PSI - Issue 18

Yaroslav Dubyk et al. / Procedia Structural Integrity 18 (2019) 630–638 Yaroslav Dubyk et al. / Structural Integrity Procedia 00 (2019) 000–000

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Fig. 4. Pressure+Bending (a) Outer diameter; (b) Inner diameter.

Fig. 2-4 shows the distribution of axial and circular stresses (dimensionless) along a slanting trajectory in the mitred bend, which are reduced to pR / h values. The stresses were removed from the outer and inner diameters, for each case of loads: internal pressure, bending moment and their combined action. The character of the distribution of curves is very similar to each other and the results, apparently, have good convergence. The small difference in data at the beginning and end of the trajectory is due to the stress concentration in the numerical model. 4. Conclusions The paper presents the possibility of combined use of the theory of shells short and long solutions, where all variables are the sum of the results of these solutions, and the need for their combined use arises from the satisfaction of boundary conditions. For convenient use of these solutions, we obtained an explicit equations for all the main variables: radial displacement w , angle x  , lateral force x Q and moment x M (main variables of the short solution), as well as circumferential displacement v , axial displacement u , lateral force L and axial force x N (main variables long solution). The dependences obtained are used to analyze the stress-strain state of the pipeline with angular misalignment. It is shown that in some simple cases of loads, for example, actions on a pipeline with angular misalignment of internal pressure only, we can use only short solution. For a more complex case, for example, the action of a bending moment, for satisfaction the boundary conditions, it becomes necessary to use both short and long solutions. The obtained analytical solution agrees well with the numerical and experimental data. The results of this numerical-analytical study will be useful for development of the FEM models, when it is important to know the limits of straight pipe sections of a mitered bend, which are influenced by the boundary conditions of a finite element model. As well as to estimate the stress concentration from the presence of angular misalignment of welds, which can be used in the regulatory documents of the oil and gas industry. Appendix A. Equations Explicit equations for short and long solutions are written for convenience, and possible use in other applications. A.1. Complete set of the equation for short solution The resulting equations for the main variables of short solution ( , , , x x x w M Q  ) , and additional variables written in the explicit form:   1 1 2 2 ( ) ( ) cos( ) w C F x C F x n    (A.1)