Plate Problems Research Articles

Vibration and Buckling Characteristics of Sandwich FGM Plate with Conventional and Auxetic Honeycomb Core

The vibration and buckling characteristics of the sandwich plate with a honeycomb core and functionally graded material (FGM) face sheet have been evaluated in this paper. The honeycomb core, also known as conventional honeycomb and auxetic honeycomb, is modeled based on positive and negative Poisson’s ratio. Material properties of face sheets varied according to simple power-law functionally graded material (P-FGM). Hamilton’s principle has been employed to derive the equation of motion, and Navier’s method is used to solve the plate problem. Three different plate configurations are used to study the effect of the thickness layer. In addition, the outcome based on span-to-thickness ratio, aspect ratio, geometric parameters, and volume fraction exponent of honeycomb structure on frequency parameter and critical buckling load is examined and exhibited for three different plate configurations. The validation of the present formulation is ascertained by comparing it with other available results. Some novel results are presented for different angles of the unit cell that can be useful as a validation study for the forthcoming research on sandwich honeycomb rectangular plates. It is observed that the thickness of the honeycomb layer plays a significant role in affecting the behavior of the sandwich plate. Besides, the auxetic structure is highly sensitive to high excitation frequency applications compared to the conventional honeycomb structure.

A new approach to problems of thermoelasticity in stresses

It is known that the plane problems of the theory of elasticity and thermoelasticity are usually reduced to solving the biharmonic equation with respect to the Airy stress function. Solutions to spatial problems can also be defined using the Maxwell and Morera stress functions, etc. In this work, for solving a spatial thermoelastic problem, in the framework of the Beltrami-Michell equations, Poisson-type differential equations are found for each component of the stress tensor. In this case, the right-hand sides of these equations depend on temperature and elastic constants. The boundary conditions consist of the usual surface forces and “additional” boundary conditions derived from the equilibrium equations. The differential equations and boundary conditions of the plane problem of thermoelasticity in stresses are discussed in more detail. Grid equations for plane and spatial problems of thermoelasticity were compiled by the finite-difference method and solved by the iterative method. The problems of a thermoelastic rectangle and a parallelepiped located in a given temperature field are solved numerically. The validity of the formulated boundary value problems and the reliability of the results obtained are substantiated by comparing the numerical results with the solutions of the problems of a thermoelastic plate and a parallelepiped known in the literature.

Damage identification technique by model enrichment for structural elastodynamic problems

Structural Health Monitoring (SHM) techniques are key to monitor the health state of engineering structures, where damage type, location and severity are to be estimated by applying sophisticated techniques to signals measured by sensors. However, very localized damage detection algorithms applied to dynamics problems when dealing with rigid structures at low-frequency range remains still a big challenge. The last due to the low influence of very localized damage on the overall response of the structure (Saint-Venant principle). In this context, in the present work, we propose a methodology for locally correcting the models from collected data for elastodynamics problems at low-frequency range which is able to predict very localized damage. The proposed technique consists in enriching the structural model in a sparse way and solving the identification problem in the frequency domain, where the influence of damage over a large frequency band is exploited to improve the prediction of the damage location. The advantages and potential of the proposed technique are illustrated for the damage detection in a plate problem, demonstrating the advantages of the method in detecting very localized damage. The proposed technique is limited to a methodological description, and further developments should be considered to approach its applicability in an industrial scenario.

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.

Analysis of Rectangular Plate with Opening by Finite Difference Method

Steel plates are commonly used to support lateral or vertical loads. Before the design of such a plate, analysis is performed to check the stability of plate for the proposed load. There are several methods for this analysis. In this research, a finite difference method (FDM) is proposed to analyze a rectangular steel plate. The plate is considered to be subjected to an arbitrary transverse uniformly distributed loading and is considered to be clamped at the two opposite edges and free at the other two edges. At first the plate is analyzed considering it to be completely solid and then it was re-analyzed considering a rectangular opening at its centre. The ordinary finite difference method is used to solve the governing differential equation of the plate deflection. The proposed method can be easily programmed to readily apply on a plate problem. The work covers the determination of displacement components at different points of the plate and checking the result by software (STAAD.Pro) analysis.

