Search for an equal-strength contour inside a viscoelastic rectangle
DOI:
https://doi.org/10.32347/2411-4049.2023.3.154-162Keywords:
Kelvin-Voigt model, Kolosov-Muskhelishvili formulas, Riemann-Hilbert problems, Volterra equationAbstract
Irregularity of geometric and physical parameters in thin-walled structures leads to significant concentrations of stresses and creates dangerous zones for the spread of cracks or plastic deformations. Under the influence of a tense state, they are similar to gills. Stress concentration zones in areas of irregularity have a significant impact on the tensile strength and durability of thin-walled structures. Traditional analytical and numerical methods known at this time are less effective in investigating the stress-strain condition of corrugated thin-walled structures. It is, therefore, necessary to develop new effective methods for solving the tasks of this class. Currently, for engineering calculations, there is virtually no comparison of simple and convenient formulas for determining the critical compressive load taking into account the peculiarities of the design. The scientific novelty of the paper is that to achieve the set goal, it will be used for the first time in the general theory developed for the calculation of buildings and structures, known as the "Theory of elasticity in ordinary differential equations." The paper will show that the accuracy of this new theory is adequate to the classical elongation theory and at the same time dramatically simplifies the solution of any problem in the calculation of tiles, which is achieved by converting them to conventional differential equations. The general methods of compiling differential equations, the methods of its simplification, for the calculation of membranes with cross-sectional incisions, and the calculation of plates under conditions of nonlinear deformation are discussed. Methods for solving differential equations with variable and momentum coefficients are specified. An algorithm and a program for the analysis of the stress-strain state of spatial structures and their elements are developed. The practical value of the paper lies in the possibility of using developed methods and programs for the design and construction of buildings, as well as for the stability tasks of slabs with holes, and panels used in construction as typical assembly elements. The given mathematical algorithm and program for specific tasks, which are distinguished by simplicity, can be used by design and research organizations in the calculation and design of plates and membranes.
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