Numerical Modeled Static Stress-Deformed State of Parallel Pipes in the Deformable Environment

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1 Open Acce Library Journal 2018, Volume 5, e4671 ISSN Online: ISSN Print: Numerical Modeled Static Stre-Deformed State of Parallel Pipe in the Deformable Environment Imail Ibrahimovich Safarov ahkent Intitute of Chemitry and echnology, ahkent, Republic of Uzbekitan How to cite thi paper: Safarov, I.I. (2018) Numerical Modeled Static Stre-Deformed State of Parallel Pipe in the Deformable Environment. Open Acce Library Journal, 5: e Received: May 21, 2018 Accepted: July 8, 2018 Publihed: July 11, 2018 Copyright 2018 by author and Open Acce Library Inc. hi work i licened under the Creative Common Attribution International Licene (CC BY 4.0). Open Acce Abtract he paper conider the tatic preure of the environment on the parallel pipe. he environment i elatic and homogeneou bodie. o determine the ambient preure, the finite element method i ued. An algorithm wa developed and a computer program wa compiled. Baed on the compiled program, numerical reult are obtained. he numerical reult obtained for two to five parallel pipe are compared with known theoretical and experimental reult. Subject Area Continuum Mechanic Keyword Pipe, Finite Element Method, Static Preure, Environment, Poible Diplacement, Calculation Area 1. Introduction At preent, and in the coming decade, enuring the operational reliability of the linear part of multi-thread underground pipeline i and will continue to be a complex cientific and engineering problem. In the modern deign, variou oftware package of automated deign are widely ued, allowing to carry out the engineering analyi of computer model without reorting to real experiment. he mot common and efficient calculation method i the finite element method (FEM). When determining the preure of the oil on the pipe, it i neceary to take into account uch factor a: the number of thread, the topography of the DOI: /oalib Jul. 11, Open Acce Library Journal

2 embankment, the condition of upporting the pipe and other factor encountered in deign practice. Accounting for other factor in analytical olution i either extremely complex, or in general i impoible becaue of the difficultie that arie in thi cae of a mathematical nature. Variou factor encountered in project practice can be accounted for uing numerical method. Recently, when olving variou kind of applied problem, the finite element method (FEM) i widely ued. A number of work are known in which dometic [1] [2] [3] [4] and foreign author [5] [6] uccefully apply FEM to determine the oil preure on a ingle laid extended pipe, under variou condition of it upport, taking into account the heterogeneity of the oil compoing the body mound of contant height (flat deformation). 2. Statement of the Problem by the Finite Element Method he mot common method for calculating complex tructure i the finite element method (FEM). It peculiarity conit in the fact that a deign repreenting a continuou medium i replaced by it analog, compoed both of cube and of a finite number of element block, the behavior of each of which can be determined in advance. he interaction of the element make it poible to preent an overall picture of the deformation of the ytem. In Figure 1, the cylindrical bodie in the deformed pace are depicted. he tiffne characteritic of each of thee element are determined in advance. he tre-train tate of uch a complex tructure can be determined with the help of FEM. he advantage of the method in it univerality: the poibility of uing element of different type, the arbitrarine of the region under conideration, imple method for contructing element of high accuracy. In the variant of the method conidered below, the method of diplacement, when joining element, the requirement of atifying natural boundary condition i not neceary. hi mot famou verion of the FEM ue the formulation of the principle of poible diplacement: δa= δa + δa = In matrix form for a three-dimenional body, it can be repreented a follow: { σ } { δε } ddd xyz= { q} ddd xyz+ { p} { δ u} ds. V V S he ame tate can have the form: { σ} { δε} = { δ } + { }{ δ } ddd xyz u ddd xyz p u ds V V S Vector of volume force, urface force and mixing of point of the body are a follow: { q} { xyz,, },{ p} { p, p, p},{ u} { uvw,, } = = =. (1) x y z he equilibrium condition (1) do not depend on which material propertie and are valid for both linear and nonlinear ytem. For a linearly elatic body having initial deformation, the phyical relationhip take the form: DOI: /oalib Open Acce Library Journal

