In-plane and shear buckling analysis of FG-CNTRC annular sector plates based on the third-order shear deformation theory using a numerical approach

The buckling analysis of thick composite annular sector plates reinforced with functionally graded carbon-nanotubes (CNTs) is presented under in-plane and shear loadings based on the higher-order shear deformation theory. It is considered that the plate is resting on the Pasternak-type elastic foundation. The overall material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are estimated through the micromechanical model. The governing equations are derived on the basis of the higher-order shear deformation plate theory, and the quadratic form of the energy functional of the system is presented. An efficient numerical method is presented in the context of variational formulation to obtain the discretized version of stability equations. The validation of the present study is demonstrated through comparisons with the results available in the literature and then comprehensive numerical results are given to investigate the impacts of model parameters on the stability of CNT-reinforced composite annular sector plates.     

Seminumerical solution for buckling of rectangular plates

A semianalytical, seminumerical method of solution is presented for the governing partial differential equation of rectangular plates subjected to in-plane loads. The basic functions in the y-direction are chosen as the eigenfunctions for straight prismatic beams. The classical method of separation of variables is employed to obtain an ordinary differential equation. The resulting equation is solved by a one-dimensional finite difference technique. The problem is then reduced to a typical eigenvalue problem which on solution yields the buckling coefficient of the plate. The method is applied on plates with different edge conditions and under various loading conditions. The results are compared with those of existing solutions. Results for the case when one loaded edge is fixed and the other simply supported were reported in the literature for the first time.     

On the elasto-plastic large deflection response of stocky annular steel plates

An incremental full-range analysis for pressure loaded annular plates is presented in outline. Von Karman large deflection plate theory coupled with the Prandtl-Reuss associated flow rule and a layered Von Mises yield criterion form the cornerstones of the analysis. The numerical solution of the governing equations is achieved via a finite-difference implementation of the DR algorithm. The DR analysis is used to carry out a numerical parameter study of the full-range response of uniformly loaded stocky steel annular plates. The study is primarily concerned with the effects of the edge support conditions in controlling the response and dimensionless results for deflections, etc. are presented for a total of nine inner and outer edge condition combinations.     

Non-linear bending of annular sector plates using a matrix method

This paper deals with the large-deflection analysis of isotropic annular sector plates clamped at the edges. The plate is subjected to a uniformly distributed lateral load. The non-linear governing equations are converted into a set of algebraic equations using integral equations associated with beam theory. These equations are solved by the modified Newton-Raphson method. The results are compared to those available for a clamped square plate by taking suitable values of sector angle and inner radius of the annular sector plate. For parametric study, the sector angle and radii ratio are varied. The results are presented in the form of graphs.     

Optimal orientation of orthotropic materials for plates designed against buckling

The paper deals with the optimization of anisotropic plates loaded by in-plane forces and designed against buckling. The internal structure of the plate is seen as a twodimensional locally orthotropic solid and the orientations of the axes of orthotropy are taken as the design variables. The problem of optimization consists in determining the best orientation of the axes of orthotropy from the condition of the maximum behaviour of the critical buckling loads. General properties of the problem, such as multiplicity of the critical parameters and splitting of multiple eigenvalues, are studied. Optimization algorithms are developed to improve the anisotropic properties of the plate. The results of numerically finding the optimal orientation of orthotropic properties are compared with conventional layouts for square and rectangular plates loaded by normal in-plane forces.     

Nonlinear axisymmetric static analysis of orthotropic thin circular plates with elastically restrained edge

This study deals with the nonlinear axisymmetric static analysis of elastic orthotropic thin circular plates with elastically restrained edges for rotation as well as in-plane displacement. Von-Karman equations have been employed and spatial discretisation has been done by using the method of orthogonal point collocation. Numerical results have been presented for orthotropic plates with hinged edge, simply supported edge, movable clamped edge and immovable clamped edge. Results are given for the whole range of elastic edge restraints from the movable simply supported edge to the immovable clamped edge. The effect of a prescribed inplane force or an inplane displacement at the edge, on the static response under a uniformly distributed transverse load has been investigated. Several new results are presented. The present results are in good agreement with the available results.     