Solution of nonlinear vibration problem of shear deformable multilayer nonhomogeneous orthotropic plates using Poincare-Lindstedt method

In this study, one of the first attempts is made to solve the nonlinear (NL) vibration problem of shear-deformable multilayer plates consisting of nonhomogeneous orthotropic layers (NHOLs) using the Poincaré-Lindstedt method. First, the shear deformation theory (SDT) for homogeneous plates is extended to multilayer plates composed of NHOLs. In the framework of von-Karman type nonlinear theory, the basic relations of the plates in question are established and then NL equations of motion based on four functions such as rotation angles, Airy stress and deflection functions are derived. Then, NL-partial differential equations (NL-PDEs) are reduced to NL-ordinary differential equations (NL-ODE) and the solution of NL-ODE is performed for the first time by the modified Poincaré −Lindstedt method, yielding new amplitude dependent expressions for NL frequency, and for the ratio of NL frequency to linear (NL/L) frequency for multilayer plates consisting of NHOLs. Finally, detailed parametric studies are carried out to gain insight into the effects of various factors such as shear strains, non-homogeneity, number and array of layers on the NL frequencies under different rectangular plate characteristics.

Seismic performance of the WCFTC-beams joint with high-strength double-side plates

This research presents a novel joint design for wall-type concrete filled steel tube (WCFTC) beams, integrated with high-strength double-side plates, resolving the problem of traditional side plate extending from the wall and interfering with wall panel installation. The study synthesizes a reliable finite element (FE) model corroborated by previous experimental analyses, enabling a detailed parametric study on the dimensions of the side plates. Comprehensive analysis assessed the impact of steel plate thickness and material strength on the joint's seismic properties, uncovering that the innovative joint design reduces steel usage while maintaining load capacity. Subsequent full-scale testing of prototypes under cyclic loading revealed the failure modes and hysteresis responses, which exhibited notable energy dissipation reflected in ductility coefficients exceeding 3.0. The study also identifies the plastic hinge location to be at a distance ranging from 1.0 to 1.5 times the beam's height from the column's edge, validating the efficacy of the joint design categorized as full-strength or rigid. Furthermore, comparative assessments of Q690D and Q460C steel plates indicated similar performance, suggesting a flexibility in high-strength material choice without loss of joint integrity. The culmination of the study is the proposal of a clear and actionable design method, predicated on the synthesized findings, offering a pragmatic reference for designers and engineers in the field. This method aims to streamline the adoption of this advanced joint construction for enhanced seismic resilience in structural applications.

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

Development of a general algorithm for solving the stability problem of anisotropic plates

This paper is devoted to the development of general algorithm for solving to the stability problem of anisotropic plates using the additional load discretization method. The study of the stability problem is relevant for all types of structural elements and machine parts, and its importance is especially increasing with respect to anisotropic thin plates. This is due to the fact that with the use of new structures and materials, the material intensity is reduced, the area of application of thin-walled systems with low stiffness, for which the danger of elastic loss of stability increases, and, therefore, the importance and relevance of the theory and methods of practical solution of problems of elastic stability of such structures increases. In many works, analytical expressions for determination of critical load are given. At present, the determination of critical loads causes great difficulties in their numerical determination. Therefore, the article presents the most effective numerical and analytical solution of this problem. As a rule, to solve stability problems of anisotropic plates, different representations of the bending deflection function in different rows are used. But the use of such representations is justified only under certain boundary conditions and under the condition of uniformly distributed load. The study described in this paper offers a way to overcome these difficulties, allowing the numerical values of critical forces to be determined without much difficulty. With increasing grid density, the accuracy of the critical load value increases rapidly and with an 8×8 grid, the deviation from the exact solution equal to is 1 %. From a practical point of view, the discovered mechanism of numerical realization of this problem allows to improve engineering design calculations of stability of anisotropic plates with different conditions on supports and with different loading

Corrosion failure analysis and solutions of aluminum alloy gas path plate used in sanding device of high-speed train