3 { σ} [ ]{ ε} [ ]{ } where [ D ] i the matrix of elatic contant, [ ] = D D e 0 (2) e 0 i the vector of initial deformation. Move are given in the form of polynomial in power of х, у, z: { u} [ A]{ α} =, (3) where [ A ] i the matrix depending on the coordinate of the element, { } α i the vector of the coefficient of the polynomial expanion of the diplacement function. he number of coefficient correpond to the number of degree of freedom of the element, and the coefficient themelve are aociated with nodal diplacement. If we denote the vector of node nodal diplacement through { u n }, then the diplacement field i divided by the dependence: { u} = [ f ]{ u n } (4) We ue the relation between deformation and diplacement. hen we get: { ε } = [ B]{ u n } (5) he matrix [B], which connect deformation with nodal diplacement, i important in the further calculation (Figure 1). he tre vector i defined by Equation (2), and taking into account (5) it will look like: { σ} [ D]{ B}{ u } [ D]{ ε } = (6) Let u conider eparately the left and right ide of the equilibrium condition (1). After ubtituting the deformation vector into the left ide of the Equation (1), it will be expreed in term of nodal diplacement and ome integral indicated n 0 Figure 1. he calculation cheme of the FEM. DOI: /oalib Open Acce Library Journal

4 by the ymbol [ K ]: { δε} [ D]{ ε} ddd xyz= δ { un} { B} [ D][ B] ddd xyzu { n} = δ { un} [ K]{ un} V V Here [ K ] i a matrix containing the baic information on the behavior of a mall region of a deformed ytem. It i called the element tiffne matrix and i the main characteritic of the ytem in the FEM. On the right-hand ide of Equation (1), the integral over the volume and over the urface can be repreented a follow: ({ δε} [ ]{ ε 0} + { δ n} { }) ddd + { δ } { } V D u q xyz u pds { n} [ ] [ ]{ 0} ddd { n} [ ] { } = δ u B D ε xyz+ δ u f qddd xyz V { n} [ ] { } + δ u f p ds S hee relation determine the vector (P) of external force, reduced to the node of external force. hu, conidering the matrix [ f ] connecting the diplacement at any point of the element with nodal diplacement and the matrix [ B ] correponding to the relation between the deformation and diplacement of the node of the element according to the Formula (6), the tiffne matrix [ K ] and the vector of external nodal force (F): [ ] = [ ] [ ][ ] K B D Bddd xyz V { } = [ ] [ ]{ ε } + [ ] { } + [ ]{ } F B D 0 ddd xyz f qddd xyz f pds V V S For each element, the equilibrium condition take the form: [ K]{ u } = { F} 3. Methodology for Calculating the Static Preure of Soil on Pipe n A a computational model, by analogy with [7] [8], a weighty elatic medium (Figure 1) i ued that contain hole and other incluion upported by circular cylinder and other incluion (foundation, heterogeneity of the ground, etc.). For pipe according to [9], we aume that the cylinder i welded to the medium (there i no lippage of the oil along the urface of the pipe). On the external contour of the medium, the boundary condition have the following form [9] (Figure 1): on vertical boundarie, hear tree and horizontal diplacement are either zero or thee boundarie are free; on the lower horizontal boundary adjacent to the bae of the embankment there are no vertical and horizontal movement; the upper urface i either free from external influence, or loaded with a urface load. he dimenion of the choen area for the calculation hould be optimal, be- V V DOI: /oalib Open Acce Library Journal

