Underground polymeric l-shaped pipeline vibrations under seismic effect

The simultaneous equations of longitudinal and transverse vibrations of an underground polymeric L-shaped pipeline under the arbitrary direction of seismic load were derived in the paper. A computational scheme of the problem was constructed using central finitedifference relations. The analysis of the results obtained on the simultaneous longitudinal and transverse vibrations of underground polymeric L-shaped pipelines under seismic loading was conducted. The stress-strain state of the L-shaped polymeric pipeline subjected to seismic effect was determined, and the axial forces and bending moments arising in curved pipelines during an earthquake were determined.


Introduction
The issues of increasing the seismic resistance of underground pipelines, mitigating seismic effect, and possible damage during earthquakes are of great importance worldwide. Improving the seismic resistance, reliability, and safety of pipeline transport is an urgent task. Pipelines laid in difficult geotechnical conditions experience such loads as the weight of the pipeline conveying a product, the weight of soil surrounding the pipeline, the response of soil to the pipeline strain, internal working pressure and temperature stresses, the loads related to the changes in physical and mechanical characteristics of the soil. To avoid pipeline failures and accidents, it is necessary to establish the influence of design features, the changes in operating conditions, and parameters on the strength and stability of the pipeline. To ensure the earthquake resistance of the above-ground and underground sections of the pipeline, it is necessary to study its stress-strain state (SSS), taking into account the naturalclimatic loads and the operating parameters and design features of the pipeline, the influence of various forms of bending. The grounds of the dynamic theory of earthquake resistance of complex systems of underground pipelines were developed in [1][2][3]. It was assumed that any structure is elongated, branching one both along the strike and in-depth, with complex rigid and flexible pipe joints in a complex node. In [4][5][6][7][8][9][10][11][12][13], the seismodynamics of the underground structures under the harmonic and real seismic impacts are numerically studied. *Corresponding author: nematilla81@mail.ru In [14], the principles of calculation and design of underground structures in seismic regions were considered. The stresses at the underground pipeline bend investigated experimentally. It was determined that when two pipelines are connected at a right angle in a node, its displacements are approximately equal to the displacements of the surrounding soil. As a result, axial forces and bending moments appear in the curved sections of the pipelines during an earthquake. The test results showed that the axial forces were initiated in the pipeline in the direction of forced vibrations, and the bending moments appear at the right angle junction of two pipelines.
When solving the problems of seismodynamics of underground pipelines, the main issue is the interaction modeling in the pipe-soil system. In this paper, we proposed a calculation method to determine the SSS of a polymeric L-shaped pipeline (figure 1), subjected to seismic effects, the front of which forms angle α with the axis Ox directed along the pipeline I axis. The results of theoretical studies allow estimating the stress-strain state of L-shaped polymeric pipelines under seismic load, directed arbitrary relative to the principal axes of the structure (figure 1) [15]. Recently, much attention has been paid to the calculation of pipelines of various configurations (L-, Т-, U-and V-shaped ones) [15].

Methods
To derive a system of differential equations of motion with boundary and initial conditions, the Hamilton -Ostrogradsky variational principle was used [16][17][18] where δТ, δП are the variations of kinetic and potential energy, δА is the variation of work of external forces, t is time.
Consider the forces and moments of two pipelines I and II connected in a node , , 12 12 . ( The stresses-strains relationship for underground polymeric pipelines are obtained from [19]    , as I→x for the first pipeline, as II →y for the second pipeline, the equations are written iteratively (see figure 1).
Considering relationships (3), the system of equations (1) in displacements has the form: The boundary conditions are: Here, for the pipe I, the boundary conditions are formed at the junction of pipelines I and II. The boundary conditions (5) and relationships are used, where the junction node of pipelines I and II is indicated by B.
The initial conditions for pipelines I and II (figure 1) have the form: Here I q 1 , I q 2 are the seismic forces acting on pipelines I and II (Fig. 1); they have the form (see monographs [14]): where μ soil is the Poisson's ratio of soil, k x is the coefficient of longitudinal interaction with soil, l is the length that corresponds to experimental study [12], u 0x and u 0y are the laws of ground motion. If we take into account relationships (8) Proceed to dimensionless displacements and coordinates In calculations, the weakly singular three-parameter Rzhanitsyn-Koltunov kernel [19] was used in expressions (5) and (9), The three-parameter kernel (11) has a weak Abel-type singularity. The kernels of this type have a weak singularity. To eliminate it, the transforms in the integrand were conducted, according to [20].
After introducing the vectors of external force displacements, we performed some transforms, then from the system of equations (9), we obtained a differential equation of motion, boundary, and initial conditions in a general vector form Boundary conditions in a vector dimensionless form for the pipe I are: (13) for pipe II are: where are the third-order matrices. Initial conditions are: To solve the boundary value problem (12), (13), and (14), we use the method of finite differences of the second order of accuracy.
The problem is solved based on the algorithm of computer implementation. Mechanical and geometrical parameters are selected as follows:

Results and Discussion
Numerical results are obtained for the displacements and force factors considering boundary conditions. The results are presented in graphs. When calculating the pipeline for earthquake resistance, the options were used for setting the ground motion during earthquakes in the form of a sinusoid (harmonic law).
The changes in longitudinal u (figure 2,a) and transverse v (figure 2,b) displacements of underground pipelines at time t at the point of their intersection (x=0, y=0) at the incidence angle α=30° of seismic load are given below.

Conclusions
Thus, the simultaneous equations of longitudinal and transverse vibrations of an underground polymeric pipeline of the L-shaped configuration were derived for an arbitrary direction of seismic load. A computational scheme of the problem was constructed using central finite-difference relations. The analysis of the results obtained on the simultaneous longitudinal and transverse vibrations of underground L-shaped polymeric pipelines under seismic loading was carried out. The SSS of the L-shaped polymeric pipeline subjected to seismic effect was determined, and the axial forces and bending moments arising in curved pipelines during an earthquake were determined. An analysis of the above problems shows