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.     

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.     

Buckling of multi-annular plates

Buckling of multi-annular plates is considered. The plate is loaded by axisymmetric radial in-plane forces, either at the outer edge or at one of the circumferential joints or a combination of the two. The plate is simply supported or clamped at the outer boundary or at one of the common joints. The various annular sections are fully connected, and they differ either in geometry or in material properties. Each annular section is homogeneous. A power series solution is used because of its applicability to various boundary conditions. A computer code is employed for the solution. Numerical examples are given for a two-part plate. The procedure is general and can be employed in various problems, such as buckling of annular plates resting on an elastic foundation, vibration analysis of annular plates, etc.     

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.     

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.     

On the stability of plates under combined compression and in-plane bending

The conventional stability analysis of plates under combined compression and in-plane bending is based on the assumption that the plate is free to move laterally and, hence, the restraints imposed by the attached elements against this motion are ignored. The paper explores the influence of these restraints on the plate under this type of loading. The unloaded edges are assumed to be partially restrained against in-plane translation while remaining straight and the distributions of the resulting forces acting on the plate are shown. The stability analysis is done numerically using the Galerkin method and various strategies that economize the numerical implementation are presented. The results are obtained showing the variation of the buckling load, from free edge translation to fully restrained, for simply supported and clamped unloaded edges for various plate aspect ratios and stress gradient coefficients. An apparent decrease in the buckling load is observed due to these destabilizing forces acting in the plate and changes in the buckling mode are observed by increasing the intensity of the lateral restraint. A comparison is made between the buckling loads predicted from various formulae in stability standards based on free edge translation and the values derived from the present investigation. A difference of about 34% in the predicted buckling load and different buckling load were found.     

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.     

Optimization of coupled thermal-structural problems of laminated plates with lamination parameters

The behaviour of a laminated plate with given boundary temperatures and displacement constraints is optimized and the optimization problem is expressed in terms of lamination parameters. Because the thermal conductivity and structural properties of a laminate depend on the lamination parameters of the laminate, the analysis of the plate consists of solving a coupled-field problem. The strain energy, or certain displacements of the laminated plate due to given boundary temperatures and displacement boundary conditions, is optimized with respect to in-plane lamination parameters, and also buckling of the plate is considered. The buckling factors for thermal loading are expressed as a function of four in-plane and four bending lamination parameters, and the smallest factor is maximized with respect to these parameters. In addition to these thermal problems, the natural frequencies of the laminated plate are studied. Since transverse shear deformations are taken into account,the natural frequencies can be expressed as functions of two in-plane and four bending lamination parameters, with respect to which the lowest natural frequency of the plate is maximized. The lay-up for the laminate, corresponding to four optimal in-plane or bending lamination parameters, consists of three layers at most and can be determined using explicit equations. Explicit equations are derived for creating a lay-up having optimal bending lamination parameters. Received May 12, 1999     

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented.     

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.     

Elasto-plastic shear buckling of square plates with circular holes

With in-plane stresses calculated by finite element analysis, critical loads are obtained by the Rayleigh-Ritz method for a square plate subjected to uniform edge shear stress and containing centrally located circular holes. Elastic and elasto-plastic buckling is examined for clamped and simply supported plates, and results are compared with previous analyses and experiments for various sized holes. The range of hole sizes considered is extended to include larger holes than previously examined, and for small holes, the results suggest that the critical stress is higher than previously thought. For elasto-plastic buckling, critical shear stresses are given for the full range of appropriate slenderness. Experimental results for the cases of simply supported plates support the analytical results, whereas verification for clamped plates remains inconclusive on account of limited reliable test data.     

Stability analysis of anisotropic laminated composite plates by finite strip method

The finite strip method has been applied to the stability analysis of rectangular shear-deformable composite laminates. However, for the plates with two opposite simply supported sides, the existing analysis was restricted to the symmetrical cross-ply laminates under compression loading.In the present study, by selecting proper displacement functions and including the coupling between different series terms, the finite strip method is extended to the stability analysis of any anisotropic laminated plates under arbitrary in-plane loading. Furthermore, a number of numerical results are presented to show the effects of thickness, fibre orientation and stacking sequence on the buckling loads.     

Layup optimization against buckling of shear panels

The object of the study was to optimize the shear buckling load of laminated composite plates. The laminates lacked coupling between bending and extension (B _ij=0) but had otherwise arbitrary selection of the ply angle variation through the thickness. The plates were rectangular and either simply supported or clamped on all edges. For orthotropic plates, it was seen that there is only one parameter necessary for finding the optimal design for different materials and plate aspect ratios. This parameter can be interpreted as the layup angle in a (+/–) orthotropic laminate. When bendingtwisting coupling is present, the buckling strength depends on the direction of the applied load. A laminate with non-zero bending-twisting coupling stiffnesses can be described with four lamination parameters. The allowable region of these parameters was investigated, and an optimization of the buckling load within this region was performed. It was seen that even this is a one parameter problem. This parameter can be interpreted as the layup anlge in an off-axis unidirectional laminate ().Notations A _ij in-plane stiffnesses of anisotropic plates, Tsai and Hahn (1980) - B _ij coupling stiffnesses of anisotropic plates - D _ij bending stiffnesses of anisotropic plates - D _ij ^* normalized bending stiffnesses - a, b, h length, width and thickness of the plate - x, y in-plane coordinates - z through-the-thickness coordinate - z ^* normalized through-the-thickness coordinate - w (x, y) out-of-plane deformation - N _xy shear buckling load - W ₁ ^* toW ₄ ^* lamination parameters - U ₁ toU ₅ linear combinations of the on-axis moduli - (z) layup angle - f _k functional of(z)     

