Analysis Of Thin Plates Research Articles

A Finite Integral Transform-Based Generalized Eigenvalue Formulation for Buckling of Orthotropic Rectangular Plates with Rotational Restraints

Although the finite integral transform (FIT) method has been developed for buckling analysis of rectangular thin plates, the existing formulation involves solving complex nonlinear determinantal equations, leading to potentially questionable numerical results. To address this challenge, a new FIT-based formulation is established that transforms the solution of a complex nonlinear equation into that of a straightforward generalized eigenvalue problem. New analytical solutions for buckling of orthotropic rectangular thin plates with rotational restraints are obtained. The solution procedure is implemented via the following four steps: imposing the two-dimensional FIT on the governing equation; inputting deflection conditions of four edges, by which the relations between the transformed deflections and specific unknowns are provided; imposing the one-dimensional FIT on the elastic conditions to replace the unknowns with the transformed deflections, a generalized eigenvalue problem is constructed; determining the final analytical solutions by solving the generalized eigenvalue problem. Comprehensive highly accurate buckling load/mode solutions for typical rotationally restrained orthotropic plates are presented as benchmarks. The effects of boundary conditions, rotational fixity factors, and loading ratios on the buckling behaviors of orthotropic plates are expediently investigated adopting the present solutions.

Galerkin’s Method and Multivariate Newton’s Method for the Nonlinear Deformation of the Magneto-Electro-Elastic Bi-Layered Laminates

In this paper, mathematical modelling for the large deformation of a magneto-electro-elastic rectangular bi-layered laminate with general boundary conditions is presented. Constitutive equations involving the magneto-electro-elastic (MEE) material properties are introduced, Maxwell equations accounts for the electric and magnetic effects are also utilized. First-order shear deformation theory (FSDT) considering the von Karman nonlinear strain is adopted, and the plain strain/stress assumption applicable for thin plate analysis is used. A rather compact set of governing equations related to kinematical variables, electric/magnetic potentials and the Airy stress function is obtained as a consequence of the preliminary condensation for the electro-magnetic state to the plate kinematics. Semi-analytic solution for a bi-layered BaTiO3-CoFe2O4 laminate with specified boundary conditions subjected to various external applied loads is performed. By employing the Bubnov-Galerkin method, the set of nonlinear partial differential equations is transformed to a set of third-order nonlinear algebraic equations for the static deformation due to applied load. Numerical results are carried out by using the multivariate Newton's method with respect to various volume fractions indicating the volume ratio between piezoelectric BaTiO3 layer and piezomagnetic CoFe2O4. From the result, the nonlinearity of the von Karman strain appears to enhance system rigidity as smaller deformations will be detected when external load is applied. Also, some other interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.

Implementation of a non-conforming finite element in abaqus for dynamic thin plate analyses

Implementation of a non-conforming finite element in abaqus for dynamic thin plate analyses

A semi-analytical method using auxiliary sine series for vibration and sound radiation of a rectangular plate with elastic edges

This paper proposes an efficient semi-analytical method using auxiliary sine series for transverse vibration and sound radiation of a thin rectangular plate with edges elastically restrained against translation and rotation. The formulation, constructed by two-dimensional sine and/or cosine series, can approximately express the bending displacement, and calculate vibration and sound radiation under excitation of point force, arbitrary-angle plane wave, or diffuse acoustic field with acceptable accuracy. It is also applied for baffled or unbaffled conditions. A post-process program is developed to predict vibrating frequencies and modes, mean square velocity spectrum, and sound transmission loss via reduced-order integrals of radiation impedances. The method is validated by experiment and simulation results, demonstrating accurate and efficient computation using a single program for transverse vibration and sound radiation of a plate under different elastic boundary conditions and different excitations. Formulas given in this paper provide a basis for the code development on transverse vibration and sound radiation analysis of thin plates.

Finite element analysis of plates by an accurate strain-based element

Plates analysis has been a very important research area for a long time. In this perspective, in this paper, finite element analysis of thin plates in bending is carried out using a new four node strain-based plate element. The strain-based approach and Kirchhoff theory for thin plates have been used to create the strain field of this element. This approach consists in having a displacement field from a proposed strain field. This developed model contains essential degrees of freedom at each corner node which are three, one displacement and two rotations. The displacement field is based on the assumed functions for various deformations which satisfy the compatibility equations, and the displacement functions are obtained by integrating these deformation functions. The numerical validation of this developed element was held through several examples from the literature. The obtained results from these numerical tests prove the good accuracy and performance of this element.

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.

