eISSN: 2577-8307 FREIJ

Forestry Research and Engineering: International Journal
Research Article
Volume 2 Issue 6

Definition of damageability of the sea buried pipeline by method of spectral summation of tension at vibrations caused by technological and casual seismic loadings
Liudmila Muravyeva,1 Nikolai Vatin2
1Leading designer in the direction of linear pipelines, Russia
2St Petersburg State Polytechnical University, Russia
Received: March 24, 2018 | Published: November 22, 2018

Correspondence: Liudmila Muravyeva, Leading designer in the direction of linear pipelines, 195251, Russia

Citation: Muravyeva L, Vatin N. Definition of damageability of the sea buried pipeline by method of spectral summation of tension at vibrations caused by technological and casual seismic loadings. Forest Res Eng Int J. 2018;2(6):313‒318. DOI: 10.15406/freij.2018.02.00065

Abstract

Based on the finite element analysis a method to determine damages and fatigue zones of a pipeline has been suggested. The purpose of this study is to evaluate damages of the natural gas pipelines due to fatigue caused by cyclic fluctuations of transportation temperature which contribute to defect growth. Offshore pipeline system operation must be ensured in case of an earthquake without interruptions for any repairs. This is very important in view of the widely varying extreme loads, combined pressure and temperature effects as well as extreme environmental impacts, and inspires to solving a number of tasks related to the evaluation of the stress-strain state of the pipeline. The aim of this calculation is to analyze safety of these a buried pipeline at random operating and environmental impacts as well as cyclic fluctuations of the transportation parameters.

Keywords: sea buried pipeline, fatigue, cyclic fluctuations, random operating, seismic loadings

Main text

Designed loads on the sea buried pipeline include internal pressure of the product (natural gas), temperature of the transported product, and weight load of the medium. Certain operating conditions may lead to strength-threatening tension in the subsea pipeline, which is instantaneous us under static and dynamic random exposures. Load analysis of the main combination is shown in Figure 1 (note: sea buried pipeline is an object of the analysis). The purpose of this study is to evaluate damages of the natural gas pipelines due to fatigue caused by cyclic fluctuations of transportation temperature which contribute to defect growth.

Figure 1 Analysis of sea buried pipeline loads effect on specific combination; where p is a working pressure load, w is a dead weight load, temp–- is temperature impact, al64 is seismic load.

Mathematic model

There is a linear relationship between the input impacts combination and the output process

x(t)=[ i=1 k C i ξ i (t) ]= i=1 k C i w[ ξ i (t) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhaca GGOaGaamiDaiaacMcacqGH9aqpdaWadaqaamaaqahabaGaam4qamaa BaaajuaibaGaamyAaaqabaqcfaOaeqOVdG3aaSbaaKqbGeaacaWGPb aabeaajuaGcaGGOaGaamiDaiaacMcaaKqbGeaacaWGPbGaeyypa0Ja aGymaaqaaiaadUgaaKqbakabggHiLdaacaGLBbGaayzxaaGaeyypa0 ZaaabCaeaacaWGdbWaaSbaaKqbGeaacaWGPbaajuaGbeaacaWG3bWa amWaaeaacqaH+oaEdaWgaaqcfasaaiaadMgaaeqaaKqbakaacIcaca WG0bGaaiykaaGaay5waiaaw2faaaqcfasaaiaadMgacqGH9aqpcaaI XaaabaGaam4AaaqcfaOaeyyeIuoaaaa@5EA4@ (1)

where С -may be constant or random values.

Mathematic model of the subsea pipeline vibrations under random operating and seismic loads can be described by a linear stochastic operator

( Ei L 2 T ) 2 4 w x 4 +[ T 0 T +( α E A 0 TL )( 1γ )+P P 0 A 0 T γ ] 4 w x 2 + 4 w t 2 +[ α E θ 0 TL γθP T 0 T γ ] w t + k c w= F ˜ (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaSaaaeaacaWGfbGaamyAaaqaaiaadYeadaahaaqabKqbGeaacaaI YaaaaKqbakaadsfaaaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG OmaaaajuaGdaWcaaqaaiabgkGi2oaaCaaajuaibeqaaiaaisdaaaqc faOaam4DaaqaaiabgkGi2kaadIhadaahaaqcfasabeaacaaI0aaaaa aajuaGcqGHRaWkdaWadaqaaiabgkHiTmaalaaabaGaamivamaaBaaa juaibaGaaGimaaqcfayabaaabaGaamivaaaacqGHRaWkdaqadaqaai abeg7aHnaalaaabaGaamyraiaadgeadaWgaaqcfasaaiaaicdaaeqa aaqcfayaaiaadsfacaWGmbaaaaGaayjkaiaawMcaamaabmaabaGaaG ymaiabgkHiTiabeo7aNbGaayjkaiaawMcaaiabgUcaRiaadcfadaWc aaqaaiaadcfadaWgaaqaaKqbGiaaicdajuaGcaWGbbWaaSbaaKqbGe aacaaIWaaajuaGbeaaaeqaaaqaaiaadsfaaaGaeq4SdCgacaGLBbGa ayzxaaWaaSaaaeaacqGHciITdaahaaqabKqbGeaacaaI0aaaaKqbak aadEhaaeaacqGHciITcaWG4bWaaWbaaKqbGeqabaGaaGOmaaaaaaqc faOaey4kaSYaaSaaaeaacqGHciITdaahaaqcfasabeaacaaI0aaaaK qbakaadEhaaeaacqGHciITcaWG0bWaaWbaaKqbGeqabaGaaGOmaaaa aaqcfaOaey4kaSYaamWaaeaacqaHXoqydaWcaaqaaiaadweacqaH4o qCdaWgaaqcfasaaiaaicdaaeqaaaqcfayaaiaadsfacaWGmbaaaiab eo7aNjabeI7aXjabgkHiTiaadcfadaWcaaqaaiaadsfadaWgaaqcfa saaiaaicdaaKqbagqaaaqaaiaadsfaaaGaeq4SdCgacaGLBbGaayzx aaWaaSaaaeaacqGHciITcaWG3baabaGaeyOaIyRaamiDaaaacqGHRa WkcaWGRbWaaSbaaKqbGeaacaWGJbaabeaajuaGcaWG3bGaeyypa0Ja bmOrayaaiaGaaiikaiaadshacaGGPaaaaa@978F@  (2)

