Vibration and buckling of skew plates with edges elastically restrained against rotation

This paper deals with vibration and buckling analyses of skew plates with edges elastically restrained against rotation using the spline strip method. The effect of rotational stiffnesses, skew angles and aspect ratios on these problems is analysed, and its characteristic charts are also presented.     

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.

     

Vibrations of skew plates immersed in water

Vibrations of skew plates immersed in water have been studied in the present work. Two skew prism fluid elements have been developed for the above study. As there is no information available on them, a few experiments have been conducted to measure the frequencies of vibration of these plates immersed in water. The results of both the theoretical and experimental investigations have been compared and found to be agreeing reasonably well. The general findings of these plates regarding their aspect ratios, thickness ratios and boundary conditions are discussed. In addition, the influence of skew angles on the vibration of skew plates have been investigated. It has been noticed that the effect of added mass is found to decrease with the increase of skew angle.     

Vibration of stiffened skew plates by using B-spline functions

This paper presents a general procedure for calculating the free vibration of stiffened skew plates by the Rayleigh-Ritz method with B-spline functions as coordinate functions. The stiffened skew plates are modelled as the skew plate with a number of stiffening beams.The results are compared with existing values based on other numerical methods. Vibration characteristics of stiffened skew plates are also studied with changing the arrangements of stiffening beams, the stiffness parameters of beams, skew angle and aspect ratio.     

Vibration studies on skew plates: Treatment of internal line supports

A comprehensive literature survey on the vibration of thin skew plates is presented and a few virgin areas on this subject are identified. As an initial part of a research plan to fill these gaps, the paper focuses on vibrating skew plates with internal line supports. For analysis, the pb-2 Rayleigh-Ritz method is used. The Ritz function is defined by the product of (1) a two-dimensional polynomial function, (2) the equations of the boundaries with each equation raised to the power of 0, 1, or 2 corresponding to a free, simply supported or clamped edge and (3) the equations of the internal line supports. Since the pb-2 Ritz function satisfies the kinematic boundary conditions at the outset, the analyst need not be inconvenient by having to search for the appropriate function; especially when dealing with any arbitrary shaped plate of various combinations of supporting edge conditions. Based on this simple and accurate pb-2 Rayleigh-Ritz method, tabulated vibration solutions are presented for skew plates with different edge conditions, skew angles, aspect ratios and internal line support positions.     

Elastic limit loads of skew plates

An elastic limit load of skew plates is analyzed on the basis of the Rayleigh-Ritz method with B-spline functions and the Huber-Mises yield criterion. Dimensionless elastic limit loads, associated maximum deflections and the positions of the first yield points are presented for different skew angles, aspect ratios and boundary conditions.     

Stability analysis of skew orthotropic plates by the finite strip method

A stability analysis based on the Finite Strip Method is presented for skew orthotropic plates subjected to in-plane loadings. The straight sides of the plate are simply supported and the other two skewed sides are supported with any combination of fixed, free and simply supported boundaries. The plate is divided into strips, in contradistinction to elements in the Finite Element Method, and the displacement function is so chosen that it satisfies the boundary conditions and also the inter-strip compatibility conditions of an elemental strip. The energy expressions required to formulate the stiffness and stability coefficient matrices are formulated using smalldeflection theory. The buckling load intensity factor is evaluated for different aspect ratios of isotropic and orthotropic skew plates and the results of certain rectangular isotropic cases are compared with earlier investigations.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

Finite element free flexural vibration analysis of plates having various shapes and varying rigidities

Natural frequencies of skew, curved and tapered plates have been determined using the isoparametric quadratic plate bending element. Plates having linearly varying thickness in one direction and also those having parabolically varying thickness in orthogonal directions have been analysed. A simple stepped plate approach for a linearly tapered plate has also been considered to compare the results. Two approaches have been adopted for the solution of skew plates. In one case, the boundary conditions have been exactly satisfied by transforming the rotational displacements at the boundary nodes along and normal to the edge. In the other case, only the vertical deflection was locked. A curved plate has also been analysed by the first method. Results thus obtained from both cases have been compared.     

Large deflection analysis of skew plates by lumped triangular element formulation

A finite difference scheme with triangular mesh is presented for the analysis of skew plate problems with large deflections. The suggested formulation is independent of the boundary condition and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using Newton-Raphson iterative method with incremental loading. The investigation is concerned with the behaviour of constant thickness clamped and simply supported isotropic skew plates with immovable edges and subjected to uniformly distributed transverse load. The effects of skew on plates with large deflections are investigated and comparisons are made with existing results; good agreement is shown.     