Nonlinear vibration analysis of thin plate with periodically distributed tunable-stiffness nonlinear oscillators in subsonic flow

This paper provides a new passive control method for nonlinear vibration of subsonic thin plates. The control method uses periodically distributed tunable-stiffness nonlinear oscillators (NLO). By using von Karman’s nonlinear plate theory and subsonic aerodynamic model, the nonlinear dynamic model of thin plate is established based on Hamilton’s principle. Through analyzing the system’s nonlinear dynamics bifurcation, the impact of periodic distribution on the stability is discussed. The amplitude-frequency responses are discussed under different NLO installation modes and design parameters by using Matcont software. The results show that the periodically distributed tunable-stiffness nonlinear oscillators can improve the stability of the system and have a remarkable suppression effect. The optimal control scheme for vibration suppression by periodically distributed tunable-stiffness nonlinear oscillators is obtained. The results obtained from this research can offer a valuable understanding of nonlinear vibration control, thereby enabling the identification of various control approaches that can be implemented in practical applications during future design endeavors.

Hygrothermal buckling analysis of thin rectangular FGM plate with variable thickness

PurposeThis paper aims to contribute novel insights into the analysis of thin functionally graded material (FGM) plates with variable thickness, considering both temperature-dependent and independent material properties, focusing on critical linear buckling temperature rise and the effect of critical linear moisture for various moisture concentrations.Design/methodology/approachThe study derives stability and equilibrium equations for thin rectangular FGM plates under hygrothermal loading, employing classical plate theory (CPT). Buckling behavior is examined using Galerkin’s method to obtain pre-buckling force resultants.FindingsThe findings highlight significant increases in critical buckling temperature with aspect ratio, distinct temperature sensitivity between materials and increasing moisture susceptibility with larger aspect ratios. These insights inform material selection and design optimization for FGM plates under hygrothermal loading, enhancing engineering applications.Research limitations/implicationsThis research primarily focuses on hypothetical scenarios and mathematical model development and analysis.Originality/valueThis paper presents original contributions in the field by addressing the hygrothermal buckling analysis of thin FGM rectangular plates with variable thickness, utilizing CPT, thereby enriching the understanding of structural behavior in varying environmental conditions.

Ritz Method-Based Formulation for Analysis of FGM Thin Plates Undergoing Large Deflection with Mixed Boundary Conditions

Ritz Method-Based Formulation for Analysis of FGM Thin Plates Undergoing Large Deflection with Mixed Boundary Conditions

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Frequency dependent exact trial functions in Galerkin boundary method for free vibration analysis of thin plate

Method of frequency dependent exact trial functions for thin plate vibration based on fundamental solutions of Voigt is suggested. Contrary to other methods, where one of the functions is chosen as the scale and specific boundary condition dependent, it employs only the frequency dependent functions for both space coordinates. Thus they are scale independent, so the same functions can be used for different boundary conditions and plate dimensions. General rule for the choice of exact trial functions is formulated. The boundary conditions are satisfied according to the Galerkin boundary method, which actually minimizes the energy residuals with respect to each weight function (fundamental solution).The verification of method is performed on examples of rectangular plate with various boundary conditions. Several arbitrarily chosen sets of trial functions were employed in calculation of the same tasks, and it is shown that accuracy of the method is almost independent from the choice of them. Even the small number of trial functions provides the very accurate results for natural frequencies.

Mass lumping schemes fitted to MLS-based numerical manifold method in vibration of plates with cutouts using CPT and FSDT

It is critical to accurately extract the natural frequencies in the transverse vibration of plates using the classical plate theory (CPT) and the first-order shear deformation plate theory (FSDT). The moving least squares (MLS) based numerical manifold method (MLS-based NMM) in conjunction with a suitable diagonally lumped mass matrix is a reasonable choice because it can naturally treat plates with cutouts and easily formulate an H2 regular approximation based on MLS interpolation, as well as mitigate shear locking issues for vibration analysis of thin plates based on FSDT. Moreover, the mass lumping techniques involving the row-sum method, the diagonal scaling method, and the mathematically rigorous manifold-based method are extended and derived in the unified framework of MLS-based NMM for transverse vibration analysis of plates. Furthermore, the positive-definiteness of the lumped mass matrix (LMM) generated by the manifold-based mass method is explained in MLS settings, and the row-sum method is proven to be a special case of the manifold-based method. A comprehensive comparative study of different LMMs is performed in accordance with the consistent mass matrix (CMM) on extensive numerical benchmarks. Numerical results demonstrate the convergence properties of LMMs compared with CMM in MLS-based NMM for plate vibration based on CPT and FSDT. It can be observed that LMMs yield comparable performance compared with CMM. The row-sum method obtains accurate results in most cases but can not guarantee the positivity. In contrast, the manifold method can guarantee the positive-definiteness of LMM and is more accurate than the diagonal scaling method in most cases.