After dividing the variables we have two independent differential equations. The first equation determines free vibrations of the system.1 The second is equation of pipeline vibrations in generalized coordinates under seismic load and operating parameters of the transported product;

2 w t 2 +[ ( α E θ 0 TL γθP T 0 T γ ) L 2 T EIm ] w t + ( ω 2 +kc) ) L 2 T mEI w= F ˜ (t) m . L 2 T EI MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGcaWG3baabaGaeyOa IyRaamiDamaaCaaabeqcfasaaiaaikdaaaaaaKqbakabgUcaRmaadm aabaWaaSaaaeaadaqadaqaaiabgkHiTiabeg7aHnaalaaabaGaamyr aiabeI7aXnaaBaaajuaibaGaaGimaaqabaaajuaGbaGaamivaiaadY eaaaGaeq4SdCMaeqiUdeNaeyOeI0IaamiuamaalaaabaGaamivamaa BaaajuaibaGaaGimaaqcfayabaaabaGaamivaaaacqaHZoWzaiaawI cacaGLPaaacaWGmbWaaWbaaeqajuaibaqcfa4aaWbaaKqbGeqabaGa aGOmaaaaaaqcfaOaamivaaqaaiaadweaciGGjbGaaiyBaaaaaiaawU facaGLDbaadaWcaaqaaiabgkGi2kaadEhaaeaacqGHciITcaWG0baa aiabgUcaRmaalaaabaWaaeWaaeaacqaHjpWDdaahaaqabKqbGeaaca aIYaaaaKqbakabgUcaRiaadUgacaWGJbGaaiykaaGaayjkaiaawMca aiaadYeadaahaaqcfasabeaajuaGdaahaaqcfasabeaacaaIYaaaaa aajuaGcaWGubaabaGaamyBaiaadweacaWGjbaaaiaadEhacqGH9aqp daWcaaqaaiqadAeagaacaiaacIcacaWG0bGaaiykaaqaaiaad2gaaa GaaiOlamaalaaabaGaamitamaaCaaajuaibeqaaKqbaoaaCaaajuai beqaaiaaikdaaaaaaKqbakaadsfaaeaacaWGfbGaamysaaaaaaa@7D85@ (3)

Let us analyze the pipeline operating loads (internal pressure, temperature effect) as random processes. Here we should determine spectral density of all random processes from operating and seismic loads:

S σ ( ω )=Su( ω )+ S t ( ω )_2 ξ ut 0 ( ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadofada Wgaaqaaiabeo8aZbqabaWaaeWaaeaacqaHjpWDaiaawIcacaGLPaaa cqGH9aqpcaWGtbGaamyDamaabmaabaGaeqyYdChacaGLOaGaayzkaa Gaey4kaSIaam4uamaaBaaabaGaamiDaaqabaWaaeWaaeaacqaHjpWD aiaawIcacaGLPaaacaGGFbGaaGOmaiabe67a4naaDaaajuaibaGaam yDaiaadshaaeaacaaIWaaaaKqbaoaabmaabaGaeqyYdChacaGLOaGa ayzkaaaaaa@5326@  (4)

where Su( ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadofaca WG1bWaaeWaaeaacqaHjpWDaiaawIcacaGLPaaaaaa@3BA1@ is response spectrum under seismic load, S t ( ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadofada WgaaqaaiaadshaaeqaamaabmaabaGaeqyYdChacaGLOaGaayzkaaaa aa@3BC1@ is vibration spectrum of temperature effects. The

third summand in the equation (4) can be treated as an interference element, which makes additional contribution due to correlation. Let us write the equations of pipeline vibrations when exposed to a sum of loads a used by a random seismic load and variation of the parameters of the transported

T i ( t )+b w t + ( ω i 2 +kc) ) m . L 2 T EI = u( t ) m s+ m np MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadMgaaeqaaKqbaoaabmaabaGaamiDaaGaayjkaiaa wMcaaiabgUcaRiaadkgacqGHflY1daWcaaqaaiabgkGi2kaadEhaae aacqGHciITcaWG0baaaiabgUcaRmaalaaabaWaaeWaaeaacqaHjpWD daWgaaqaaiaadMgaaeqaamaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSIaam4AaiaadogacaGGPaaacaGLOaGaayzkaaaabaGaamyBaaaa caGGUaWaaSaaaeaacaWGmbWaaWbaaKqbGeqabaqcfa4aaWbaaKqbGe qabaGaaGOmaaaaaaqcfaOaamivaaqaaiaadweacaWGjbaaaiabg2da 9maalaaabaGaamyDamaabmaabaGaamiDaaGaayjkaiaawMcaaaqaai aad2gadaWgaaqaaKqbGiaadohajuaGcqGHRaWkcaWGTbWaaSbaaKqb GeaacaWGUbGaamiCaaqabaaajuaGbeaaaaaaaa@623B@  (5)

b = [ α E θ 0 TL γθP T 0 T γ m L 2 T EI ] ,α is a coefficient. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkgaqa aaaaaaaaWdbiaacckapaGaeyypa0ZdbiaacckapaWaamWaaeaadaWc aaqaaiabgkHiTiabeg7aHnaalaaabaGaamyraiabeI7aXnaaBaaaju aibaGaaGimaaqabaaajuaGbaGaamivaiaadYeaaaGaeq4SdCMaeqiU deNaeyOeI0IaamiuamaalaaabaGaamivamaaBaaajuaibaGaaGimaa qabaaajuaGbaGaamivaaaacqaHZoWzaeaacaWGTbaaaiabgwSixpaa laaabaGaamitamaaCaaajuaibeqaaKqbaoaaCaaajuaibeqaaiaaik daaaaaaKqbakaadsfaaeaacaWGfbGaamysaaaaaiaawUfacaGLDbaa peGaaiiOa8aacaGGSaGaeqySde2dbiaacckapaGaamyAaiaadohape GaaiiOa8aacaWGHbWdbiaacckapaGaae4yaiaab+gacaqGLbGaaeOz aiaabAgacaqGPbGaae4yaiaabMgacaqGLbGaaeOBaiaabshacaqGUa aaaa@6D78@