Post-buckling behavior of thin steel plates using computational models

Thin plates loaded in plane will buckle at very small loads, and due to unavoidable out-of-plane imperfections, the theoretical buckling load cannot be observed experimentally. If the plate is adequately supported along its boundaries, it will be able to carry a much higher load than the theoretical buckling load.Computational models can be used to study the post buckling behaviour of thin plate structures up to failure. Failure of such structures is usually due to large out-of-plane deflections, yielding, and rupture. Therefore, the computational model should include the effects of geometric and material nonlinearities. In this paper, the nonlinear finite element analysis program NONSAP and ANSR-III were modified and used in the analysis. Since these programs did not include the suitable elements and material properties to conduct the subject study, new elements and new material properties were added to the programs. In particular, a thin shell element was added and the solution routines were modified to improve its accuracy and efficiency.The modified programs were used on a Super Computer to calculate the post buckling strength of stiffened and unstiffened plates subjected to uniaxial compression, and plates subjected to in plane bending or shear. Crippling of plates subjected to in-plane or eccentric edge compressive loads was also examined. The results from the computational models were compared with test results and reasonable agreements were obtained. A computational model was developed for a multi-story thin steel plate shear wall subjected to cyclic loading and the results from the model were compared with experimental results, and again agreement was achieved.     

Aeroelastic tailoring considering uncertainties in material properties

The robustness of aeroelastic design optimization with respect to uncertainties in material and structural properties is studied both numerically and experimentally. The model consists of thin orthotropic composite wings virtually without fuselage. Three different configurations with consistent geometry but varying orientation of the main stiffness axis of the material are investigated. The onset of aeroelastic instability, flutter, is predicted using finite element analysis and the doublet-lattice method for the unsteady aerodynamic forces. The numerical results are experimentally verified in a low-speed wind tunnel. The optimization problem is stated as to increase the critical air speed, above that of the bare wing by massbalancing. It is seen that the design goals are not met in the experiments due to uncertainties in the structural performance of the wings. The uncertainty in structural performance is quantified through numerous dynamic material tests. Once accounting for the uncertainties through a suggested reformulation of the optimization problem, the design goals are met also in practice. The investigation indicates that robust and reliable aeroelastic design optimization is achievable, but careful formulation of the optimization problem is essential.     

Stress wave calculations in composite plates using the fast Fourier transform

The protection of composite turbine fan blades against impact forces has prompted the study of dynamic stresses in composites due to transient loads. The mathematical model treats the laminated plate as an equivalent anisotropic material. The use of Mindlin's approximate theory of crystal plates results is five two-dimensional stress waves. Three of the waves are flexural and two involve in-plane extenisonal strains. The initial value problem due to a transient distributed transverse force on the plate is solved using Laplace and Fourier transforms. A fast computer program for inverting the two-dimensional Fourier transform is used. Stress contours for various stresses and times after application of load are obtained for a graphite fiber-epoxy matrix composite plate. Results indicate that the points of maximum stress travel along the fiber directions.     

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.     

A systematic topology optimization approach for optimal stiffener design

A systematic topology optimization based approach is proposed to design the optimal stiffener of three-dimensional shell/plate structures for static and eigenvalue problems. Optimal stiffener design involves the determination of the best location and orientation. In this paper, the stiffener location problem is solved by a microstructure-based design domain method and the orientation problem is modelled as an optimization orientation problem of equivalent orthotropic materials, which is solved by a newly developed energy-based method. Examples are presented to demonstrate the application of the proposed approach.     

Natural frequency and buckling optimization of laminated hybrid composite plates using genetic algorithm and simulated annealing

In this study, genetic algorithm and simulated annealing are used to maximize natural frequency and buckling loads of simply supported hybrid composite plates. The aim of the study is to use two different techniques of optimization on the frequency and buckling optimization of composite plates, and compare the techniques for their effectiveness. The composite plate is made of carbon/epoxy and glass/epoxy hybrid plies, and assumed to be symmetric and balanced. The effect of hybridization is investigated using both techniques. The buckling problem has many maxima in the vicinity of local maxima. The best configurations are identified for different plate aspect ratios. The performance of both techniques is compared in terms of number of function evaluations as well as the capability of finding best configurations.     

Minimum cost design of a cellular plate loaded by uniaxial compression

Cellular plates are constructed from two base plates and an orthogonal grid of stiffeners welded between them. Halved rolled I-section stiffeners are used for fabrication aspects. The torsional stiffness of cells makes the plate very stiff. In the case of uniaxial compression the buckling constraint is formulated on the basis of the classic critical stress derived from the Huber’s equation for orthotropic plates. The cost function contains the cost of material, assembly and welding and is formulated according to the fabrication sequence. The unknown variables are the base plate thicknesses, height of stiffeners and numbers of stiffeners in both directions. The cellular plate is lighter and cheaper than the plate stiffened on one side. The Particle Swarm Optimization and the IOSO techniques are used to find the optimum. PSO contains crazy bird and dynamic inertia reduction criteria, IOSO is based on a response surface technology.