5 caue thi affect the time pent on the calculation of the FEC and, conequently, the efficiency of the program baed on it. If the oil i an iotropic material or the ytem of the pipe-oil in quetion ha an axi of ymmetry (both in geometry and in material), it i poible to reduce the deign area by taking only a ymmetrical half of it. he breakdown of the choen calculation area i carried out in the form of tetrahedral finite element. In thi cae, the center meh hould thicken a it approache the pipe; it i around the pipe that the greatet concentration of oil preure occur. o etimate the convergence of the reulting approximate olution correponding to thi breakdown, it i neceary to make a finer diviion of the computational domain into an exact olution. hen a comparion of the olution correponding to both breakdown hould be made. If they differ from each other by an amount greater than the predetermined accuracy of the computation, it i neceary to make an even maller third partition of the domain and the correponding olution compare with the olution for the econd breakdown, etc. It hould be noted that with a dene arrangement of pipe in the place of their contact, ingular point arie, in a mall neighborhood of which it i impoible to achieve the neceary accuracy of calculation for any mallet breakdown (elaticity theory i inapplicable at thee point). he ame point arie in the place where the pipe ret on a flat bae. When determining the oil preure on rigid round pipe, uch a ferroconcrete pipe in particular [10] [11], thi difficulty i eaily overcome by the following method: with the help of FEM [12], the vertical and horizontal oil preure at all point of the pipe, except for the pecial one, i determined; a concentrated force i applied at a particular point, directed vertically at the point of upport of the pipe or horizontally at their point of contact, equal in magnitude to the area of the diagram of the vertical and horizontal preure of the oil acting on the pipe, repectively. We ditribute the proper weight of the oil of the embankment according to [13] [14] along the breaking point a follow: at each node of thi triangular finite element, we apply a downward concentrated force equal in weight to the part of the oil bounded by thi element divided by the number of node. he urface load i ditributed along the node of the upper boundary in the form of concentrated force. If it i neceary to obtain the influence matrice (Green function), then it i neceary to calculate the unit concentrated force, applying it conitently at each node of the upper boundary. Modeling of material of oil, pipe and other incluion i carried out with the help of the correponding value of elatic contant ( Е, ν ) and pecific gravity. hi make it poible to take into account the condition of upporting the pipe, the heterogeneity and verboity of the oil of the embankment and the bae, and the multitude of laying. 4. Parametric Analyi of Stre-Strain State of Reinforced Concrete Underground Round ube Uing the program MSK-1, the influence of the following factor on the preure DOI: /oalib Open Acce Library Journal

6 ditribution of the oil of the embankment around the round reinforced concrete underground pipe wa invetigated: the number of thread, the ditance between the pipe, the location of the pipe (extreme, middle), the Poion coefficient of the embankment oil, the type of pipe upport, the change of the relief of the embankment along the pipe, length of pipe. he influence of the number of thread in Figure 2-4 how the dependence of the imum oil preure on the pipe on the number of thread and the Figure 2. Graph of the dependence of the imum oil preure on the pipe on the number of thread and the Poion ratio (v) (D = 4 m, Н = 12 mм, d = 0, γ = 16.7 kg/m 3 ). Figure 3. Graph of the dependence of the imum oil preure on the five-laying pipe (σ ) on the poition of the pipe and Poion ratio (v) (D = 3 m, Н = 12 m, d = 0, γ = 16.7 kg/m 3, n = 4). DOI: /oalib Open Acce Library Journal

7 ingle output; extreme development of multiline tacking; average production of multiline tacking. Figure 4. Dependence of the imum preure of rock on the working (σ ) on the number of thread (n) (D = 3 m, Н = 8 m, d = 0, γ = 16.7 kg/m 3, ν = 0.3). Poion ratio of the oil. At the ame time, the upport wa firmly upported on a flat olid bae. From Figure 2-4 it follow that the value of σ for pipe laid in everal tring i 10% - 30% le than the correponding value for a ingle laid pipe, which i determined by [15]. In thi cae, the imum oil preure depend on location of the pipe. On an average pipe it i 15% - 25% le than on an extreme one. he fact that the outer tube i unloaded i le due to the fact that only one nearby middle pipe exert a ignificant influence on it unloading, and the other i the outer tube, firt, far from it (1.0D), and econdly, between the two outer tube lie the middle tube, which i a kind of creen, reducing the mutual unloading effect of the two outer tube. herefore, in particular, the imum preure of the oil on the edge pipe i practically independent of the number of thread (in Figure 4, the value of σ for pipe of two yarn tacking and the outer tube of multi-threading i hown in one curve. two pipe are located on both ide of it, and not one, a in the cae of an extreme tube). From Figure 3 and Figure 4 it follow that for a number of thread greater than three, the value of σ on the middle tube i practically independent of the number of thread with the concept of a period of pipe and explained in [16]. In Figure 2, it i clear that a the Poion ratio rie ( ν = 0.1,,0.4 ), the imum ground preure on the middle tube decreae, and with a decreae in the number of thread thi decreae i tronger and amount to, for example, 7% DOI: /oalib Open Acce Library Journal

