Abstract
The paper studies the stress-strain state of flat elastic isotropic thin-walled shell structures in the framework of the S. P. Timoshenko shear model with pivotally supported edges. The stress-strain state of shell structures is described by a system of five second-order nonlinear partial differential equations under given static boundary conditions with respect to generalized displacements. The system of equations under study is linear in terms of tangential displacements, rotation angles, and nonlinear in terms of normal displacement. To find a solution to the system that satisfies the given static boundary conditions, integral representations for generalized displacements containing arbitrary holomorphic functions are used. Finding holomorphic functions is one of the main and difficult points in the proposed study. The integral representations constructed in this way allow us to reduce the original problem to a single nonlinear operator equation with respect to the deflection, the solvability of which is established using the principle of compressed maps.
Highlights
There are a large number of works devoted to the strength of thin-walled shell structures, taking into account the geometric and/or physical nonlinearity [1,2,3,4,5,6,7,8,9]
Questions related to the qualitative study of the stress-strain state within the framework of more general models of the theory of thin-walled shell structures that do not rely on the Kirchhoff-Love hypotheses were included in the well-known list of unsolved problems of the mathematical theory of shells by I.I
There are a number of works [18,19,20,21,22,23,24] devoted to the study of the stress-strain state in the framework of the Timoshenko shear model
Summary
There are a large number of works devoted to the strength of thin-walled shell structures, taking into account the geometric and/or physical nonlinearity [1,2,3,4,5,6,7,8,9]. The problem of convergence of the numerical solution to the exact (real) solution of the problem always comes to the fore The solution to this problem, as is known, is based on a rigorous mathematical study of the stress-strain state of thin-walled shell structures. At present, this problem is sufficiently fully studied in the framework of the simplest Kirchhoff-Love model [10,11,12,13,14,15,16,17]. The conformal mapping method is used to study a nonlinear problem for arbitrary flat shells under different boundary conditions
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