On the non-classic models of beams, plates and shells

For the problems of statics, of free vibrations, and of buckling of beams, plates and shells the Timoshenko–Reissner's model with shear is compared with the classic Kirchhoff–Love model and with the 3D theory of elasticity. By using some test examples the formal asymptotic character of 1D and 2D models is established and their field of application is found. The asymptotic expansions based on the small shell or plate thickness compared with the length of wave are used. The special attention is paid to the buckling or vibration modes localized near the free surface.