Numerical analysis of the stress‐strain state of inhomogeneous plates and shallow shells under local loading based on the 3D elasticity theory

Abstract To solve complex problems of the theory of elasticity of plates and shallow shells subject to local dynamic loading, two versions of the semi‐analytical finite element method are proposed. In the first variant, a model for calculating plates and shallow shells when taking into account shear and compression is used to construct dynamic equilibrium equations. Its distinctive feature is the assignment of the sought functions on the outer surfaces of the layers, which allows dividing the layers into sublayers with a corresponding increase in the accuracy of the calculation results. All known models for the calculation of plates and shallow shells taking into account shear and compression have one reduction surface. For such models, it is difficult to divide layers into sublayers. The original distinctive feature of the second variant is finding the distribution of the sought functions by the thickness of the structure by means of an exact solution of the corresponding system of differential equations. The authors did not find similar variants of the semi‐analytical finite element method in publications. Both x ‐axis options use finite element approximation. Trigonometric functions are used to approximate the y ‐axis. The necessity of developing two approaches is attributed to the fact that they have inaccuracies of approximation and arithmetic calculations, which are especially seen in the calculation of forced vibrations at the frequencies close to resonance. The coincidence of the calculation results obtained by the two proposed variants of the semi‐analytical finite element method can serve as a confirmation of their reliability. An analysis of the behavior of plates and flat shells under forced vibrations at frequencies close to resonance was carried out. The conducted studies have shown that for the class of problems under consideration, using the option with a polynomial approximation of the required functions over the thickness of the structure when considering a layer within one sublayer can lead to a large error.