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Effect of pressure and defects on the pipe flattening factor
Engineering Failure Analysis 94 (2018) 469–479
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Effect of pressure and defects on the pipe flattening factor ⁎
W. Edjeou, G. Pluvinage , J. Capelle, Z. Azari
T
École Nationale d'Ingénieurs de Metz (ENIM), Université de Lorraine, France
A R T IC LE I N F O
ABS TRA CT
Keywords: Pipe Ovalization Flattened factor Working pressure
Starting from an actual case of an oil pipe exhibiting significant ovalization, maximum admissible flattened factor fad proposed by codes (3%) is discussed. The weak point of this rule is its geometric character. It is shown that the fad decreases with pressure and with geometrical defects. It is not very sensitive to the bent shape of the pipe, or the location between ovalization and orientation supports. A simple elastic model based on a safety factor of two (2) is proposed
1. Introduction During service, pipes are subjected to deformations. There are two types of deformation which occur in the operating service, one of them is a curvature deformation. The other one is the ovalization which can be like a vertical or an horizontal deformation of initially circular section that takes an oval shape. For example when a curved pipe is subjected to an internal pressure, a resultant outward force tends to straighten this curvature according to its rigidity. Due to the difference in area between the extrados and the intrados in pipe curvature, a resultant force acts by outwardly attempting to straighten the bend and deforms the cross section into an oval shape, resulting in stress levels higher than in a straight pipe. This phenomenon is known as the “Bourdon” effect. Abdulhameed [1] has shown that in-plane bending moments cause ovalization. Depending on the direction of the moment, the shape of the ovality is different. When the curved pipe is subjected to opening bending moment, the resulting axial forces are tensile forces below the neutral axis and compression forces above. The situation is opposite for a closing bending moment. The stress intensification factor kA is defined as the ratio between the stress for a curved and a straight pipe in service. According to Abdulhameed [1], the internal pressure causes an increase in the pipe stress up to 33% higher than the CSA-Z662 code [2] estimated values because the code does not consider the Bourdon effect in evaluating the hoop and the longitudinal stresses of the pipe elbow. The closing bending moment results in higher stresses for unpressurized pipe bends than the opening bending moment. However, for pressurized pipes, opening bending moments result in higher stresses than closing bending moment. This depends on the level of applied internal pressure. The internal pressure has a reduction effect in the case of pipe bends subjected to closing bending moment. However, in the case of pipe bends subjected to in-plane opening bending moment, the internal pressure tends to increase the pipe stresses. The stresses on the pipe elbow are affected by the bend angle. Therefore, the bend angle should be one of the parameters considered in the stress intensification factor. Orynyak et al. [3] have proposed analytical solutions for determining stresses and strains in curved pipe by taking into account the conditions of clamping and connections between straight and curved pipes subjected to internal pressure and a bending moment. Orynyak et al.'s work [3] is based on Kirchhoff-Love's assumptions for thin shells. The analytical method is based on simplifying hypotheses and the approximate solution is written in the form of Krylov functions. To check the analytical solution and its applicability, numerical solutions were proposed based on the finite difference method. The unknown parameters were developed in Fourier series. ⁎
Corresponding author. E-mail address: [email protected] (G. Pluvinage).
https://doi.org/10.1016/j.engfailanal.2018.08.017 Received 19 July 2018; Received in revised form 17 August 2018; Accepted 17 August 2018 Available online 28 August 2018 1350-6307/ Published by Elsevier Ltd.
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Fig. 1. a: Geometrical dimensions of a curved pipe. b non-dimensional hoop stress
σφ kz σ 0
versus angular position
The in-plane bending moment Mz has the following form:
Mz = k zσ0πtR2
(1)
Where kz is a parameter to estimate the order of magnitude of the bending moment, σ0 is the flow stress, 2R the pipe diameter and t the wall thickness. σ Fig.1b: non-dimensional hoop stress φ distribution on the outer surface of the bend at the point θ = 45° (half of the curvature) k z σ0
for a curvature B = 250 mm over the angular position π/2 − φ (°). The pipe diameter is 2R =262.5 mm and wall thickness is t = 12.5 mm [2]. Starting from the equilibrium equations for a toroidal shell in a nonlinear geometric formulation, they established the equilibrium equations for stresses and strains. In analysis of thin shells, for the expression of the flexural deformations, the linear displacements are often neglected; this makes it possible to simplify the procedure to find the solution. Fig. 1 gives the non-dimensional hoop stress σφ/kzσ0 distribution on the outer surface of the bend at the point θ = 45° (half of the curvature) versus the angular positionπ/2 − φ (°). The curvature is equal to B = 250 mm. The pipe diameter is 2R = 262.5 mm and wall thickness is t = 12.5 mm. One notes that the stress intensification is significant. Numerous codes take into account ovalization such as API 579-1/ASME FFS-1 2007 Fitness-For-Service sample code, [4]. Section distortion is also an element to consider for the functionality of the pipeline. A simple way of measuring this distortion is the flattening factor f = Dmax-Dmin/D where D is the initial diameter, ovalization is this factor multiply by 100? Fig. 2. Table 1 summarizes the different boundary values of this factor considered in some codes. This paper starts from an inspection report concerning a major oil pipeline which conveys oil from western Canada to eastern Canada via the Great Lakes states. This pipe exhibits an important ovalization (8.8%).
