_{1}

The paper considers the static pressure of the environment on the parallel pipe. The environment is elastic and homogeneous bodies. To determine the ambient pressure, the finite element method is used. An algorithm was developed and a computer program was compiled. Based on the compiled program, numerical results are obtained. The numerical results obtained for two to five parallel pipes are compared wit h known theoretical and experimental results.

At present, and in the coming decades, ensuring the operational reliability of the linear part of multi-thread underground pipelines is and will continue to be a complex scientific and engineering problem. In the modern design, various software packages of automated design are widely used, allowing to carry out the engineering analysis of computer models without resorting to real experiments. The most common and efficient calculation method is the finite element method (FEM). When determining the pressure of the soil on the pipes, it is necessary to take into account such factors as: the number of threads, the topography of the embankment, the conditions of supporting the pipes and other factors encountered in design practice. Accounting for other factors in analytical solutions is either extremely complex, or in general is impossible because of the difficulties that arise in this case of a mathematical nature. Various factors encountered in project practice can be accounted for using numerical methods. Recently, when solving various kinds of applied problems, the finite element method (FEM) is widely used. A number of works are known in which domestic [

The most common method for calculating complex structures is the finite element method (FEM). Its peculiarity consists in the fact that a design representing a continuous medium is replaced by its analog, composed both of cubes and of a finite number of element blocks, the behavior of each of which can be determined in advance. The interaction of the elements makes it possible to present an overall picture of the deformation of the system. In

In matrix form for a three-dimensional body, it can be represented as follows:

The same state can have the form:

Vectors of volume forces, surface forces and mixing of points of the body are as follows:

The equilibrium conditions (1) do not depend on which material properties and are valid for both linear and nonlinear systems. For a linearly elastic body having initial deformations, the physical relationships take the form:

where

where

We use the relations between deformations and displacements. Then we get:

The matrix [B], which connects deformations with nodal displacements, is important in the further calculation (

Let us consider separately the left and right sides of the equilibrium condition (1). After substituting the deformation vector into the left side of the Equation (1), it will be expressed in terms of nodal displacements and some integral indicated

by the symbol

Here

On the right-hand side of Equation (1), the integrals over the volume and over the surface can be represented as follows:

These relations determine the vector (P) of external forces, reduced to the nodes of external forces.

Thus, considering the matrix

For each element, the equilibrium conditions take the form:

As a computational model, by analogy with [

・ on vertical boundaries, shear stresses and horizontal displacements are either zero or these boundaries are free;

・ on the lower horizontal boundary adjacent to the base of the embankment there are no vertical and horizontal movements;

・ the upper surface is either free from external influences, or loaded with a surface load.

The dimensions of the chosen area for the calculation should be optimal, because this affects the time spent on the calculation of the FEC and, consequently, the efficiency of the program based on it. If the soil is an isotropic material or the system of the pipe-soil in question has an axis of symmetry (both in geometry and in material), it is possible to reduce the design area by taking only a symmetrical half of it. The breakdown of the chosen calculation area is carried out in the form of tetrahedral finite elements. In this case, the center mesh should thicken as it approaches the pipes; it is around the pipes that the greatest concentration of soil pressure occurs. To estimate the convergence of the resulting approximate solution corresponding to this breakdown, it is necessary to make a finer division of the computational domain into an exact solution. Then a comparison of the solutions corresponding to both breakdowns should be made. If they differ from each other by an amount greater than the predetermined accuracy of the computations, it is necessary to make an even smaller third partition of the domain and the corresponding solution compare with the solution for the second breakdown, etc. It should be noted that with a dense arrangement of pipes in the places of their contact, “singular points” arise, in a small neighborhood of which it is impossible to achieve the necessary accuracy of calculations for any smallest breakdown (elasticity theory is inapplicable at these points). The same points arise in the places where the pipes rest on a flat base. When determining the soil pressure on rigid round pipes, such as ferroconcrete pipes in particular [

We distribute the proper weight of the soil of the embankment according to [

Using the program MSK-1, the influence of the following factors on the pressure distribution of the soil of the embankment around the round reinforced concrete underground pipes was investigated: the number of threads, the distance between the pipes, the location of the pipe (extreme, middle), the Poisson coefficient of the embankment soil, the type of pipe support, the change of the relief of the embankment along the pipes, length of pipes.

