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Postbuckling of a Uniformly Compressed Simply Supported Plate with Free In-Plane Translating Edges
Myntiuk V.B.
Q2
The postbuckling of a Kirchhoff isotropic simply supported plate is considered in detail. The in-plane displacements on the edges of the plate are not constrained. The solution is obtained using the principle of the total potential energy stationarity. The expression for energy is written in the three versions: in terms of the Biot strains, the Cauchy-Green strains, and the strains corresponding to Füppl-von Kármán plate theory. Some approximate solution is constructed by the classical Ritz method. The basis functions are taken in the form of Legendre polynomials and their linear combinations. The obtained diagram of equilibrium states is rather similar to the classical diagrams of compressed shells. We show the failure of Föppl-von Kármán theory under large deflections. Using the Biot strains and the Cauchy-Green strains leads to the discrepancy between the results of at most 5 %. We demonstrate the high accuracy and convergence of the approximate solution.
Biot Stress and Strain in Thin-Plate Theory for Large Deformations
Myntiuk V.B.
Q2
We propose a theory of nonlinear deformation of a plate on the basis of an energetically conjugate pair of the Biot stress tensors and the right stretch tensor. When the dimensionality of the problem is reduced from three to two, the classical Kirchhoff conjectures are used, the linear part is retained in the expansion of the right stretch tensor with respect to a degenerate coordinate, and no additional simplifications are assumed. Connection is obtained between the asymmetric and symmetric components of the Biot tensor; the equivalence is demonstrated of the virtual work principle with the equilibrium equations, the natural boundary conditions, and additional conditions for the dependence of asymmetric stress moment resultants on symmetric moments.
Postbuckling Analysis of Flexible Elastic Frames
Khalilov S.A., Myntiuk V.B.
Q2
completely geometrically nonlinear beam model based on the hypothesis of plane sections and expressed in terms of engineering strains and apparent stresses is applied to the structural analysis of frames. The numerical results are obtained by the Raley–Ritz method with a representation of solutions as a sum of analytical basis functions which were previously proposed by the authors. The convergence of approximate solutions is investigated. High degree of accuracy is demonstrated for both determination of the solution components and the fulfillment of equilibrium equations. It is shown that the limit values of external loads can substantially differ from those predicted by the Euler buckling analysis, which may lead to catastrophic consequences in designing thin-walled structures.
The semi-chromatic number of a graph
Vizing V.G.
Q2
We introduce the notion of the semi-chromatic number of a graph with a nonempty number of edges. Then we prove that the difference between the semi-chromatic number and the half of the chromatic number is at most 1.
Multicoloring the incidentors of a weighted undirected multigraph
Vizing V.G.
Q2
Under consideration are the undirected multigraphs with weighted edges. In multicoloring incidentors, to each incidentor there is assigned a multicolor; i.e., an interval of colors whose length is equal to the weight of the incidentor. A multicoloring is proper if the multicolors of every two adjacent or mated incidentors are disjoint. We give some lower and upper estimates for the minimal number of colors necessary for a proper multicoloring of all incidentors of a graph.
Multicriteria graph problems with the MAXMIN criterion
Vizing V.G.
Q2
Under study is the r-criteria problems for the r-weighted graphs (r ≥ 2). Certain kinds of subgraphs are called admissible. Solving some problem means choosing a Pareto optimal admissible subgraph from the complete set of alternatives (CSA). The main result of this paper is as follows: Suppose that a criterion denoted by MAXMIN requires maximization of the minimal first edges’ weight of the admissible subgraph and there is an effective procedure constructing the CSA for a (r − 1)-criteria problem without this MAXMIN criterion. Then the CSA for the initial r-criteria problem is created effectively.
Coloring the vertices of a graph with majority restrictions on colors
Vizing V.G.
Q2
The problem of coloring the vertices of a graph is under consideration assuming that the majority (maximal admissible) color is specified for each vertex. A criterion given for the chromaticity of this prescription generalizes the Vitaver theorem. An estimate of the greatest value of a majority color can be required for the chromaticity of the prescription. Some analogs of the Nordhaus-Gaddum theorem are proved concerning the relations among the chromatic characteristics of a graph and its complement.
Bounds for the incidentor chromatic number of a weighted undirected multigraph
Vizing V.G., Pyatkin A.V.
Q2
A proper incidentor coloring of an undirected weighted multigraph is called admissible if the absolute value of the difference between the colors of the incidentors of each edge is at least the weight of this edge. The minimum number of colors necessary for an admissible incidentor coloring is called the incidentor chromatic number of the multigraph. The problem of finding this number is studied in the paper. The NP-hardness of this problem is proved for Δ colors. Some upper and lower bounds are found for the incidentor chromatic number.
On bounds for the incidentor chromatic number of a directed weighted multigraph
Vizing V.G.
Q2
An incidentor coloring of a directed weighted multigraph is called admissible if: (a) the incidentors adjoining the same vertex are colored by different colors; (b) the difference between the colors of the final and initial incidentors of each arc is at least the weight of this arc. The minimum number of colors necessary for an admissible coloring of all incidentors of a multigraph G is bounded above and below. The upper and lower bounds differ by ┌Δ/2┐ where Δ is the degree of G.