By solving the equation (5), let us determine the roots of the standard equation:

λ 1 = [ b ] 2 ( [ b ] 2 2 ( ω i 2 +kc m ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaBaaajuaibaqcfa4aaSbaaKqbGeaacaaIXaaabeaaaeqaaKqbakab g2da9iabgkHiTmaalaaabaWaamWaaeaacaWGIbaacaGLBbGaayzxaa aabaGaaGOmaaaacqGHsisldaGcaaqaaiabgkHiTmaabmaabaWaaSaa aeaadaWadaqaaiabgkHiTiaadkgaaiaawUfacaGLDbaaaeaacaaIYa aaamaaCaaajuaibeqaaiaaikdaaaqcfaOaeyOeI0YaaeWaaeaadaWc aaqaaiabeM8a3naaBaaajuaibaGaamyAaaqabaqcfa4aaWbaaKqbGe qabaGaaGOmaaaajuaGcqGHRaWkcaWGRbqcfaIaam4yaaqcfayaaiaa d2gaaaaacaGLOaGaayzkaaaacaGLOaGaayzkaaaabeaaaaa@55F9@  (6a)

λ 2 = [ b ] 2 ( [ b ] 2 2 ( ω i 2 +kc m ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaBaaajuaibaqcfa4aaSbaaKqbGeaacaaIYaaabeaaaeqaaKqbakab g2da9iabgkHiTmaalaaabaWaamWaaeaacaWGIbaacaGLBbGaayzxaa aabaGaaGOmaaaacqGHsisldaGcaaqaamaabmaabaWaaSaaaeaadaWa daqaaiaadkgaaiaawUfacaGLDbaaaeaacaaIYaaaamaaCaaajuaibe qaaiaaikdaaaqcfaOaeyOeI0YaaeWaaeaadaWcaaqaaiabeM8a3naa BaaajuaibaGaamyAaaqabaqcfa4aaWbaaKqbGeqabaGaaGOmaaaaju aGcqGHRaWkcaWGRbqcfaIaam4yaaqcfayaaiaad2gaaaaacaGLOaGa ayzkaaaacaGLOaGaayzkaaaabeaaaaa@5420@  (6b)

Calculate transfer function of the equation

Let us calculate transfer function of the equation (5), assuming that у= Ф(λ)  e λt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qeiabg2da9iaabccacaWGKqWdaiaacIcapeGaeq4UdW2d aiaacMcapeGaaeiiaiaadwgapaWaaWbaaeqajuaibaWdbiabeU7aSj aadshaaaaaaa@41AF@  and solving the resultant equation:

Transfer function is described by the equation

Φ( λ )= u( t ) k c EI λ 2 + ω i 2 k c +[ b ] S t ( ω ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agn aabmaabaGaeq4UdWgacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaWc aaqaaiabgkHiTiaadwhadaqadaqaaiaadshaaiaawIcacaGLPaaacq GHsislcaWGRbWaaSbaaKqbGeaacaWGJbaajuaGbeaaaeaacaWGfbGa amysaaaaaeaacqaH7oaBdaahaaqcfasabeaacaaIYaaaaKqbakabgU caRiabeM8a3naaDaaajuaibaGaamyAaaqaaiaaikdaaaqcfaOaam4A amaaBaaajuaibaGaam4yaaqabaqcfaOaey4kaSYaamWaaeaacaWGIb aacaGLBbGaayzxaaaaaiaadofadaWgaaqaaiaadshaaeqaamaabmaa baGaeqyYdChacaGLOaGaayzkaaGaaiilaaaa@5AE6@  (7)

Joint spectral density of random functions (t) and s t ( ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada WgaaqaaiaadshaaeqaamaabmaabaGaeqyYdChacaGLOaGaayzkaaaa aa@3BE1@ can be calculated on the basis of the following assumption:

s ut( ω ) =Φ( iω ) s x ( ω )={ i s x ( ω )( ω<0 ). i s x ( ω )( ω>0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada WgaaqaaKqbGiaadwhacaWG0bqcfa4aaeWaaeaacqaHjpWDaiaawIca caGLPaaaaeqaaiabg2da9iabfA6agnaabmaabaGaamyAaiabeM8a3b GaayjkaiaawMcaaiaadohadaWgaaqcfasaaiaadIhaaeqaaKqbaoaa bmaabaGaeqyYdChacaGLOaGaayzkaaGaeyypa0Zaaiqaaeaadaqhaa qaaiaadMgacaWGZbWaaSbaaKqbGeaacaWG4baabeaajuaGdaqadaqa aiabeM8a3bGaayjkaiaawMcaamaabmaabaGaeqyYdCNaeyipaWJaaG imaaGaayjkaiaawMcaaiaac6caaeaacqGHsislcaWGPbGaam4Camaa BaaajuaibaGaamiEaaqabaqcfa4aaeWaaeaacqaHjpWDaiaawIcaca GLPaaadaqadaqaaiabeM8a3jabg6da+iaaicdaaiaawIcacaGLPaaa caGGSaaaaaGaay5Eaaaaaa@676C@  (8)

