Non-rectangular Plates Research Articles

Non-rectangular plates with irregular initial imperfection subjected to nonlinear static and dynamic loads

Non-rectangular plates with irregular initial imperfection subjected to nonlinear static and dynamic loads

A Novel Rigid Finite Element Formulation for Dynamic Analysis of Flexible Plates

This paper introduces a new rigid finite element method (RFEM) formulation for dynamic analysis of plates. In RFEM, a flexible body is divided into several rigid elements which are interconnected by spring and damping elements. Mainly, RFEM has been used to model systems with beam-like slender components, such as ropes and cables, and few RFEM formulations were developed to model plates and shells. In this study however, by means of the Timoshenko beam theory and cuboid rigid elements, a novel RFEM formulation is developed to model flexible flat plates with generic geometries of the surface, considering large deformation. For this purpose, an RFEM formulation is presented for straight beams with rectangular cross-section by means of the Timoshenko beam theory. Next, this formulation is expanded to model flexible rectangular plates with uniform thickness and then, it is elaborated on how to model non-rectangular plates with uniform thickness using only cuboid elements and consequently, the development of the proposed RFEM formulation is completed. After investigating various case studies, the competence of the proposed formulation is evaluated thoroughly. The results of RFEM came on a proper agreement with the FEM results in a way that the maximum difference between these results in dynamic analysis was about 3%.

Dynamic and Chaotic Responses of Porous Nanocomposite Nonrectangular Plates with Single-Variable-Edge

The study introduces the dynamical characteristics of functionally graded porous nanocomposite new plates reinforced with graphene platelets (GPLs). The profiles of the structures are defined by four edges, in which an edge changes according to arbitrary functions such as sine function, power function, or arc function. Three types of porous distribution are combined arbitrarily with three types of the graphene platelet reinforcement through the Halpin–Tsai micromechanical model. The theory of elasticity and mechanics of materials are used to derive the governing equations, and then the Galerkin’s procedure is used to achieve the dynamic and chaotic characteristics of the complex-shaped plates. In addition, the obtained results are compared with previous literature and finite element method analyses including fundamental frequencies and nonlinear dynamic responses to confirm accuracy and reliability of the calculating process. The research promises significant contribution to modern science and technology.

A new general analytical method for dynamical behaviors of functionally graded graphene nanoplatelet reinforced non-rectangular plates with many different shapes subjected to thermo-electro-mechanical load

This article studied a general analytical method for dynamical behaviors of various FG-GPLRC non-rectangular plates with a piezoelectric layer. For the first time, classical plate theory and Galerkin’s method are used to scrutinize the nonlinear vibration of plates with many different shapes. Profiles of new plates are defined by four edges, specifically two parallel opposite edges, and others are assumed to horizontal change according to arbitrary functions. Linear and nonlinear vibration of plates are confirmed with results obtained by Finite Element Method. Effects of environment, material, and geometrical parameters are studied. Besides, periodic and chaotic motions of plates are investigated.

Dynamical Behaviors of Sinusoidal Nanocomposite Plates Subjected to Thermomechanical Loads

In this paper, a new plate structure has been found with the change of profile according to the sine function which we temporarily call as the sinusoidal plate. The classical plate theory and Galerkin’s technique have been utilized in estimating the nonlinear vibration behavior of the new non-rectangular plates reinforced by functionally graded (FG) graphene nanoplatelets (GNPs) resting on the Kerr foundation. The FG-GNP plates were assumed to have two horizontal variable edges according to the sine function. Four different configurations of the FG-GNP plates based on the number of cycles of sine function were analyzed. The material characteristics of the GNPs were evaluated in terms of two models called the Halpin–Tsai micromechanical model and the rule of mixtures. First, to verify this method, the natural frequencies of new non-rectangular plates made of metal were compared with those obtained by the Finite Element Method (FEM). Then, the numerical outcomes are validated by comparing with the previous papers for rectangular FGM/GNP plates — a special case of this structure. Furthermore, the impacts of the thermal environment, geometrical parameters, and the elastic foundation on the dynamical responses are scrutinized by the 2D/3D graphical results and coded in Wolfram-Mathematica. The results of this work proved that the introduced approach has the advantages of being fast, having high accuracy, and involving uncomplicated calculation.

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

Geometric Mapping for Non-Rectangular Plates with Micro/Nano or Macro Scaled under Different Effects

The main purpose of this study is to give a perspective via discrete singular convolution, differential quadrature (DQ) and harmonic differential quadrature (HDQ). For this purpose, DQ and HDQ methods are developed for the buckling, analysis of non-rectangular plates. Plates of, skew, shape is considered under axial loads. Furthermore, transformation formulations and some perspective for nano or macro scaled many problems with different effects discussed via discrete singular convolution and differential quadrature methods.