8 for three-laying pipe and 1. hi i explained by the fact that the greater the value of the coefficient v, the greater the ditribution capacity of the ground environment. Conequently, we can aume that for a number of thread four or more, the value of σ on the average i practically independent of the coefficient v. An explanation of thi phenomenon i given in [17] [18]. A can be een from Figure 3, a the coefficient ν increae, the difference in the value of σ for pipeline with different number of thread decreae. hu, the imum ground preure on the pipe of multi-thread tacking i le than the ingle one. At the ame time, the imum ground preure on the outer tube i greater than the average preure. he preure of σ on the edge pipe i practically independent of the number of thread. he imum oil preure decreae with increaing number of thread and with a number of thread greater than three, thi decreae become inignificant. Hence it follow that the difference between the imum oil preure on the outer and middle pipe of multiline tacking n 3 i practically independent of the number of thread and for denely laid pipe i 15% - 20%. In addition, with an increae in the Poion ratio of oil, the value of σ on the edge pipe i reduced. With the number of thread n 4, the value of σ on the middle tube i practically not from the coefficient v. Effect of the ditance between the pipe, the reult of the analyi of the imum ground preure on two and three-tranded laying pipe (a) between them are hown in Figure 5. he graph in Figure 5 how that a the ditance between the pipe increae, the value of σ increae. At 0 d D 0.5, the increae in σ i inignificant (3%), and at 0.5 d D 2.0 a ignificant increae in the imum ground preure i oberved, decaying at d D 2.0. At d D 3.0, the imum ground preure per pipe laid in everal trand correpond to the imum Figure 5. Graph of the dependence of the imum oil preure (v ) on the pipe from the ditance in the light (d) between the filament. (Н = 8 m, γ = 16.7 kg/m 3, ν = 0.3). DOI: /oalib Open Acce Library Journal

9 preure per ingle laid pipe and coincide with the value determined by [19]. hu, the mutual influence of multifilament tacking pipe take place at a ditance between u d < 3D and lead to a decreae in the imum ground preure on them compared to a ingle tacked pipe. he preure of σ on the middle and outer tube reache a minimum value when the d = 0 pipe are laid cloely and are repectively 0.74 and 0.85 of the imum preure on a ingle tacked pipe. On the bai of the obtained dependence of the magnitude of the ditance between the pipe, the following formula are derived for determining the oil preure coefficient for pipe of multiline tacking: for 0 d D 2.5 k 2 ( ) ( ) ( ) c K = 0.1 d D ; K = 0.01 d D d D + 0.9, (7) M M c k where K M and K M are the coefficient that take into account the reduction of the imum oil preure, repectively, on the edge and middle multiline tack a compared to a ingle tacked pipe. Analyi of the influence of the ditance between the pipe on the horizontal preure of the oil ( σ ) at the horizontal diameter wa carried out by double-laying pipe, the theoretical and experimental tudie carried out have hown that the quantity σ doe not depend on the number of thread. In thi cae, it i neceary to ditinguih the horizontal preure of the oil on the pipe from the ide of the adjacent pipe ( σ ) and from the oppoite (pipe-free ide) ( σ ). From Figure 6, the horizontal preure σ Γ for d D 0 i a contant value Figure 6. Graph of the dependence of the imum ground preure (v ) on the pipe from the ditance in the light (d) between the trand (Н = 6 m, D = 2 m, γ = 16.7 kg/m 3, ν = 0.3). DOI: /oalib Open Acce Library Journal

10 and coincide with the correponding σ = 0. At 0 d D 2.0, it increae intenively and at d D= 0 tend to infinity. hi i due to the appearance of a ingular point, in which the theory of elaticity i not applicable. he harp increae in d with decreaing ditance d i explained by the convergence of the two tre concentrate, which are the pipe. he influence of the Poion ratio on the horizontal preure of the oil i hown in Figure 7. It follow from the graph that the value of σ and σ r increae with increaing coefficient ν, and the horizontal preure on the ide of the adjacent tube increae more intenively, i.e. increae the coefficient 4.1. Stre State of the Soil around the Pipe σ by a factor 2.8, and σ r in 2.3 time. For a more complete analyi of the oil preure on the pipe of multi-tranding, the diagram of radial ( σ ) and tangential ( σ ) ground preure are conidered r for variou parameter of laying multi-thread pipe on a flat olid bae. Figure 8-10 how the diagram (σ г ) for pipe which laid in one and two trand at a ditance of d of 0.0D, 0.5D, 1.0D, 2.0D, 3.0D. All the diagram of the ame ign correpond to the compreion preure. It i een from the diagram that for d < 3.0D they are aymmetric, and for d = 3.0D are ymmetric. he preence of the aymmetry of the diagram i due to the mutual influence of the multiline tacking tube on the preure ditribution of the oil around each pipe. With an increae in ditance 6, thi effect gradually weaken and doe not affect when d 3.0D, i.е. in thi cae the tube of multi-trand folding of the diagram σ i practically ymmetrical. herefore, in the deign of pipe, the deviation of the ordinate σ from the vertical diameter can be ignored for d > 2.0D. he ordinate of the imum radial preure deviate from the pipe lock in the oppoite direction to the location of the adjacent pipe. he analyi how that thi effect i manifeted epecially in multiline tacking at d < 2.0D. hi i due to the fact that on one ide of the end pipe i located next to it a pipe that unload Figure 7. Dependence of the horizontal preure (σ г ) on Poion ratio (ν) (D = 2 m, d = 2 m, H = 6 m, γ = 16.7 kg/m 3, n = 2). DOI: /oalib Open Acce Library Journal

