Development of Methodology for Main Pipelines Linear Section Stress-Strain State Сhanges Assessment

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1 International Journal of Industrial and Manufacturin Systems Enineerin 7; (6): doi:.648/j.imse.76. ISSN: (Print); ISSN: (Online) Development of Methodoloy for Main Pipelines Linear Section Stress-Strain State Сhanes Assessment Liubomyr Zhovtulia, Andriy Oliyny, Andriy Yavorsyi, Masym Karpash, Iryna Vashchysha Enery Manaement and Technial Dianostic Department, National Technical University of Oil and Gas, Ivano-Franivs, Uraine address: (L. Zhovtulia) To cite this article: Liubomyr Zhovtulia, Andriy Oliyny, Andriy Yavorsyi, Masym Karpash, Iryna Vashchysha. Development of Methodoloy for Main Pipelines Linear Section Stress-Strain State Сhanes Assessment. International Journal of Industrial and Manufacturin Systems Enineerin. Vol., No. 6, 7, pp doi:.648/j.imse.76. Received: Auust 9, 7; Accepted: September 6, 7; Published: December, 7 Abstract: The linear section of main oil and as pipelines, which are potentially the most danerous type of pipeline networs, was selected as the object of investiation. A technique for assessin the stress-strain state of the pipeline was developed. Underround pipeline deformation process was mathematically modeled accordin to the set of points displacement. The process of the underround pipeline points position determination process, estimation of the interpolation accuracy of the underround pipeline spatial position, interpolation step and smoothin parameters determination were described. Keywords: Pipeline Networs, Stress-Strain State, Methodoloy, Ris Assessment, Mathematic Model, Axis Coordinates. Introduction Pipeline linear part in Uraine has considerable lenth and is under constant movement of soil. Statistics of emerencies [] on pipelines confirm that one of the most danerous riss of this ind - the riss eodynamic oriin, untimely detection of which may result in an emerency. Movement of the pipe axis causes the chane of the stress strain state, critical values of which lead to the destruction of metal.. Method Mathematical modellin of the underround section deformation process While modellin the process of main pipelines underround sections deformation, based on data of chane the spatial confiuration of their axis, the approach, suested for the on-surface pipeline part, is applied. In such a case, usin the experimental methods [, 3, 4], the eometrical confiuration of pipeline axis is determined, includin the specific accuracy at the control moment of time. It is assumed that the initial position of the pipeline axis is nown (for example, accordin to the desin documentation). Thus, for the radius vector of the pipeline point the followin relationship is written. r( s, φ, r, t) = rl ( s, φ, r, t) + ρ( s, φ, r, t) (cos ω( s, φ, r, t) b + sin ω( s, φ, r, t) n ) + D +Ψ( s, φ, r, t) τl nl l where s, ϕ, r - related to the investiated area of underround pipeline, which is simulated as the curved cylindrical body with coordinates respectively: s - alon the pipeline axis; ϕ - by vectorial anle; r l - radius vector of the point on the upper eneratrix of the pipeline; D - exernal diameter of the pipeline; ρ(s, ϕ, r, t); ω(s, ϕ, r, t); Ψ(s, ϕ, r, t) - functions that describe the eometry chane of the investiates site, respectively in radial, transverse and lonitudinal directions and are either iven, or those that are expressed in the process of solvin the problem; Τ l; b l; n l - vectors of the tanent binormal and normal to the upper enerator. At the initial time, when the pipeline is considered an object with a rectilinear axis, the dependence () in coordinate form can be written as follows: l ()

