Vibration of plates immersed in hot fluids

The vibration of structures immersed in water has been studied by Lamb and others. The influence of thermal stresses on the vibration characteristics of plates has been investigated and reported by several researchers. The work reported here deals with the determination of natural frequencies of plates immersed in water and subjected to six types of temperature distributions. The analysis is based on finite element method. A comment on thermal buckling of plates immersed in water is also reported.     

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.     

Vibration and buckling analysis of plates of abruptly varying stiffness

The application of B-spline functions and the Rayleigh-Ritz procedure to analyze vibration and buckling of plates with abruptly varying stiffnesses is presented. Numerical examples of stepped thickness plates and perforated plates are presented and the results are discussed in comparison with those of other approximate methods.To demonstrate the versatility of the present method, vibration and buckling of skew plates of stepped thickness are also studied for various skew angles, ratios of thickness and ratios of width.     

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.     

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.

     

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.     

Nonlinear analysis of skew plates using the finite element method

In this paper finite element analysis of the large deflection behaviour of skew plates has been done. A high precision conforming triangular plate bending element has been used. The central deflection, bending and membrane stresses have been reported for simply supported and clamped rhombic plates. The variations of these quantities have been studied for different skew angles.     

An integral-equation solution for the finite deflection of clamped skew sandwich plates

The large-deflection behaviour of skew sandwich plates is governed by a system of five coupled nonlinear partial differential equations which are highly complex in nature. In the reported study, this problem is analysed using an integral-equation approach. The integral equations of beams along the skew directions is used with appropriate boundary conditions to transform the governing nonlinear partial differential equations into a set of nonlinear algebraic equations. These equations are then solved using an iterative scheme suggested by Brown. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load-deflection characteristics have been presented for clamped skew sandwich plates.     

On the forced vibration of the fiber partially immersed in fluid

In this paper an experimental and theoretical investigation of the forced vibration of a partially immersed fiber has been carried out. An optical method utilizing a forward light scattering pattern has been used to detect a small (<1.0 μm) vibration amplitude of the fiber. The physical and mathematical model of the partially immersed fiber vibration has been put forward. Based on an analytical solution of the model, natural frequencies of the partially immersed fiber vibration have been found; they are consecutive positive roots of the transcendental equation. An “effective” speed of the wave propagation over the fiber has been introduced, which allows one to find the physical meaning of normal modes of the partially immersed fiber vibration. Theoretical predictions agree well with experimental data. The sensor exhibits an excellent sensitivity and could be used for measuring physical properties of fluids and liquid level.     

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.     

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.     

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.     

Geometrically non-linear free vibrations of clamped simply supported rectangular plates. Part I: the effects of large vibration amplitudes on the fundamental mode shape

The geometrically non-linear free vibrations of thin isotropic and laminated rectangular composite plates with fully clamped edges have been successfully investigated in previous series of works using a theoretical model based on Hamilton’s principle and spectral analysis. The objective of this work is the extension of the above model to the case of clamped clamped simply supported simply supported rectangular plates, denoted by CCSSSSRP, in order to determine their fundamental non-linear mode shape, and associated amplitude-dependent resonant frequencies, and flexural stress distribution. Numerical data are given for both linear and non-linear analysis, for various plate aspect ratios and vibration amplitudes. Good agreement was found with previous published results.     

Large deflection analysis of clamped skew sandwich plates by parametric differentiation

The differential equations governing the large deflection behavior of skew sandwich plates are highly complex in nature and do not lend themselves for easy solution. In the study reported herein, this problem is solved by a numerical technique known as “parametric differentiation”. The non-linear differential equations in terms of displacements are transformed into a set of linear differential equations with variable coefficients. These are solved to give the gradients of the displacements in the load direction. The subsequent solution of a set of initial value problems yield the displacements proper. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load deflection characteristics have been presented for clamped skew sandwich plates.     

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.     

Optimal locations of point supports in plates for maximum fundamental frequency

This paper investigates the optimal locations of rigid point supports to maximize the fundamental frequency of free vibration of plates. The computational method uses the Rayleigh-Ritz method for the vibration analysis and the simplex method of Nelder and Mead for the optimization search of support locations. Optimal results have been obtained for various common shapes of plates with a few point supports. The results show that the frequency of the plate is sensitive to the locations of the point supports. Moreover, these new optimal results provide useful information to designers seeking to exploit the position of point supports in their plate designs. It has been found that multiple solutions are a common feature of this plate optimization problem.     

Performance study of a simple finite element in the analysis of skew rhombic plates

An exhaustive study of skew rhombic plates in transverse bending is carried out using a simple three-noded element. Uniformly distributed, as well as concentrated, loads are considered for various support conditions. It is interesting to note that even for Morley's acute skew plate with singularity, very good convergence is achieved for deflection and moment values with the usual mesh pattern. Because of the inadequacy of published works, two other elements have been deployed to facilitate comparison of results. It is seen that the element under discussion fits excellently into the analysis of skew plates. Because of the availability of the element stiffness matrix in the explicit form, the use of the element is straightforward and involves less complexity. A minicomputer has been employed for the analysis and the CPU time for different elements has been reported in the convergence study for the sake of comparison.     

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.     

Finite element flutter analysis of composite skew panels

A finite element method is used to study the dynamic instability of laminated composite skew panels subjected to supersonic flow. The FEM employs eight-noded isoparametric elements which take into account transverse shear deformation. The linearized Piston theory is applied to assess the aerodynamic loads. The effects of skew angle on critical aerodynamic parameters are investigated for different aspect ratios, boundary constraints, fibre orientations and lamination schemes. It is observed that the skew angle has a stabililzing effect on the flutter boundary, whereas couplings have a destabilizing effect. The higher aspect ratio is also found to exhibit a stabilizing effect on the flutter boundary.     

The free vibration of skew plates using the hierarchical finite element method

The hierarchical finite element method is used to determine the natural frequencies and modes of flat, isotropic skew plates. A number of such plates with different boundary conditions—including free edges and point supports—are considered in this paper. The dependence of frequency on skew angle, aspect ratio and Poisson's ratio is investigated, though succinctness prohibits a complete study exploring the full interrelation of these parameters. Extensive results are presented in diagrammatic, graphical, and tabular format; these are shown to be in very good agreement with the work of other investigators, and should prove a valuable source of data for use by engineers and scientists.