotating fixed coefficients. Consequently, we present more than 400 comprehensive free vibration results for plates with classical/non-classical boundaries, all the present results are confirmed by FEM/analytical solutions and can be used as benchmark data for further research.