2 67 Liubomyr Zhovtulia et al.: Development of Methodoloy for Main Pipelines Linear Section Stress-Strain State Сhanes Assessment x = s P φ rl = y = rsinφ R r R z = rcosφ D s L where R ; R - respectively inner and outer radii of the pipeline; L - lenth of the investiated area. In controlled time moment the dependence () is written as: D x = s αn( s) + ( αn( s)sin φ_ αb( s)cos φ) r D rt = y = y( s) βn( s) + ( βn( s)sin φ + βb( s)cos φ) r D z = z( s) γ n( s) + ( γ n( s)sin φ + γ n( s)cos φ) r where positions of s, ϕ, r ain the same meanin as in (), s; y( s ); z( s) - points position of the upper simulated area, D - pipeline diameter; α n( s ) ; β n ( s) ; γ n ( s) - positions of normal vector to the upper eneratix; α b( s ) ; β b ( s) ; γ b ( s) - position of binormal vector. When constructin the (3) the followin suppositions were used: Thus, the only output information concernin the eometry chane of the underround sections are the coordinates of its deformed axle, then in () it is assumed that: rl ( s, φ, r, t) = s; y( s); z( s) ρ( s, φ, r, t) = r ω( s, φ, r, t) = φ ψ( s, φ, r, t) = () (3) That is due to the fact, that, coordinates of the upper eneatrix are experimentally defined and are set in the form of points position si; y( si); z( s i), and for oriination of ( s; y( s); z( s )) interpolation or approximation procedures [6, 7] are used, while there is no information about behavior of ρ( s; φ; r; τ ); ω( s, φ, r, τ ) та ψ( s, φ, r, τ ), which maes their record in such a form in which it was written for undeformed areas. If the representation of (3) leads to physically unrealistic results, these functions are simulated by the techniques, listed in [5], where the cross-section confiuration chane is counted towards the various types of it presentation - ellipticity, pearshape, ellipticity parameter spacin of axle deree of deformability - thus the mentioned methods are justified for surface areas, when the information concernin the crossection damaes is available at least visually. In case of underround sections, the representation of (3) is justified with the lac of information concernin the cross-section deformation. This explains the choice of ψ( s, φ, r, τ ) =, as tain into account the underround areas it is also not impossible to perform the visual inspection of the hypothesis justification concernin the flat sections. If the (4) same methods are used as for the investiation of underround and for surface areas, it is thus at different ways of settin ρ( s; φ; r; τ ); ω( s, φ, r, τ ) ; ψ( s, φ, r, τ ) there is one more problem -it is difficult to mae the balance equation for underround areas, as it is impossible to tae into account in these equations the action of mass forces (weiht of the pipe, weiht of the product, weiht of the soil actin on each section of the pipeline) Thus, tain into account () and (3), the followin sequence of calculations is performed:. In controlled and initial moment of time the vectors of local basis are defined in each point of simulated area [8]: r Эi = ξ (5) rt Эi = ξ ξ = s; ξ = φ; ξ 3 = r; i =,,3, where r is calculated accordin to (), а r t - accordin to (3). Calculatin the derivatives are carried out by direct differentiation of () and (3) correspondin to the coordinates.. Based on (5) the components of metric tensor are defined: = Эi Э j, j, i =,,3 (6) i, i, j =,,3 = Э Э j 3. Components q and q form matrix, and for the correctness of the calculations the hypothesis should be carried out: G G = det ; = det. Performin of (7) based on (6) allow to provide calculations of matrix contravariant components G and G as matrix components, inverse to these: It is obvious, that accordin to () = =. ; (7) (8)

3 International Journal of Industrial and Manufacturin Systems Enineerin 7; (6): G = r ; dαn dβn G = r ( II ( αβ + βb + ds ds dγ n + γ )( R Rrsin φ + r )). ds Thus, in case of small strains, the statements of (7) are performed, as in this case: dα n dβ <<, n dγ <<, n << ds ds ds 4. Components of tensor strain are calculated by the [8]: ( ) i, j,,3 (9) ε = = () 5. Based on (5) - () the tensor strain components are defined accordin to Hoo s Law by applyin the linear elastic theory device [8]: σ = λi ( ε ) + µ ε () I The listed calculations can be performed within the model of anisotropic body: σ з Сl ε (), l= = where Cl - tensor components of material elastic modulus, althouh () in used only if the pipeline material is substantially anisotropic, and coeficients Cl are nown. For enineerin calculations () is typically used, where rl and xl - material Lame parameters, related to Youn's modulus and Poisson coefficient of the material in the followin way: E µ = ( + σ) σ E λ = ( σ)( + σ) (3) As for pipeline steels, it is enerally taen Е= MPa, σ =,3. In represented () function Ι,( ε) is the first strain invariant and is calculated by the formula: 3 3 j= i= where ε і is calculated by (), and Ι,( ε) = ε (4) - accordin to (8). Determination of σ components allows to identify the most danerous investiatin areas, concernin the sector stress state chanes, and if at the initial time the pipeline stress is equal zero, then () allows to assess the true strain values. Stress acceptance criterion may be elastic strenth value ( σ 35 МПа), or yield strenth ( σ 44 МПа), пр in case when the iven values are different for various types of pipin steels and are determined from the reference literature [9]. It should be mentioned that the described approach to the assessment of the underround stress state is interal, and it does not require the detailed information on loadin and stress, the impact of which on the areas is due to displacement measurements. In case, when some tensions (for instance, due to pressure impact, temperature chanes, etc.) are acquainted, it is possible to use the superposition principle of elastic theory: Н B σ = σ + σ (5) where σ B - tensions, determined by (), σ - acquainted tensions, Н σ - tensions of unnown nature. 3. Position Measurement for Underround Pipeline Section Position measurement of pipeline layin, to et its actual position in space and curvature, can be performed by be carried out by non-contact methods by measurin the component of the electromanetic field created by the alternatin current flowin throuh the pipeline from the low-frequency enerator. Fiure. Generalized non-contact determination chart for attitude sensin of underround sinular oil and as pipeline. - round; oil and as pipeline; 3 insulation coatin; 4 damaes of insulation coatin; 5 operator; 6 receivin set of control device; 7 sinal-enerator; 8 round connection; 9 collar; - connector; operational sump The eneralized non-contact determination chart for attitude sensin of underround sinular oil and as pipeline is iven in Fiure. For control procedure (Fiure ) sinal-enerator (7) or cathodic protection stations, connectin to the oil and as pipeline, are used as the current source, and round connection 8. Location of pipeline axis is determined by the pipelines т