During the maintenance of a high-speed train, it was frequently found that the anodized aluminum alloy gas path plates in the sanding device had different degrees of corrosion. In this paper, SEM, EDS, XRD and LAD-EDTEST spectrometer were used to analyze the macro and micro morphology, elements and phases of corrosion product, element compositions of the matrix and assembly problem of the gas path plates. The mechanism of delamination spalling corrosion caused by local pitting corrosion of anodized aluminum alloy gas path plate was described. In addition, the properties of the coatings produced by different surface treatment methods and different substrate materials were compared. The test results indicate that the corrosion protection level of the micro-arc oxide coating is higher than that of the anodic oxide coating, which is mainly composed of γ-Al2O3 and contains a certain amount of α-Al2O3. Compared with 2A12-T4 aluminum alloy, the micro arc oxidation (MAO) coating of 6082-T6 aluminum alloy has lower porosity, higher hardness, higher corrosion potential, lower corrosion current, and better corrosion resistance, and 6082-T6 aluminum alloy is more suitable for use as the gas plate material. Meanwhile, it is proposed that the MAO coating with 35 μm can obviously improve the corrosion resistance of the gas path plate.

Sharp spectral gap estimates for higher-order operators on Cartan–Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

New Accurate Flexural Analysis for Different Types of Plates in a Rectangular Sewage Tank by Utilizing a Unified Analytic Solution Procedure

In the present paper, a modified Fourier series approach is developed for new precise flexural analysis of three different types of concrete plates in a rectangular sewage tank. The bending problems of the bottom plate, side-plate, and the fluid-guiding plate are not easily solved via using the traditional analytic approaches. Based on the Fourier series theory, the present approach provides a unified semi-inverse solving procedure for the above plates by means of choosing three different kinds of Fourier series as the trial functions. Although all the trial functions are quite similar to the classical Navier-form solution, new, precise analytic flexural solutions for plates without Navier-type edge conditions (all edges simply-supported) are achieved, which is mainly attributed to employing the Stoke’s transform technique. For each case, the plate-bending problems are finally altered to deal with linear algebra equations. Furthermore, owing to the orthogonality and completeness of the Fourier series, the obtained solutions perfectly satisfy both the edge conditions and the governing partial differential equation of plates, which paves an easily implemented and rational way for engineers and researchers to provide new, exact designs of plate structures. The main contribution of this study lies in the provision of a unified solution procedure for addressing complex plate-bending problems across diverse boundary conditions. By employing a range of Fourier series types, this approach offers a comprehensive solution framework that accommodates the complexities inherent in plate analysis. The correctness of the present analytic solutions is verified against precise finite element method (FEM) results and ones available in the literature. Finally, the influences of foundation, edge conditions, and aspect ratio on flexural behaviors of plates are discussed in detail.

A T-splines-oriented isogeometric topology optimization for plate and shell structures with arbitrary geometries using Bézier extraction

Recently, the non-uniform rational B-splines (NURBS) have been considerably employed in modeling plate and shell structures, or developing the related size, shape or topology optimization methods for their design. However, the NURBS with the tensor product feature strongly hinders the effectiveness of the optimization on complex structures. The primary intention of the current research is to propose a new T-splines-oriented Isogeometric Topology Optimization (T-ITO) method and then address the critical design problem of plate and shell structures with arbitrary shapes in practical applications. Firstly, the Bézier extraction is utilized in the T-splines to map the geometrical model into a family of Bézier elements, each of which is presented by a local Density Distribution Function (DDF) for elementary topology. A global DDF is constructed by an assembly of all local DDFs with a natural connection to the present structural topology. Secondly, the T-splines-based IsoGeometric Analysis (T-IGA) formulation with the Bézier extraction for complex plate and shell structures is developed using the Kirchhoff-Love theory, where a universal numerical implementation framework is constructed, including the Rhino, MATLAB, export modules with all geometric information and import modules for the analysis. Thirdly, a mathematical formulation for plate and shell structures with arbitrary geometries is developed using the T-ITO method to improve structural loading-capability, and the sensitivity analysis with respect to design variables is rigorously derived in detail. Finally, several numerical examples of plate and shell structures with both regular and elaborate geometries are tested to demonstrate the validity, effectiveness, superiorities and indispensability of the T-ITO method.

