In engineering approaches to calculating the seismic resistance of underground structures, in particular pipelines, it is important to find experimentally the forces of interaction with the ground, as this frees us from the need to solve a difficult dynamic problem for the ground (elastic or having more complex mechanical properties). In this work inaccuracies are corrected and further development of the known from literature theory of quasistatic experiment for determining the coefficient of longitudinal interaction of soil and underground pipeline in seismic problems is given. It is shown that only the second approximation for the named coefficient in the expansion with a small parameter, equal to the ratio of the length of the pipe sample to the length of the seismic wave, takes into account the longitudinal deformation of the pipe; the first approximation corresponds to the experiment with an absolutely rigid pipe.

A
strict reduction of the seismic problem to the problem for relative pipeline
movements is presented. Conditions are found on the external (remote from the
pipe) boundaries of the soil, the fulfillment of which ensures the correctness
of determining the coefficient of longitudinal interaction for seismic problems
in experiments

Engineering methods for finding the average (averaged) velocity of propagation of longitudinal waves in pipelines with flexible joints are presented. By accurate analysis of the problem of oscillations of a one dimensional periodically inhomogeneous structure it is shown that the results of engineering approaches for rod velocity are the first or long-wave asymptotic approximation which valid when the period of external influence (the length of the seismic wave) significantly exceeds the size of the periodicity cell of the pipeline (the length of the pipe with a joint). Thus, it is established that when this condition is met, the problem of pipeline dynamics with joints is reduced to a much simpler problem of vibrations of a homogeneous pipeline, the velocity of wave propagation in which is equal to the found average value.
Numerical examples are given that demonstrate a significant (sometimes by an order of magnitude) decreasing of the rod velocity in the presence of flexible joints.

The existing engineering approaches in seismodynamics of long underground structures are presented: the theory of full contact without relative displacement at soil-structure interface, the approaches are related to modeling of the soil-structure interaction by simple deformable elements, and the approaches based on hypotheses about nature of the "laws" describing this interaction. Based on the exact statement of the problems, ranges of applicability of the engineering approaches are clarified. It is shown, that translation of the methods developed for above ground constructions to underground structures is not competent and may introduce errors.

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).