11 Figure 8. Diagram of radial ground preure on pipe of two-thread tyling. DOI: /oalib Open Acce Library Journal

12 Figure 9. Diagram of radial oil preure on pipe of a two-thread tyling. Figure 10. Diagram of radial oil preure on pipe of a two-wire tyling. DOI: /oalib Open Acce Library Journal

13 the firt. he oppoite ide i free and there i no unloading effect from thi ide the outer tube receive. Due to thi unbalanced unloading of the outer tube, the value of σ i hifted. Figure 11 how a graph of the dependence of the deviation of the ordinate σ (angle β) on the parameter d. he ordinate of thi biconvex curve decreae with increaing d. At 0 < d < 0.5D and d > 2.0D, the angle β varie inignificantly (actually from 15 to 14 30' and from 2 to 0 ). he main change in angle β occur at 0.5 < d < 2.0D. he imal value of the ordinate σ reache at d = 0 (tube laid cloe), minimal (00) at d < 3.0D, when each pipe work a a ingle tacked one. he analyi in Figure 8-10 how that the diagram of σ on the half of the pipe oppoite the location of the adjacent pipe (in the Figure the left half of the diagram to the ordinate σ ) in all cae the effect of two pipe at d D< 3.0 on the preure ditribution of the oil around of them i local and extend to a ection up to >15 and <180 (in the figure, the right half of the diagram). Figure 8, Figure 12, and Figure 13 how the diagram of the radial preure of the oil σ г on a ingle laid pipe and on pipe laid cloely in two, three, four and five thread. In all cae, the preure diagram on the outer tube are aymmetric, and the average tube (for n > 3) are explained by ymmetrical unloading by two adjacent pipe. he upper part of the diagram of the multiline tacking i lightly flattened in comparion with the diagram for a ingle laid pipe (Figure 8). hi oblatene i greater for medium pipe than for extreme tube, which indicate a more uniform ditribution of preure, their greater load. It hould alo be noted that the diagram σ for the outer tube of multiline tacking practically do not differ from the σ diagram for double-laying pipe. hu, when contructing the diagram σ for the end pipe of multiline tacking, we can ue the reult of calculation of double-laying pipe. Figure 11. Change of parameter β from d/d. DOI: /oalib Open Acce Library Journal

14 Figure 12. Diagram of radial oil preure on three-threaded pipe tyling. Figure 13. Diagram of radial ground preure on a five-pipe tacking. It follow from Figure 13, b (n = 5), the diagram for the central and neighboring middle tube practically coincide. he diagram for the medium pipe for n = 4 (Figure 13(a)) and for n = 5 are alo mall from each other. hu, when determining the preure of the oil on the pipe of pipe of four-tranding, the concept of pipe period wa alo introduced there. It mean a minimum number of pipe, in which the addition of another pipe from the edge practically ha no effect on the tre-train tate of the oil around the central pipe. Conequently, the value of the period for the leeve i four. Analogouly, the DOI: /oalib Open Acce Library Journal

15 Figure 14. Diagram of radial oil preure σ г on a ingle (a), gutter (b) and three-tranded pipe (D = 4 m, H = 8 m, υ = 0.3, d = 1 m, γ = 16.7 kh/m 3 ). DOI: /oalib Open Acce Library Journal