Computer generated elasto-plastic design data for pressure loaded circular plates

Two finite-difference computer programs for the non-axisymmetric elasto-plastic large deflection analysis of circular plates are described. One of the programs uses a constitutive model based on the Ilyushin full-section yield criterion, whereas the other uses the von Mises layered yield criterion. The programs have been used to model the full-range response including multi-stage unloading, of uniformly loaded circular plates with various boundary conditions for a range of slendernesses. From the computed results design charts of first yield over-pressure versus maximum and residual centre deflection have been compiled. A numerical example of the use of these charts in the design of an end closure to a cylindrical pressure vessel is presented. Finally, sample charts for ‘pre-dished’ plates are presented as an illustration of the further potential of these programs for design data generation.     

Geometrically nonlinear analysis of a Reissner type plate by the boundary element method

The boundary element method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates taking into account the effects of transverse shear deformation. This paper presents the general equations for finite deformation of a Reissner type plate, and also gives an integral formulation of the Von Kármán type nonlinear governing equations which involve coupling between in-plane and out-of-plane deformation. The boundary and the domain of the plate are discretized to solve nonlinear plate bending problems. All unknown variables are at the boundary. An iterative procedure is applied to achieve linearization of the nonlinear equations. Some numerical results of the computation are compared with the analytical solutions and other numerical techniques, and good agreement is obtained.     

A parametric study of the axisymmetric full-range response of thickness-tapered circular steel plates

The incremental equations governing the full-range axisymmetric flexural response of thickness-tapered circular plates and their numerical solution via a finite-difference implementation of the Dynamic Relaxation (DR) technique are briefly outlined. A computer implementation of the DR analysis is used to carry out a parametric study of the response of thickness-tapered plates subjected to uniform normal pressure loading. Dimensionless results (deflections etc.) spanning the practical range of linear thickness tapers are presented for simply supported and clamped, slender and stocky plates.     

Nonlinear axisymmetric vibration of orthotropic thin circular plates with elastically restrained edges

This paper deals with the large-amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates of varying thicknesses with edge elastically restrained against rotation and in-plane displacement. Geometric nonlinearity due to moderately large deflections is included. Linear, parabolic and cubic variations of thickness are considered. Harmonic vibrations are assumed and time is eliminated from the von Kármán-type governing equations by the Kantorovich averaging method. The orthogonal point collocation method is used for spatial discretisation. Results are presented for the linear frequency of first axisymmetric mode and for the amplitude-period response. The effect of taper ratio, orthotropic parameter and rotational and in-plane stiffness of the support on the nonlinear vibration behaviour is investigated.     

Effect of shear deformation on the yielding of circular plates

Shear Deformation Plate Theory (SDPT) is used to analyse the first yield response of axisymmetric circular plates. Expressions for the first yield pressure and the associated plate centre deflection of uniformly loaded, simply supported and clamped plates are presented. Corresponding Classical Thin Plate Theory (CPT) expressions are also deduced. It is demonstrated that the SDPT and CPT expressions are related via correction factors which are functions of the plate thickness-ratio β, and Poisson's ratio. Graphs of the correction factors plotted against β are presented. They illustrate that inherent shear flexibility causes a slight reduction in the first yield pressure compared with the CPT value when the plate edges are simply supported and a slight increase when they are clamped. Plate centre deflections are, however, increased significantly for both support conditions.     

