Optimal locations of internal line supports for rectangular plates against buckling

This paper concerns the optimal locations of straight line internal supports for rectangular thin plates, under uniaxial compression, so as to maximize the elastic buckling loads. The optimization problem can be readily solved by using the newly developed pb-2 Rayleigh-Ritz method for the buckling analysis. Since the method avoids any discretization, the problems associated with the need to move the nodal points so as to coincide with the internal line support are eliminated. Moreover, the optimization exercise may be performed on a personal computer since the method requires a relatively smaller memory and computational effort as a result of a lower number of unknown variables involved when compared to other discretization methods such as the finite difference or finite element methods. Optimal results will be presented for plates with various combinations of edge conditions, aspects ratios and one or two internal line supports. The results indicate large sensitivity of buckling loads to supporting conditions.     

Free vibrations of rectangular unsymmetrically laminated composite plates with internal line supports

Free vibrations of rectangular unsymmetrically laminated composite plates with internal line supports in one or two directions are studied. The static beam functions, which are the complete solutions of a unit width strip taken from the plate in the direction parallel to the edges of the plate and under a series of transverse static sinusoidal loads, are used as the trial functions of the transverse bending deflection of the plate. Besides, the vibrating bar functions are used as the trial functions of the in-plane stretching deformation of the plate. The Rayleigh–Ritz method is applied to arrive at the governing eigenfrequency equation in which the coupling effect of bending and stretching is contained. Different transverse boundary conditions combining with various in-plane constraints are considered in the analysis. The effects of fiber orientation angles and locations of internal line supports on eigenfrequencies of cross-ply and antisymmetric angle-ply laminated composite plates of two layers are investigated in detail. Convergence and comparison study show the correctness and effectiveness of the proposed method. Sets of valuable results are presented for the first time, which can serve as benchmarks for future reference.     

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 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.     

Large deflection of clamped skew plates

A theoretical analysis is presented for the large deflection elastic behaviour of clamped, uniformly loaded orthotropic skew plates. The governing nonlinear partial differential equations are transformed into a set of nonlinear algebraic equations. For this purpose a network of discrete points over the domain is considered, and the integral equation of beams along the skew directions is used with appropriate boundary conditions. The resulting nonlinear algebraic equations are then solved by a Newton-Raphson procedure.A convergence study of the solution has been made, and numerical results for a few cases of orthotropy and skew angles are presented in graphical form. The results obtained by this method are compared with available results of other investigators.     

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.     

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.     

Bending of skew plates by spline-finite-strip method

Spline finite strip was devised by Cheung et al. in 1982 [1]. Unlike the standard finite element method, this method employs B-3spline function for interpolation in one direction and local Hermite cubic polynomial in the other direction. The general form of displacement function is given as their product. Extensive numerical examples on right plates and shells were well documented by Cheung et al. [1]. but the applicability of this method in the analysis of skew plates remains unexplored. The main theme of the present paper is to generalize the technique to include parallelogram plates. As this extension still retains the banded nature, only a small amount of extra computing effort is required. The convergence of the method is established and it is supported by numerous examples of different loading and support conditions. It has shown that spline finite strip method, which requires less variables for interpolation, can achieve the same order of accuracy as the conforming finite element.     

Thermal post-buckling characteristics of clamped skew plates

Thermal post-buckling characteristics of clamped skew plates with restrained edges subjected to planar temperature distributions are studied. The problem is formulated in terms of Von Kármán equations expressed in terms of displacements generalised to include thermal effects. A perturbation method is employed to obtain a linear set of partial differential equations. The solution to this linear set is then obtained by using the Galerkin method. Numerical results are presented for different skew plate configurations for both uniform and two-dimensional temperature distributions. Plots of central deflection, membrane and bending stress distributions are presented and the post-buckling characteristics are discussed in detail.     

Vibration of quadrilateral plates

A novel method for the solution of vibration problems of quadrilateral plates is proposed. With this method, the boundary conditions at the four edges and the differential equation are exactly satisfied. The frequency of vibration is found by satisfying the continuity conditions along a diagonal. The method also yields the mode shape of vibration of the plate under any of the classical boundary conditions and has been implemented on a digital computer.     

Thermal stress analysis of skew plates by finite element method

Thermal stresses are induced in general due to nonuniform temperature distribution or due to the boundary restriction. Most of the work reported so far deals with either plates with edges clamped in plane of the plate or plates with stress free edges. While studying buckling or post-buckling problems, one should ideally analyse the plates with mixed in-plane boundary conditions. Hence, in the present analysis, thermal stress analysis of skew plates with mixed in-plane boundary conditions using finite element approach is attempted. In addition, the effect of in-plane boundary conditions on the thermal stresses is also discussed.     

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.     

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.     

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.     

Estimation of skew angle in text-image analysis bySLIDE: Subspace-based line detection

A new signal processing method is developed for estimating the skew angle in text document images. Detection of the skew angle is an important step in text processing tasks such as optical character recognition (OCR) and computerized filing. Based on a recently introduced multiline-fitting algorithm, the proposed method reformulates the skew detection problem into a special parameter-estimation framework such that a signal structure similar to the one in the field of sensor array processing is obtained. In this framework, straight lines in an image are modeled as wavefronts of propagating planar waves. Certain measurements are defined in this virtual propagation environment such that the large amount of coherency that exists between the locations of the pixels on parallel lines is exploited to enhance a subspace in the space spanned by the measurements. The well-studied techniques of sensor array processing (e.g., the ESPRIT algorithm) are then exploited to produce a closed form and high-resolution estimate for the skew angle.     

Flexural vibration of shear deformable circular and annular plates on ring supports

This paper presents an analysis of free flexural vibration of circular and annular Mindlin plates lying on multiple internal concentric ring supports. The Rayleigh-Ritz method with an admissible displacement function expressed in terms of a set of simple polynomials is used to obtain the governing eigenvalue equation. The natural frequencies of circular and annular Mindlin plates with one or two ring supports having inner and outer peripheries with different combinations of free, simply supported and clamped conditions are determined. A comprehensive set of vibration results is presented and, where possible, these have been verified with existing solutions published in the open literature. No previous theoretical results for thick annular plates supported with internal rings are known to exist. Thus the study should be useful and valuable to engineers and researchers.     

Vibration of triangular plates of variable thickness

The problem of the natural frequencies and mode shapes of cantilevered triangular plates with variable thickness and arbitrary planform is solved using the finite element technique. This is done for various combinations of four non-dimensional geometric parameters, namely, the aspect ratio, the two thickness ratios along the two coordinate directions and the sweepback angle. The frequencies for the various cases are tabulated and a few typical mode shapes have been presented graphically.     

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.     

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.