Beginning with the solutions of two-dimensional Kelvin and Mindlin problems about concentrated forces acting in elastic space and inside of elastic half-space, analytical dependences between forces and average displacements in the direction of the action of forces on a circle, centered at the points of application of forces, are constructed. Based on the obtained ratios, models of the interaction of the pipeline with the ground during its bending vibrations under the influence of seismic waves are constructed. This approach is similar to the one used earlier by D.D. Barkan to find the connection between the forces acting on the foundations of buildings and ground structures and the vertical or shear displacements of the structures under dynamic influences based on classical elastic solutions of Boussinesq and Cerutti.

The solution of the problem of bending seismic vibrations of an extended pipeline obtained by the described method provides, apparently, the first theoretical justification and assessment of the accuracy of the engineering theory of "complete pinching" of an underground pipeline in the ground for the case of its bending vibrations (this theory is accepted in calculations for seismic resistance in current regulatory documents).