Abstract
The study examined the construction of the fundamental solution for the equations of statics {1,2} – approximation for transversely isotropic plates under bending with the action of concentrated force. Equations {1,2} -approximation were obtained by the decomposition method in the thickness coordinate using the Legendre polynomials. These equations take into account all the components of the stress tensor, including the transverse shear and normal stresses. Since the classical theory of Kirchhoff-Love doesn’t take account of these stresses, the study on the basis of refined theories of stress-strain state of transversely isotropic plates under the action of concentrated force effects is an important scientific and technical problem. The fundamental solution of obtained equations results using a two-dimensional Fourier integral transform and inverse treatment techniques, built with the help of a special G-function. This method allows reducing the system of resolving differential equations for statics of flat plates and shells to a system of algebraic equations. After that, the inverse Fourier transform restores the fundamental solution. The work was carried out numerical studies that demonstrate patterns of behavior of components of the stress-strain state, depending on the elastic constants of transversely isotropic material. The results play a decisive role in the study of boundary value problems in the mechanics of thin-walled elements of constructions, including under the influence of concentrated and local diverse forces.
Highlights
In modern technology, engineering structures with thin-walled structural elements are widely used
The use of such materials makes it necessary to build a refined theory of plates and shells, taking into account the phenomena associated with the transverse shear and compression
Generalized theory of plates and shells in the variant of {1,2}-approximation is used in the article for derivation of static equations for transversely isotropic plates under bending with action of concentrated force
Summary
In modern technology, engineering structures with thin-walled structural elements are widely used. In modern engineering new composite materials are intensively used for creating protective coatings on the friction surfaces and for the manufacture of various items of equipment [2] The use of such materials makes it necessary to build a refined theory of plates and shells, taking into account the phenomena associated with the transverse shear and compression. The selected method is the most appropriate for this task, because it is not based on any hypotheses, and uses the method of Vekua decomposition of unknown functions in Fourier series using Legendre polynomials [3] This approach allows considering the thin plate, but the plate of medium. Generalized theory of plates and shells in the variant of {1,2}-approximation is used in the article for derivation of static equations for transversely isotropic plates under bending with action of concentrated force. Bending problem for transversely isotropic plates using the equations of statics {1,2}approximation is solved below
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