Postbuckling behavior of rectangular moderately thick plates and sandwich plates

ＰＯＳＴＢＵＣＫＬＩＮＧＢＥＨＡＶｉＯＲＯＦＲＥＣＴＡＮＧＵＬＡＲＭＯＤＥＲＡＴＥＬＹＴＨＩＣＫＰＬＡＴＥＳＡＮＤＳＡＮＤＷＩＣＨＰＬＡＴＥＳＣｈｅｎｇＺｈｅｎ－ｑｉａｎｇ（成振强）ＷａｎｇＸｉｕ－ｘｉ（王秀喜）ＨｕａｎｇＭａｏ－ｇｕａｎｇ（黄茂光）...     

Junction of thin plates

The elasticity problem of a thin plate with an edge is considered using asymptotic methods. The small parameter ε describes the relative thickness of the plate. In the case when the elasticity coefficients are everywhere of the same order of magnitude, the asymptotic behaviour of the plate is such that the angle of the edge remains constant under the deformation (the junction is called `rigid'). We also consider a junction mode of a narrow filet (the order of its width is O(ε)) of a `soft' elastic material, the elasticity coefficients being O(ε) with respect to those of the plates. In this case (called `elastic junction'), the asymptotic modelling contains an energy bilinear form associated with the filet which involves the variation of the angle and the sliding along the junction.     

Nonlinear three-dimension analysis for axially symmetrical circular plates and multilayered plates

Analytic nonlinear three-dimension solutions are presented for axially symmetrical homogeneous isotropic circular plates and multilayered plates with rigidly clamped boundary conditions and under transverse load.The geometric nonlinearily from a moderately large deflection is considered.A developmental perturbation method is used to solve the complicated nonlinear three-dimension differential equations of equilibrium.The basic idea of this perturbation method is using the two-dimension solutions as a basic form of the corresponding three-dimension solutions,and then processing the perturbation procedure to obtain the three-dimension perturbation solutions.The nonlinear three-dimension results in analytic expressions and in numerical forms for ordinary plates and multilayered plates are presented.All of the plate stresses are shown in figures.The results show that this perturbation method used to analyse nonlinear three-dimension problems of plates is effective.     

Vibration of rectangular orthotropic plates

Summary An approximate analytical procedure has been given to solve the problem of a vibrating rectangular orthotropic plate, with various combinations of simply supported and clamped boundary conditions. Numerical results have been given for the case of a clamped square plate.Nomenclature 2a, 2b sides of the rectangular plate - h plate thickness - E _x , E _y , E, G elastic constants of te orthotropic material - D _x E _x h ³/12 - D _y E _y h ³/12 - H _xy Eh ³/12+Gh ³/6 D _x , D _y and H _xy are rigidity constants of the orthotropic plate - mass of the plate per unit area - Poisson's ratio - W deflection of the plate - p circular frequency - b/a ratio - X _m , Y _n characteristic functions of the vibrating beam problem - p ² a ² b ²/H _xy the frequency parameter.     

Isopachic contouring of opaque plates

Contour maps of change of thickness of opaque plates subjected to external loads are obtained using holographic interferometry in conjunction with the moiré effect. A simple holographic-interferometry arrangement is used first to obtain contour maps of the out-of-plane displacements of the two sides of the object. Carrier patterns of equal magnitude but opposite sings are added to these contours. Superposition of the reconstructed holograms of the two sides produces a pattern of additive-moiré fringes, which are contours of change of thickness. Effects of midplane warpage of the loaded specimen are cancelled. Sensitivity is /2 per fringe order, contrast of the isopachic-fringe pattern is excellent, and the process is compatible with a mechanical-testing-machine environment.Paper was presented at 1983 SESA Spring Meeting held in Cleveland, OH on May 15–20.     

Anticlastic behavior of flat plates

The purpose of this paper is to show that the readings from strain gages can be used effectively to compute small transverse deflections in a rectangular plate and, further, show that the theory developed by Lamb for the rectangular-plate problem agrees with experiment. A numerical procedure is developed, based on the trapezoidal rule, which determines the transverse deflections from the readings of strain gages mounted to the top and bottom surface of a rectangular plate subjected to large longitudinal curvatures. It is shown using the strain-gage technique that experiment agrees with Lamb's theory forb ² /Rt ratios up to 50.     