Considering the transfer function (7), the joint spectral density can be defined as

s ut( ω ) =Φ( iω ) s t ( ω )= ( ( u 10 ) )( t ) k c )/ ( λ 2 + k c ω i 2 +[ α( Eθ/ TL )γθp( T 0 /T )γ ] ) ( s t( ω ) ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada WgaaqaaKqbGiaadwhacaWG0bqcfa4aaeWaaeaacqaHjpWDaiaawIca caGLPaaaaeqaaiabg2da9iabfA6agnaabmaabaGaamyAaiabeM8a3b GaayjkaiaawMcaaiaadohadaWgaaqcfasaaiaadshaaeqaaKqbaoaa bmaabaGaeqyYdChacaGLOaGaayzkaaGaeyypa0ZaaSGbaeaadaqada qaaiabgkHiTmaabmaabaGaamyDamaaBaaajuaibaGaaGymaiaaicda aeqaaaqcfaOaayjkaiaawMcaaaGaayjkaiaawMcaamaabmaabaGaam iDaaGaayjkaiaawMcaaiabgkHiTmaabiaabaGaam4AamaaBaaajuai baGaam4yaaqcfayabaaacaGLPaaaaeaadaqadaqaaiabeU7aSnaaCa aajuaibeqaaiaaikdaaaqcfaOaey4kaSIaam4AamaaBaaajuaibaGa am4yaaqabaqcfaOaeqyYdC3aa0baaKazfa4=baGaamyAaaqaaiaaik daaaqcfaOaey4kaSYaamWaaeaacqGHsislcqaHXoqycqGHflY1daqa daqaamaalyaabaGaamyraiabeI7aXbqaaiaadsfacaWGmbaaaaGaay jkaiaawMcaaiabgwSixlabeo7aNjabgwSixlabeI7aXjabgkHiTiaa dchacqGHflY1daqadaqaamaalyaabaGaamivamaaBaaajuaibaGaaG imaaqcfayabaaabaGaamivaaaaaiaawIcacaGLPaaacqGHflY1cqaH ZoWzaiaawUfacaGLDbaaaiaawIcacaGLPaaaaaWaaeWaaeaacaWGZb WaaSbaaeaajuaicaWG0bqcfa4aaeWaaeaacqaHjpWDaiaawIcacaGL PaaaaeqaaaGaayjkaiaawMcaaiaacYcaaaa@931E@  (9)

where < s t ( ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada WgaaqcfasaaiaadshaaeqaaKqbaoaabmaabaGaeqyYdChacaGLOaGa ayzkaaaaaa@3C9D@ is spectral density of temperature fluctuations of the product.

Stress-strain state of the pipeline shell can be evaluated using a finite element method. Internal stresses are associated with the loads on the pipeline wall shown in Figure 2.

Figure 2 Loads and stresses in the section of the pipeline shell: where N are longitudinal stresses in the pipeline wall, М,Q - bending moments and shearing stresses are distributed along the pipeline wall symmetric with respect to mid-surface of the shell.

Let us analyze random stationary external impact on the wall of the offshore pipeline. A relation linking tensor of the random strain with equivalent stress is called von Mises equation.2–4

σ ke 2 ( t )= 1 2 [ ( σ x σ y ) 2 + ( σ y σ z ) 2 + ( σ z σ x ) 2 +σ ( τ xy 2 + τ yz 2 + τ zx 2 ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaDaaajuaibaGaeyydICIaam4AaiaadwgaaeaacaaIYaaaaKqbaoaa bmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaa qaaiaaikdaaaWaamWaaeaadaqadaqaaiabeo8aZnaaBaaajuaibaGa amiEaaqcfayabaGaeyOeI0Iaeq4Wdm3aaSbaaKqbGeaacaWG5baaju aGbeaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakab gUcaRmaabmaabaGaeq4Wdm3aaSbaaKqbGeaacaWG5baabeaajuaGcq GHsislcqaHdpWCdaWgaaqcfasaaiaadQhaaKqbagqaaaGaayjkaiaa wMcaamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSYaaeWaaeaacq aHdpWCdaWgaaqcfasaaiaadQhaaeqaaKqbakabgkHiTiabeo8aZnaa BaaajuaibaGaamiEaaqabaaajuaGcaGLOaGaayzkaaWaaWbaaKqbGe qabaGaaGOmaaaajuaGcqGHRaWkcqaHdpWCdaqadaqaaiabes8a0naa Baaajuaibaqcfa4aa0baaKqbGeaacaWG4bGaamyEaaqaaiaaikdaaa aajuaGbeaacqGHRaWkcqaHepaDdaWgaaqcfasaaKqbaoaaDaaajuai baGaamyEaiaadQhaaeaacaaIYaaaaaqabaqcfaOaey4kaSIaeqiXdq 3aaSbaaKqbGeaajuaGdaqhaaqcfasaaiaadQhacaWG4baabaGaaGOm aaaaaeqaaaqcfaOaayjkaiaawMcaamaaCaaajuaibeqaaiaaikdaaa aajuaGcaGLBbGaayzxaaaaaa@816F@  (10)

We can evaluate probabilistic characteristics of there and nonequivalent stressed state in time and spectral ranges.