Stochastic analysis of thin beams and plates incorporating von-Kármán nonlinear strains using meshless method

The current paper proposes methods for stochastic meshless analysis of thin beams and plates, by considering von-Kármán nonlinear strains. Here, Young’s modulus is assumed to have a spatial variation over the domain and is modeled as a homogeneous random field. Further, it is discretized utilizing moving least square based shape functions. Gaussian and lognormal properties are presumed to model the randomness related to Young’s modulus. Element-free Galerkin method is used as the meshless tool. The present study suggests employing perturbation method if the quantities of interest are limited to the first or second probabilistic moments of the responses. The nonlinear equations in the proposed perturbation formulations are solved using Newton-Raphson iterative scheme for finding out the deterministic part of the response, as well as its initial two derivatives with respect to random variables. Additionally, if there is an interest in higher probabilistic moments, probability density functions, cumulative distribution functions, and so forth, the study proposes a method using high-dimensional model representation (HDMR) in the meshless framework. In HDMR, the nonlinear connection between input variables and the output response is represented in terms of hierarchical correlated function expansions. These hierarchical functions are approximated over the response values evaluated using deterministic analysis on selected number and combinations of control points of random variables involved. Lagrange interpolation method is employed for the construction of response surfaces. Unlike Monte Carlo simulation (MCS) which needs numerous deterministic nonlinear analyses to be performed during each simulation, HDMR requires only a few deterministic nonlinear analyses to construct the component functions. Hence, simulations performed on the reduced order models of high-dimensional response can be used to estimate the probabilistic moments as well as density functions and cumulative distributions of response, with high computational efficiency compared to MCS. The probabilistic moments and distributions produced for a range of coefficient of variation of input random field up to 30%, for both the normal and lognormal input random fields are compared with those provided by crude MCS on the system of equations and are observed to be matching well.

Geometrically Nonlinear Analysis of Beam-Reinforced Thin Plates Using the Methodology of Groebner Bases

This paper illustrates the utility of the methodology of Groebner bases computations combined with the energy method in the analysis of geometric nonlinear beam-reinforced thin rectangular isotropic plates (BRP) for modeling rectangular duct system deflections under internal positive pressure. The governing integro-partial differential equation is derived based on Kirchhoff/von Karman plate theory. With Rayleigh/Ritz methodology, a system of coupled polynomial algebraic equations is generated by using the exact solution of the BRP plate from linear analysis as a shape function. Then Groebner basis methodology is employed to decouple these equations. Under certain conditions, an analytical expression for the lateral displacements of the BRP plate under pressure can be obtained in a fully symbolic form, in terms of such parameters as geometric and material properties of the beams and panels. The analytical solutions have been compared with the results using the finite element software, ANSYS. The comparative study indicates that for commonly used duct panels with the aspect ratio (L:W) less than 1:4, the analytical solutions for displacements are in very close agreement. Finally, the study is found to be a unique alternative, worthy of further investigation, and potentially effective in the analysis of similar problems occurring in a variety of engineering applications.

Buckling analysis of thin plates with circular cut-outs for sustainable ventilated food packaging design

Thin plates with circular cut-outs are widely used in engineering structures such as bridges, construction, and ventilated corrugated paperboard boxes in food packaging. This study investigates the critical buckling loads of thin elastic plates comprised of a single circular hole at different locations and with different sizes. To determine the critical loads, experimental tests and finite element method simulations are conducted. The vertical location and the diameter of the hole impact the buckling results significantly. The results indicate a significant correlation between the experimental and finite element data. The study develops second and third order polynomial formulas that predict buckling loads. In this study, corrugated paperboard plates were used as samples to gain insights into ventilated food packaging designs, aiming to enhance their buckling strength while reducing packaging material consumption for packaging sustainability purposes. The solution proposed in this study can be applied to predict the buckling of other elastic thin plate structures with a circular hole, relevant to industries such as construction and aerospace.

Free vibration analysis of thin plates with side cracks by the weak form quadrature element method

The weak form quadrature element method is applied to vibration analysis of thin plates with side cracks. Eigenfunctions are introduced into the inner square domain around the crack tip, while the rest of the problem domain is analyzed by the weak form quadrature element method (QEM). The continuity of displacements and slopes on the interface between the inner and outer domains is enforced. A generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first ten nondimensional frequency parameters for several typical problems are obtained and compared with those available in the literature. The corresponding modes of free vibration are plotted, and the influence of the crack on vibration frequencies and modes is discussed at length.

