Exact solutions of the problem of flexural and cross-section-shear buckling modes and free oscillations of the rectangular orthotropic plate with loose edges

The article regards linearized problems about elastic stability of an orthotropic rectangular plate with loose edges under the influence of running forces of the invariable directions which cause in a plate either a single-sided and double-end compression, or pure shear. For stating the problem we use the known equations of the specified theory of plates of Timoshenko type considering the cross-section shears. On the basis of double trigonometrical base-load functions such analytical solutions of the specified problems are presented which fulfill all static boundary conditions. Depending on the structure of these solutions in order to conform to the equations of perturbed equilibrium of a plate the corresponding equations of Bubnov method are made, proceeding from which the bifurcation values of operating forces are defined and buckling modes corresponding to them are ultimately determined. On the basis of the stated method some analytical solutions are also discovered for the problem of small free oscillations of a plate with loose edges.