4 69 Liubomyr Zhovtulia et al.: Development of Methodoloy for Main Pipelines Linear Section Stress-Strain State Сhanes Assessment finder operation manual. Each determines control point on the pipeline axis is set by the temporary spill or steel pill with iven number for further position determination. Location of determined control points shows the pipeline location in sectional view. Pipeline occurrence depth is set in place of pipeline axis determination, usually by pipelines finder, allowin to perform direct occurrence depth measurement. Pipeline occurrence depth h ф is set by the formula: hф = H D (6) where H - distance from the round surface to the pipeline axis in meters, determined by the pipelines finder. In the absence of pipelines finders, allowin to perform direct pipeline occurrence depth measurement. summin with the parameter level position of pipeline occurrence depth. For iven investiations the pipelines finder SeeTech SR- 6 is used, allowin to determine the location of pipelines axis and occurrence depth with precise accuracy. The measures data of upper eneratix point position and initial pipeline point positions allow to assess the effective stress values. 4. Accuracy Evaluation for Spatial Attitude Interpolation of Above- Ground Pipeline Deformed Axle For realization of the method of stress-strain state assessment, expressed by the dependencies ()-(3) it is necessary, by means of experimentally measured upper eneratix points positions ( s c, ycs i ), z( s i)) to et the expression for radius vector for every eneratix position as r = ( S; ycs); z( s)), where y( s ) and z( c ) are continuous functions. For this purpose the widely nown interpolation device is used, applyin interpolative cubic spline [5, 6] or interpolative cubic spline with test data smoothin. [4]. For interpolative cubic spline the interpolation rid settins are set [, ], characterized by the relationships between the minimum and maximum distances between interpolation nodes: h h max ε 8 zε 3 = h 3 min 3 f 3 max (8) Fiure. Determination of oil and as main pipeline occurrence depth. manetic antenna of pipelines finder receivin device, pipeline, 3 manetic field lines The manetic antenna of the pipeline finder receiver is placed perpendicular to the axis of the pipeline and at the anle of 45 from the vertical axis. By movin the manetic antenna from the projection of the pipeline axis alternately in different directions, the location of points A and C is determined (Fiure ) by means of minimum volume of the sound sinal and/or by the indicator level of pipeline finder receivin device. Distances AB and BC are determined. Section averae АВ and ВС will be equal the distance from the round surface to the pipeline axis. Oil pipeline occurrence depth h ф is determined by the formulae: AB + BC hф = D = ( AB + BC D ) (7) where D - oil pipeline diameter. Next step is the determination of space conditions by means of hih-precision GPS transmitters. Measurements are performed at fixed checpoint. The process peculiarity is where ε - accuracy necessary for interpolatin of the function f( x ) by spline S f ( x), value of ε sets the accuracy level of function value assinment at interpolation nodes; f - norm of function f ( x) at iven metric space []. Dependence (7) can be written in more compact form, tain into account that for main pipelines the radius of axis curvature should meet the hypothesis: R( x) C DTP (9) where DTP - pipeline diameter, C - the constant iven by the value C [ 9;], R( x ) - radius of pipeline curvature, which for enineerin calculations can be written in the form: R( x) = f ( x) () Tain into account (9) and () the dependence (8) for equally spaced rid for nodes coordinates measurin in increments h can be written in the form:

