An Asymptotic Method for Solving Three-Dimensional Boundary Value Problems of Statics and Dynamics of Thin Bodies

The equations of the three-dimensional problem of elasticity for thin bodies (bars, beams, plates, shells) in dimensionless coordinates are singularly perturbed by a small geometrical parameter. The general solution of such a system of equations is a combination of the solutions of an internal problem and a boundary-layer problem. The asymptotic orders of the stress tenser components and of the displacement vector in the second and mixed boundary value problems for thin bodies are established; the inapplicability of classical theory hypothesis for the solution of these problems is proved. In the case of a plane first boundary value problem for a rectangular strip a connection of the asymptotic solution with the Saint-Venant principle is established and its correctness is proved. Free and forced vibrations of beams, strips and possibly anisotropic and layered plates are considered by an asymptotic method. The connection of free-vibration frequency values with the propagation velocities of seismic shear and longitudinal waves is established. In a three-dimensional setting forced vibrations of two-layered, three-layered and multi-layered plates under the action of seismic and other dynamic loadings are considered and the resonance conditions are established. At theoretical justification for the expediency of using seismoisolators in an aseismic construction is given.