16 value of the period () of the pipe laid at ome ditance from each other wa analyzed. he reult of thi analyi are preented in able 1. From able 1 it can be een that the value of decreae with increaing ditance between the pipe. hi i due to a decreae in the mutual influence of the pipe a the ditance between them increae. In order to preent the general picture of the ditribution of the radial preure of the embankment on the pipe, Figure 14 how the line of equal radial preure for pipe laid in one, two and three thread repectively. Symmetrical arrangement of the line i typical for a ingle laid pipe (Figure 14). In addition, the line are alo ymmetric for central tube for odd multicultural packing in the vicinity of 1.5D from the center of the pipe in both direction (for example, for n = 3, Figure 14). For the outermot tube, aymmetry and diplacement of the vertice are oberved b in the oppoite direction from the adjacent pipe (Figure 12). In addition, σ r at n = 2 and n = 3 have le ordinate and are more flattened than for a ingle laid pipe (n = 1). hi flattening indicate a more uniform ground preure on multi-threaded pipe compared to a ingle-laid σ r for double-laying pipe and the three-threaded outer tube are almot identical. Figure 15 how the diagram of m for a ingle pipe and double-laying pipe with a ditance in the light d = 0,,3. It i characteritic that on the half of the pipe free from the influence of the adjacent one (in the figure, the left half) τ doe not depend on the parameter d and the diagram τ i the ame a for the ingle pipe. he ordinate of the right half of the diagram τ for 0 < d < 3.0D are maller than the left one due to the unloading effect of the neighboring pipe. For d > 3.0D, the effect i no longer affected and the diagram τ i imilar to the diagram for a ingle pipe. he imum of the tangential preure for any half of the diagram i achieved at θ = 60 from the vertical axi in both direction. he larget value of τ in all cae for the outer tube occur on the left-hand half of the tube (free from the influence of the neighboring pipe) and exceed the mall tangent preure on the right half of the tube by a factor of 2.2, d = 0 ; by 1.55 time at d = 0.5D and by 1.1 time at d = 1.0D Influence of the ype of Support of Pipe Figure 11 how graph of the dependence of the value of σ on the upport of pipe and the Pauon coefficient ν. In the calculation, the following condition for upporting the pipe were ued at the uggetion of the hydropower plant: - bae with angle of capture 2α 0 = 90 ; bae with an angle of coverage 2α = ; able 1. Dependence of the pipe period on the ditance in the light between them. D/D DOI: /oalib Open Acce Library Journal

17 - a foundation with an angle of coverage 2α 0 = 120, while the height of the foundation from the ground to the bottom point of the pipe wa aumed to be 0.2D. In addition, upport wa conidered on a flat bae. A can be een from Figure 16, the larget value of σ correpond to upport on the foundation, and the mallet on the bae with a coverage angle of 2α = 120. For example, at ν = 0.1, the value of σ for pipe upporting 0 Figure 15. Diagram of radial oil preure on pipe of a two-thread tyling. Figure 16. Graph of the dependence of the imum ground preure (v ) on the pipe from the ditance in the light (d) between the trand (Н = 6 m, D = 2 mм, γ = 16.7 кг/m 3, v = 0.2). DOI: /oalib Open Acce Library Journal

18 the foundation i larger than the correponding value for pipe that upport a flat olid bae by 3%. olid bae with an angle of coverage 2α 0 = 90 by 6%, the bae with an angle of coverage 2α 0 = 120 by 8%. hi phenomenon i explained by the fact that the larger the I pipe protrude above the urface of the bae (together with the foundation). he more the preure of the oil acting on thi pipe i concentrated. We alo note that, regardle of the type of upport, the quantity σ decreae with increaing coefficient υ. With an increae in the coefficient υ by a factor of 4, σ decreae depending on the type of upport of the pipe in time. Given the light change in σ, depending on the method of upport (2% - 8%) in deigning pipe baed on a olid bae, thi factor can be ignored. able 2 how the dependence of the coefficient of imum vertical oil preure ( K = σz γh, h imum embankment height) on reinforced concrete pipe from the number of thread and the profile of the embankment. he pipe are upported by a reinforced concrete foundation with an angle of coverage up to 120. he wall thickne of the pipe i 0.1D. he ditance in the light between the pipe i 0.5D. Pipe are made of concrete of cla B 25; ν = 0.15; E = 30,000 MPa; oil of the mound with elatic contant ν = 0.3; E = 30 MPa. In the firt row, able 2 how the reult for long pipe laid over a bulk of contant height (flat deformation). In the econd row, able 2 how the reult for long pipe laid under the mound with the applied length of pipe laid under the mound with a variable longitudinal profile in the form of a triangle with an angle lope β = up to 30. From able 2 it follow that the coefficient K decreae with the number of thread. And thi fact i true both for a flat problem (the firt line) and for a volume one (econd line). For example, the value of K for an average three-threaded pipe (n = 3) i 35% maller than the correponding value for a ingle pipe (n = 1) in the cae of a flat problem ( β = 0 ), and by 37%, in the cae of patial problem (30 ). o analyze the influence of the longitudinal relief of the embankment on the oil preure on the pipe and compare the reult of the planar and patial problem, the imum height of the embankment ( β = 30 ) wa aumed equal to the height of the mound of contant height ( β = 0 ). From able 2, the value in the firt line differ from the correponding value of the econd row by an average of 30%. From thi it follow that taking into account the variable length of the embankment height along the length of the pipe reduce the etimated ground able 2. Dependence of the coefficient K on the number of thread and the angle of the longitudinal profile of the mound. n β β = DOI: /oalib Open Acce Library Journal

