Abstract Based on the bending shells theory the pipeline equilibrium equations with a complex profile were derived, and variants of setting the boundary conditions were suggested. The detailed algorithm for deriving the equations was given, which allows building mathematical models for different classes of problems. The proposed approach allows us to investigate the stress-strain state of a wide class of curved pipes without imposing additional conditions on the lines of interface of domains with different geometry. The corresponding boundary problems were formulated, provides a list of newly emerging application tasks. The approaches to solving these problems were suggested.