Influence of Geometrical Parameters on the Critical Load in Three-Dimensional Stability Problems for Rectangular Plates and Beams

Kokhanenko, Yurii Vasil'evich; Zelenskii, V. S.

doi:10.1023/b:inam.0000008216.30794.f6

Cited by 8 publications

(4 citation statements)

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“…Discrete problems are written using a base scheme, which corresponds to the class of plane elastic problems for piecewise-homogeneous media [5,[9][10][11][12]. The base scheme is a difference scheme constructed on an arbitrary qth cell, q Q = 1, .…”

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Edge effects in a sandwich plate: a plane problem

Kokhanenko

2004

Int Appl Mech

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Edge effects in a rectangular sandwich plate with isotropic components are studied. The mathematical model is represented by the homogeneous equations of linear elasticity, which is indicative of an approximate approach in edge-effect theory. The initial equations are reduced to inhomogeneous ones and an exact problem is formulated. Approximate solutions are found by the mesh method. Discrete problems are based on the concept of base scheme. The mesh equations are written in an explicit form and then solved using a computation optimization procedure. As an example, edge-effect zones in a real composite are analyzed.Keywords: edge effect, exact approach, free strains, sandwich plate, base scheme Introduction. Edge effects in deformable media can be studied using either exact or approximate approach. The exact approach employs exact mathematical and mechanical models and an exact (quantitative) criterion of edge-effect zones [3]. Other approaches are approximate.For the problem under consideration, the exact approach is characterized by the inhomogeneous equations of linear elasticity (exact mathematical model), a piecewise-homogeneous medium model (exact mechanical model), and a quantitative (exact) criterion of edge-effect zones. The exact approach allows determining, with prescribed accuracy, the boundaries of stress-concentration zones and analyzing the geometry and stress state within these zones.Almost all approximate approaches in edge-effect theory employ homogeneous (eigen)solutions, which are used in the case of self-balanced loading [1,2,4,8]. These solutions do not allow analyzing the stress state within the edge-effect zones.The criterion of edge effect is decrease in the surface load at a point of interest by a given factor. Since quantitative and qualitative characteristics of loading are disregarded, this criterion is called approximate (qualitative). Eigensolutions allow us to estimate quantitatively (to admit or reject) Saint Venant's principle.The present paper analyzes edge effects in a sandwich plate under the conditions of uniaxial loading and plane strain. The governing equations describe a plane homogeneous problem of elasticity for piecewise-homogeneous media and imply, as mentioned above, an approximate approach in edge-effect theory.We will set up inhomogeneous equations of elasticity. To derive inhomogeneous equations and to formulate an exact problem for edge effects, we will use the concept of free strains. The inhomogeneous boundary-value problem will be solved approximately by the mesh method based on the concept of base scheme [5,[9][10][11]. Discrete problems will be formulated and methods for their solution selected. Since the stresses σ ij may exhibit singularity near the free edge [2], we will use a computation optimization procedure to improve the accuracy of solving the mesh equations. An example will be given.1. Problem Formulation. Consider a three-layer rectangular plate with isotropic layers. The face layers are identical, which results in the geometrical symmetry of the s...

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Edge effects in a sandwich plate: a plane problem

Kokhanenko

2004

Int Appl Mech

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“…To solve approximately the problems posed, we will use the finite-difference method, with variational difference discretization based on the concept of base scheme [3,[9][10][11][12]. To this end, the infinite region is replaced with a finite rectangular region whose dimensions are determined from a computational experiment.…”

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confidence: 99%

Two-dimensional stability problem for two interacting short fibers in a composite: In-line arrangement

2004

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A solution is found to a plane problem for a composite material reinforced with two in-line fibers and subjected to longitudinal compression. The problem formulation is based on the piecewise-homogeneous model and the three-dimensional theory of stability. The dependence of the critical strain and buckling mode on the distance between the fibers is studied for various mechanical and geometrical characteristics of the composite components.

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“…It is presented in works [7][8][9][10][11]; 4) the shell and ribs can be considered as solid deformable bodies, i.e., as a three-dimensional medium. This is a general, accurate and complex approach [12][13][14][15].…”

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Experimental justification of the analytical method for determining the upper and lower bounds of the critical loads in ribbed shells

Gavrilenko

Matsner

2006

Strength Mater

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The method developed for determining the upper and lower bounds of the critical load parameters in elastic ribbed shells is described. The critical parameters for cylindrical shells with three types of stiffening, namely, by cross ribs, by stringers and by rings only, are justified experimentally. It is shown that the lower bounds of the critical loads agree with the minimum experimental parameters much better than the upper bounds of the critical loads determined from the linear momentless theory.Keywords: experiment, analytical method, critical loads, upper and lower bounds, ribbed shells.Introduction. The analytical methods are more convenient in investigations than the numerical ones as they make it possible to carry out the analysis of the critical loads from analytical expressions.The present work involves the approach [1] that assumes the subcritical state of the stiffened shell to be momentless and uniform. A similar approach is presented in [2] wherein the procedure using equations of a mixed type is described, eight cases of buckling are analyzed, the minimum upper critical load is determined and its comparison with that obtained in experiments is made. It is shown in [2] that the minimum design parameters of the critical loads turned out to be much higher than the experimental ones, which is typical of the usual classical theory of stability of momentless shells at a uniform subcritical state.In contrast to the above-mentioned works, the critical loads corresponding already to 17 different cases of buckling are refined below. The upper critical loads [2, 3] are compared with those obtained by the method proposed by the present authors. The estimation of the lower bounds for the load-carrying capacity of stiffened shells is made and their comparison with the experimental results obtained both in Ukraine (Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine) and in the USA (Californian Technological Institute) and also in other scientific centers is made. Particular attention has been given to those shells that were calculated for stability taking into account the initial field of the shape imperfections. In this case, special-purpose equipment for their measurements and also the procedures for analyzing and describing the initial deflections along the whole surface of the shells under study were developed.In developing the methods for calculating the stability of thin-walled stiffened shells, the following approaches are used: 1) the structurally orthotropic scheme for a momentless shell when the stiffness characteristics of ribs are distributed uniformly along the sheath. The approach is used in [4][5][6];2) the theory of ideal elastic thin-walled ribbed shells at a momentless uniform subcritical state. Its assumptions and hypotheses coincide completely with usual assumptions of the classical theory of smooth shells. The difference consists merely in the taking into account the discreteness of rib location. The main results and techniques are described in [2,3];

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Influence of Geometrical Parameters on the Critical Load in Three-Dimensional Stability Problems for Rectangular Plates and Beams

Cited by 8 publications

References 6 publications

Edge effects in a sandwich plate: a plane problem

Edge effects in a sandwich plate: a plane problem

Two-dimensional stability problem for two interacting short fibers in a composite: In-line arrangement

Experimental justification of the analytical method for determining the upper and lower bounds of the critical loads in ribbed shells

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