Fig. 2. Definition of minimum and maximum diameter for the flattening parameter. 470
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Table 1 Values of the flattening factor limit according to different codes. Codes
Flattening factor f (%)
CSA-Z662-07 App.C [4] DNV-0S-F101 (2000) [5] API 1111-(1999) [6] Murray et al. [7]
3.0 3.0 5.5–6.2 4.3
Validity of codes flattening factor is limited to elastic behavior of the pipe. In the elastic plastic regime, local buckling of pipes has been described by Murray et al. [7] using the double modulus theory. Its harmfulness was studied by examination of local maximum stresses obtained by the Finite element method. This study offers the opportunity to discuss code rules about ovalization which is based only on the cross section of pipe and doesn't take into account the service pressure. For that the induced stress intensification factor is determined in order to limit the local maximum stress to half of the yield stress. Extension of code rules to combine ovalization with weld joint or corrosion defects is made. 2. Pipe geometry and material The studied pipe which is used for oil transportation was installed during the 50’s. It lies at the bottom of an American Great lake, fig. 3. Pipe diameter is 508 mm and its thickness is 20.6 mm. It has been detected that in a particular place, pipe deflection is 2.5° and ovalization is about 8.8% [8]. Characteristics of the pipe are given in Table 2. The pipe line was built in 1953 therefore using a “vintage” pipe steel. The mechanical properties are given in Table 3. 3. Results Stress and displacements were calculated using Finite Element Method (FEM) method. Abaqus™ version 6.13 software was used in our work. The following parameters were studied - The pipe is slightly bent but the study on a straight tube will be done for comparison between horizontal and vertical ovalization, - It is assumed to be localized in the middle of two supports and the result will be compared to that obtained at a location on embedding, - Ovalization length is assumed to be equal to 2 m, - Boundary conditions take into account the existence of support where the pipe is clamped - loading conditions incorporate
Fig. 3. Underwater picture of the studied pipe showing deflection and girth weld (http://media.mlive.com) 471
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Table 2 Characteristics of the studied pipe. Diameter
D (mm)
508
Thickness Working pressure Linear weight Ovality CZAZ 662 Bending angle Radius of curvature Distance between supports
t (mm) p (MPa) Kg/m % degree 91D (mm) S (m)
20.6 4.1 247.8 8.80 2.5 46,228 22.86
Table 3 Mechanical properties of the studied pipe steel. Material
Yield stress σy (MPa)
Ultimate strength σul (MPa)
Elongation at failure %
Young's modulus (GPa)
Steel A33
185
227
15
210
internal working pressure and the linear weight of the pipe is (2431 N/m) which is not negligible (fig. 4), - Material behavior is assumed to be elastic, further results indicate that the material yield stress is only exceeded by 6.1% for horizontal ovalization of 8.8%. Therfore, in the limits fixed by codes, the material remains elastic. 3.1. Results for the studied pipe The results in Fig. 5 concern the curved pipe with an angle of 2.5° and horizontal ovalization located at mid-span. They indicate that the maximum local (l) hoop stress σθθ, l increases linearly with ovalization Ov according to the following simple equation:
σθθ,l (MPa) = AB,ms,H ∗Ov + σθθ,0
(2)
where AB, H is a constant (subscript B indicating a curved pipe, ms mean the mid-span and H is the position of horizontal ovalization). σθθ, 0 is the hoop stress for the circular pipe cross section under working pressure, Fig. 5. One notes the hurtful effect of ovalization. Hoop stress is 2 times σθθ, 0 for 3% ovalization and 4.3 times σθθ, 0 for 8.8% ovalization. Central deflection d is also a linear function of ovalization according to: (3)
d(mm) = B∗Ov + d 0
where B is a new constant (B = 0.104) and d0 central deflection or a pipe with circular cross section at working pressure (d0 = 3.55 mm). Central deflection increases by 9% for 3% of ovalization, Fig. 6. 3.2. Influence of curvature Hoop stress versus ovalization was computed on straight pipe. A similar linear relationship was obtained between maximum hoop stress and horizontal ovalization.
σθθ,l (MPa) = AS,ms,H ∗Ov + σθθ,0
(4)
Subscript S indicates a straight pipe. The constant AS, ms, H = 16.55 and σθθ, 0 = 47.30 MPa. One notes that hoop stress for a circular cross section is lower for a bent pipe by 3.4%. For 8.8% ovalization, bending of the pipe increases hoop stress by 5.8%. Local stress amplification due to curvature is significantly less than ovalization. 3.3. Influence of ovalization orientation A comparison was made between horizontal and vertical ovalization for a straight pipe. Again a linear relationship was found between maximum local hoop stress and ovalization
Fig. 4. FEM pipe model with boundaries conditions 472
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Fig. 5. The hoop stress versus the ovalization for the bent pipe with horizontal ovalization.
Fig. 6. Displacement vs ovalization for the bent pipe with horizontal ovalization.
σθθ, l (MPa) = AS, ms, V ∗Ov + σθθ,0 The constant AS, V = 17.78 and σθθ, horizontal ovalization.
(5) 0
= 49.1 MPa. For 3% vertical ovalization, local hoop stress is greater by 3.2% than the
3.4. Influence of ovalization location A comparison been made between horizontal ovalization located at mid span and at embedment for a straight pipe. The following linear relationship was obtained:
σθθ,l (MPa) = AS,em,H ∗Ov + σθθ,0
(6)
The subscript (em) means location at embedment. The constant AS, em, H = 15.35 and σθθ, 0 = 47.7 MPa. For 3% horizontal ovalization, local hoop stress is greater by 3.2% at mid span than at embedment. Table 4 summarizes the different results for constant Aij (with i = ms or em and j = H or V)) and σθθ, 0. One notes a small difference of values of σθθ, 0 for the straight pipe; this is due to the value at origin by linear fit of data. From Table 4, it can be seen that ovalization location and orientation have little effect on local stress amplification neither does the curved shape of the pipe; these effects have about 5% of influence. However the major effect in stress amplification is ovalization. Local hoop stress for 8.8% ovalization is always close to the yield stress (σy = 185 MPa) and justify stress computing with elastic behavior. However pipe design never allows the design stress higher than yield stress divided by a safety factor. Consequently the actual ovalization cannot be accepted and the associated pipe part needs to be replaced where working pressure reduced. 473
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Table 4 Values of constant Aij (with i = ms or em and j = H or V)) and σθθ,
0.
Pipe shape
Orientation
Location
Ai,
Straight Straight Straight Bent Bent
Horizontal Vertical Horizontal Horizontal Horizontal
Mid-span Mid-span Embedment Embedment Mid-span
16.55 17.72 15.35 15.36 16.67
j
σθθ,
0
(MPa)
47.3 49.1 47.7 47.7 46.4
σθθ, l (8.8%) 195.4 MPa 205.6 MPa 184.9 MPa 185.47 MPa 196.4 MPa
Local stresses have been computed using elastic behavior, however for 8.8% ovalization, yield stress is exceeded of a few percent.
4. Discussion 4.1. On 3% value of admissible ovalization The CSA Z 662-07 App.C [4] and DNV-OS-F101 [5] codes prescribe a maximum value for ovalization. This value is independent of ovalization orientation and working pressure and it is a pure geometrical criterion. We have seen that vertical orientation induces higher local hoop stress but the difference is small and can be considered as negligible therefore it is justified that codes don't take into account the position of the major axis of the oval cross section. The value of working pressure is also not taken into account. For a working pressure ps = 4.1 MPa, Table 5 gives the local maximum hoop stress for 3% of ovalization and the ratio between this stress and yield stress σy One notes that for each case, a safety factor close to 2 is obtained on yield stress but depends on circumstances. The safety factor is defined as the ratio of yield stress over maximum local stress. For each working pressure, the local maximum hoop stress σθθ, l is a linear function of the circular cross section hoop stress (σθθ, 0)
σθθ,l = kA σθθ,0
(7)
kA (Ov) is the stress intensification factor which depends on the ovalization. (8)
kA = k Ov ∗Ov + 1 The hoop stress for circular cross section is a function of the working pressure ps
σθθ,0 = ps
D 2t
(9)
where t is the pipe thickness and D the internal diameter. Assuming that local maximum hoop stress cannot exceed half the yield stress:
σθθ, l = kA ps
σy D = 2t 2
(10)
By replacing kA by Eq. 7:
( kOv ∗OvL + 1) ∗ps
σy D = 2t 2
(11)
Kov is a parameter which depends on pipe geometry. The maximum admissible ovalization OvL is a function of working pressure according to:
OvL =
σy. t − D . ps D . ps . kov
(12)
Eq. 12 indicates that the admissible ovalization OvL allow a local maximum hoop stress lower than yield stress divided by 2 decreases as a power function with working pressure. It depends on pipe geometry through diameter, thickness and parameter kov. Ovl versus the working pressure is presented in Fig. 7. One notes that the admissible ovalization for the present working pressure (4.1 MPa) is 3.09% which is considerably less than the studied case (8.8%). The maximum working pressure in order to never violate the assumption on maximum local hoop stress less than yield stress divided by 2 is the pressure of 6 MPa. Beyond this value, no Table 5 Local maximum hoop stress for 3% of ovalization and values of ratio σθθ,l/σy Pipe shape
Orientation
Location
σθθ, l (3%)
σθθ, l/σy
Straight Straight Straight Curved
Horizontal Vertical Horizontal Horizontal
Mid-span Mid-span Embedment Mid-span
93.91 MPa 97.07 MPa 93.91 MPa 96.45 MPa
0.51 0.52 0.51 0.52
474
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Fig. 7. Admissible ovalization to allow a local maximum hoop stress less than yield stress divided by 2 versus the working pressure.