Wave propagation analysis in pre-stressed incompressible hyperelastic multi-layered plates using a plate theory

Wave propagation analysis of pre-stressed incompressible hyperelastic thin multi-layered plates is developed based on the first-order shear deformation plate theory (FSDT) with second order thickness extensibility. Variables describing the thickness-direction deformation are obtained from the incompressibility condition. The incremental theory of nonlinear elasticity is employed in conjunction with a plate theory. The multi-layered plate is assumed to be symmetric in the thickness direction. Since the plate theory has not been used for the wave propagation analysis of hyperelastic plates, the results of flexural wave modes are compared with results of the straight-crested flexural waves based on the three-dimensional (3D) elasticity theory available in the literature and the results for in-plane wave modes are compared with those developed in this study using the plane-stress elasticity theory. The dispersion curves of both extensional and flexural waves for single-layer and three-layer plates are presented and the effects of different parameters including the homogenous pre-stretch, the wave propagation direction and the plate initial thickness on the dispersion curves are discussed in detail. The results indicate that the phase speed of the extensional waves depends on the wave propagation angle due to the induced anisotropy. As the overall shear modulus of the three-layer plate increases, the phase velocity increases. Also, the effect of pre-stretch and thickness on phase speed of flexural waves is not pronounced for high frequencies.

Buckling analysis of thin plates using direct interpolation boundary element method

The present study employs the direct integration boundary element method (DIBEM) to analyze the buckling behavior of thin plates. Unlike conventional approaches, this method eliminates the need for domain discretization or specialized solutions. Furthermore, it constitutes a continuation from the previous research in Doval et al. (2013), which employed the radial integration method for the same investigation.To showcase the efficacy of the proposed approach, it is utilized in the stability analysis of both perforated and non-perforated plates. The findings are then compared against analytical responses, in the case of non-perforated plates, and finite element method results, for perforated plates.Moreover, the use of boundary elements reduces the size of the eigenvalue problem involved in stability analysis, as compared to finite elements, provided that only the domain and a few internal points are needed for discretization.Overall, this paper presents a novel and effective approach to the analysis of buckling plates.

Two-dimensional orthotropic plate problems in a thermal environment: Refined crack modelling

Three two-dimensional (2D) refined models are devised for the analysis of orthotropic thin plates with part-through surface and internal cracks under in-plane forces due to temperature differences. The refined models are used to formulate the governing equations for cracked orthotropic thin plates in a thermal environment. Subsequently, the refined models and their ensuing modified forms can be directly applied to all-over and finite-length cracks, respectively. Using an interface-fitted numerical scheme (i.e., matched interface and boundary technique), the resulting equations of motion for the buckling and free vibration of cracked orthotropic thin plates are solved. The effects of various crack types and dimensions on the critical buckling temperatures and vibration frequencies of orthotropic thin plates are investigated. The present models are also validated by comparing their numerical results with the line-spring model and three-dimensional finite-element solutions. Illustrative examples, including boron fiber/epoxy and glass-fibre-reinforced composite materials, are considered to show the superiority of the proposed models over the conventional one, especially in terms of understanding the effect of in-plane forces arising from temperature differences at the cracked location.

The DISCS method for perforation simulation of thin metallic plates by a rigid projectile

The DISCS method had been developed for penetration analysis of an axisymmetric projectile with a slender nose into a semi-infinite medium. Recently, the authors developed an approximate approach to assess perforation of concrete slabs of finite thickness, using the DISCS method. The present paper extends the approximate model to the perforation analysis of ductile metallic thin plates and determines the key parameters of the projectile residual velocity, the target perforation limit, and the ballistic limit. The present investigation focuses on thin metallic plates, the thickness of which is considerably smaller than that of the concrete slabs and smaller in comparison to the projectile nose length. The method is rather simple, easy to implement and provides a fast solution. An idealized target is characterized by a set of discs of small thickness responding in the plate plane, disregarding the local damage around the penetration ductile hole. The in-plane response of the discs during penetration and perforation of an idealized target are calculated. The Residual Velocity upon perforation of the Idealized Target (RVIT) is lower than the residual velocity of the real plate since it disregards the local damage, hence, the RVIT is the lower bound of the true residual velocity of the real perforated thin plate. Therefore, a positive RVIT for the thin plate with a given thickness indicates right away that the real target will be perforated. Using the RVIT parameter, the proposed DISCS method turns into a series of RVIT-perforation problems of plates with thicknesses that are smaller than the real thickness, to determine the decreasing RVIT with increasing thickness, until the full thickness is considered. The analysis is carried out using the modified (hybrid) DISCS method, that aims at analyzing penetration depth into a thick target. The analysis is based on the theoretical field equations and the constitutive equations of the plate material. The proposed method is based on a theoretical model and on engineering insight. Neither empirical constants nor any calibrations are required. Validation of the proposed method is carried out thorough comparisons with numerous test data on ductile aluminum and steel plates of different thicknesses that are struck by rigid projectiles at different impact velocities. All the comparisons demonstrate very good correspondence between the proposed method predictions and the test data.