Matrix representation of stress on pipeline

Let us have matrix representation of an expression for σ ke 2 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaDaaajuaibaGaeyydICIaam4AaiaadwgaaeaacaaIYaaaaKqbaoaa bmaabaGaamiDaaGaayjkaiaawMcaaaaa@3F14@ :

σ( t )=( σ x σ y σ z τ xy 2 τ yz 2 τ zz 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maabmaaeaqabeaa cqaHdpWCdaWgaaqcfasaaiaadIhaaKqbagqaaaqaaiabeo8aZnaaBa aajuaibaGaamyEaaqabaaajuaGbaGaeq4Wdm3aaSbaaKqbGeaacaWG 6baabeaaaKqbagaacqaHepaDdaWgaaqcfasaaKqbaoaaDaaajuaiba GaamiEaiaadMhaaeaacaaIYaaaaaqabaaajuaGbaGaeqiXdq3aaSba aKqbGeaajuaGdaqhaaqcfasaaiaadMhacaWG6baabaGaaGOmaaaaaK qbagqaaaqaaiabes8a0naaBaaajuaibaqcfa4aa0baaKqbGeaacaWG 6bGaamOEaaqaaiaaikdaaaaajuaGbeaaaaGaayjkaiaawMcaaaaa@5ADC@ (11)

Then , where according to5

M= 1 2   ( 2   1   1   0   0   0 1   2   1   0   0   0 1   1   2   0   0   0    0    0   0   6   0   0    0    0   0   0   6   0    0    0   0   0   0   6 ) * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eacq GH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaabaaaaaaaaapeGaaiiO a8aadaqadaabaeqabaGaaGOma8qacaGGGcGaaiiOaiaacckapaGaey OeI0IaaGyma8qacaGGGcGaaiiOaiaacckapaGaeyOeI0IaaGyma8qa caGGGcGaaiiOaiaacckapaGaaGima8qacaGGGcGaaiiOaiaacckapa GaaGima8qacaGGGcGaaiiOaiaacckapaGaaGimaaqaaiabgkHiTiaa igdapeGaaiiOaiaacckacaGGGcWdaiaaikdapeGaaiiOaiaacckaca GGGcWdaiabgkHiTiaaigdapeGaaiiOaiaacckacaGGGcWdaiaaicda peGaaiiOaiaacckacaGGGcWdaiaaicdapeGaaiiOaiaacckacaGGGc WdaiaaicdaaeaacqGHsislcaaIXaWdbiaacckacaGGGcGaaiiOa8aa cqGHsislcaaIXaWdbiaacckacaGGGcGaaiiOa8aacaaIYaWdbiaacc kacaGGGcGaaiiOa8aacaaIWaWdbiaacckacaGGGcGaaiiOa8aacaaI WaWdbiaacckacaGGGcGaaiiOa8aacaaIWaaabaWdbiaacckacaGGGc GaaiiOa8aacaaIWaWdbiaacckacaGGGcGaaiiOaiaacckapaGaaGim a8qacaGGGcGaaiiOaiaacckapaGaaGima8qacaGGGcGaaiiOaiaacc kapaGaaGOna8qacaGGGcGaaiiOaiaacckapaGaaGima8qacaGGGcGa aiiOaiaacckapaGaaGimaaqaa8qacaGGGcGaaiiOaiaacckapaGaaG ima8qacaGGGcGaaiiOaiaacckacaGGGcWdaiaaicdapeGaaiiOaiaa cckacaGGGcWdaiaaicdapeGaaiiOaiaacckacaGGGcWdaiaaicdape GaaiiOaiaacckacaGGGcWdaiaaiAdapeGaaiiOaiaacckacaGGGcWd aiaaicdaaeaapeGaaiiOaiaacckacaGGGcWdaiaaicdapeGaaiiOai aacckacaGGGcGaaiiOa8aacaaIWaWdbiaacckacaGGGcGaaiiOa8aa caaIWaWdbiaacckacaGGGcGaaiiOa8aacaaIWaWdbiaacckacaGGGc GaaiiOa8aacaaIWaWdbiaacckacaGGGcGaaiiOa8aacaaI2aaaaiaa wIcacaGLPaaadaahaaqabeaacaGGQaaaaaaa@D5BA@  (12)

At tri axial compression or tension σ kB=0,| M | =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaabaqcfaIaeyydICIaam4AaiaadkeajuaGcqGH9aqpcaaIWaGa aiilamaaemaabaGaamytaaGaay5bSlaawIa7aaqabaGaeyypa0JaaG imaaaa@43D6@

Stressed state in a point σ kB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaabaqcfaIaeyydICIaam4AaiaadkeaaKqbagqaaaaa@3BB2@  of the pipeline is a multidimensional random process with the six time-varying components. Equivalent stresses are considered to be strength criteria of the pipeline design as per von Mises criterion.4

The equivalent stress σ ( r ) kB ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaCaaabeqcfasaaKqbaoaabmaajuaibaGaamOCaaGaayjkaiaawMca aaaajuaGdaWgaaqcfasaaiabg2GiNiaadUgacaWGcbaabeaajuaGda qadaqaaiaadshaaiaawIcacaGLPaaaaaa@424E@ in the point n of the pipeline under review determines fatigue life of the pipeline.

In practice, the pipelines operated in seismic areas are often exposed to random loads, insofar as the external impact parameters are stochastic here. The distributed static load leading to a dangerous stressed state in the pipeline wall is restricted by the maximum allowable load.

Let us describe a sea buried pipeline as a linear system. Seismic damage is other than local damage, as an increase in the pipe curvature is observed along the fixed sections.

This study focuses on designed pipeline risk assessment by virtue of the theory of runs.

Average number of runs U( Q 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada qadaqaaiaadgfadaWgaaqcfasaaiaaicdaaeqaaaqcfaOaayjkaiaa wMcaaaaa@3B49@ within level Q 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgfada Wgaaqcfasaaiaaicdaaeqaaaaa@3858@ is calculated by formula:

U( Q 0 )= 1 2Π D F D F e ( Q m F ) 2 2 D F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada qadaqaaiaadgfadaWgaaqcfasaaiaaicdaaKqbagqaaaGaayjkaiaa wMcaaiabg2da9maalaaabaGaaGymaaqaaiaaikdacqqHGoauaaWaaO aaaeaadaWcaaqaaiaadseadaWgaaqcfasaaiaadAeaaKqbagqaaaqa aiaadseadaWgaaqcfasaaiaadAeaaeqaaaaaaKqbagqaaiaadwgada ahaaqabeaacqGHsisldaWcaaqaamaabmaabaGaamyuaiabgkHiTiaa d2gadaWgaaqcfasaaiaadAeaaKqbagqaaaGaayjkaiaawMcaamaaCa aabeqcfasaaiaaikdaaaaajuaGbaGaaGOmaiaadseadaWgaaqcfasa aiaadAeaaKqbagqaaaaaaaaaaa@510C@  (13)

Failures may be treated as independent accidental events and estimated using a rare-event probability equation.

