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Analytical and numerical investigation using limit analysis on the ductile failure of pipes containing surface cracks
Engineering Failure Analysis 104 (2019) 480–489
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Analytical and numerical investigation using limit analysis on the ductile failure of pipes containing surface cracks
T
⁎
M. Mouwakeha, , S. Masria, M. Hadj Melianib,d, R.K. Suleimanc, G. Pluvinaged, M. Nait-Abdelazize a
Department of Applied Mechanics, Faculty of Mech. Eng., Aleppo University, Aleppo, Syria LPTPM, Hassiba Benbouali University of Chlef, Chlef 02000, Algeria c Center of Research Excellence in Corrosion (CoRE-C), King Fahd University of Petroleum & Minerals (KFUPM), Dhahran 31261, Saudi Arabia d University of Lorraine, LEM3 Laboratory, Ecole d'Ingénieur de Metz, France e University of Lille, Unité Mécanique de Lille, Av. Paul Langevin, 59650 Villeneuve d'Ascq, France b
A R T IC LE I N F O
ABS TRA CT
Keywords: Limit analysis Constraint factor Surface crack Finite element analysis Polyethylene pipe
Using cracked pipes design standards and finite element analysis, the limit load analysis for pipes containing surface cracks was determined. The study was performed on five pipes of different diameters with a constant crack length and depth. The crack geometry is a semi-elliptical surface crack. The cracked pipes are subjected to internal pressures which are obtained from formulas of cracked pipes design standards. Due to the ductile behavior of polyethylene pipes, the failure occurs when the critical stress reaches a value equal to the ultimate tensile strength multiplied by a constraint factor. In this work, the constraint factor was calculated and its evolution with the pipe diameter was analyzed. Three different definitions of a constraint factor based on global or local approaches were also compared, so that a new failure criterion can be obtained. The new failure criterion makes the prediction of the pipe residual life possible which, in turn, facilitates a systematic approach to maintenance and replacement of pipes.
1. Introduction The first appearance of the theory of limit analysis was in the late 1930s of the last century [1]. The theory constitutes a branch of the plasticity theory that is related to elastic/perfectly plastic behavior. In the mid-1950s, a large number of analytical solutions were firstly found to calculate the ultimate load of beams and shells, leading to more realistic values of the capability to resist plastic collapse. This led progressively to the design of concepts based on a limit state that is replacing those based on allowable stress. The introduction of Linear Elastic Fracture Mechanics in the mid-1950s [2] led designers to consider principally the risk of brittle fracture governed by the global stress in apparent opposition to the theory of plastic collapse governed by the nominal stress. In the early1970s, Dowling and Towley [3] showed that the theories of limit analysis and fracture mechanics can be combined to form a novel approach, i.e., the two criteria approach which is the basis for the method of integrity-failure diagrams. The newly developed idea in the method of two criteria is to consider an interaction between the two mechanisms so that the ultimate load Pult is less than that given by limit analysis Pl.a and higher than that given by elastic fracture mechanics Pf.m: Pf.m < Pult < Pl.a. Numerous works have been conducted to obtain limit load solutions in pipes containing surface cracks subjected only to internal pressure, or to combined loads (internal pressure and bending) [4–10]. Since pipes generally fail in a ductile manner due to the
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Corresponding author at: LPTPM, Hassiba BenBouali University of Chlef, P.O.Box. 151, Hay Salem, 02000 Chlef, Algeria. E-mail address: moufi[email protected] (M. Mouwakeh).
https://doi.org/10.1016/j.engfailanal.2019.06.007 Received 14 March 2019; Received in revised form 19 May 2019; Accepted 4 June 2019 Available online 05 June 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Failure Analysis 104 (2019) 480–489
