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The stability of a rectangular plate under two different compressive loading scenarios applied to the free edges of the plate has been investigated. It is assumed that under the applied load the edges have translational and rotational degrees of freedom. In the first case, a conservative external force, that is assumed to have a fixed direction with respect to the deforming body, is applied while in the second case the load is directed to be tangential to the deflected middle surface, the follower force. It is found that for a sufficiently narrow plate the buckling is localized in the neighborhood of free edges as the conservative load is applied. A similar phenomenon is known for the case when a plate is uniformly compressed along hinged edges. In the case of a follower force, the static formulation of the problem leads to the conclusion that localized buckling does not exist. However, depending on the parameters of the problem, the usual (global) buckling is predicted. [ABSTRACT FROM AUTHOR]

A two-dimensional linearized problem on plane buckling modes (BMs) of a rectangular plate with free edges, made of an elastic orthotropic material, underbiaxial tension-compression is considered. With the use of double trigonometric basis functions, displacement functions exactly satisfying all static boundary condition on plate edges are constructed. It is shown that the exact analytic solutions found describe only the pure shear BMs, and if the normal stress in one direction is assumed equal to zero, an analog of the solution given by the kinematic Timoshenko model can be obtained. Upon performing the limit passage to the zero harmonic in the displacement functions of one of the directions, the solution to the problem of biaxial compression can be obtained by equating the Poisson ratio to zero; in the case of uniaxial compression, this solution exactly agrees with that following from the classical Bernoulli-Euler model. [ABSTRACT FROM AUTHOR]

The linear problem on plane modes of free oscillations of a rectangular orthotropic plate with free unloaded edges is considered. A procedure for constructing displacement functions exactly satisfying the boundary conditions, with the use of double-trigonometric basis functions, is offered. Exact and approximated analytical solutions to the problem formulated are found, which presumably describe all plane modes of free oscillations of the plate in the class of the functions indicated. It is established that, in the use of variational principles, the variations of required functions must be considered not only arbitrary, but also mutually independent. Therefore, the solutions constructed give physically reliable results for the frequencies and modes of free oscillations only if the problem is stated in the form of Bubnov variation equations, which depend on the structure of displacement functions. It is found that the exact analytical solutions of the problem correspond to oscillation modes without shear strains. It is shown that it is possible to select such solutions from them which correspond to trigonometric functions with a zero harmonic in one direction. These solutions describe only flexural oscillation modes of the plate, and the results obtained are equivalent to those given by the classical Kirchhoff model known in the theory of rods, plates, and shells. [ABSTRACT FROM AUTHOR]