Plate Problems Research Articles

Quasinormal mode expansion in thin elastic plates

Elastic wave manipulation using large arrays of resonators is driving the need for advanced simulation and optimization methods. To address this we introduce and explore a robust framework for wave control: quasinormal modes (QNMs). Specifically, we consider the problem for thin elastic plates, where the Green's function formalism is well known and readily exploited to solve multiple scattering problems. By studying the associated nonlinear eigenvalue problem we derive a dispersive QNM expansion, providing a reduced-order model for efficient forced response computations which reveals physical insight into the resonant mode excitation. Furthermore, we derive eigenvalue sensitivities with respect to resonator parameters and apply a gradient-based optimization to design quasi-bound states in the continuum and position eigenfrequencies precisely in the complex plane. Scattering simulations validate our approach in structures such as graded line arrays and quasicrystals. Drawing on QNM concepts from electromagnetism we demonstrate significant advances in elastic metamaterials, highlighting their potential for tailored wave manipulation. Published by the American Physical Society 2024

Nonlinear combined resonance of magneto-electro-elastic plates

The existing articles on the nonlinear resonances of magneto-electro-elastic (MEE) structures are mainly devoted to the parametric resonance and primary resonance, neglecting the coupling effect between parametric and forced excitations. To reveal this issue, the coupled longitudinal and transverse excitations are considered to investigate the combined resonance of MEE plates, in which the simply supported ends is considered. The expressions of magnetic-, electric- and displacement- fields are determined using the Maxwell's equation and first order shear deformation theory (FSDT). Applying the Galerkin method, the nonlinear ordinary differential equations are derived. The combined resonance problem of MEE plates with simply supported ends is solved by developing the method of varying amplitude (MVA). And the resonance trajectory is depicted by amplitude-frequency diagram in the follow-up analysis, in which the complicated dynamical phenomena with jump, hysteresis and multi-stable solution can be observed. To reveal the dynamic mechanism, comprehensive numerical analyses including electric potential, magnetic potential, excitations, temperature variation and other factors are conducted, the bifurcation and chaotic dynamic behaviors are also analyzed. The results elucidate that these parameters under discussion play significant roles.

Research and Application of Euler Cutting Path Based Optimisation Algorithm with Greedy-RRT Algorithm

With the development of social economy, mould processing enterprises have an increasingly strong demand for reducing costs and improving economic efficiency. The optimisation of steel plate cutting path is one of the key links to improve production efficiency. In this paper, the optimal cutting path problem of steel plate under different cutting tasks is investigated, and the optimal cutting path under complex geometries is found through various optimisation algorithms. For cutting task N3, the cutting path is divided into outer contour cutting, inner hole cutting and inner hole part cutting. According to the problem, all rectangular parts inside the ellipse need to be cut before the ellipse. Through the Eulerian cutting path optimisation algorithm, we solve the air-travel problem between the inner holes and the outer contour as well as the air-travel problem between the inner holes of the ellipse and the parts inside the ellipse. The total length of the finally determined optimal air-travel path is 97 units. By combining the greedy algorithm and the RRT algorithm, we determine the number and location of the bridges and design the optimal cutting path starting from the lower right corner of the steel plate. The final clearance of the bridges is 4 units, the total length of the combined optimal path clearance is 35.56 units, and the number of bridges is 8, which are located between adjacent rectangles.

Study of Optimal Cutting Paths for Steel Plates Based on Enumeration and Nonlinear Programming

With the development of society and economy, mould processing enterprises have an increasingly strong need to reduce costs and increase economic efficiency. Improving the efficiency of the cutting process is one of the key links to achieve this goal, and the planning of the cutting path occupies a major position in it. Under the condition that the actual cutting distance is fixed, the shortest air distance becomes the goal pursued by enterprises. In this paper, the optimal cutting path problem for steel plates is studied, and the optimal paths under different cutting tasks are found by enumeration method and nonlinear planning model.