The conditional probability of the structural strain φ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@3AB8@ exceeding the level a within the duration of an earthquake 0 oτt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+gacq GHKjYOcqaHepaDcqGHKjYOcaWG0baaaa@3D95@  t at least once is equal to:

H t =1exp[ 0 t U( a|T )dT ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaadshaaeqaaKqbakabg2da9iaaigdacqGHsislciGG LbGaaiiEaiaacchadaWadaqaaiabgkHiTmaapehabaGaamyvamaabm aabaGaamyyaiaacYhacaGGubaacaGLOaGaayzkaaGaamizaiaadsfa aKqbGeaacaaIWaaabaGaamiDaaqcfaOaey4kIipaaiaawUfacaGLDb aaaaa@4D27@  (14)

The probability of the load F(t) to exceed the value Q0 within the duration Т at least once is equal to:

H t =1exp[ 0 T U ¯ ( Q 0 )dt ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqaaiaadshaaeqaaiabg2da9iaaigdacqGHsislciGGLbGaaiiE aiaacchadaWadaqaaiabgkHiTmaapehabaGabmyvayaaraWaaeWaae aacaWGrbWaaSbaaeaacaaIWaaabeaaaiaawIcacaGLPaaacaWGKbGa amiDaaqcfasaaiaaicdaaeaacaWGubaajuaGcqGHRiI8aaGaay5wai aaw2faaiabgwSixdaa@4DC0@  (15)

The required safety level of a structure P( φ>a*|t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada qadaqaaiabeA8aQjabg6da+iaadggajuaicaGGQaqcfaOaaiiFaiaa cshaaiaawIcacaGLPaaaaaa@3FE4@  which supports the design seismic risk value Р* over the rated life Т0 is calculated by formula:

P( φ>a*|t )= P 1exp( T 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada qadaqaaiabeA8aQjabg6da+iaadggajuaicaGGQaqcfaOaaiiFaiaa cshaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadcfadaahaaqcfa sabeaacqGHxiIkaaaajuaGbaGaaGymaiabgkHiTiGacwgacaGG4bGa aiiCamaabmaabaGaeyOeI0Iaey4jIKTaamivamaaBaaajuaibaGaaG imaaqabaaajuaGcaGLOaGaayzkaaaaaaaa@4EB3@  (16)

where φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQb aa@3836@  is the earthquake event frequency.

Reasonable structure reliability level is established on the basis of performance and reliability analysis carried out for existing structures; contingency analysis (for accidents occurred and simulated); and also on the basis of material resource efficiency considerations and safety requirements.

Cyclic variations of the transported product parameters can be recorded using linear addition hypothesis or Miner’s rule.5 This method has been developed to determine total damager ate П over a period of time Т caused by all loading cycles on the pipeline:

Π= k=0 N m 1 j=0 N a 1 P kj N pkj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfc6aqj abg2da9maaqadabaWaaabmaeaadaWcaaqaaiaadcfadaWgaaqcfasa aiaadUgacaWGQbaajuaGbeaaaeaacaWGobWaaSbaaKqbGeaacaWGWb Gaam4AaiaadQgaaeqaaaaaaeaacaWGQbGaeyypa0JaaGimaaqaaiaa d6eajuaGdaWgaaqcfasaaiaadggaaeqaaiabgkHiTiaaigdaaKqbak abggHiLdaajuaibaGaam4Aaiabg2da9iaaicdaaeaacaWGobqcfa4a aSbaaKqbGeaacaWGTbaabeaacqGHsislcaaIXaaajuaGcqGHris5aa aa@5399@  (17)