Error analysis of the element-free Galerkin method for a nonlinear plate problem

In this work, we derive a geometrically nonlinear plate model based on the Kirchhoff hypothesis and the large deflection hypothesis, and provide an error analysis of the corresponding element-free Galerkin method. The penalty method is employed to enforce boundary conditions. The error estimate explicitly depends on nodal spacing, number of monomial basis functions, continuity of weight functions, and penalty factors, which provides some practical choices among these key parameters in engineering applications. In addition, we offer guidance on selecting appropriate penalty factors to improve numerical accuracy. Numerical experiments, involving clamped square and circle plates with uniform and nonuniform nodes, as well as a plate model with a corner singularity, are presented to validate the theoretical results.

Thermo-mechanical coupled peridynamics simulation of concrete failure under fire scenarios

A coupled thermo-mechanical bond-based peridynamical (PD) method is developed to simulate thermal cracking processes in concrete. Based on the bond-based PD heat conduction and motion equations, the PD differential operator (PDDO) is used to express the classical computational fluid dynamics basic equations in a non-local integral form, establishing a PD model for the thermo-mechanical coupling problems. In this model, concrete is considered a heterogeneous material composed of aggregates and mortar, with material parameters assumed to be temperature-dependent. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. The model is applied to simulate the transient heat transfer problem of a homogeneous plate. The results show good agreement with the analytical solution, providing evidence for the correctness and accuracy of the proposed coupled numerical method. The damage behavior of the borehole heated concrete plate is analyzed, followed by the simulation of the damage to the concrete plate under a fire scenario. The numerical results accurately predict the temperature distribution within the heterogeneous concrete and the resulting material damage. This study provides new theoretical basis for the fire resistance design and high-temperature performance improvement of concrete materials and structures.

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.

New Natural Frequency Studies of Orthotropic Plates by Adopting a Two-Dimensional Modified Fourier Series Method

The free vibration behavior of orthotropic thin plates, which are clamped at three edges and free at one edge, is a matter of great concern in the engineering field. Various numerical/approximate approaches have been proposed for the present problem; however, lack precise analytic benchmark solutions are lacking in the literature. In the present study, we propose a modified two-dimensional Fourier series method to effectively handle free vibration problems of plates under various edge conditions. In the given solution, the adopted trial function automatically satisfies several boundary conditions. After imposing Stoke’s transformation in the trial function and letting it satisfy the remaining boundary conditions, we can change the present plate problem into calculating several systems of linear algebra equations which are easily handled. The present method can be regarded as an easily implemented, rational, and rigorous approach, as it can exactly satisfy both the governing equation and the associated edge conditions. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through the utilization of different types of Fourier series, and it can be extended for the further dynamic/static analysis of plates under different shear deformation theories. Finally, all the novel analytical solutions are confirmed to be sufficiently accurate since they match well with the FEM results. The new analytic solution obtained may serve as a benchmark for validating other numerical and approximate methods.

Application of a hierarchical quadrature element method to interface crack analysis in bi-material systems

This paper presents an extension of the hierarchical quadrature element method (HQEM) with p-convergence to evaluating fracture mechanical parameters of bi-material interface cracks. The numerical implementation of HQEM in combination with the virtual crack closure method (VCCM) is comprehensively illustrated, and the formulas for calculating the strain energy release rate (SERR) are rederived. The relationship between components of SERR and the complex stress intensity factors (SIFs) is established. In addition, the effectiveness of an auxiliary parameter related to the dual energy release rates in measuring the accuracy of the present method is examined.The feasibility of applying HQEM to the interface crack problems is demonstrated by investigating the benchmark problem of an infinite bi-material plate with a central interface crack. Two loading cases are considered, including pure tensile loading and mixed tensile-shear loading. The influence of the crack tip element size Δa on the convergence characteristics and computational accuracy of fracture parameters is quantitatively investigated to provide guidance for choosing the most reasonable value of Δa. Numerical results show that even with very coarse meshes, together with a virtual crack extension consisting of only one element, HQEM yields highly accurate results for the total SERR and the complex SIFs, making it suitable for complex, large-scale interface crack problems in dissimilar media.

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

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.