Post-buckling behaviour of moderately thick cylindrically orthotropic annular plates

A finite element formulation including the effects of shear deformation and cylindrically orthotropic material properties is described for studying the post-buckling behaviour of annular plates. Numerical results for the buckling load parameter and ratios of nonlinear load parameter to buckling load parameter for various values of orthotropic properties, thicknesses and radii ratios of the plates are presented.     

Static,vibration and buckling analysis of axisymmetric circular plates using finite elements

The static, vibration, and buckling analysis of axisymmetric circular plates using the finite element method is discussed. For the static analysis, the stiffness matrix of a typical annular plate element is derived from the given displacement function and the appropriate constitutive relations. By assuming that the static displacement function, which is an exact solution of the circular plate equation ?²?²W = 0, closely represents the vibration and buckling modes, the mass and stability coefficient matrices for an annular element are also constructed. In addition to the annular element, the stiffness, mass, and stability coefficient matrices for a closure element are also included for the analysis of complete circular plates (no center hole). As an extension of the analysis, the exact displacement function for the symmetrical bending of circular plates having polar orthotropy is also given.     

Large elastic deformations of clamped skewed plates by dynamic relaxation

This investigation is concerned with the nonlinear behaviour of clamped Isotropic skew plates of constant thickness subjected to a uniformly distributed transverse load. The recently developed numerical technique of dynamic relaxation has been adopted for the analysis. A detailed study of the large deflection behaviour of skew plates has been made by varying three parameters, viz. skew angle, load, and aspect ratio. Numerical results have been compared with the available solutions. Representative nondimensional solutions are presented in the form of graphs to elucidate the nonlinear effect due to large deflection at higher loads.     

Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis

This and a companion paper (Computers and Structures 26, 915–923, 1987) present a local finite element model based on a refined approximate theory for thick anisotropic laminated plates. The three-dimensional problem is reduced to a two-dimensional case by assuming piecewise linear variation of the in-plane displacements u and ρ and a constant value of the lateral displacement w across the thickness. By using a substructuring technique the present model is demonstrated to be practical and economical. The static bending stresses, transverse shearing stresses and in-plane displacements are predicted in the present paper. The vibration and buckling analyses will be presented in the second paper. Comparison with both exact three-dimensional analysis and a high-order plate bending theory shows that this model provides results which are accurate and acceptable for all ranges of thickness and modular ratio.     

Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM

In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)₂]_s and angle-ply [(15°/−75°)₂]_s, [(30°/−60°)₂]_s, [(45°/−45°)₂]_s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates.     

Natural vibrations of shear deformable cantilevered skewed trapezoidal and triangular thick plates

In a recent companion paper, the efficacy of higher-order shear deformable, C⁰ continuous, Lagrangian isoparametric plate finite element analyses has been demonstrated on cantilevered skewed (parallelogram) thick plates. The present work extends the method to include skewed thick plates having trapezoidal and triangular planforms, and is the first known vibrational study of such plates. Extensive and accurate nondimensional frequency tables and graphical charts are presented for a series of trapezoidal plates showing the effect of aspect ratio, chord ratio, thickness ratio, and skew angle. The need for the present higher-order shear deformable plate finite element method for skewed trapezoidal plate vibrations increases as the skew angle increases and as the aspect ratio, chord ratio, and thickness ratio decreases. Some theoretical and experimental data hitherto published for delta and skewed triangular cantilevered plates are compared with results obtained using the present finite element method. No published theoretical results for cantilevered skewed trapezoidal and triangular thick plates are known to exist.     

Numerical optimum design of elastic annular plates with respect to buckling

Annular plates of fixed volume, under uniform radial edge compression are considered. The plate is divided into N segments of linearly variable or constant thickness. Geometrical constrains for the segment thickness and for the effective stress are assumed. Two types of objective functions are explored: (I) maximum value of critical load at the fixed number of circumferential buckling half-waves j, (II) unimodal optimum design for such a number j that buckling load equals max

. The shooting method is applied to compute the distribution of axial forces in the prebuckling state and the basic buckling load for the plate of variable thickness. The optimal distribution of segment thicknesses Xk for k = 1, …, N are computed by means of Rosenbrock's method with internal penalty function. Results of numerical analysis are reported for the both optimum design problems I and II.     

Post-buckling of cylindrically orthotropic circular plates on elastic foundation with edges elastically restrained against rotation

The post-buckling behaviour of cylindrically orthotropic circular plates is investigated through a finite element formulation, with the plates resting on an elastic foundation and their edges are elastically restrained against rotation. Results are presented in the form of linear buckling load parameters and empirical formulae for radial load ratios for various values of spring stiffness, foundation stiffness and orthotropy parameter.