scholarly journals BUCKLING A COMPRESSED ELASTIC RECTANGULAR PLATE WITH FREE EDGES

2020 ◽  
Vol 82 (2) ◽  
pp. 244-251
Author(s):  
I.M. Peshkhoev ◽  
B.V. Sobol
Keyword(s):  
Boundary Value Problem ◽  
Rectangular Plate ◽  
Compressive Load ◽  
Elastic Base ◽  
The Other ◽  
Internal Stresses ◽  
Schmidt Method ◽  
Linearized Problem ◽  
Elastic Rectangular Plate ◽  
Nonlinear Elastic

The influence of a small transverse load on the stability of an elastic rectangular plate compressed in one direction with free longitudinal edges, lying on a nonlinear elastic base, is studied. The plate contains internal stress sources in the form of continuously distributed edge dislocations and disclinations, or other sources, such as heat. The reaction of a nonlinear elastic base is taken into account in the form of a polynomial of the third degree of deflection. The compressive load is evenly distributed along the two opposite edges of the plate, which are freely pinched or pivotally supported. The other two edges of the plate are free. We consider a nonlinear boundary value problem for a modified system of nonlinear Karman equations that take into account internal stresses. By replacing variables, the problem is reduced to a sequence of two problems, of which the linear boundary value problem determines the function of stresses caused by the presence of internal sources, and the other, nonlinear, determines the deflection of the plate and the function of stresses caused by the compressive load. Using the Lyapunov-Schmidt method, the branching of solutions of a modified system of nonlinear Karman equations is investigated. The critical values of the load parameter are determined from a linearized problem based on a trivial solution. In this case, the variational method in combination with the finite-difference method is used to solve the linearized problem. The coefficients of the system of branching equations of the Lyapunov-Schmidt method are calculated numerically. The post-critical behavior of the plate is investigated and asymptotic formulas for new equilibria in the vicinity of critical loads are derived. For different parameter values of compressive loads and internal stresses the relations between the values of the parameters of the base, which preserve its load-bearing capacity in the vicinity of the classical critical load.

scholarly journals BUCKLING AND POSTBUCKLING BEHAVIOR COMPRESSED OF THE RECTANGULAR PLATE WITH INTERNAL STRESSES, LYING ON NON-LINEAR ELASTIC FOUNDATION

2019 ◽  
Vol 81 (2) ◽  
pp. 137-145
Author(s):  
I. M. Peshkhoev ◽  
B. V. Sobol
Keyword(s):  
Rectangular Plate ◽  
Elastic Foundation ◽  
Asymptotic Formulas ◽  
Internal Stresses ◽  
Postbuckling Behavior ◽  
Non Linear ◽  
Compressive Loads ◽  
Elastic Rectangular Plate ◽  
Non Linear Operator ◽  
Edge Dislocations

The problem of the effect of initial imperfections in the form of small transverse loads on the loss of stability and the post-critical behavior of a compressed elastic rectangular plate lying on a non-linearly elastic foundation is considered. The plate contains in a flat state continuously distributed edge dislocations and wedge disclinations or other sources of internal stresses. The research is conducted on the basis of a modified system of non-linear Karman equations for an elastic plate with dislocations and disclinations which additionally takes into account the reaction of the base in the form of a second or third degree polynomial in deflection. Two cases of boundary conditions are considered: free pinching and movable hinged support of the edges. The problem is reduced to solving a non-linear operator equation which is investigated by the Lyapunov-Schmidt method. The linearized equation is a multiparameter boundary value problem for eigenvalues which is solved by a finite-difference method. The coefficients of the system of ramification equations are calculated numerically. The post-buckling behavior of the plate is investigated and asymptotic formulas are derived for new equilibria in the neighborhood of critical loads. For different values of the parameters of compressive loads and the parameter of internal stresses, relations have been established between the values of the parameters of the base, at which its bearing capacity is preserved in the neighborhood of the classical value of the critical load.

scholarly journals Stress distribution in a rectangular plate having two opposing edges sheared in opposite directions

1923 ◽  
Vol 103 (723) ◽  
pp. 598-610 ◽  
Keyword(s):  
Shear Stress ◽  
Stress Distribution ◽  
Rectangular Plate ◽  
Royal Society ◽  
External Stress ◽  
The Other ◽  
Optical Methods ◽  
Centre Line ◽  
Distribution Of Stress

The type of deformation under investigation is indicated by fig. 1. A rectangular plate ABCD is deformed into the shape A'B'C'D'. The two opposing edges AB, CD are shifted horizontally without alteration of length into the position A'B', C'D', the other boundaries AD, BC being kept free from external stress. In a paper which appeared in the 'Proc. Royal Society', December 28, 1911, Prof. E. G. Coker investigated this same type of deformation using optical methods to determine the distribution of stress along the centre line OX. He found that if the plate was square the shear stress along OX was distributed in a munner which was approximately parabolic. As the ratio of AD to AB decreased the curve of distribution first of all became flat-topped, and for yet smaller ratios two distinct humps made their appearance.

Tandem Conveyor Queues

1989 ◽  
Vol 3 (4) ◽  
pp. 517-536
Author(s):  
F. Baccelli ◽  
E.G. Coffman ◽  
E.N. Gilbert
Keyword(s):  
Boundary Value Problem ◽  
Joint Distribution ◽  
Queueing System ◽  
The Other ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Special Cases ◽  
A Cell ◽  
Service Times ◽  
Input Position

This paper analyzes a queueing system in which a constant-speed conveyor brings new items for service and carries away served items. The conveyor is a sequence of cells each able to hold at most one item. At each integer time, a new cell appears at the queue's input position. This cell holds an item requiring service with probability a, holds a passerby requiring no service with probability b, and is empty with probability (1– a – b). Service times are integers synchronized with the arrival of cells at the input, and they are geometrically distributed with parameter μ. Items requiring service are placed in an unbounded queue to await service. Served items are put in a second unbounded queue to await replacement on the conveyor in cells at the input position. Two models are considered. In one, a served item can only be placed into a cell that was empty on arrival; in the other, the served item can be placed into a cell that was either empty or contained an item requiring service (in the latter case unloading and loading at the input position can take place in the same time unit). The stationary joint distribution of the numbers of items in the two queues is studied for both models. It is verified that, in general, this distribution does not have a product form. Explicit results are worked out for special cases, e.g., when b = 0, and when all service times are one time unit (μ = 1). It is shown how the analysis of the general problem can be reduced to the solution of a Riemann boundary-value problem.

Bifurcations near a multiple eigenvalue of the rectangular plate problem

1986 ◽  
Vol 103 (1-2) ◽  
pp. 35-62
Author(s):  
Zoltán Sadovský
Keyword(s):  
Aspect Ratio ◽  
Rectangular Plate ◽  
Small Perturbation ◽  
Perturbation Parameter ◽  
Multiple Eigenvalue ◽  
Bifurcation Problem ◽  
Operator Equations ◽  
Double Eigenvalue ◽  
Elastic Rectangular Plate ◽  
Degenerate Cases

SynopsisWe consider the bifurcation problem of the Föppl–Kármán equations for a thin elastic rectangular plate near a multiple eigenvalue allowing for a small perturbation parameter related to the aspect ratio of the plate. The first step in the study is to introduce equivalent operator equations in the energy spaces of the problem which explicitly contain the perturbation parameter. By dealing partially with a general formulation, we obtain the main results for the double eigenvalue and Z2 ⊓ Z2 symmetry of bifurcation equations. We are chiefly interested in the degenerate cases of bifurcation equations.

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