Solution of the External Pochhammer-Chree Problem and Bending Seismic Vibrations of the Pipeline in Infinite Elastic Continuu

Abstract

The possibility of partial splitting of the dynamic equations of the linear theory of elasticity for bulk expansion and the components of the rotation vector of medium particles in cylindrical coordinates in the general case of a nonstationary problem is proved. This result is a generalization of a similar fact established by A. Love with a simple harmonic dependence of the named functions on time and two spatial coordinates. An exact analytical solution of the problem on bending vibrations of an elastic space with a circular cylindrical cavity (external Pochhammer-Chree problem) is given. It is shown that the study published by K. Toki and Sh. Takada on this problem does not provide a solution to the posed problem. On the basis of the solution obtained for the external Pochhammer-Chree problem, bending vibrations of an underground pipeline caused by the action of a seismic wave are studied. The results obtained in this case provide, apparently, the first theoretical substantiation of the engineering theory of rigid jamming of the pipeline in the soil for the case of bending vibrations, widely used in regulatory documents on seismic construction.