Abstract. An initial stability of Kirchhoff plates by the Boundary Element Method (BEM) is presented in the paper. A plate is subjected by external in-plane normal and tangential conservative loadings acting in two perpendicular directions. The Betti's theorem is used to derive the boundary-domain integral equations. The direct version of the Boundary Element Method is presented with combination to simplified boundary conditions. The singular and non-singular approach of the boundary integrals derivation is used.

The main motivation of this work was to present a semi-analytical extension of the correspondence principle in stochastic dynamics. It is demonstrated for the stochastic structural free vibrations of Kirchhoff–Love elastic, isotropic and rectangular plates supported by viscoelastic generalized Maxwell dampers. The ambient temperature of the plate affects the dampers only and is included in a mathematical model using the frequency–temperature correspondence principle. The free vibration problem of the plate–viscoelastic damper system is solved using the continuation method and also the Finite Element Method (FEM). The stochastic approach begins with an initial deterministic sensitivity analysis to detect the most influential parameters and numerical FEM recovery of the polynomial representation for lower eigenfrequencies versus these parameters. A final symbolic integration leads to the first four basic probabilistic characteristics, all delivered as functions of the input uncertainties.

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The paper is concerned with free vibrations of geometrically non-linear elastic Timoshenko beams with immovable supports. The equations of motion are derived by applying the Hamilton principle. The approximate solutions are based on the negligence of longitudinal inertia forces but inclusion of longitudinal deformations. The Ritz method is used to determine non-linear modes and the associated non-linear natural frequencies depending on the vibration amplitude. The beam is discretized into linear elements with independent displacement fields. Consideration of the beams divided into the regular mesh enables one to express the equilibrium conditions for an arbitrary large number of elements in form of one difference equation. Owing to this, it is possible to obtain an analytical solution of the dynamic problem although it has been formulated by the finite element method. Some numerical results are given to show the effects of vibration amplitude, shear deformation, thickness ratio, rotary inertia, mass distribution and boundary conditions on the non-linear natural frequencies of discrete Timoshenko beams.