ovalization is tolerated. A safety factor of 2 is traditionally used in the design. The present pipe steel is a low strength steel for which an empirical relationship exists between fatigue endurance σD limit and yield stress:
σD = σy /2
(13)
The choice of this safety factor value ensures in addition the protection against fatigue induced by the variation of working pressure. However, it is necessary to correct the endurance limit by the stress ratio. 4.2. On the admissible value for combined ovalization and welding joint Another aspect which is not taken into account in codes is ovalization combined with a defect (welded joint or corrosion defect). For conservative reasons, these defects are assumed to be located at the same place of the ovalization. A welded joint introduces the local stress concentration. This formula is valid if the material remains elastic. The elastic assumption allows additively of stress and Eq. 8 becomes:
kA = kOv, w ∗Ov + kt
(14)
The condition of maximum value of local hoop stress is rewritten as:
(kOv ∗OvL, w + kt ) ∗ps
σy D = 2t 2
(15)
The maximum admissible ovalization combined with a weld joint OvL,w is a function of working pressure according to:
OvL, w =
σy. t − (kt ) D . ps D . ps . kov, w
(16)
In the following case, the welded joint has an angle Ψ = 45° and a weld toe radius ρ = 0.5 mm. Again a linear relationship was found between maximum local hoop stress and ovalization Ov
σθθ, l (MPa) = AB, H , ms, W ∗Ov + σθθ,0, w
(17)
Subscript W is used for a welded joint. Coefficient AB,H,msW =19.22 and hoop stress under working pressure for a circular cross section with a welded joint σθθ, 0, w = 61.9 MPa,see Fig. 8. This value allows to compute an experimental value of stress concentration factor kt = 1.225 which is close to the numerical value [9]. The coefficient kov, w was found as kov,w = 0.38,. The admissible ovalization OvL,w for ovalization combined with welded joint is 1.59%. This value for the same conditions, but without a welded joint is 3.09%. 4.3. On the admissible value for combined ovalization and corrosion defect Corrosion defects are characterized by 3 dimensions: depth (a), length (2c) and width (W). The failure of a corrosion defect which considered as a semi-elliptical, is controlled by its size and the flow stress σ0 of the material. The input parameters include pipe inner diameter (D), wall thickness (t), specified minimum yield strength (σy), Maximum Allowable Operating Pressure (MAOP), longitudinal extent of corrosion (2c) and defect depth (d). According to several codes and particularly ASME B 31 [10], a corrosion defect in a pipe is treated as a loss of material. Therefore the effective thickness tef is considered to be bearing the working pressure: 475
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Fig. 8. The maximum local hoop stress versus the ovalization for the bent pipe with horizontal ovalization and welded joint.
(18)
t ef = t − a
Numerous solutions of limit pressure (pL) for a corrosion defect are proposed in codes and limit load analysis literature (ASME B31G [10], modified ASME B31G [11], DNV RP-F101 [12]. The limit pressure (pL) which is less than the burst pressure pc is generally expressed as:
() () a
⎛ 1−α t pL = 2t . σ0 . ⎜ D ⎜1 − α a . t ⎝
1 M
⎞ ⎟ ⎟ ⎠
(19)
M is the Folias correction taking into account the pipe curvature, flow stress σ0, and α a geometrical parameter and (a/t) the relative defect depth. M and α take different values according to the method applied. Flow stress is generally a simple function of yield stress and ultimate strength and a is the defect depth. Condition on maximum value of local hoop stress is rewritten as:
σy D = 2(t − a) 2
(20)
σθθ,l (MPa) = AS,H,ms,cor ∗Ov + σθθ,0,cor
(21)
σθθ,l = kA ps
The subscript cor means corrosion defect. For a relative defect depth (a/t) = 0.5, coefficient AS,H,ms,cor = 59.87 and hoop stress under working pressure for a circular cross section with corrosion defect of relative depth a/t = 50% σθθ, 0, cor = 105.4 MPa. These two parameters depend on relative defect value, see Table.6 The stress intensification factor depends on ovalization according to:
kA = k Ov,cor ∗Ov + 1
(22)
The Finite Element Method results indicate that kOv,
cor
is a linear function of relative corrosion defect depth:
k Ov,cor = 0.49(a/t) + 0.31
(23)
For ovalization larger than 8.1%, computing is done with elasto-plastic behavior of the material which is assumed to follow Ludwik's law: (24)
σ = Kεn
where n is the strain hardening exponent (n = 0.1 for pipe steel A33) and K is the strain hardening coefficient (K = 344 MPa for the Table 6 Values of parameters As,H,ms,cor
a/t = 0.1 a/t = 0.5
andσθθ, 0, cor
with relative defect depth (a/t)
As,H,ms,cor
σθθ,
33.93 59.87
84.38 105.4
476
0, cor
(MPa)
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Fig. 9. The maximum admissible ovalization versus to the relative defect depth (a/t) for different working pressures
same steel). The combination of ovalization and a weld joint creates local plasticization in the worst cases. The maximum admissible value of ovalization combined with a corrosion defect OvL,cor is a function of working pressure and defect depth according to:
OvL,cor =
σy . (t − a) − D. ps D. ps . k ov,cor
(25)
The maximum admissible ovalization decreases linearly with the relative defect depth (a/t) and with the working pressure see Fig. 7. For a pipe with ovalization the limit pressure is given by modifying the Eq. 18:
() () a
pL =
2t . σ . KA D 0
⎡ 1−α t ⎢ ⎢1 − α a . t ⎣
1 M
⎤ P ⎥ = L,0 KA ⎥ ⎦
(26)
The defect assessment method used, ASME B31G code is based on the estimated repair factor (ERF) in which the safe operating pressure associated with each corrosion defect is the limit pressure. ERF is the ratio of the Maximum Allowable Operating Pressure (MAOP) and the safe operating pressure pL. Necessity for repair is given by the following criterion:
ERF =
MAOP ≤ 1 or 0.95 pL
(27)
ERF greater than one requires repair or lowering of the MAOP. The evolution of the estimated repair factor ERF is entered in Fig. 9 for a corrosion defect of relative depth a/t = 0.1 and aspect ratio c/a = 10 and for the working pressure of 4.1 MPa. The ERF increases linearly with ovalization and the maximum admissible ovalization with this criterion is Ov,L = 4.8%. This indicates that the pipe failure occurs for a greater ovalization and greater corrosion depth. One notes that the criterion of ERF = 1 for admissible defect depth is less conservative that the criterion of local hoop stress equal to yield stress divided by 2. 4.4. On fatigue risk induced by ovalization It has been seen that ovalization increases stress locally. For admissible ovalization or less the strain remains elastic but for large ovalizations (Ov > 4.8%) the strain must be plastic. Working pressure in a pipe is subject to variation with time. Data about such variations is not available for the studied pipe. Therefore, in the following we used the fluctuation of gas pressure inside pipelines provide by M. Dadfarnia et al. [13]. Fig. 11 shows the pressure data for a period of one year. It is assumed that such pressure fluctuations will be repeated every year. Since the pressure fluctuation range is not constant, a proper cyclic counting method was adopted to obtain the pressure range under these variable amplitude loading conditions. One notes that the stress ratio R is different to zero Figs. 10 and 11. Therefore the fatigue law needs to be corrected to take into account the R ratio effect on fatigue. In the elastic state, the general Basquin law is used:
σa, R = σ ′f (Nr )b ′
(28)
where σf is the fatigue resistance and b the Basquin’ exponent.σa,
R
477
is the equivalent stress amplitude given for stress ratio R:
Engineering Failure Analysis 94 (2018) 469–479
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Fig. 10. Evolution of the estimated repair factor ERF for a combined ovalization-corrosion defect (relative depth a/t = 0.1 and aspect ratio c/ a = 10, working pressure of 4.1 MPa) vs ovalization.