5 International Journal of Industrial and Manufacturin Systems Enineerin 7; (6): h 3 8 ε zε CD TP 3 = () L 8 ε zε CD TP 3 h = L 3 () For pipeline section with the lenth L = м, pipe diameter DTP =, м with measurement accuracy level sm, step value h, with which it is necessary to measure the coordinates of the points of the upper eneratix, with step h = 6 meters, which is quite acceptable in critical. The interpolation cubic spline peculiarity is the followin: at its development the accuracy of interpolation settin affects sinificantly the axis interpolation accuracy. As a rule, the sinificant deviation from real data yields results that do not correspond to the actual physical picture of the process. The way out is possible by means of implementation of two approaches:. Applyin the other implementation methods (Larane, Chebyshev and Hermite polynomial) or approximation by the LS method, with the resultin curves can differ sinificantly from real in some cases (insufficient number of interpolation nodes, their inappropriate placement, etc.);. Use of approaches, related to embeddin the smoothin spline device, allowin to reduce the error of points position measurin by means of some correction coeficients that depend on the accuracy of measurin these points position by testin methods. While embeddin the smoothin splice device, the desired W a b smoothin function minimizes on class [ ; ] interated on [ ab ] function interval with their square of functional in the form of [6]: n ɶ = Φ ( u) = U ( x) dx + P ( U( x ) f ) a (3) Formula () requires detailed explanation: f ɶ - positions of actually measured points; U( x ) - points positions on the curvature, describin the spline; P - weihtin coefficients. Minimization problem () is solved for different values P. In extreme cases, if P for any K, then the constructed spline will not actually be a smoothin, it will pass throuh all nodes with point positions ( x; f ɶ ). If P, then the actually obtained line will be straihtforward, since it delivers the extremum of a functional in the form of D Φ ( u) = U ( x) dx (4) which, obviously, will have a minimum for U ( x) = U( x) = Ax + B - that is, U( x ) - straihtforward a line. With the nowin of performed measurements accuracy f, it is possible to et the values P, where in the function confiuration U( x ) will, from the one hand, smooth the effect of measurements error, and from the other hand, will not allow to lose the features of the real section confiuration. This can be depicted as simulation in the followin way (Fiure 3) I absolute smoothin; II non-smoothed spline; III smoothed curvature Fiure 3. Test data smoothin. Optimizin methods (4) with parameters P, which characterize the level of data smoothin dependin on the measurement accuracy, are well-nown, and they are used for above-round sections [5], for this reason their application for underround sections is well-reasoned. In particular, the procedure of functional minimizin (3) is used, by implementation of iteration procedure, at each step of which the coefficients are based on the formula: ( j+ ) P ( j) U ( x ( j) ) yɶ = P ε (5) which is realized until the fulfillment of the condition is achieved ( j U ) ( x ) yɶ ε (6) In formulae (5), (6) j - iteration process step number; ε - accuracy of node points position measurement, ( j) P - smoothin coefficient value at iteration process step j, j U ( x ) - smoothed positions of node point X after minimizin procedure (3) at iteration process step under number j ; y - initial non-smoothed positions of this node point. Test calculations within implementation of these method show, that by the embeddin this smoothin iteration procedure, the error of stress evaluation is ± 5MPa for the operatin pipeline section, displacement measurement of which is carried out with the accuracy of mm for the pipeline section with the L=м. 5. Conclusion Authors conducted the theoretical research of simulation

6 7 Liubomyr Zhovtulia et al.: Development of Methodoloy for Main Pipelines Linear Section Stress-Strain State Сhanes Assessment the process of pipeline deformation for the assessment, namely assessment of the existin values and prediction of stresses. The method of evaluation of the pipeline stressstrain state is developed. The process of the underround section of the pipeline deformation is mathematically developed accordin to the data of the displacement of a certain set of points. The determination process of underround pipeline underround pipeline points position, interpolation accuracy evaluation of the of the above-round pipeline deformed axis spatial and determination process of interpolation step and smoothin parameters were described. The conducted investiations allow to estimate the underround section of main pipeline stress-strain state by the non-contact method. The authors plan to continue the study the pipelines stresses assessment, subject to soil landslide References [] Gas pipeline incidents. 8-th Report of the European Gas Pipeline Incident Data Group (97-). Access mode: ( [] V. Y. Bash. Stress and deformation analysis by thermoelectric technique / V. Y. Bash. - Kyiv: Nauova duma, p. [3] V. A. Zolochevsy. Experiments in development mechanic / V. A. Zolochevsy. - Moscov: Stryizdat, p. [4] V. V. Kluev. Non-destructive testin and dianostics / V. V. Kluev. - Moscov: Mashynostroenie, p. - (3). [5] A. P. Oliyny Mathematical models for process of quasystationary deformation of pipline and industrial systems under their spatial confiuration chane [Text] / A. P. Oliyny // Scientific publication. - Ivano-Franivs: IFNTUOG,. - 3 p. [6] A. A. Samarsy. Numerical procedures: Collee-level study uide / A. A. Samarsy, A. V. Gulin. - Moscov: Naua. MEO of physical and mathematical literature, p. [7] G. I. Marchu. NUMERICAL METHEMATICS PROCEDURES / G. I. Marchu. - Moscov: Naua, p. [8] L. I. Sedov. Continuum mechanics / L. I. Sedov. - Moscov: Naua, p. [9] SNiP Main pipelines / Gosstriy SSSR. - М.: CPTE Gosstroy SSSR, p. [] N. P. Korneychu. Extremal problems in approximation theory / N. P. Korneychu. - Moscov: Naua, p. [] N. P. Korneychu. Splines in approximation theory / N. P. Korneychu. - Moscov: Naua, p. [] N. S. Bahvalov. Numerical procedures/ N. S. Bahvalov. - Moscov: Naua, p.

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