M. Mouwakeh, et al.
σN,c σm σeff σg.c. E L Pf σyy
Nomenclature a l D t P σu (Rm)
crack depth crack length pipe diameter pipe thickness internal pressure in pipe ultimate tensile strength
critical net stress mean or average stress effective stress critical gross stress elastic Young modulus constraint factor critical failure pressure opening stress
behaviour of the constitutive material, failure prediction tools are mostly based on the limit analysis. This failure criterion assumes that failure occurs when critical net stress σNc reaches ultimate tensile strength σu. We note that ductile failure is sensitive to critical net stress σNc(load divided by the ligament cross section) whatever brittle failure is sensitive to gross stress σgc (load divided by the entire section). The above-mentioned criterion needs to be modified to take into account constraints due to loading mode and geometry effects in the following manner:
σN,c = L. σu
(1)
where L is called the constraint factor. Design standards for pipes with defects such as ASME B31G, Modified ASME B31G [11], DNV RF101 [12] and CHOI formulas [13] are based on limit analysis and allow the calculation of the critical failure pressure for corroded pipes. They also incorporate a safety factor which defines a lower bound of the experimental results [14–16]. The basic question is the adopted value of the constraint factor and its closeness to values generally obtained. In addition, it is interesting to capture the evolution of constraint factor with ligament size, defect and pipe geometries as follows:
a a L = L⎛ , ⎞ ⎝D t ⎠
(2)
where D is the pipe diameter, t is the thickness, and a is the defect depth. In this work, the determination of the critical failure pressure using the above-mentioned standards has firstly been conducted for five cases by varying both D and t parameters. The geometry of the defect was remained unchanged. After calculating the critical failure pressures Pf, values of the critical net stresses and the corresponding constraint factors L can be consequently calculated. Indeed, according to Eq. (1), the constraint factor can be expressed as follows [17–19]:
L = σN,c/σu
(3)
Moreover, a second way to obtain the constraint factor was investigated. Using a finite element method (FEM), the calculation of stress distribution along the ligament was achieved, which allows us to obtain the average values of maximal principal stresses σm. Another calculation of the constraint factor deduced from FEM analysis can be therefore obtained as follows:
L∗ = σm/σu
(4)
Finally, the volumetric method [10] was also used to offer an alternative way to compute the constraint factor [20–24]. This method requires one to determine, from the variation of stress distribution away from crack tip, the fracture process zone (FPZ) using the volumetric method which is a local fracture criterion. It assumed that the fracture process requires a certain volume which is assumed to be a cylinder in which the diameter of the cylinder is the effective distance. The physical meaning of this fracture process volume is “the high stress region” where the necessary fracture energy release rate is stored. Inside the fracture zone, the effective stress σeff, which is the average value of the stress distribution, acts as a local fracture stress. Consequently, a third definition of the constraint factor can be assumed:
L∗∗ = σm/σu
(5)
This work aims to compare and discuss these different constraint factors in order to propose a new failure criterion through the designing of pressurized pipes, taking into account the following remarks:
• Limit analysis formulas (ASME B31G, Modified ASME B31G, DNV RF101, and CHOI) consider the defect as a lack of material. They neglect the stress concentration induced by the defect, • Similar to limit analysis formulas, they refer to the nominal stress, i.e., the average stress in the ligament surface under defect, • The volumetric method takes into account the stress concentration induced by the defect, i.e., the stress concentration relative to the gross stress to separate from the stress increase due to the reduction of the ligament surface, • This stress concentration is evaluated by the constraint factor, • The comparison between the methods makes it possible to estimate the error using a much more conservative limit analysis. This