Fig. 11. Working pressure evolution for a gas pipe over approximatively one year ()
σa σ − σm = ul σa, R σul
(29)
Where σa is the stress amplitude (half of the stress range): Therefore Basquin's fatigue law is rewritten as:
σ σa = ⎛1 − m ⎞ σ ′f (Nr )b σul ⎠ ⎝ ⎜
⎟
(30)
where σm is the mean stress and σul the ultimate strength. For low cycle fatigue, the Manson-Coffin low cycle fatigue law is written as: c/b σ ′f ∆ϵt ⎛1 − σm ⎞ (Nr )b + ϵ′ ⎛1 − σm ⎞ (Nr )c = f 2 E⎝ σul ⎠ σul ⎠ ⎝ ⎜
⎟
⎜
⎟
(31)
′
where ϵf is the fatigue ductility and c the Coffin exponent. Δϵt is the total strain range. Fatigue properties of Pipe steel A33 are given by an empirical relationship from static mechanicals properties and entered in Table 7. The working pressure evolution presented in Fig. 11 indicates that the average load ratio is about 0.75 and the mean pressure is Table 7 : Low cycle fatigue parameters for pipe steel A33 Parameter
Empirical relationship
value
Basquin's exponent b Coffin's exponent c Fatigue ductility in ϵf′ Fatigue resistance σf′ (MPa)
b = −0.12 c = − 0.6 ϵf′ = (ϵf′)0.6 σf′ = 3.5σul
0.12 0.6 0.32 795 MPa
478
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Table 8 Low cycle fatigue condition for a combined ovalization-corrosion defect (relative depth a/t = 0.1 and aspect ratio c/ a = 10, working pressure of 4.1 MPa) Pressure conditions
Working pressure variations
Stop and go
Mean pressure/working pressure Strain range Number of cycles to failure
0.75 0.0008 6.1.106
0.5 0.0025 247,000
equal to 0.88 of working pressure. In the next we assume that for the studied pipe, loading conditions are similar. If we consider only pressure variation during operating service, the use of the equation indicates that the pipe is operating in conditions of local hoop stress below the endurance limit. If we consider the particular case where working pressure drops to zero and increases to his service value (stop and go procedure), the load ratio is zero. Ultimate strength is reached for ovalization equal to 4.8% for a corrosion defect with relative depth a/t = 0.1, aspect ratio c/a = 10 and for the working pressure of 4.1 MPa (the worst case). For limit ovalization (3%), the total strain amplitude is 0.0025 and the number of cycles allowed for stop and go is 247,000, Table 8. One notes that the presence of limit ovalization (3%) and corrosion defect (relative depth a/t = 0.1) tolerate a large number cycles of pressure cycles. However, attention needs to be given to stop and go procedure and therefore maintenance. 5. Conclusion Ovalization in a pipe under working pressure greatly increases the local hoop stress. The stress intensification factor of the studied pipe with ovalization of 8.8% under working pressure of 4.1 MPa is 3.5 times the hoop stress of a circular cross section under same working pressure and is not admissible. The location and orientation of the ovalization has a little effect on the local hoop stress value as well as the curvature of the pipe. Codes such as CSA and DNV allow a maximum ovalization of 3%. This recommendation is only geometric and does not take into account working pressure. Combined weld joint or corrosion defects are totally ignored by the codes. Using the simple criterion of a maximum admissible hoop stress less than the yield stress divided by the safety factor of 2, a model assuming that the pipe remains elastic under working pressure shows that admissible ovalization Ov,ad decreases as a power function of working pressure. Ov,ad depends on the pipe geometry through diameter, thickness and parameter kov with governs stress intensification. For a welded joint combined with ovalization, the admissible value depends on the stress concentration factor of the welded joint. For a corrosion defect combined with ovalization, the admissible value depends on relative defect depth. References [1] D. Abdulhameed, R. C, The influence of bourdon effect and ovalization effect on the Stress distribution on pipe elbows, 11th Pipeline Technology Conference 2016, EITEP Institute, 2016. [2] CSA Z662-15 Canadian code for oil and gas pipeline systems. [3] I.V. Orynyak, Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect, Int. J. Solids Struct. 44 (5) (2007) 1488–1510. [4] API 579-1/ASME FFS-1, Fitness-For-Service, 2007, 2007. [5] DNV-0S-F101 (2000) [6] API 1111-(1999) [7] N.W. Murray, P. Bilston, Local buckling of thin-walled pipes being bent in the plastic range, Thin-Walled Struct. 14 (5) (1992) 411–434. [8] Private communication @garretellison [9] G. Glincka, The weld profile effect on stress intensity factors in weldments, Int. J. Fract. 35 (1987) 3–20. [10] American National Standard Institute (ANSI)/American Society of Mechanical Engineers (ASME), Manual for determining strength of corroded pipelines, ASME B31G, 1984. [11] J. Kiefner, P. Vieth, A modified criterion for evaluating the strength of corroded pipe, Final Report for PR 3-805 Project to the Pipeline Supervisory Committee of the American Gas Association, 1989 Battelle, Ohio. [12] Recommended Practice, DET Norske Veritas, DNV-RP-F101, Corroded Pipelines, (2004). [13] M. Dadfarnia, B.P. Somerday, P. Sofronis, I.M. Robertson, Interaction of hydrogen transport and material elastoplasticity in pipeline stee1s, J. Press. Vessel. Technol. 131 (2009) 041404.
