BUCKLING A COMPRESSED ELASTIC RECTANGULAR PLATE WITH FREE EDGES

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.

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

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.