is why limit analysis is used to establish a relative value. This paper is organized as follows: in Section 2, we will present the material and geometries before describing the calculation of critical failure pressures and stresses in Section 3 according to the standards previously introduced and the corresponding constraint factors L expressed by Eq. (3). Sections 4 and 5 are dedicated to the calculation of the additional constraint factors L* and L**. In Section 6, the results obtained through the different methods are discussed and followed by the concluding remarks. 481
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2. Material, pipes and crack geometry The studied material is a high-density polyethylene whose main properties are: σu = 25 MPa, ρ = 960 kg/m3, E = 450 MPa, υ = 0.45, where σu, ρ, E, and υ are the ultimate tensile stress, mass density, Young's modulus and Poisson's ratio, respectively. In this study, five pipes with different wall thicknesses are chosen: (D = 75 mm, t = 6.8 mm), (D = 90 mm, t = 8.2 mm), (D = 110 mm, t = 10 mm), (D = 125 mm, t = 11.4 mm), and (D = 160 mm, t = 14.6 mm). The crack geometry is assumed to be a semi-elliptical surface crack as shown in Fig. 1, where l = 2c is the crack length and a is the crack depth. The crack dimensions remain constant for all pipes. The crack dimensions are the following: l = 2c = 100 mm, a = 5 mm. 3. Calculation of constraint factor L using design standards 3.1. Critical failure pressure The critical failure pressure is calculated for different pipes using the formula given by the various standards. 3.1.1. ASME B31G standard This semi-empirical approach was developed in the late 1960s and early 1970s and is generally used to calculate the residual strength of the pressurized defected pipes [11]. Based on a wide and comprehensive series of experiments conducted on different sections of defected pipes, the failure pressure was formulated, taking into account the following two conditions: 1. The maximum value of the circumferential stress should not exceed the yield stress, i.e., σθθ < σy. 2. A relatively short defect can be represented in the form of a parabola while a long one is represented in the form of a semielliptical shape. According to this standard, critical failure pressure is given by the following relationship:
Pf =
⎡ 2(1.1σy ) t ⎢ 1 − ⎢ D ⎢1 − ⎣
( )( ) ⎤⎥ ⎥ ( ) ( ) ⎥⎦ 2
3
a
a
2
3
t
t
M
where
M=
l 2 D L 2 D 1 + 0.8 ⎛ ⎞ ⎛ ⎞ for 0.8 ⎛ ⎞ ⎛ ⎞ ≤ 4 ⎝D⎠ ⎝ t ⎠ ⎝D⎠ ⎝ t ⎠
(6)
where D, a, t, M, σy and l are the outer diameter, crack depth, wall thickness, bulging factor, yield stress and longitudinal corrosion defect length, respectively. 3.1.2. Modified ASMEB 31G standard The modified ASME B31G standard incorporates the new bulging factor M, so that the critical failure pressure Pf is given by the following relationship:
Pf =
M= For:
( ) ⎤⎥
a ⎡ 2(1.1σy ) t ⎢ 1 − 0.85 t ⎢ D (a t) ⎢ 1 − 0.85 M ⎣
⎥ ⎥ ⎦
(7)
l 2 D l 4 D 2 1 + 0.6275 ⎛ ⎞ ⎛ ⎞ − 0.003375 ⎛ ⎞ ⎛ ⎞ ⎝D⎠ ⎝ t ⎠ ⎝D⎠ ⎝ t ⎠
l 2 D D t
( ) ( ) ≤ 50
Fig. 1. Pipe and crack geometry. 482
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3.1.3. DNV RP-F101 standard The DNV RP-E101 standard is the first general and comprehensive standard for the calculation of pipes containing surface cracks subject to solely internal pressure loads, or combined loads consisting of internal pressure and bending loads [12]. According to this standard, the critical failure pressure is given by the following relationship:
Pf =
For: Q =
2(σU) × D−t
t ⎡ 1 − a/t ⎤ ⎢ ⎣ 1 − (a/t)/Q ⎥ ⎦
1 + 0.31
( ) where D, a t, and σu are identical to the terms previously introduced, and Q is the bulging factor. 1 Dt
(8)
2
,
3.1.4. CHOI standard Combining limit analysis and finite element analysis of a pipe containing a surface crack [13], the equation of critical failure pressure is derived and given by the following relationship: 2
Pf = 0.9 ×
2(σU) × t ⎡ ⎛ L ⎞ ⎛ L ⎞⎤ ⎥ ⎢C0 + C1 Rt + C2 Di ⎝ ⎠ ⎝ Rt ⎠ ⎦ ⎣ ⎜
⎟
⎜
⎟
(9)
where σU, Di, t, and R are the ultimate tensile strength, inside diameter, wall thickness and average pipe radius, respectively and C0, C1, and C2 are given in terms of a/t. Fig. 2 shows the variation of critical failure pressure as a function of a/D ratio. We note that the critical failure pressure decreases linearly with the increase of a/D ratio or with decreasing the diameter. As shown in Fig. 2, the standards ASME 31G, modified ASME 31G, and CHOI exhibit the same trends, i.e., the critical failure pressure decreases linearly versus a/D ratio, while it remains constant for DNV standard.