A novel solution for dynamic behaviors of multi-span bridge plates

The present paper draws a comprehensive analysis of the free vibration characteristics and discusses modal localization phenomena of multi-span bridge plate structures through an analytical method, numerical simulations, and experimental measurements. A novel analytical solution using the generalized superposition segment method (GSSM) is first proposed to investigate the transverse vibration of a multi-span bridge, which offers complete flexibility in describing arbitrarily classical boundary conditions. Furthermore, a new experimental setup achieves precisely simply supported boundary conditions, and scaled physical models of two- and three-span bridge plates are employed to validate the analytical solution. The effect of span length mismatch is studied by analyzing the resonant frequencies and mode shapes of a multi-span bridge plate with varying span lengths. The good consistency of results by theoretical analysis, numerical simulation, and experimental measurements indicate that the proposed analytical solution can solve the vibration problem of the multi-span bridge plate efficiently and reflect the real-world boundary condition. Finally, the two modal localization behaviors are observed in theoretical analysis and experimental measurement. Through analytical solutions, numerical simulations, and experimental measurements, this study enhances the understanding of the dynamic behavior of multi-span bridges. It provides a new theoretical benchmark for predicting vibrations of multi-span bridge systems and useful insights into the transient behavior of multi-span bridge plates, suggesting several fruitful avenues for future research, including exploring passive and active control systems and vibration suppression techniques.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Meshless Error Recovery Parametric Investigation in Incompressible Elastic Finite Element Analysis

The meshless displacement error-recovery parametric investigation in finite element method-based incompressible elastic analysis is presented in this study. It investigates key parameters such as interpolation schemes, patch configurations, dilation indexes, weight functions, and meshing patterns. The study evaluates error recovery effectiveness (local and global), convergence rates, and adaptive mesh improvement for triangular/quadrilateral discretization schemes. It uses meshless moving least squares (MLS) interpolation with rectangular and circular support regions and solves benchmark plate and cylinder problems. It is observed that a circular influence region, a cubic spline weight function, and regular mesh patterns yield a better performance of than an MLS-based error recovery method. The study also concludes that lower dilation index values with rectangular influence regions are preferable for regular meshes, while higher dilation index values with radial influence regions are suitable for preferable meshes to enhance MLS error recovery.

Semi-Analytical Closed-Form Solutions of the Ball–Plate Problem

Semi-Analytical Closed-Form Solutions of the Ball–Plate Problem

A semi-analytical solution for critical buckling loads of orthotropic stiffened rectangular thin plates

Stiffened plates are widely used in various engineering fields as main load-carrying components. Semi-analytical methods are effective for studying the eigenbuckling problems of stiffened rectangular plates, but there are no semi-analytical solutions available for stiffened plates with arbitrary homogeneous boundary conditions. This work aims to develop a semi-analytical method for solving the eigenbuckling problems of orthotropic stiffened thin plates. In this method, base functions for constructing the mode functions of stiffened plate are the mode functions of a simple plate (a plate without stiffener), and the base functions are obtained with the extended separation-of-variable method, which is a closed-form solution method for the eigenvalue problems of plates. The critical buckling load is achieved by substituting the mode functions into the Rayleigh's principle. The present method can deal with arbitrary homogeneous boundary conditions without anticipating the forms of the base functions for different boundary conditions. The accuracy can be improved by using more superposition terms. Besides, an exact closed-form solution of simply supported stiffened plates is achieved, serving as benchmark solutions. Numerical experiments validate the accuracy of the present solutions, and the study on the minimum stiffener stiffness is conducted for different boundary conditions.