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Effect of pressure and defects on the pipe flattening factor ⁎
W. Edjeou, G. Pluvinage , J. Capelle, Z. Azari
T
École Nationale d'Ingénieurs de Metz (ENIM), Université de Lorraine, France
A R T IC LE I N F O
ABS TRA CT
Keywords: Pipe Ovalization Flattened factor Working pressure
Starting from an actual case of an oil pipe exhibiting significant ovalization, maximum admissible flattened factor fad proposed by codes (3%) is discussed. The weak point of this rule is its geometric character. It is shown that the fad decreases with pressure and with geometrical defects. It is not very sensitive to the bent shape of the pipe, or the location between ovalization and orientation supports. A simple elastic model based on a safety factor of two (2) is proposed
1. Introduction During service, pipes are subjected to deformations. There are two types of deformation which occur in the operating service, one of them is a curvature deformation. The other one is the ovalization which can be like a vertical or an horizontal deformation of initially circular section that takes an oval shape. For example when a curved pipe is subjected to an internal pressure, a resultant outward force tends to straighten this curvature according to its rigidity. Due to the difference in area between the extrados and the intrados in pipe curvature, a resultant force acts by outwardly attempting to straighten the bend and deforms the cross section into an oval shape, resulting in stress levels higher than in a straight pipe. This phenomenon is known as the “Bourdon” effect. Abdulhameed [1] has shown that in-plane bending moments cause ovalization. Depending on the direction of the moment, the shape of the ovality is different. When the curved pipe is subjected to opening bending moment, the resulting axial forces are tensile forces below the neutral axis and compression forces above. The situation is opposite for a closing bending moment. The stress intensification factor kA is defined as the ratio between the stress for a curved and a straight pipe in service. According to Abdulhameed [1], the internal pressure causes an increase in the pipe stress up to 33% higher than the CSA-Z662 code [2] estimated values because the code does not consider the Bourdon effect in evaluating the hoop and the longitudinal stresses of the pipe elbow. The closing bending moment results in higher stresses for unpressurized pipe bends than the opening bending moment. However, for pressurized pipes, opening bending moments result in higher stresses than closing bending moment. This depends on the level of applied internal pressure. The internal pressure has a reduction effect in the case of pipe bends subjected to closing bending moment. However, in the case of pipe bends subjected to in-plane opening bending moment, the internal pressure tends to increase the pipe stresses. The stresses on the pipe elbow are affected by the bend angle. Therefore, the bend angle should be one of the parameters considered in the stress intensification factor. Orynyak et al. [3] have proposed analytical solutions for determining stresses and strains in curved pipe by taking into account the conditions of clamping and connections between straight and curved pipes subjected to internal pressure and a bending moment. Orynyak et al.'s work [3] is based on Kirchhoff-Love's assumptions for thin shells. The analytical method is based on simplifying hypotheses and the approximate solution is written in the form of Krylov functions. To check the analytical solution and its applicability, numerical solutions were proposed based on the finite difference method. The unknown parameters were developed in Fourier series. ⁎
Corresponding author. E-mail address: [email protected] (G. Pluvinage).
https://doi.org/10.1016/j.engfailanal.2018.08.017 Received 19 July 2018; Received in revised form 17 August 2018; Accepted 17 August 2018 Available online 28 August 2018 1350-6307/ Published by Elsevier Ltd.
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Fig. 1. a: Geometrical dimensions of a curved pipe. b non-dimensional hoop stress
σφ kz σ 0
versus angular position
The in-plane bending moment Mz has the following form:
Mz = k zσ0πtR2
(1)
Where kz is a parameter to estimate the order of magnitude of the bending moment, σ0 is the flow stress, 2R the pipe diameter and t the wall thickness. σ Fig.1b: non-dimensional hoop stress φ distribution on the outer surface of the bend at the point θ = 45° (half of the curvature) k z σ0
for a curvature B = 250 mm over the angular position π/2 − φ (°). The pipe diameter is 2R =262.5 mm and wall thickness is t = 12.5 mm [2]. Starting from the equilibrium equations for a toroidal shell in a nonlinear geometric formulation, they established the equilibrium equations for stresses and strains. In analysis of thin shells, for the expression of the flexural deformations, the linear displacements are often neglected; this makes it possible to simplify the procedure to find the solution. Fig. 1 gives the non-dimensional hoop stress σφ/kzσ0 distribution on the outer surface of the bend at the point θ = 45° (half of the curvature) versus the angular positionπ/2 − φ (°). The curvature is equal to B = 250 mm. The pipe diameter is 2R = 262.5 mm and wall thickness is t = 12.5 mm. One notes that the stress intensification is significant. Numerous codes take into account ovalization such as API 579-1/ASME FFS-1 2007 Fitness-For-Service sample code, [4]. Section distortion is also an element to consider for the functionality of the pipeline. A simple way of measuring this distortion is the flattening factor f = Dmax-Dmin/D where D is the initial diameter, ovalization is this factor multiply by 100? Fig. 2. Table 1 summarizes the different boundary values of this factor considered in some codes. This paper starts from an inspection report concerning a major oil pipeline which conveys oil from western Canada to eastern Canada via the Great Lakes states. This pipe exhibits an important ovalization (8.8%).
