ВИКОРИСТАННЯ СИМЕТРІЇ ДЛЯ РОЗВ’ЯЗУВАННЯ ЗАДАЧ ДЕФОРМУВАННЯ ПЛИТ НА ЖОРСТКІЙ ОСНОВІ

Abstract

The optimization of the design scheme of deformation of a rectangular multilayer plate with transversally isotropic layers resting on a rigid (non-deformable) foundation is proposed. The essence of optimization is to consider such a design diagram of the plate, in which the stress-strain state (SSS) of plate would be fully described by only one component, namely the unflexural component of symmetrical SSS relative to the middle surface of plate which is bilaterally symmetrically loaded. To do this, instead of the actual design of the multilayer plate, which is deformed without separation from the foundation, it is suggested to consider the design diagram of the plate, which is formed by supplementing it with a symmetric one about the contact surface of the foundation. In this case, the plate will be symmetrically loaded with respect to the middle surface of the plate, and the thickness of the plate will double. On the middle surface, the conditions of sliding contact of the upper and lower parts of the symmetrical plate, i.e. sliding contact of the plate with foundation are fulfilled. To model absolutely rigid contact at the border with the foundation, an additional thin layer of high rigidity ("non-deformable layer") is introduced into the supplemented plate. This does not change the essence of the calculation model of plate. The SSS of plate will be unflexural, which significantly simplifies its modeling. A two-dimensional model of deformation of multilayer rectangular plates on a rigid foundation with isotropic and transversally-isotropic layers is constructed in an elastic formulation for a unflexural SSS, with a high degree of iterative approximation, but three-dimensional by the nature display of the SSS. This model sufficiently takes into account transverse shear deformations and of transverse compression of the plate under transverse loading. The model is continuous, that is, the number of equations and the order of differentiation of the solving system of equations does not depend on the number of layers in the plate. This order of differentiation and the number of solving equations can depend only on the order of iterative approximation of the model. The derivation of the solving differential equations in the generalized forces and displacement functions is given, as well as the boundary conditions are obtained by the variational Lagrange method. The results of the analytical solution of the problem of deformation of homogeneous rectangular plate in sliding contact with a rigid foundation whith Navier-type boundary conditions under the action of a transverse sinusoidal load are given. A comparison of the calculation results with the exact three-dimensional solution ones was made. Keywords: multilayered plate, rigid foundation, transverse shear, transverse compression, continual model