19 able 3. Dependence of the coefficient K on the length of the pipe l. u K preure a compared with the calculation performed on the flat-deformed cheme. hi effect wa obtained for the firt time. In thi cae, a follow from able 1 thi effect i lightly le pronounced for a ingle pipe (29%) and lightly tronger for two-thread (32%) and three-thread (30%) tacking pipe. Influence of length of pipe. able 3 how the dependence of the coefficient K for reinforced concrete pipe of two-trand packing from their length l ( β = 0 ). From able 3 it follow that with a decreae in pipe length the coefficient K kill. In thi cae, when the length l = 10.0D, it effect on K i inignificant. hu, the length l = 10.0D i that boundary of applicability of the plane theory of elaticity (plane deformation) for extended pipeline at a contant height of the embankment. In the work [20] the concept of the core, which in our cae i equal to 10.0D, i derived, and i the boundary between the hort and length pipe, i.e. at l = 10.0D, the plane-deformed cheme give an overetimate of the K coefficient even at a contant height of the embankment. hi overetimate i 38% at l = 6.0 and 55% at l = 4.0. hu, taking into account the length of the pipe reduce the deign ground preure in comparion with the calculation uing a flat-deformed cheme, if l = 10.0D. 5. Concluion 1) he imum tatic preure of the oil ( σ ) per pipe i omewhat le than that of a ingle pipe, an average of 10% for the outer pipe and 20% for the middle pipe. In thi cae, the quantity c σ increae with increaing parameter d, having a minimum at d = 0 (tube tacked cloely) and a imum at d = 3.0, which coincide with the correponding value for a ingle tube. 2) he preure σ decreae with increaing Poion ratio ν of the oil. he greatet value of σ correpond to upporting on the foundation, and the mallet to a profiled bae with a large angle of coverage. Preure σ on the outer tube and on the middle pipe i practically independent of the number of thread. 3) he horizontal tatic preure ( σ ) of the oil located between the pipe decreae with increaing parameter d, reaching a minimum at d = 3.0D, equal to the correponding value for a ingle pipe. he horizontal preure of the oil ( σ r ) on the outermot pipe from the oppoite ide of the adjacent pipe i equal to the horizontal preure on a ingle pipe and doe not depend on the parameter d. With a pipe pacing of 0.5D d 3.0D, the value of σ r 28% i more than σ. he value of σ r and σ increae directly in proportion to DOI: /oalib Open Acce Library Journal

