Open Access proceedings Journal of Physics: Conference series

Introduction

In the study of the seismic resistance of underground pipelines [1-7, 10-11] attention is focused mainly on modeling the interaction in the pipe-soil system. The stress-strain state of life-support systems is determined in terms of the coefficients of interaction of these structures with the surrounding soil, including, the coefficient k_x of uniform shear of a pipeline relative to the soil (coefficient of shear of the pipeline).

Analysis of the consequences of strong earthquakes shows that the seismic resistance of underground structures depends on the direction of the seismic wave. Since during earthquakes underground structures can be exposed to waves from arbitrary directions, a determination of the stress-strain state of underground pipelines in the presence of longitudinal, torsional, and transverse oscillations is relevant and serves to determine the possible seismic hazard [3-7].

In [4] a system of equations of motion of a linear underground pipeline subject to a seismic wave in the plane of the axis of the pipeline is constructed on the basis of the Hamilton-Ostrogradskii variational principle. Modern computing facilities make it possible to take account of numerous factors and solving more difficult problems for underground structures by means of mathematical models and schemes for longitudinal and transverse oscillations of a pipeline [4].

Pipeline life support systems consist of straight-line sections connected by joints and orthogonally and non-orthogonally coupled together. A seismic wave initiated during an earthquake affects such a system of pipelines at an arbitrary angle of attack in space. For an underground system of arbitrarily located pipelines with an arbitrary angle of attack of seismic effect in space, it is necessary to develop new computational mathematical models and software for determining the stress-strain state.

In this paper, we proposed an approach for determining the stress-strain state of a pipeline exposed to an arbitrarily directed plane seismic wave, the normal vector to the wave front with axis Ox makes an angle α, and β is the angle between the projection of this vector on Oyz plane and the pipeline axis Oy [1-3].

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   Methods
   

Consider an orthogonal piping system located on Oxy plane and interacting with surrounding soil. Consider an underground pipeline of a length 180 m and two П-shaped sections on Oxy plane with dimensions 2 m wide and 5 m long (Figure 1). Let the left and right ends of the pipeline be fixed to the ground, and the seismic wave is set as a harmonic function with incidence angles α=45^◦, β=30^◦. In general, the task is an unsteady-state spatial task for studying processes in underground pipelines under the action of seismic waves.

Figure-1. Section of an underground pipeline of complex orthogonal configuration

The system of differential equations for linear sections of underground pipelines, considering their viscoelastic interaction with soil, arbitrary direction of seismic action and corresponding boundary conditions at the ends (2) and initial conditions (3) has the form [2]

, (1)

, (2)

, at t=0, (3)

where М, А, В, С, D, F, K, L – are the sixth-order matrices, U – pipeline displacements, U₀ – given ground displacements during an earthquake in the form of seismic waves depending on time and coordinates.

As a numerical method for solving the equation of motion (1), taking into account (2) and (3), the finite element method (FEM) in spatial coordinates and the implicit finite difference method (MKR) in time to discretize the problem [9] of the pipeline under arbitrary seismic waves are used.

The stresses in the underground pipeline due to axial force N and the combined action of the axial force and the moment of force M under arbitrary action are calculated by the following formulas

, (4)

(5)

The mechanical and geometrical parameters of an underground pipeline and soil are taken as follows: E=2·10⁵ MPa; ρ=7.8·10³ kg/m³; l=180 m; a₀=0.008 m; u₀=a₀·sinω(t-x·cosα/C_p)·H(t-x·cosα/C_p); ω=2π/T; T=0.3 s; C_p=700 m/s; u_0х=u₀·cosα; u_0y=u₀·sinα·cosβ; u_0z=u₀·sinα·sinβ; in a straight-line section k_x=1.5·10⁴ kN/m³; k_y,z=3.9·10⁴ kN/m³; in a complex section k_x=0.5·10⁴ kN/m³; k_y,z=1.3·10⁴ kN/m³; μ_soil=0.2; μ_pipe=0.3; D_H=0.5 m; D_B=0.49 m. The maximum value of acceleration of a given wave is 3.50 m/s², which corresponds to a 9 point earthquake on the MSK-64 scale.

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