Fig. 2. Definition of minimum and maximum diameter for the flattening parameter. 470
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Table 1 Values of the flattening factor limit according to different codes. Codes
Flattening factor f (%)
CSA-Z662-07 App.C [4] DNV-0S-F101 (2000) [5] API 1111-(1999) [6] Murray et al. [7]
3.0 3.0 5.5–6.2 4.3
Validity of codes flattening factor is limited to elastic behavior of the pipe. In the elastic plastic regime, local buckling of pipes has been described by Murray et al. [7] using the double modulus theory. Its harmfulness was studied by examination of local maximum stresses obtained by the Finite element method. This study offers the opportunity to discuss code rules about ovalization which is based only on the cross section of pipe and doesn't take into account the service pressure. For that the induced stress intensification factor is determined in order to limit the local maximum stress to half of the yield stress. Extension of code rules to combine ovalization with weld joint or corrosion defects is made. 2. Pipe geometry and material The studied pipe which is used for oil transportation was installed during the 50’s. It lies at the bottom of an American Great lake, fig. 3. Pipe diameter is 508 mm and its thickness is 20.6 mm. It has been detected that in a particular place, pipe deflection is 2.5° and ovalization is about 8.8% [8]. Characteristics of the pipe are given in Table 2. The pipe line was built in 1953 therefore using a “vintage” pipe steel. The mechanical properties are given in Table 3. 3. Results Stress and displacements were calculated using Finite Element Method (FEM) method. Abaqus™ version 6.13 software was used in our work. The following parameters were studied - The pipe is slightly bent but the study on a straight tube will be done for comparison between horizontal and vertical ovalization, - It is assumed to be localized in the middle of two supports and the result will be compared to that obtained at a location on embedding, - Ovalization length is assumed to be equal to 2 m, - Boundary conditions take into account the existence of support where the pipe is clamped - loading conditions incorporate
Fig. 3. Underwater picture of the studied pipe showing deflection and girth weld (http://media.mlive.com) 471
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Table 2 Characteristics of the studied pipe. Diameter
D (mm)
508
Thickness Working pressure Linear weight Ovality CZAZ 662 Bending angle Radius of curvature Distance between supports
t (mm) p (MPa) Kg/m % degree 91D (mm) S (m)
20.6 4.1 247.8 8.80 2.5 46,228 22.86
Table 3 Mechanical properties of the studied pipe steel. Material
Yield stress σy (MPa)
Ultimate strength σul (MPa)
Elongation at failure %
Young's modulus (GPa)
Steel A33
185
227
15
210
internal working pressure and the linear weight of the pipe is (2431 N/m) which is not negligible (fig. 4), - Material behavior is assumed to be elastic, further results indicate that the material yield stress is only exceeded by 6.1% for horizontal ovalization of 8.8%. Therfore, in the limits fixed by codes, the material remains elastic. 3.1. Results for the studied pipe The results in Fig. 5 concern the curved pipe with an angle of 2.5° and horizontal ovalization located at mid-span. They indicate that the maximum local (l) hoop stress σθθ, l increases linearly with ovalization Ov according to the following simple equation:
σθθ,l (MPa) = AB,ms,H ∗Ov + σθθ,0
(2)
where AB, H is a constant (subscript B indicating a curved pipe, ms mean the mid-span and H is the position of horizontal ovalization). σθθ, 0 is the hoop stress for the circular pipe cross section under working pressure, Fig. 5. One notes the hurtful effect of ovalization. Hoop stress is 2 times σθθ, 0 for 3% ovalization and 4.3 times σθθ, 0 for 8.8% ovalization. Central deflection d is also a linear function of ovalization according to: (3)
d(mm) = B∗Ov + d 0
where B is a new constant (B = 0.104) and d0 central deflection or a pipe with circular cross section at working pressure (d0 = 3.55 mm). Central deflection increases by 9% for 3% of ovalization, Fig. 6. 3.2. Influence of curvature Hoop stress versus ovalization was computed on straight pipe. A similar linear relationship was obtained between maximum hoop stress and horizontal ovalization.
σθθ,l (MPa) = AS,ms,H ∗Ov + σθθ,0
(4)
Subscript S indicates a straight pipe. The constant AS, ms, H = 16.55 and σθθ, 0 = 47.30 MPa. One notes that hoop stress for a circular cross section is lower for a bent pipe by 3.4%. For 8.8% ovalization, bending of the pipe increases hoop stress by 5.8%. Local stress amplification due to curvature is significantly less than ovalization. 3.3. Influence of ovalization orientation A comparison was made between horizontal and vertical ovalization for a straight pipe. Again a linear relationship was found between maximum local hoop stress and ovalization
Fig. 4. FEM pipe model with boundaries conditions 472
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Fig. 5. The hoop stress versus the ovalization for the bent pipe with horizontal ovalization.
Fig. 6. Displacement vs ovalization for the bent pipe with horizontal ovalization.
σθθ, l (MPa) = AS, ms, V ∗Ov + σθθ,0 The constant AS, V = 17.78 and σθθ, horizontal ovalization.
(5) 0
= 49.1 MPa. For 3% vertical ovalization, local hoop stress is greater by 3.2% than the
3.4. Influence of ovalization location A comparison been made between horizontal ovalization located at mid span and at embedment for a straight pipe. The following linear relationship was obtained:
σθθ,l (MPa) = AS,em,H ∗Ov + σθθ,0
(6)
The subscript (em) means location at embedment. The constant AS, em, H = 15.35 and σθθ, 0 = 47.7 MPa. For 3% horizontal ovalization, local hoop stress is greater by 3.2% at mid span than at embedment. Table 4 summarizes the different results for constant Aij (with i = ms or em and j = H or V)) and σθθ, 0. One notes a small difference of values of σθθ, 0 for the straight pipe; this is due to the value at origin by linear fit of data. From Table 4, it can be seen that ovalization location and orientation have little effect on local stress amplification neither does the curved shape of the pipe; these effects have about 5% of influence. However the major effect in stress amplification is ovalization. Local hoop stress for 8.8% ovalization is always close to the yield stress (σy = 185 MPa) and justify stress computing with elastic behavior. However pipe design never allows the design stress higher than yield stress divided by a safety factor. Consequently the actual ovalization cannot be accepted and the associated pipe part needs to be replaced where working pressure reduced. 473
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Table 4 Values of constant Aij (with i = ms or em and j = H or V)) and σθθ,
0.
Pipe shape
Orientation
Location
Ai,
Straight Straight Straight Bent Bent
Horizontal Vertical Horizontal Horizontal Horizontal
Mid-span Mid-span Embedment Embedment Mid-span
16.55 17.72 15.35 15.36 16.67
j
σθθ,
0
(MPa)
47.3 49.1 47.7 47.7 46.4
σθθ, l (8.8%) 195.4 MPa 205.6 MPa 184.9 MPa 185.47 MPa 196.4 MPa
Local stresses have been computed using elastic behavior, however for 8.8% ovalization, yield stress is exceeded of a few percent.
4. Discussion 4.1. On 3% value of admissible ovalization The CSA Z 662-07 App.C [4] and DNV-OS-F101 [5] codes prescribe a maximum value for ovalization. This value is independent of ovalization orientation and working pressure and it is a pure geometrical criterion. We have seen that vertical orientation induces higher local hoop stress but the difference is small and can be considered as negligible therefore it is justified that codes don't take into account the position of the major axis of the oval cross section. The value of working pressure is also not taken into account. For a working pressure ps = 4.1 MPa, Table 5 gives the local maximum hoop stress for 3% of ovalization and the ratio between this stress and yield stress σy One notes that for each case, a safety factor close to 2 is obtained on yield stress but depends on circumstances. The safety factor is defined as the ratio of yield stress over maximum local stress. For each working pressure, the local maximum hoop stress σθθ, l is a linear function of the circular cross section hoop stress (σθθ, 0)
σθθ,l = kA σθθ,0
(7)
kA (Ov) is the stress intensification factor which depends on the ovalization. (8)
kA = k Ov ∗Ov + 1 The hoop stress for circular cross section is a function of the working pressure ps
σθθ,0 = ps
D 2t
(9)
where t is the pipe thickness and D the internal diameter. Assuming that local maximum hoop stress cannot exceed half the yield stress:
σθθ, l = kA ps
σy D = 2t 2
(10)
By replacing kA by Eq. 7:
( kOv ∗OvL + 1) ∗ps
σy D = 2t 2
(11)
Kov is a parameter which depends on pipe geometry. The maximum admissible ovalization OvL is a function of working pressure according to:
OvL =
σy. t − D . ps D . ps . kov
(12)
Eq. 12 indicates that the admissible ovalization OvL allow a local maximum hoop stress lower than yield stress divided by 2 decreases as a power function with working pressure. It depends on pipe geometry through diameter, thickness and parameter kov. Ovl versus the working pressure is presented in Fig. 7. One notes that the admissible ovalization for the present working pressure (4.1 MPa) is 3.09% which is considerably less than the studied case (8.8%). The maximum working pressure in order to never violate the assumption on maximum local hoop stress less than yield stress divided by 2 is the pressure of 6 MPa. Beyond this value, no Table 5 Local maximum hoop stress for 3% of ovalization and values of ratio σθθ,l/σy Pipe shape
Orientation
Location
σθθ, l (3%)
σθθ, l/σy
Straight Straight Straight Curved
Horizontal Vertical Horizontal Horizontal
Mid-span Mid-span Embedment Mid-span
93.91 MPa 97.07 MPa 93.91 MPa 96.45 MPa
0.51 0.52 0.51 0.52
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Fig. 7. Admissible ovalization to allow a local maximum hoop stress less than yield stress divided by 2 versus the working pressure.
