The key factor determining the strength, reliability, service life and fail-safe operation of the main pipeline is its stress-strain state. The purpose of this article is to develop a mathematical framework and methodology for calculating the stress-strain state of a pipeline section laid in complex geotechnical conditions, taking into account all planned and altitude changes and impacts at various points of operation, as well as during repair and after its completion. The mathematical framework is based on differential equations reflecting the equilibrium state of the pipeline, taking into account the features of the sections (configuration, size, initial stress state, acting forces, temperature conditions, interaction with soil, supports, and pipe layers). The equilibrium equations are drawn up in a curvilinear coordinate system – the same one that is used for in-pipe diagnostics. According to the results of the solution, all stress components are determined at each point both along the length of the pipeline and along the circumference of any section. At the same time, transverse and longitudinal forces, bending moments, shearing forces, pipeline displacements relative to the ground and soil response to displacements are determined. As an example, a solution is given using the developed mathematical framework. During the course of calculation, the places where the lower form of the pipe does not touch the ground and the places where the support reaction becomes higher than a predetermined limit are determined. A comparative analysis was accomplished, and the optimal method for section repair has been selected.
Mathematical framework for analysis of internal energy dynamics and spectral distribution in compressible turbulent flows