Vibration of functionally graded carbon nanotube reinforced quadrilateral plates using geometric transformation discrete singular convolution method

SummaryFree vibration characteristics of thick skew plates reinforced by functionally graded carbon nanotubes (CNTs) reinforced composite are presented. Discrete singular convolution (DSC) method is used for the numerical solution of vibration problems via geometric mapping technique. Using the geometric transformation via a four‐node element, the straight‐sided quadrilateral physical domain is mapped into a square domain in the computational space. Then the method of discrete singular convolution with some singular kernels such as Regularized Shannon's delta (RSD) and Lagrange's delta kernels (LDK) have been used for spatial discretizing of the resulting governing equation of motion. Calculated results have been presented in order to show the effects of volume fraction of CNT, skew angles, CNT distribution types, plate aspect ratio and length‐to‐thickness ratio on the frequency of CNT reinforced skew plate. The current results are compared with the related results available in the literature and obtained by different methods. It is shown that reasonable accurate results are obtained for free vibration of nanocomposite plates with less computational effort for higher modes. Several test examples for different plate have been selected to demonstrate the convergence properties, accuracy, and simplicity in numerical implementation of DSC procedures. This approach has verified the accuracy and applicability of DSC method to the class of problem considered in this study. Furthermore, in the numerical examples in the scope of the study, the results obtained with DSC method using a coarser grid are more accurate than the values obtained by finite elements and differential quadrature (DQ) methods. It is also revealed that the method of discrete singular convolution is a promising and potential approach for computational mechanics of nonrectangular plates with nanocomposite reinforced.

Free vibration analysis of non-rectangular plates in contact with bounded fluid using element free Galerkin method

Free vibration characteristics of a flexible non-rectangular bottom of a fluid-filled rectangular tank are investigated using a semi-analytical technique. Element free Galerkin method (EFG) based on moving kriging (MK) shape functions is used along with the Mindlin plate theory to model the flexible plate. Two different types of non-rectangular structural domains, viz., skew and trapezoidal configurations are considered in the present study. The fluid is assumed to be incompressible, inviscid and the effect of surface waves is ignored. Potential flow theory is assumed and the velocity potential is used to calculate the kinetic energy of the fluid. The results obtained are first validated with available solutions in the literature for elastic rectangular plates in contact with the fluid. Thereafter, the effects of various structural and fluid domain parameters on natural frequencies of thin isotropic and laminated composite skew/trapezoidal bottom of a rectangular tank filled with fluid are investigated in detail. New results on the vibration of non-rectangular plates in contact with the fluid, hitherto not found in the literature are presented here for the first time.

A Hybrid Analytical and Finite Element Method for Mid-Frequency Vibration Analysis of Plate Structures with Discontinuities

A hybrid analytical and numerical method is presented for the mid-frequency vibration analysis of a class of plate structures with discontinuities based on the concept of structural partitioning. The type of structures considered includes rectangular plates with internal property discontinuity, homogeneous but non-rectangular plates, or built-up structures composed of rectangular homogenous plates with complex joints. Compared with the conventional finite element (FE) method, the present method has the advantage of high accuracy and high efficiency in the analysis of mid-frequency vibration of the structures of concern. The main idea of the proposed approach is to divide the whole structure into uniform rectangular plate regions and other non-rectangular regions. The vibration behavior of the rectangular regions is accurately and efficiently described by analytical wave solutions so that the FE modeling for these regions is not necessary. The other non-rectangular regions are modeled by the conventional FE method. The analytical waves used to describe the rectangular regions are obtained by the symplectic method, thereby avoiding the limitation of the conventional analytical method in dealing with plates having two opposite edges simply supported. By enforcing the displacement continuity and force equilibrium at the connection interfaces, the dynamic coupling is established between the rectangular regions described in terms of the analytical waves and the regions modeled by FEs. Furthermore, the hybrid solution formulation for the mid-frequency vibration of the entire structure is proposed. The high accuracy and efficiency of the present method are demonstrated by several numerical examples, with the effect of element size of the FE regions investigated. Finally, the applicability of the proposed method is analyzed.

On the implementation of the conjugate gradient Fourier transform method for scattering by planar plates

The accuracy of two conjugate gradient fast Fourier transform formulations for computing the electromagnetic scattering by resistive plates of an arbitrary periphery is discussed. One of the formulations is based on a discretization of the integral equations prior to the introduction of the Fourier transform, whereas the other is based on a similar discretization after the introduction of the Fourier transform. The efficiency and accuracy of these formulations are examined by comparison with measured data for rectangular and nonrectangular plates. The latter method is found to provide a more accurate computation of the plate scattering by eliminating aliasing errors (other than those due to undersampling). It is also found to be substantially more efficient. Its greatest advantage is realized when solving large systems generated by convolutional operators not yielding Toeplitz matrices, as is the case with plates having nonuniform resistivity. >

Free vibration analysis of simply supported symmetrical trapezoidal plates

In this paper, superposition techniques developed by Gorman for the free vibration analysis of rectangular plates are extended to the free vibration analysis of quadrilateral non-rectangular plates. A highly accurate analytical solution is provided for the simply supported symmetrical trapezoidal plate. Results are shown in both graphical and tabulated forms for a wide range of plate aspect ratios. Computer generated shape forms are also shown for the first six free vibration modes of the plate shown in Figure 1. Good agreement is noted when comparison of eigenvalues is made with reliable previously available data.