Where Nm is a number of intervals composing the measurement range σ m ,; N а MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaWdbiaad2gaaKqba+aabeaapeGa aiilaiaacUdacaWGobWdamaaBaaajuaibaWdbiaadcdba8aabeaaaa a@3DE1@  is a number of intervals composing the measurement range σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaWdbiaadggaaKqba+aabeaaaaa@3A4D@ , σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaWdbiaadggaaKqba+aabeaaaaa@3A4D@  is amplitude of stresses; Рki is repeat ability of full cycles that are in k range for σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaWdbiaadggaaKqba+aabeaaaaa@3A4D@  and j range for
σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaGaamyBaaqcfayabaaaaa@3A3A@ ; k=0.1,…, ( Na1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaeaacaWGobqcfaIaamyyaKqbakabgkHiTiaaigdaaiaa wIcacaGLPaaaaaa@3C3F@ c; j=0.1,…, ( Nm1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaeaacaWGobqcfaIaamyBaKqbakabgkHiTiaaigdaaiaa wIcacaGLPaaaaaa@3C4B@ ; kΔ σ a < σ a ( k+1 )Δ σ a ;  ( σ m ) min +jΔ σ m < σ m ( σ m ) min +( j+1 )Δ σ m . σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabgs5aejabeo8aZ9aadaWgaaqcfasaa8qacaWGHbaa paqabaqcfa4dbiabgYda8iabeo8aZ9aadaWgaaqcfasaa8qacaWGHb aajuaGpaqabaWdbiabgsMiJ+aadaqadaqaa8qacaWGRbGaey4kaSIa aGymaaWdaiaawIcacaGLPaaapeGaeyiLdqKaeq4Wdm3damaaBaaaju aibaWdbiaadggaa8aabeaajuaGpeGaai4oaiaabccapaGaaiika8qa cqaHdpWCpaWaaSbaaKqbGeaapeGaamyBaaWdaeqaaKqbakaacMcada Wgaaqcfasaa8qacaWGTbGaamyAaiaad6gaaKqba+aabeaapeGaey4k aSIaamOAaiabgs5aejabeo8aZ9aadaWgaaqcfasaa8qacaWGTbaapa qabaqcfa4dbiabgYda8iabeo8aZ9aadaWgaaqcfasaa8qacaWGTbaa paqabaqcfa4dbiabgsMiJ+aacaGGOaWdbiabeo8aZ9aadaWgaaqcfa saa8qacaWGTbaajuaGpaqabaGaaiykamaaBaaajuaibaWdbiaad2ga caWGPbGaamOBaaqcfa4daeqaa8qacqGHRaWkpaWaaeWaaeaapeGaam OAaiabgUcaRiaaigdaa8aacaGLOaGaayzkaaWdbiabgs5aejabeo8a Z9aadaWgaaqcfasaa8qacaWGTbaajuaGpaqabaWdbiaac6cacqaHdp WCpaWaaSbaaKqbGeaapeGaamyBaaqcfa4daeqaaaaa@7CAF@  is a constant component of the cycle, where ( σ m ) min MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaqa aaaaaaaaWdbiabeo8aZ9aadaWgaaqcfasaa8qacaWGTbaapaqabaqc faOaaiykamaaBaaajuaibaWdbiaad2gacaWGPbGaamOBaaqcfa4dae qaaaaa@3F81@  is a minimum value σ m ;Δ σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaWdbiaad2gaaKqba+aabeaapeGa ai4oaiabgs5aejabeo8aZ9aadaWgaaqcfasaa8qacaWGTbaajuaGpa qabaaaaa@404F@  is a step for σ m ;Δ σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaWdbiaad2gaaKqba+aabeaapeGa ai4oaiabgs5aejabeo8aZ9aadaWgaaqcfasaa8qacaWGHbaajuaGpa qabaaaaa@4043@  is a step for σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4Wdm3damaaBaaajuaibaWdbiaadggaaKqba+aabeaaaaa@3A4D@ .

Flow chart of formula (17) implementation is shown in Fig.3.

Flow chart of total damage on pipeline
Flow chart of formula (17) implementation is shown in Figure 3.

Figure 3 Flow chart of total damage rate determination of the pipeline design.

Pipelines having diameter to wall thickness ratio of higher than 20 mm are called thin walled, distribution of normal stresses that are perpendicular to the surface is uniform over the entire wall thickness. For isotropic materials stress-strain dependence is represented as follows under plane stress: follows:

( ε H ε L )= 1 E [ 1    v1 v    1 ]( σ H σ L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaea qabeaacqaH1oqzdaWgaaqaaiaadIeaaeqaaaqaaiabew7aLnaaBaaa baGaamitaaqabaaaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaig daaeaacaWGfbaaamaadmaaeaqabeaacaaIXaaeaaaaaaaaa8qacaGG GcGaaiiOaiaacckacaGGGcWdaiabgkHiTiaadAhacaaIXaaabaGaey OeI0IaamODa8qacaGGGcGaaiiOaiaacckacaGGGcWdaiaaigdaaaGa ay5waiaaw2faamaabmaaeaqabeaacqaHdpWCdaWgaaqaaiaadIeaae qaaaqaaiabeo8aZnaaBaaabaGaamitaaqabaaaaiaawIcacaGLPaaa aaa@5830@ (18)

For determination of internal stresses that appear in the walls of the offshore subsea pipeline under design loads a finite element model of the pipeline has been developed using solid finite elements. The internal stresses are calculated using finite element method and ANSYS software is present in Figure 1.

Pressure and temperature variation cycles associated with changes in natural gas transportation modes were simulated using a technique represented in Figure 3.

Calculations previously made for the non-buried pipeline as shown in demonstrated that the total damage rate Di for all wave loads was D=0.026,А=1.5741014, considering that service lifeТ=38.4years.6

At the stage of designing the subsea pipelines in the Caspian Sea, the decision was taken to bury the pipelines with consideration of seismic hazard. Based on the calculations it was decided to bury the subsea pipelines in the landfall sections to ensure protection from cycling waves.5 Nore searches of the fatigue parameters of the buried offshore subsea pipelines were made. We can determine total damage rate of the pipelines using technique from Figure 3. Combination of the subsea pipeline loads present on Figure 1 (shown as a percentage in the diagram).6–10

It is necessary to perform researches to determine pipeline fatigue. This article covers evaluation of the fatigue of the buried subsea pipelines. The fatigue calculations of the sea buried pipelines are made using simplified formulas to evaluate the fatigue rate of the underground pipelines. This method is not fully applicable to the operating mode of the buried offshore subsea pipelines (Figure 1). Simplified method of fatigue staring the valuation using Weibull distribution for simulation of the long-term fatigue stress distribution is described in the guidelines.2

Cumulative stress distribution function can be expressed as follows:8,9

Q( Δσ )=exp[ ( Δσ q ) h ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgfada qadaqaaiabgs5aejabeo8aZbGaayjkaiaawMcaaiabg2da9iGacwga caGG4bGaaiiCamaadmaabaGaeyOeI0YaaeWaaeaadaWcaaqaaiabgs 5aejabeo8aZbqaaiaadghaaaaacaGLOaGaayzkaaWaaWbaaKqbGeqa baGaamiAaaaaaKqbakaawUfacaGLDbaaaaa@4A46@  (19)