An incremental galerkin method for plates and stiffened plates

In order to perform a detailed analysis of large deflection behavior of a rectangular plate or stiffened plate, an efficient semi-analytical method is developed. First, incremental forms of the governing differential equations of plates and stiffened plates with initial deflection are derived. These equations are linearized and may be easily solved. Secondly, these equations are solved for each load increment by the Galerkin method with a special consideration of simply supported boundaries.A procedure of equilibrium correction at intermediate load steps is presented such that good accuracy of the solution may be maintained with larger load steps.This method is successfully applied to plates with initial deflection subjected to in-plane as well as out-of-plane loads to obtain the whole histories of the behavior of these plates. Application of this method to stiffened plates with initial deflection is also presented. Comparisons of results obtained by this method with those obtained by other methods are made and the validity of the method is demonstrated.This incremental version of the Galerkin method is found to be extremely advantageous in certain types of plate and stiffened plate problems. These types are identified and the efficiency of the method is demonstrated.     

A numerical scheme for integrating the rate plasticity equations with an “a priori” error control

An alternative scheme for the integration of incremental plasticity constitutive equations is presented and discussed. The method is based on a linear inequality theory approach and exploits the idea of the “a posteriori” local linearization of the yield surface. Results of both the flow theory of plasticity and the deformation theory can be obtained with a predetermined maximum violation of both the yield condition and the flow rule. Numerical results are shown, and compared to analytical solutions as well as to the results of other methods for the simple case of the von Mises yield function.     

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.     

On choice of optimal anisotropy of composite plates against buckling, with special attention to bending-twisting coupling

The lay-up optimization problems of composite plates against buckling are studied. The plate has symmetric lay-up and is loaded by an in-plane loading. The Classical Lamination Plate Theory is used. The necessary optimality conditions for the plate orthotropy/anisotropy optimization problems are considered, with special attention to bending-twisting coupling. The purpose of the optimization is to maximize the (lowest) buckling eigen value. The varied lay-up/layer orientation angles are considered both as smooth functions of the location coordinates and as having the same values at every point. It is shown that the twisting moment (calculated in the principal curvature axes) plays an important role in the conditions. Two example problems for a thin composite plate, loaded by shear, are considered. The first one corresponds to the case of one shear loading direction. The details of some known numerical solutions are studied. The obtained numerical results are in agreement with the theoretical results. The maximal lowest eigen values are 2-folded ones. A schematic model for treatment of the known optimal numerical solution (60° unidirectional lay-up for a long plate loaded by shear) is proposed. The second example problem corresponds to the case of shear loading acting in two opposite directions. The ways of equalizing the lowest buckling values corresponding to both loading directions are proposed. Numerical results for various aspect ratio and plate thickness values are presented. The potential of weight saving is demonstrated.     

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigen mode of the panel.In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths.The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.     

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.     

Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method

A general finite strip method of analysis is presented for the geometrically nonlinear analysis of laterally loaded, rectangular, isotropic plates. The analysis is based on the use of Mindlin plate theory and therefore includes the effects of transverse shear deformation. The nonlinearity is introduced via the strain-displacement equations and correspondingly the analysis pertains to problems involving moderate displacements but small rotations. The principle of minimum potential energy is used in the development of the strip and the complete plate stiffness equations and the latter equations are solved using the Newton-Raphson method. In numerical applications a particular type of finite strip is used in which all five reference quantities (three displacements and two rotations) are represented by cubic polynomial interpolation across the strip whilst the ends of the strip are simply supported for bending/shearing behaviour and immovable for membrane behaviour. These applications are concerned with uniformly loaded plates of both thin and moderately-thick geometry and detailed presentation is given of both displacement- and force-type quantities.     

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.     

Static and dynamic analysis of stiffened plates

In this paper a method for the static and dynamic analysis of eccentrically stiffened annular sector plates is presented. The plate is clamped on all the edges. The integral equation technique is adopted for the solution. In the static analysis the deflection and stresses at centre and the stresses at the edges are obtained and they are presented graphically. The results are compared for particular cases with those of other investigators who have used different analytical methods. The natural frequencies of stiffened clamped plates are also obtained for plates with different sector angles.