20 the height of the embankment and decreae with decreaing Poion ratio of the oil. 4) Diagram of the radial and tangential tatic preure of the oil for the outer tube of multiline tacking ( n 3 ) and double-laying pipe practically coincide. he diagram of the radial oil preure are aymmetric for the outer tube, and are practically ymmetrical for the middle one. he angle of thi deviation depend on the parameter d and at 0 d 3D which varie repectively from 15 to 0. 5) he value of the imum radial and horizontal preure of the oil on a ingle pipe, obtained in accordance with the [21] and the FEM for an extended embankment, having a contant height (flat deformation) are in good agreement. hi give ground for uing FEM to calculate multicell pipe in the deign practice. 6) he account of the variable along the length of the pipe of the height of the embankment reduce the deign ground preure a compared with the calculation performed on the flat-deformed cheme. hi effect i more pronounced for multi-threaded pipe and weaker for a ingle pipe [22]. 7) Allowance for the length of the pipe reduce the etimated ground preure a compared with the calculation uing a flat-deformed cheme, if their length i l 10.0D [23]. Reference [1] Avliyakulov, N.N. and Safarov, I.I. (2007) Modern Problem of Static and Dynamic of Underground Pipeline. Science and echnology, ahkent, 306 p. [2] Agapkin, V.M., Boriov, S.N. and Krivohein, B.L. (1987) Reference Guide for the Calculation of Pipeline. Nedra, Mocow, 190 p. [3] Ainbinder, A.B. and Kamerhtein, A.G. (1982) Calculation of runk Pipeline for Strength and Stability. Nedra, Mocow, 343 p. [4] Alehin, V.V., et al. (2003) Numerical Analyi of the Strength of Underground Pipeline. URSS, Mocow, 320 p. [5] Borodavkin, P.P. and Sinyukov, A.M. (1984) Strength of the Main Pipeline. Nedra, Mocow, 245 p. [6] Erzhanov, N.S., Aitaliev, J.M. and Maanov, Z.K. (1980) Seimic Stre of Underground Structure in an Aniotropic Maif. Science, Alma-Ata, 211 р. [7] Kamerhtein, A.G., Rozhdetvenkiy, V.V. and Ruchimky, M.N. (1963) Calculation of Pipeline for Strength: A Reference Book. Houe Gotoptekhizdat, Mocow, 424 p. [8] Zenkevich, O. (1975) Finite Element Method in Engineering. Mir, Mocow, 623 р. [9] Lee, S.H. (2006) Application of the Perfectly Matched Layer for Seimic Soil-Structure Interaction Analyi in the ime Domain. Univerity of Hawaii, Manoa. [10] Samarky, A.A. (2005) Mathematical Modeling: Idea, Method, Example. Fizmatlit, Mocow, 316 p. [11] Seleznev, V.Y., Alehin, V.V. and Pryalov, S.N. (2005) Fundamental of Numerical Simulation of Main Pipeline. KomKniga, Mocow, 496 p. [12] Potnov, V.A. and Kharkhur, I.Y. (1974) he Finite Element Method in Calcula- DOI: /oalib Open Acce Library Journal

21 tion of Ship Structure. Science, Leningrad, 342 р. [13] Vinogradov, S.V. (1980) Calculation of Underground Pipeline for External Load. Stroiizdat, Mocow, 135 p. [14] Priekin, V.A. and Rotorguev, G.I. (2010) he Bai of the Finite Element Method in the Mechanic of Deformable Bodie. Novoibirk Univerity Pre, Novoibirk, 238 p. [15] Safarov, I.I. and Auliyakulov, N.N. (2005) Method of Increaing the Seimic Reitance of Underground Platic Pipeline. Uzbek Journal of Oil and Ga, No. 44S, [16] Seleznev, V.Y., Alehin, V.V. and Klihin, G.S. (2002) Method and echnologie of Numerical Simulation Ga Pipeline Sytem. Publihing Houe of the URSS, Mocow, 448 p. [17] Seleznev, V.Ye., Alehin, V.V. and Pryalov, S.N. (2009) Mathematical Modeling of the Main Pipeline Sytem. Additional Chapter. MAKS Pre, Mocow, 356 p. [18] Chichelov, V.A., et al. (2006) Calculation of the Stre-Strain State of Pipeline Operated under Difficult Condition in a Nonlinear Setting. Gazprom, Mocow, 80 р. [19] Shammazov, A.M., et al. (2004) Development of a Method for Calculating the Stre-Strain State of Ga Pipeline Laid in Complex Engineering-Geological Condition. Oil and Ga, 2, [20] Safarov, I.I., ehaev, M.K. and Boltayev, Z.I. (2016) Propagation of Linear Wave in Extended Lamellar Bodie. Lambert Academic Publihing, Saarbrücken, 315 p. [21] Safarov, I.I., ehaev, M.H. and Boltaev, Z.I. (2012) Wave Procee in Mechanical Waveguide. Lambert Academic Publihing, Saarbrücken, 217 p. [22] Bozorov, M.B., Safarov, I.I. and Shokin, Y.I. (1966) Numerical Simulation of Vibration, Diipative Homogeneou and Heterogeneou Mechanical Sytem. Science, Mocow, 188 p. [23] Safarov, I.I. (1992) Ocillation and Wave in Diipatively Underbred Environment and Structure. Science, ahkent, 250 p. DOI: /oalib Open Acce Library Journal