Vibration of continuous circular plates

The general development of the theory given here considers the material to be orthotropic and continuous over (n ? 1) elastic or rigid supports. The effect of rotatory inertia and in-plane loads are also included while formulating the equations of motion. Double and triple series solutions are given for orthotropic continuous plates. By matching the continuity conditions at the intermediate supports and satisfying the boundary conditions at the outer edge, the frequency determinant is obtained. For the purpose of numerical computations, an isotropic plate continuous over an intermediate-rigid or elastic-support and free and with no in-plane loads at the outer edge is considered. It is found that the influence of Poisson's ratio on the frequency parameter is significant only for the first symmetric or asymmetric modes. The rotatory inertia influences the frequency parameter when the radius to thickness ratio is less than 80, viz, when the plate is thick. Moreover, the elasticity of the support influences considerably the free vibration of plates.     

Stability of annular orthotropic plates

It is assumed that the orthotropy of the plate material is rectangular or polar, and a uniformly distributed, external compressive or tensile load is applied to the internal boundary of the plate. Stability is analyzed by the Ritz method with the use of Alfutov-Balabukh and Bryan energy criteria. Diagrams of the critical external load and buckling modes as a function of the plate dimensions are given. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 166–170, March–April, 2000.     

Vibration of thick auxetic plates

This short communication investigates the effect of negative Poisson's ratio on the natural frequency of thick plates of arbitrary shape. Using the Mindlin plate theory, it was generally found that as the plate's Poisson's ratio becomes more negative, the Mindlin-to-Kirchhoff natural frequency ratio increases with decreasing rate. Upon comparing (a) the use of the simplified constant shear correction factor and the more accurate variable shear correction factor, (b) with and without rotary inertia, it was found that all the four combinations stated in (a) and (b) do not give appreciable difference when the Poisson's ratio of the plate is positive. However in the case of plates with negative Poisson's ratio, results reveal that when at least one of the simplifying assumptions is used, the Mindlin-to-Kirchhoff natural frequency ratio is overestimated, and that the overestimation further increases when both the simplifying assumptions are used. When benchmarked against Reddy plate theory, the use of variable shear correction factor has almost the same effect as the inclusion of rotary inertia. Hence the use of either variable shear correction factor or rotary inertia is proposed for modeling the vibrational frequencies of conventional and auxetic isotropic plates.     

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].One of the methods for solving the equations with rapidly oscillating coefficients is the asymptotic method of small geometric parameter. The standard procedure of this method usually includes preparatory stages. At the first stage, as a rule, a rectangular periodicity cell is distinguished to be a typical element. At the second stage, the scale of global coordinates is changed so that the rectangular structure periodicity cells became square cells of size l × l. The third stage consists in passing to the dimensionless global coordinates relative to the plate characteristic dimension L. As a result, the dependence between the new local variables and the new scaled dimensionless variables is such that the factor 1/α, where α=l/L ? 1 is a small geometric parameter, appears in differentiating any function of the local coordinate with respect to the global coordinate. After this, the solution of the problem in new coordinates is sought as an asymptotic expansion in the small geometric parameter [1], [4–10].We note that, in the small geometric parameter method, the asymptotic series simultaneously have the form of expansions in the gradients of functions depending only on the global coordinates. This averaging procedure can be applied to linear and nonlinear boundary value problems for differential equations with variable periodic coefficients for which the periodicity cell can be affinely transformed into the periodicity cube. In the case of an arbitrary dependence of the coefficients on the coordinates (including periodic dependence), another averaging technique can be used in linear problems. This technique is based on the possibility of the integral representation of the solution of the original problem for the linear equation with variables coefficients in terms of the solution of the same problem for an equation of the same type but with constant coefficients [11–13]. The integral representation implies that the solution of the original problem can be represented in the form of the series in the gradients of the solution of the problem for the equation with constant coefficients [13].The aim of the present paper is to develop methods for calculating effective characteristics of corrugated plates. To this end, we first write out the equilibrium equations for a flexible anisotropic plate, which is inhomogeneous in the thickness direction and in the horizontal projection, with an initial deflection. We write these equations in matrix form, which allows one to significantly reduce the length of the expressions and to simplify further calculations. After this, we average the initial matrix equations with variable coefficients. The averaging procedure implies the statement of problems such that, after solving them, we can calculate the desired effective characteristics. By way of example, we consider the case of a corrugated plate made of a homogeneous isotropic material whose corrugations are hexagonal in the horizontal projection. In this case, we obtain approximate expressions for the components of the effective tensors of flexural rigidity and longitudinal compliance and expressions for the effective plate thickness.