Studies on the static and dynamic deformation of isotropic and anisotropic elastic shell-like bodies of complex shape performed using classical and refined problem statements are reviewed. To solve two-dimensional boundary-value problems and eigenvalue problems, use is made of a nontraditional discrete-continuum approach based on the spline-approximation of the unknown functions of partial differential equations with variable coefficients. This enables reducing the original problem to a system of one-dimensional problems solved with the discrete-orthogonalization method. An analysis is made of numerical results on the distribution of stress and displacement fields and dynamic characteristics depending on the loading and boundary conditions, geometrical and mechanical parameters of elastic bodies. Emphasis is placed on the accuracy of the results Keywords: shell structures, static and dynamic problems, variable parameters, models, discrete-continuum methods
Introduction.Many members of modern engineering structures have the form of intricately shaped shells fixed in various ways and subjected to various distributed and local loads. Shell elements are widely used to meet the requirements imposed by the severe operating conditions of machines, aircraft, transportation vehicles, industrial and civil facilities. The complication of the configuration of shell elements necessitates developing a theory and methods for solving static and dynamic problems for shells made of anisotropic inhomogeneous materials.A typical feature in the development of the theory of plates and shells is the relationship between the setting up of a mathematical model describing a given class of problems and the development of a method for solving them. For example, Kirchhoff's theory of thin plates [101] and the Kirchhoff-Love theory of thin shells [97] tend not only to provide a realistic description of the deformation of plates and shells, but also to make such models so simple as to solve a number of problems with available computational capability. An example of such relationship is the Donnell-Mushtari-Vlasov theory of shells [5,28,36]. Its basic equations were simplified so as to find solutions to some classes of problems over wide ranges of their characteristics, which is demonstrated by the names given to this theory: technical theory of shells, theory of shallow shells, theory of shells with highly variable stress state [2,8,9,16,20,23,24,29,37]. This relationship is even stronger nowadays when computers are widely used to solve problems in the theory of shells and mathematical models for certain classes of shells are set up so as to provide for all aspects and possibilities associated with the solution of problems [63].The solution of two-dimensional boundary-value problems of the statics and dynamics of plates, shells, and solids described by partial differential equations with variable coefficients involves severe computational difficulties. To solve them, use is sometimes made of approaches based on separation of variables and redu...