ovalization is tolerated. A safety factor of 2 is traditionally used in the design. The present pipe steel is a low strength steel for which an empirical relationship exists between fatigue endurance σD limit and yield stress:
σD = σy /2
(13)
The choice of this safety factor value ensures in addition the protection against fatigue induced by the variation of working pressure. However, it is necessary to correct the endurance limit by the stress ratio. 4.2. On the admissible value for combined ovalization and welding joint Another aspect which is not taken into account in codes is ovalization combined with a defect (welded joint or corrosion defect). For conservative reasons, these defects are assumed to be located at the same place of the ovalization. A welded joint introduces the local stress concentration. This formula is valid if the material remains elastic. The elastic assumption allows additively of stress and Eq. 8 becomes:
kA = kOv, w ∗Ov + kt
(14)
The condition of maximum value of local hoop stress is rewritten as:
(kOv ∗OvL, w + kt ) ∗ps
σy D = 2t 2
(15)
The maximum admissible ovalization combined with a weld joint OvL,w is a function of working pressure according to:
OvL, w =
σy. t − (kt ) D . ps D . ps . kov, w
(16)
In the following case, the welded joint has an angle Ψ = 45° and a weld toe radius ρ = 0.5 mm. Again a linear relationship was found between maximum local hoop stress and ovalization Ov
σθθ, l (MPa) = AB, H , ms, W ∗Ov + σθθ,0, w
(17)
Subscript W is used for a welded joint. Coefficient AB,H,msW =19.22 and hoop stress under working pressure for a circular cross section with a welded joint σθθ, 0, w = 61.9 MPa,see Fig. 8. This value allows to compute an experimental value of stress concentration factor kt = 1.225 which is close to the numerical value [9]. The coefficient kov, w was found as kov,w = 0.38,. The admissible ovalization OvL,w for ovalization combined with welded joint is 1.59%. This value for the same conditions, but without a welded joint is 3.09%. 4.3. On the admissible value for combined ovalization and corrosion defect Corrosion defects are characterized by 3 dimensions: depth (a), length (2c) and width (W). The failure of a corrosion defect which considered as a semi-elliptical, is controlled by its size and the flow stress σ0 of the material. The input parameters include pipe inner diameter (D), wall thickness (t), specified minimum yield strength (σy), Maximum Allowable Operating Pressure (MAOP), longitudinal extent of corrosion (2c) and defect depth (d). According to several codes and particularly ASME B 31 [10], a corrosion defect in a pipe is treated as a loss of material. Therefore the effective thickness tef is considered to be bearing the working pressure: 475
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Fig. 8. The maximum local hoop stress versus the ovalization for the bent pipe with horizontal ovalization and welded joint.
(18)
t ef = t − a
Numerous solutions of limit pressure (pL) for a corrosion defect are proposed in codes and limit load analysis literature (ASME B31G [10], modified ASME B31G [11], DNV RP-F101 [12]. The limit pressure (pL) which is less than the burst pressure pc is generally expressed as:
() () a
⎛ 1−α t pL = 2t . σ0 . ⎜ D ⎜1 − α a . t ⎝
1 M
⎞ ⎟ ⎟ ⎠
(19)
M is the Folias correction taking into account the pipe curvature, flow stress σ0, and α a geometrical parameter and (a/t) the relative defect depth. M and α take different values according to the method applied. Flow stress is generally a simple function of yield stress and ultimate strength and a is the defect depth. Condition on maximum value of local hoop stress is rewritten as:
σy D = 2(t − a) 2
(20)
σθθ,l (MPa) = AS,H,ms,cor ∗Ov + σθθ,0,cor
(21)
σθθ,l = kA ps
The subscript cor means corrosion defect. For a relative defect depth (a/t) = 0.5, coefficient AS,H,ms,cor = 59.87 and hoop stress under working pressure for a circular cross section with corrosion defect of relative depth a/t = 50% σθθ, 0, cor = 105.4 MPa. These two parameters depend on relative defect value, see Table.6 The stress intensification factor depends on ovalization according to:
kA = k Ov,cor ∗Ov + 1
(22)
The Finite Element Method results indicate that kOv,
cor
is a linear function of relative corrosion defect depth:
k Ov,cor = 0.49(a/t) + 0.31
(23)
For ovalization larger than 8.1%, computing is done with elasto-plastic behavior of the material which is assumed to follow Ludwik's law: (24)
σ = Kεn
where n is the strain hardening exponent (n = 0.1 for pipe steel A33) and K is the strain hardening coefficient (K = 344 MPa for the Table 6 Values of parameters As,H,ms,cor
a/t = 0.1 a/t = 0.5
andσθθ, 0, cor
with relative defect depth (a/t)
As,H,ms,cor
σθθ,
33.93 59.87
84.38 105.4
476
0, cor
(MPa)
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Fig. 9. The maximum admissible ovalization versus to the relative defect depth (a/t) for different working pressures
same steel). The combination of ovalization and a weld joint creates local plasticization in the worst cases. The maximum admissible value of ovalization combined with a corrosion defect OvL,cor is a function of working pressure and defect depth according to:
OvL,cor =
σy . (t − a) − D. ps D. ps . k ov,cor
(25)
The maximum admissible ovalization decreases linearly with the relative defect depth (a/t) and with the working pressure see Fig. 7. For a pipe with ovalization the limit pressure is given by modifying the Eq. 18:
() () a
pL =
2t . σ . KA D 0
⎡ 1−α t ⎢ ⎢1 − α a . t ⎣
1 M
⎤ P ⎥ = L,0 KA ⎥ ⎦
(26)
The defect assessment method used, ASME B31G code is based on the estimated repair factor (ERF) in which the safe operating pressure associated with each corrosion defect is the limit pressure. ERF is the ratio of the Maximum Allowable Operating Pressure (MAOP) and the safe operating pressure pL. Necessity for repair is given by the following criterion:
ERF =
MAOP ≤ 1 or 0.95 pL
(27)
ERF greater than one requires repair or lowering of the MAOP. The evolution of the estimated repair factor ERF is entered in Fig. 9 for a corrosion defect of relative depth a/t = 0.1 and aspect ratio c/a = 10 and for the working pressure of 4.1 MPa. The ERF increases linearly with ovalization and the maximum admissible ovalization with this criterion is Ov,L = 4.8%. This indicates that the pipe failure occurs for a greater ovalization and greater corrosion depth. One notes that the criterion of ERF = 1 for admissible defect depth is less conservative that the criterion of local hoop stress equal to yield stress divided by 2. 4.4. On fatigue risk induced by ovalization It has been seen that ovalization increases stress locally. For admissible ovalization or less the strain remains elastic but for large ovalizations (Ov > 4.8%) the strain must be plastic. Working pressure in a pipe is subject to variation with time. Data about such variations is not available for the studied pipe. Therefore, in the following we used the fluctuation of gas pressure inside pipelines provide by M. Dadfarnia et al. [13]. Fig. 11 shows the pressure data for a period of one year. It is assumed that such pressure fluctuations will be repeated every year. Since the pressure fluctuation range is not constant, a proper cyclic counting method was adopted to obtain the pressure range under these variable amplitude loading conditions. One notes that the stress ratio R is different to zero Figs. 10 and 11. Therefore the fatigue law needs to be corrected to take into account the R ratio effect on fatigue. In the elastic state, the general Basquin law is used:
σa, R = σ ′f (Nr )b ′
(28)
where σf is the fatigue resistance and b the Basquin’ exponent.σa,
R
477
is the equivalent stress amplitude given for stress ratio R:
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Fig. 10. Evolution of the estimated repair factor ERF for a combined ovalization-corrosion defect (relative depth a/t = 0.1 and aspect ratio c/ a = 10, working pressure of 4.1 MPa) vs ovalization.
