Analogy of a Linear Chain and Seismic Vibrations of Segmental and Viscoelastic Pipelines

Abstract

In the problem of seismic vibrations of a segmental pipeline with damping joints, deformed by the law of linear viscoelasticity, an original analogy with a linear chain of concentrated masses was put forward. The constructed discrete system generalizes the monatomic lattice model in the sense that the viscoelastic interaction between the masses of the chain is considered and, moreover, the forced (and not own) oscillations of such a system are investigated. By the transition from a discrete system to a continuous one, the integro-differential oscillation equation of a segmented pipeline with viscoelastic joints in an elastically resisting medium is obtained. This equation is a generalization of the well-known Klein–Gordon differential equation describing the “constrained” vibrations of an elastic rod or string in a medium with elastic resistance. In addition, the equation gives the problem of seismic vibrations of a continuous pipeline from a polymer (viscoelastic) material.

Joint stationary seismic vibrations of a viscoelastic pipeline and soil were studied in an exact formulation and maximum stresses in the pipeline were found by solving the resulting integro-differential equation. The same stresses were found using the “hard pinch” engineering approach, according to which displacements and deformations in the seismic wave and pipeline are the same. By analyzing the stresses found under the viscoelasticity law in the form of the Kelvin–Voigt model relationship, it is established that the generally accepted position that the engineering approach gives an upper estimate for the stresses in the pipeline is valid only in the subsonic case (when the seismic wave velocity is lower than the wave velocity in the pipeline) and is not valid in the supersonic mode, when the exact theory can lead to stresses exceeding those calculated on the basis of the engineering approach.