The influence of the number of threads in Figures 2-4 shows the dependence of the maximum soil pressure on the pipes on the number of threads and the

Poisson’s ratio of the soil. At the same time, the support was firmly supported on a flat solid base. From Figures 2-4 it follows that the value of

The fact that the outer tube is unloaded is less due to the fact that only one nearby middle pipe exerts a significant influence on its unloading, and the other is the outer tube, first, far from it (1.0D), and secondly, between the two outer tubes lies the middle tube, which is a kind of “screen”, reducing the mutual unloading effect of the two outer tubes. Therefore, in particular, the maximum pressure of the soil on the edge pipe is practically independent of the number of threads (in

From

Consequently, we can assume that for a number of threads four or more, the value of

As can be seen from

Thus, the maximum ground pressure on the pipes of multi-thread stacking is less than the single one. At the same time, the maximum ground pressure on the outer tubes is greater than the average pressure. The pressure of

Hence it follows that the difference between the maximum soil pressure on the outer and middle pipes of multiline stacking

Effect of the distance between the pipes, the results of the analysis of the maximum ground pressure on two and three-stranded laying pipes (a) between them are shown in

The graphs in

pressure per single laid pipe and coincides with the value determined by [

Thus, the mutual influence of multifilament stacking pipes takes place at a distance between us d < 3D and leads to a decrease in the maximum ground pressure on them compared to a single stacked pipe. The pressure of

On the basis of the obtained dependences of the magnitude of the distance between the pipes, the following formulas are derived for determining the soil pressure coefficients for pipes of multiline stacking: for

where

Analysis of the influence of the distance between the pipes on the horizontal pressure of the soil (

From

and coincides with the corresponding_{r} in 2.3 times.

For a more complete analysis of the soil pressure on the pipes of multi-stranding, the diagrams of radial (

Figures 8-10 show the diagrams (σ_{г}) for pipes which laid in one and two strands at a distance of d of 0.0D, 0.5D, 1.0D, 2.0D, 3.0D. All the diagrams of the same sign correspond to the compression pressure. It is seen from the diagrams that for _{s} is practically symmetrical. Therefore, in the design of pipes, the deviation of the ordinate

the first. The opposite side is free and there is no unloading effect from this side the outer tube receives.

Due to this “unbalanced unloading” of the outer tube, the value of

The analysis in Figures 8-10 shows that the diagrams of

_{г} on a single laid pipe and on pipes laid closely in two, three, four and five threads. In all cases, the pressure diagrams on the outer tubes are asymmetric, and the average tubes (for n > 3) are explained by symmetrical unloading by two adjacent pipes.

The upper part of the diagrams of the multiline stacking is slightly flattened in comparison with the diagram for a single laid pipe (

It follows from

Consequently, the value of the period for the sleeves is four. Analogously, the

value of the period (T) of the pipes laid at some distance from each other was analyzed. The results of this analysis are presented in

From

In order to present the general picture of the distribution of the radial pressure of the embankment on the pipes,

For the outermost tubes, asymmetry and displacement of the vertices are observed b in the opposite direction from the adjacent pipes (_{r} at n = 2 and n = 3 have less ordinates and are more flattened than for a single laid pipe (n = 1).

This flattening indicates a more uniform ground pressure on multi-threaded pipes compared to a single-laid σ_{r} for double-laying pipes and the three-threaded outer tubes are almost identical.

The maximum of the tangential pressures for any half of the diagram is achieved at

- base with angle of capture

D/D | 0 … 0.5 | 0.5 | 1.0 … 2.0 | 2.0 |
---|---|---|---|---|

T | 4 | 3 | 2 | 1 |

- a foundation with an angle of coverage

As can be seen from

the foundation is larger than the corresponding values for pipes that support a flat solid base by 3%. solid base with an angle of coverage

The distance in the light between the pipes is 0.5D. Pipes are made of concrete of class B 25; n = 0.15; E = 30,000 MPa; soil of the mound with elastic constants n = 0.3; E = 30 MPa.

In the first row,

From

For example, the value of

To analyze the influence of the longitudinal relief of the embankment on the soil pressure on the pipes and compare the results of the planar and spatial problems, the maximum height of the embankment (

n | 1 | 2 | 3 |
---|---|---|---|

β | 1.75 | 1.36 | 1.13 |

β = 30˚ | 1.24 | 0.92 | 0.79 |

u_{0} | 4.0 | 6.0 | 10.0 | 15.0 |
---|---|---|---|---|

K_{max} | 0.64 | 0.85 | 1.36 | 1.37 |

pressure as compared with the calculation performed on the flat-deformed scheme. This effect was obtained for the first time.

In this case, as follows from

Influence of length of pipes. _{max} for reinforced concrete pipes of two-strand packing from their length l (

From _{max} kills. In this case, when the length_{max} is insignificant. Thus, the length

Thus, taking into account the length of the pipes reduces the design ground pressure in comparison with the calculation using a flat-deformed scheme, if

1) The maximum static pressure of the soil (

2) The pressure

3) The horizontal static pressure (

4) Diagrams of the radial and tangential static pressure of the soil for the outer tubes of multiline stacking (

5) The values of the maximum radial and horizontal pressure of the soil on a single pipe, obtained in accordance with the [

6) The account of the variable along the length of the pipe of the height of the embankment reduces the design ground pressure as compared with the calculation performed on the flat-deformed scheme. This effect is more pronounced for multi-threaded pipes and weaker for a single pipe [

7) Allowance for the length of the pipes reduces the estimated ground pressure as compared with the calculation using a flat-deformed scheme, if their length is

Safarov, I.I. (2018) Numerical Modeled Static Stress-Deformed State of Parallel Pipes in the Deformable Environment. Open Access Library Journal, 5: e4671. https://doi.org/10.4236/oalib.1104671