The vibration characteristics of LFGMs plates with openings based on spectral shift Legendre method

This paper introduces a novel spectral shift Legendre method for analyzing the vibrational characteristics of laminated functionally graded materials (LFGMs) plates with openings. By employing shifted Legendre series as permissible displacement functions, the computational process is significantly simplified by transforming the integral and differential calculations in the vibration characteristic equation into matrix multiplication. The stability and accuracy of the method are ensured by expanding the displacement at shifted Gauss-Lobatto points (SGLP). The proposed method is then extended to study the dynamic performance of LFGMs plates with openings. To achieve this, a dynamic equation for LFGMs plates without openings is constructed based on the first-order shear deformation theory (FSDT) and LFGMs theory. Subsequently, by setting the weights of the SGLP corresponding to the opening locations to zero, a rapid and accurate solution to the vibration problems of LFGMs plates with openings is realized. To verify the reliability and accuracy of the method, vibrational analyses of several typical LFGMs plates with openings are conducted, and the results are compared with existing literature and results obtained using the finite element method (FEM). The comparisons indicate that the method is reliable and effective, offering a more direct and convenient modeling process. This work significantly expands the application scope of spectral methods and provides a new perspective for the vibrational analysis of LFGMs with openings.

Isogeometric Analysis for Buckling and Wrinkling of Variable-Stiffness Sandwich Plates Under Compression

In this paper, an isogeometric analysis (IGA) framework based on the high-order sandwich plate theory is developed for the first time to deal with both buckling and wrinkling problems of variable-stiffness sandwich plates under compression. The present sandwich plate model is formulated based on the extended high-order sandwich plate theory (EHSAPT). The weak-form governing equations are firstly derived from the principle of virtual work, and afterward the IGA formulations based on the Non-Uniform Rational B-Splines (NURBS) are applied to predict the instability loads and the corresponding instability modes. Both the membrane and non-membrane conditions are integrated into the linear buckling analysis. The novelty of this work lies in that the employment of high-order continuous NURBS basis functions can directly fulfill the [Formula: see text] continuity requirement inherent in the EHSAPT theory, which enables the EHSAPT-based IGA model to be formulated with fewer degrees of freedom. Besides, the developed EHSAPT-based IGA model has also the ability to capture the overall buckling or wrinkling modes of the sandwich plates. The accuracy and effectiveness of the proposed EHSAPT-based IGA model are verified by comparison with previously published results and those obtained using ABAQUS. Effects of the core thickness and fiber angle on the buckling behaviors of the sandwich plates are also studied in numerical examples. It is observed that the non-uniform in-plane pre-stress over the skin layers is the main reason for the non-periodic wrinkling of constant-stiffness sandwich constructions, whereas the non-uniformity of both the in-plane pre-stress and out-of-plane bending stiffness is responsible for the non-periodic wrinkling of variable-stiffness sandwich constructions. The results presented herein may be beneficial for the design of variable-stiffness sandwich constructions under compression.

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.

Analytic analysis of free vibration problem of the plate with a rectangular cutout using symplectic superposition method combined with domain decomposition technique

Plates with rectangular cutouts is widely seen in the field of engineering structures. Therefore, it is crucial to examine analytical solutions for free vibration (FV) of these structures. Despite the existence of approximate/numerical methods, analytical solutions are lacking in the literature. In this study, we employ the symplectic superposition method to effectively analyze the FV problems encountered in plates with rectangular cutouts while integrating the domain decomposition technique. To address the issue of irregular geometry, the rectangular cutout plate is divided into multiple sub-plates. By dividing the problem into multiple sub-problems and solving them separately using variable separation and symplectic eigen expansion, we obtain analytical solutions. Finally, we combine the sub-problem solutions to resolve the initial issue. This solution method can be considered a logical, analytical, and systematic approach as it starts with the fundamental governing equation and is derived without assuming forms of solutions. The study presents a comprehensive set of numerical results that include mode shapes (MSs) and natural frequencies (NFs). The results are rigorously validated using the finite element method (FEM) and relevant literature. The symplectic superposition method demonstrates excellent convergence and precise accuracy, making it suitable for analytically modeling more complex mechanical problems of plates.

