Dynamic Behavior of Underground Viscoelastic Pipelines Under Seismic Impact in the Form of Real Earthquake Records

MATERIAL METHOD

A rectilinear underground polymer pipeline interacting with the surrounding soil is considered under the action of seismic action of an arbitrary direction in the form of real earthquake records. To obtain differential equations of vibrations of underground polymer pipes, the Hamilton–Ostrogradsky variational principle is used [22]

, (1)
где δТ и δП – соответственно вариации кинетической и потенциальной энергий; δА – вариация работы внешних сил.
Based on the assumptions given in [22], the pipeline is modeled in the form of a rod performing joint longitudinal, transverse and torsional vibrations in the plane of the Oxy, while the displacements are selected as follows:
, (2)
where u and v are the displacements of the central line of the pipe, α₁ is the angle of inclination of the tangent to the elastic line with a pure bend.
Expressions of moving u_i(x,y,t) from (2) substituting for the sign of variation δ_ui.
When taking into account the expression ui (2), the Cauchy relations get the form
(3)
In this case, the variations of kinetic, potential energies and the work of external forces are represented as
, (4)
, (5)
. (6)
The relationship between stresses and deformations for underground polymer pipelines has the form [23-24]
; . (7)
Taking into account the ratio (3) and (7), the following longitudinal, transverse forces and bending moment of an underground polymer pipeline are formulated:
; ;
. (8)
From the variational equation (1) after performing the corresponding operations, taking into account the forces of interaction of the pipeline with the ground [1, 2], we obtain the following systems of differential equations of motion of an underground polymer pipeline with the corresponding initial and boundary conditions:
(9)
Natural boundary conditions for an underground pipeline:
(10)
Natural initial conditions:
(11)
We apply a weakly singular three-parametric Rzhanitsyn –Koltunov kernel in expressions (9) and (10) [21]
с^-1, с^-α. (12)
The three-parameter kernel (12) has a weak Abel-type singularity. This kind of kernels have a weak feature, to eliminate this, we use transformations in the integrand expression according to. Therefore, by replacing variables, we eliminate this feature [21]:
(13)
where
.
The calculation can be performed with different variations of rheological parameters A_i, α, β. After some transformations, we obtain a system of differential equations of motion, boundary and initial conditions in a general vector form
(14)
Boundary conditions:
(15)
Initial conditions:
, (16)
where
, М, А, В, С, –matrices of the third order.
To solve the boundary value problem, we use the method of finite differences of the second order of accuracy
(17)
Equation (17), divided by , we obtain
(18)
We introduce the notation:
(19)
Taking into account the introduced notation (19), we rewrite the equation
(20)
_{We solve the vector equation with respect to the desired function}
(21)
_{Consider the problem on the Oxy plane: the pipeline is loaded in the xy plane, i.e. the seismic movement of the soil occurs in the horizontal plane at an angle to the longitudinal axis of the pipeline, while the ends of the pipeline are pinched. Underground pipeline, pinched at both ends.}
(22)
_{Initial conditions:}
, (23)
<sub>according to the finite-difference scheme, we have

Then the initial conditions (23) get the form:</sub>
. (24)
_{If we take into account the initial conditions (24) at , equations (21) get the form}
, , (25)
_{When j=1 equations (21), taking into account the initial conditions (24), will take the form:}
. (26)
_{Equation (21) is solved at 2≤j≤M.
For i=1 of (21), taking into account the boundary conditions (22), we obtain the following equations:}
. (27)
_{At 2≤i≤N-1, equation (21) is solved:}
. (28)
_{For i=N-1, equation (21) is written as:}
. (29)
_{For i=1, j=1 from equation (26) we obtain the following equation}
, (30)
_{For i=2, j=1, equation (26) is solved
For i=N-1, j=1, equation (28) is written as:}
. (31)
_{After determining the displacements, stresses at the considered points of the polymer pipe according to the formula (8) by the finite difference method, taking into account the boundary conditions (22), when the stability condition is met in the form of τ=1/4h. Based on the compiled algorithm, a program for “Borland Delphi 7” is implemented on a PC.}