Asymptotic analysis of perforated plates and membranes. Part 2: Static and dynamic problems for large holes

Static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi-scale asymptotic homogenization method. The problems of bending and vibration of perforated plates are considered. Using the asymptotic homogenization method the original boundary-value problems are reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. In the present paper the perforated plates with large holes are considered, and the singular perturbation method is used to solve the pertinent unit cell problems. Matching of limiting solutions for small and large holes using the two-point Padé approximants is also accomplished, and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained.     

Static instability analysis for travelling membranes and plates interacting with axially moving ideal fluid

The out-of-plane instability of a moving plate, travelling between two rollers with constant velocity, is studied, taking into account the mutual interaction between the buckled plate and the surrounding, axially flowing ideal fluid. Transverse displacement of the buckled plate (assumed cylindrical) is described by an integro-differential equation that includes the centrifugal force, the aerodynamic reaction of the external medium, the vertical projection of membrane tension, and the bending force. The aerodynamic reaction is found analytically as a functional of the displacement. To find the critical divergence velocity of the moving plate and its corresponding buckling mode, an eigenvalue problem and variational principle are derived. Plate divergence, both within a vacuum and when submerged in an external medium, is investigated with the application of analytical and numerical techniques.     

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15–19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.     

Static fracture testing of PMMA plates having flawed fastener holes

Results of static fracture tests on PMMA plates with part-elliptical cracks at fastener holes are presented. Experimental configurations include three crack locations with respect to both open and loaded holes. During static testing, slow growth prior to specimen separation allows for calculation of the maximum stress-intensity factor and identification of the location on the crack border at which it occurs. The testing techniques developed produce a simple, economical means of experimentally validating theoretical analyses of crack problems.     

Birefringent-coating analysis of laterally loaded perforated plates

The purpose of this investigation was to obtain experimental stress and deflection data for thick, circular, simply supported plates, containing circular transverse perforations in square motif, under uniform lateral loading. The stress-concentration factor and the deflection-multiplier factor, the ratio of the maximum principal stress and the maximum deflection of the perforated plate to that of the solid-plate specimen, respectively, were obtained for each perforated specimen. These factors can be conveniently used for the design of tube sheets, perforated heads, or other similar structural components.     

Static analysis of point-supported super-elliptical plates

This paper reports the static analysis of point-supported super-elliptical plates of uniform thickness subjected to a uniformly distributed lateral load. The plate perimeter was defined by a super-elliptic function with a power, corresponding to shapes ranging from an ellipse to a rectangle. The analysis was based on the Kirchhoff–Love plate theory and the computations were carried out by the Ritz method. Lagrange multipliers were used to satisfy the boundary conditions. Isotropic and homogeneous plates with 20 different shapes were examined for two distinct aspect ratios. Convergence studies were performed for the central deflection and the central bending moments. The results were checked against those of a corner-supported square plate and good agreement was obtained.     

Asymptotic analysis of laminated plates and shallow shells

It was noted long ago [1] that the material strength theory develops both by improving computational methods and by widening the physical foundations. In the present paper, we develop a computational technique based on asymptotic methods, first of all, on the homogenization method [2, 3]. A modification of the homogenization method for plates periodic in the horizontal projection was proposed in [4], where the bending of a homogeneous plate with periodically repeating inhomogeneities on its surface was studied. A more detailed asymptotic analysis of elastic plates periodic in the horizontal projection can be found, e.g., in [5, 6]. In [6], three asymptotic approximations were considered, local problems on the periodicity cell were obtained for them, and the solvability of these problems was proved. In [7], it was shown that the techniques developed for plates periodic in the horizontal projection can also be used for laminated plates. In [7], this was illustrated by an example of asymptotic analysis of an isotropic plate symmetric with respect to the midplane.     

Elasto-plastic buckling analysis for perforated steel plates subject to uniform compression

This study investigates the elasto-plastic buckling behaviour of simply supported square and rectangular thin steel plates having elliptic cut-outs by means of finite element method. Plates with simply supported in the out-of-plane direction are applied uniform compression in long-edge direction. A50 steel was used in the analysis and the focus was on the effect of plate aspect ratio, elliptical hole size, elliptical hole angle, elliptical hole location and slenderness ratio on buckling behaviour. It was found in the study that as the plate slenderness ratio increases, the critical buckling stress decreases for all the perforated plates.     

Limit analysis of multi-layered plates. Part II: Shear effects

In the first part of this work [Dallot, J., Sab, K., 2007. Limit analysis of multi-layered plates. Part I: the homogenized Love-Kirchhoff model. J. Mech. Phys. Solids, in press, doi:10.1016/j.jmps.2007.05.005], the limit analysis of a multi-layered plastic plate submitted to out-of-plane loads was studied. The authors have shown that a homogeneous equivalent Love-Kirchhoff plate can be substituted for the heterogeneous multi-layered plate, as the slenderness (length-to-thickness) ratio goes to infinity. In fact, the out-of-plane shear stresses are shown to become asymptotically negligible when compared to in-plane stresses, as the slenderness ratio goes to infinity. Actually, failure of thick multi-layered structures often occurs by shearing in the core layers and sliding at the interfaces between the layers. Both shearing and sliding are caused by the out-of-plane shear stresses. The purpose of the present paper is to build an enhanced Multi-particular Model for Multi-layered Material (M4) taking into account shear stress effects. In this model, each layer is seen as a Reissner-Mindlin plate interacting with its neighboring layers through interfaces. The proposed model is asymptotically consistent with the homogenized Love-Kirchhoff model described in the first part of the work, as the slenderness ratio goes to infinity. Kinematic and static methods for the determination of the limit load of a thick multi-layered plate which is submitted to out-of-plane distributed forces are described. The special case of multi-layered plates under cylindrical bending conditions is studied. These conditions lead to simplifications which often allow for the analytical resolution of the Love-Kirchhoff and the M4 limit analysis problems. The benefit of the proposed M4 model is demonstrated on an example. A comparison between the heterogeneous 3D model, the Love-Kirchhoff model and the M4 model is performed on a three-layer sandwich plate under cylindrical bending conditions. Finite element calculations are used to solve the 3D problem, while both the Love-Kirchhoff and the M4 problems are analytically solved. It is shown that, when the contrast between the core and the skins strengths is high, the Love-Kirchhoff model fails to capture the plastic collapse modes that cause the ruin of the sandwich plate. These modes are well captured by the M4 model which predicts limit loads that are very consistent with the limit loads predicted by the heterogeneous 3D model (the relative error is found to be smaller than 1%).     