Fig. 11. Working pressure evolution for a gas pipe over approximatively one year ()
σa σ − σm = ul σa, R σul
(29)
Where σa is the stress amplitude (half of the stress range): Therefore Basquin's fatigue law is rewritten as:
σ σa = ⎛1 − m ⎞ σ ′f (Nr )b σul ⎠ ⎝ ⎜
⎟
(30)
where σm is the mean stress and σul the ultimate strength. For low cycle fatigue, the Manson-Coffin low cycle fatigue law is written as: c/b σ ′f ∆ϵt ⎛1 − σm ⎞ (Nr )b + ϵ′ ⎛1 − σm ⎞ (Nr )c = f 2 E⎝ σul ⎠ σul ⎠ ⎝ ⎜
⎟
⎜
⎟
(31)
′
where ϵf is the fatigue ductility and c the Coffin exponent. Δϵt is the total strain range. Fatigue properties of Pipe steel A33 are given by an empirical relationship from static mechanicals properties and entered in Table 7. The working pressure evolution presented in Fig. 11 indicates that the average load ratio is about 0.75 and the mean pressure is Table 7 : Low cycle fatigue parameters for pipe steel A33 Parameter
Empirical relationship
value
Basquin's exponent b Coffin's exponent c Fatigue ductility in ϵf′ Fatigue resistance σf′ (MPa)
b = −0.12 c = − 0.6 ϵf′ = (ϵf′)0.6 σf′ = 3.5σul
0.12 0.6 0.32 795 MPa
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Table 8 Low cycle fatigue condition for a combined ovalization-corrosion defect (relative depth a/t = 0.1 and aspect ratio c/ a = 10, working pressure of 4.1 MPa) Pressure conditions
Working pressure variations
Stop and go
Mean pressure/working pressure Strain range Number of cycles to failure
0.75 0.0008 6.1.106
0.5 0.0025 247,000
equal to 0.88 of working pressure. In the next we assume that for the studied pipe, loading conditions are similar. If we consider only pressure variation during operating service, the use of the equation indicates that the pipe is operating in conditions of local hoop stress below the endurance limit. If we consider the particular case where working pressure drops to zero and increases to his service value (stop and go procedure), the load ratio is zero. Ultimate strength is reached for ovalization equal to 4.8% for a corrosion defect with relative depth a/t = 0.1, aspect ratio c/a = 10 and for the working pressure of 4.1 MPa (the worst case). For limit ovalization (3%), the total strain amplitude is 0.0025 and the number of cycles allowed for stop and go is 247,000, Table 8. One notes that the presence of limit ovalization (3%) and corrosion defect (relative depth a/t = 0.1) tolerate a large number cycles of pressure cycles. However, attention needs to be given to stop and go procedure and therefore maintenance. 5. Conclusion Ovalization in a pipe under working pressure greatly increases the local hoop stress. The stress intensification factor of the studied pipe with ovalization of 8.8% under working pressure of 4.1 MPa is 3.5 times the hoop stress of a circular cross section under same working pressure and is not admissible. The location and orientation of the ovalization has a little effect on the local hoop stress value as well as the curvature of the pipe. Codes such as CSA and DNV allow a maximum ovalization of 3%. This recommendation is only geometric and does not take into account working pressure. Combined weld joint or corrosion defects are totally ignored by the codes. Using the simple criterion of a maximum admissible hoop stress less than the yield stress divided by the safety factor of 2, a model assuming that the pipe remains elastic under working pressure shows that admissible ovalization Ov,ad decreases as a power function of working pressure. Ov,ad depends on the pipe geometry through diameter, thickness and parameter kov with governs stress intensification. For a welded joint combined with ovalization, the admissible value depends on the stress concentration factor of the welded joint. For a corrosion defect combined with ovalization, the admissible value depends on relative defect depth. References [1] D. Abdulhameed, R. C, The influence of bourdon effect and ovalization effect on the Stress distribution on pipe elbows, 11th Pipeline Technology Conference 2016, EITEP Institute, 2016. [2] CSA Z662-15 Canadian code for oil and gas pipeline systems. [3] I.V. Orynyak, Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect, Int. J. Solids Struct. 44 (5) (2007) 1488–1510. [4] API 579-1/ASME FFS-1, Fitness-For-Service, 2007, 2007. [5] DNV-0S-F101 (2000) [6] API 1111-(1999) [7] N.W. Murray, P. Bilston, Local buckling of thin-walled pipes being bent in the plastic range, Thin-Walled Struct. 14 (5) (1992) 411–434. [8] Private communication @garretellison [9] G. Glincka, The weld profile effect on stress intensity factors in weldments, Int. J. Fract. 35 (1987) 3–20. [10] American National Standard Institute (ANSI)/American Society of Mechanical Engineers (ASME), Manual for determining strength of corroded pipelines, ASME B31G, 1984. [11] J. Kiefner, P. Vieth, A modified criterion for evaluating the strength of corroded pipe, Final Report for PR 3-805 Project to the Pipeline Supervisory Committee of the American Gas Association, 1989 Battelle, Ohio. [12] Recommended Practice, DET Norske Veritas, DNV-RP-F101, Corroded Pipelines, (2004). [13] M. Dadfarnia, B.P. Somerday, P. Sofronis, I.M. Robertson, Interaction of hydrogen transport and material elastoplasticity in pipeline stee1s, J. Press. Vessel. Technol. 131 (2009) 041404.
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