Where Q is probability of stress range exceedance σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaajuaibaGaaGimaaqcfayabaaaaa@39D3@ ; h are parameters of Weibull distribution; q is Weibull scale parameter, it is determined for the stress range, σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaajuaibaGaaGimaaqcfayabaaaaa@39D3@ :

q= Δ σ 0 ( ln n 0 ) 1 h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghacq GH9aqpdaWcaaqaaiabgs5aejabeo8aZnaaBaaajuaibaGaaGimaaqc fayabaaabaWaaeWaaeaaciGGSbGaaiOBaiaad6gadaWgaaqcfasaai aaicdaaKqbagqaaaGaayjkaiaawMcaamaaCaaabeqcfasaaKqbaoaa laaajuaibaGaaGymaaqaaiaadIgaaaaaaaaaaaa@4601@  (20)

Where n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada WgaaqcfasaaiaaicdaaKqbagqaaaaa@3903@ is stress range of n0 cycles. According to the technique,11 Weibull distribution parameters h are determined using linear interpolation of the stress range for values (0.90÷1.0) from the Table 1 for the curves S-N We can calculate duction factor of the allowable stresses from the curve F1, it’s present on Figure 4. Considering corrosion protection of the pipeline from the Table 2 we obtain14 a reducing factor of 0.19. In this case the stress reduction will be within

82.501 МPa for σ e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaajuaibaGaamyzaaqabaaaaa@3975@ =485.3.11–21

S-N

Weibull shape parameter h

 

 

 

 

 

curves

0.50

0.60

0.07

0.80

0.90

1. 00

1.10

1.20

F1

523.3

376.7

289.9

233.9

196.4

169.6

149.6

134.3

Table 1 Allowable extreme stress range during 108 cycles for components in seawater with cathodic protection21

Figure 4 Allowable extreme stress range during 108 cycles for components in seawater with cathodic protection.21

Fatigue damage Utilization η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE7aOb aa@3825@

 

Weibull shape parameter

 

 

 

 

0.5

0.6

0.7

0.8

0.9

 

1

1.1

1.2

0.1

0.497

0.511

0.526

0.54

0.552

0.563

0.573

0.582

0.2

0.609

0.62

0.632

0.642

0.652

0.661

0.67

0.677

0.22

0.627

0.638

0.648

0.659

0.668

0.677

0.685

0.692

0.27

0.661

0.676

0.686

0.695

0.703

0.711

0.719

0.725

0.3

0.688

0.697

0.706

0.715

0.723

0.73

0.737

0.743

0.33

0.708

0.717

0.725

0.733

0.741

0.748

0.754

0.76

0.4

0.751

0.758

0.765

0.772

0.779

0.785

0.79

0.795

0.5

0.805

0.81

0.816

0.821

0.826

0.831

0.835

0.839

0.6

0.852

0.856

0.86

0.864

0.868

0.871

0.875

0.878

0.67

0.882

0.885

0.888

0.891

0.894

0.897

0.9

0.902

0.7

0.894

0.897

0.9

0.902

0.905

0.908

0.91

0.912

0.8

0.932

0.934

0.936

0.938

0.939

0.941

0.942

0.944

1.00

1.000

1.000

1.000

1.000

1.000

 

1.000

1.000

1.000

Table 2 Reduction factor on stress to correspond with utilisation factor η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE7aOb aa@3825@  for C - W3 curves14

Let us analyze the sea buried pipeline laid on the bottom of the Caspian Sea. The pipeline is buried and its designed service life is 30 years. Taking into account the allowable stresses14 σ e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaajuaibaGaamyzaaqabaaaaa@3975@ =485.3 MPa, stress reduction will be as follows:

(485.3( 82.501 )=402.799MPa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaqa aaaaaaaaWdbiaaisdacaaI4aGaaGynaiaac6cacaaIZaGaeyOeI0Yd amaabmaabaWdbiaaiIdacaaIYaGaaiOlaiaaiwdacaaIWaGaaGymaa WdaiaawIcacaGLPaaapeGaeyypa0JaaGinaiaaicdacaaIYaGaaiOl aiaaiEdacaaI5aGaaGyoaiaad2eacaWGqbGaamyyaaaa@4ACB@

Fatigue damages reduce the allowable stresses by 17%.Additional distinguishing marks to be added to the character of classification of steel subsea pipelines present in Table 3.14

Pipeline class

k c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgada Wgaaqcfasaaiaadogaaeqaaaaa@38A0@

L , LI

1,5

L2

1,65

L3

1,8

G , Gl

1,4

G2

1,5

G3

1,65

Table 3 Strength factor k c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgada Wgaaqcfasaaiaadogaaeqaaaaa@38A0@  for pipeline pure buckling calculation

The stress value of 402.799 MPa is obtained from the Table 4 of the standards14 using ne(G3) coefficient of 1.33 and considering k coefficient of 0.864 from the Table 4. For the pipeline having diameter of 406.4mm and wall thickness of 14.5 mm the allowable stress range is 261.66 MPa.22–26

Pipeline

 

k σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaeq4Wdmhabeaaaaa@38CA@

class

For normal operational conditions

For short - term loads during construction and hydraulic tests

L , LI

0,8

0,95

L2

0,727

0,864

L3

0,696

0,826

G , Gl

0,8

0,95

G2

0,762

0,905

G3

0,727

0,864

Table 4 Strength factors in terms of total stresses k σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaeq4Wdmhabeaaaaa@38CA@

The allowable stress for the pipeline is 255.6 MPa.14 The result obtained does not exceed the allowable level but we still have 2.3% to reach the allowable stress level. Requirements of standards4,9 are used in the calculation. To evaluate fatigue of the buried subsea pipeline, it is required to carry out fatigue tests of the pipelines in order not to rely on standard coefficients in the calculations when evaluating strength of the pipelines during the design stage and not to contemplate about probable margin of the allowable stresses.

Acknowledgements

None.

Conflict of interest

The author declares there are no conflicts of interest.

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