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

Vectfem: a generalized MATLAB-based vectorized algorithm for the computation of global matrix/force for finite elements of any type and approximation order in linear elasticity

In this paper, we introduce a new vectorized MATLAB-based algorithm for efficient serial computation of global matrix/force arising from finite element method (FEM) for meshes of any type and approximation order in linear elasticity. Because for-loops in MATLAB are very slow, we propose a modified process that takes advantage of vectorization and sparse assembly to achieve good performance while using the same memory as the standard algorithm. For this purpose, by using good programming practices, the implementation of this scheme is succinctly described and can be integrated into any MATLAB package dealing with FEM. Specifically, attention is paid to the calculation of the triplet (row index, column index, matrix components) as well as the assembly of the global stiffness matrix, mass matrix and force vector. Additionally, an extension of the proposed approach for Mindlin plate theory and functionally graded materials is outlined. Finally, the accuracy of this strategy is verified on selected numerical tests after comparing the obtained results with those of ABAQUS. In terms of performance, the study conducted on a set of meshes considering the standard algorithm and two other well-known MATLAB vectorized algorithms revealed that: (i) for a 2D beam problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-triangle elements, a speedup of about 8 and 15 is achieved with sparse and fsparse, respectively. (ii) for a 3D plate problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-tetrahedral elements, a speedup of about 4 and 8 is achieved with sparse and fsparse, respectively. When compared to ABAQUS performance, the proposed scheme results in a computational time that is about five times smaller.

Non-coercive problems for elastic plates with thin junction

We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other.

A general solution for electromechanical analysis of electroelastic composite plates with arbitrary holes

Stroh formalism provides an elegant method of solution for two-dimensional anisotropic elasticity problems through the generalized eigen relation associated with the dual coordinate system that gives the material eigen values and eigen vectors, and also the fundamental elasticity matrix and its associated Barnett-Lothe tensors. The extended Stroh-like formalism of Hwu and Hsieh has expanded the scope of basic Stroh formalism by addressing the electro-mechanical coupling effects in electro elastic plates. This paper presents an inclusive solution that addresses the two-dimensional problems in infinite electro-elastic, anisotropic, and isotropic plates with an arbitrary hole under remote arbitrary coupled electro-mechanical loading. This is achieved by adopting the Stroh formalism and its subsequent versions to consider the anisotropic and electro-elastic plates, and further, by incorporating the arbitrary biaxial loading condition into the boundary conditions, and the generalized mapping function into the basic formulation to accommodate all types of in plane loading and a variety of hole shapes respectively. Various equations of the solution are derived explicitly for easy understanding and computer implementation. The stress function is derived explicitly to satisfy the condition of traction free hole with remote electro-mechanical loading under generalized plane stress-open circuit condition. The stresses and electric displacements around the hole boundary are obtained by taking the derivative of the stress function with respect to unit normal along the hole boundary. The solution presented can be degenerated to the anisotropic or isotropic case by choosing the corresponding material properties, or appropriate values of complex parameters respectively. This general solution has exactly reproduced the results of other solutions in the literature for piezoelectric plates given by different methods. New results of stresses and electric displacements are obtained for various arbitrary holes in [PZT5H/45/-45/PZT5H]s graphite/epoxy plate and PZT 4 plates. Results for holes in piezoelectric laminates and PZT 4 piezo layer are also presented by FEM.

Seismic performance of high-strength steel octagonal double-sided plate joints in wall-type concrete-filled steel tubular columns connected to steel beams

This study explores the use of a novel octagonal double-sided plate joint made from high-strength steel for connecting steel beams to wall-type concrete-filled steel tubular (WCFST) columns, which is a new solution to the problem of traditional side plate extending from the wall and interfering with wall panel installation. Two full-scale test specimens were developed and assessed under cyclic loading. The results illustrate that the joint specimens exhibit considerable energy dissipation capacities. After experimental verification, a reliable finite element model was obtained and used for a parametric analysis of side plate dimensions. The results suggest that the height, thickness, and strength grade of the side plate should be 1.2 times the beam height, 1.0 times the beam flange thickness, and Q460C, respectively. Hence, the well-designed proposed joint can be employed with less steel and is superior to previous joints in terms of load capacity and energy consumption.

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.