THE MATHEMATICAL MODELS AND GENERALIZED VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS FOR PERFORATED THIN PLATES

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A new approach for solving mixed boundary value problems along holes in orthotropic plates

A new analytical method for solving mixed boundary value problems along holes in composite plates is presented. This addresses problems, where a part of the hole boundary is stress-free, and the other part is subjected to displacement or/and load boundary conditions. The present approach simplifies and speeds up the numerical calculations for the implementation of the boundary conditions by deriving stress functions, which automatically satisfy the stress-free boundary condition over a part of the hole boundary. Only the boundary conditions on the loaded part of the hole need, therefore, to be enforced, typically by a numerical technique such as collocation. Application of this method to orthotropic laminates with a pin-loaded hole problem is discussed.     

A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

A two-dimensional solution is presented for bending analysis of simply supported functionally graded ceramic–metal sandwich plates. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity and Poisson’s ratio of the faces are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. We derive field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results of the sinusoidal, third-order, first-order and classical theories are presented to show the effect of material distribution on the deflections and stresses.     

Variational principles of perforated thin plates and finite element method for buckling and post-buckling analysis

On the basis of the general theory of perforated thin plates under large deflections^{[1, 2]}, variational principles with deflectionw and stress functionF as variables are stated in detail. Based on these principles, finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated, owing to the multiple connexity of the region. Project supported by National Natural Science Foundation of China.     

Finite element analysis of post-buckling dynamics in plates. Part II: A non-stationary analysis

With the secondary bifurcation and the local post-secondary buckling behavior being analyzed in Part I, Part II of this study consists of developing an adaptive non-stationary load sweeping algorithm to investigate post-buckling dynamics and mode jumping phenomena of generally (mechanically and thermally) loaded thin plates in a global context. The non-stationary sweeping procedure has the merits of adapting large load steps to capture static characteristics of stable equilibrium paths both before and after mode jumping and reduce automatically the step size to ensure a dynamic transition between the two stable branches. Thus, it is computationally effective. Furthermore, by adopting the non-stationary sweeping scheme, this procedure can avoid spurious convergence of the transient response to an unstable equilibrium.Corresponding to different post-secondary bifurcation forms, which are determined using asymptotical finite element analysis developed in Part I, subsequent buckling patterns of various complexity occurring after mode jumping are obtained using the method developed in this article. Qualitative changes in post-buckled patterns are observed after the occurrence of the secondary bifurcation or the mode jumping. Free vibration analysis using the tangent stiffness matrix obtained from the converged static or dynamic solutions shows a vibration modal shifting phenomena occurs during the process of the load sweep. The spurious convergence phenomenon caused by the application of the traditional hybrid static–dynamic method is found and explained.     

Limit analysis of multi-layered plates. Part I: The homogenized Love-Kirchhoff model

The purpose of this paper is to determine , the overall homogenized Love-Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate, and to study the relationship between the 3D and the homogenized Love-Kirchhoff plate limit analysis problems. In the Love-Kirchhoff model, the generalized stresses are the in-plane (membrane) and the out-of-plane (flexural) stress field resultants. The homogenization method proposed by Bourgeois [1997. Modélisation numérique des panneaux structuraux légers. Ph.D. Thesis, University Aix-Marseille] and Sab [2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. C. R. Méc. 331, 641-646] for in-plane periodic rigid perfectly plastic plates is justified using the asymptotic expansion method. For laminated plates, an explicit parametric representation of the yield surface is given thanks to the π-function (the plastic dissipation power density function) that describes the local strength domain at each point of the plate. This representation also provides a localization method for the determination of the 3D stress components corresponding to every generalized stress belonging to . For a laminated plate described with a yield function of the form , where σ^u is a positive even function of the out-of-plane coordinate x₃ and is a convex function of the local stress σ, two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials ( in the skins and in the core) is studied. It is found that, for small enough contrast ratios (), the normalized strength domain is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticité. Eyrolles, Paris].     

Dynamical models for plates and membranes. An asymptotic approach

We examine the asymptotic expansion of a time-dependent displacement field defined over a three-dimensional elastic body whose shape corresponds to a thin plate. We show that under simple assumptions it is possible to derive from the principles of virtual work two known plate equations and three membrane models. Our results modify the displacement-stress method used by P.G. Ciarlet to derive von-Kárman plate equations. The modified algorithm allows one to employ techniques of algebraic geometry which simplify the computational aspects of the analysis.     

Statistical analysis of fluctuations of froth pressure on perforated plates without downcomers

Statistical properties of fluctuations of pressure drop across the gas–liquid dispersions on the tray were analyzed using a 215 mm dia. three plates column with different trays. Hydrodynamic regimes of the froth were characterized in terms of the statistical properties evaluated. It was proved that the on-line estimation of the flow behavior of the froth was possible on the basis of the pattern recognition of the power spectral density function of the pressure fluctuations. The time-domain stochastic models for fluctuations of the pressure drop were identified so as to be available for its forecast and simulation.