3.2. Calculation of critical net stress Critical net stress is calculated from the critical failure pressure values for different diameters using the formula:
σN , c = Pf . D/2t∗
(10)
where Pf is the critical failure pressure and t∗ is the thickness of the ligament and equals to: (11)
t∗ = t − a
Similarly, the variation of the critical net stress versus a/D ratio is shown in Fig. 3. We note that the critical net stress increases upon increasing the a/D ratio.
3.3. Calculation of constraint factor L The constraint factor L is calculated according to the different standards by using Eq. (3). Table 1 below shows the obtained values. Since the constraint factor is obtained by dividing the critical net stress by a constant (the ultimate tensile stress σu=25 MPa), the evolution of this parameter as a function of a/D ratio is the same to that shown in Fig. 3. While the values of constraint factor obtained for ASME, Modified ASME and Choi methods exhibit the same trends with a relative difference of about 15%, the DNV standard is most conservative, the value of the constraint factor being approximately two times higher than that issued for the other standards when considering the maximum a/D ratio.
C ri tical Pressure Pf (MPa)
6
5
DNV
4
ASME
3 2
Modified ASME
1
Choi
0 0,03
0,04
0,05 a/D
0,06
0,07
Fig. 2. Variation of critical failure pressure versus a/D ratio. 483
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120 DNV
100
(Mpa)
60
n,c
80
40
ASME
Modified ASME
20 0 0,03
Choi 0,04
0,05 a/D
0,06
0,07
Fig. 3. Variation of critical net stress versus a/D ratio. Table 1 Values of constraint factors for all standards. D (mm)
L ASME
L M.ASME
L DNV
L Choi
75 90 110 125 160
2.284 1.8 1.574 1.484 1.374
1.804 1.56 1.442 1.393 1.328
4.151 2.818 2.2 1.96 1.673
1.568 1.508 1.499 1.468 1.381
4. Calculation of the constraint factor L* using fem 4.1. Calculation of the mean of maximum stresses acting on the ligament The finite element software ABAQUS® was used for the computations reported in this study. The material was assumed to be elastic/perfectly plastic obeying the Von-Mises flow criterion. Critical failure pressure values which were previously computed have been applied on the internal surface of the pipes. In terms of boundary conditions, the ends of the pipe are fixed. FE calculations were performed for different diameters to obtain the equivalent Von-Mises and maximum principal stresses. The meshing of the 3D finite element model consists of hexahedral elements. A longitudinal section of the pipe illustrating the mesh around the crack zone is given in Fig. 4. An example of the maximum principal stress distribution is shown in Fig. 5. Mean maximum stress σm, which is the average of the maximum principal stresses acting over the ligament, can be deduced as a function of the critical failure pressure.
4.2. Calculation of the constraint factor L* The corresponding constraint factor L* deduced from the mean maximum stress was obtained by using Eq. (4). Table 2 shows the constraint factor values L* according to mean maximum stress values obtained from FE computation.
Fig. 4. Mesh model used for the pipe (longitudinal section). 484
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Fig. 5. Maximum principal stress distribution around crack zone (longitudinal section). Table 2 Values of constraint factors L* for all standards. D (mm)
L٭ASME
L٭M.ASME
L٭DNV
L٭Choi
75 90 110 125 160
1.148 1.456 1.145 1.122 1.088
1.012 1.308 1.068 1.062 1.026
1.836 2.076 1.467 1.386 1.277
0.914 1.275 1.102 1.121 1.111
5. Determination of constraint factor L** using volumetric method 5.1. Calculation of the effective stress σeff Similarly, by using FEM and the ABAQUS program and applying the same boundary conditions used in the above modeling, we have calculated the stress distribution around the crack tip in both longitudinal and radial directions. A mesh refinement, shown in Fig. 6, has been performed in the region close to the crack tip in order to capture as accurately as possible the stress gradient and consequently the stress distribution in the vicinity of the crack tip. The evolution of the maximum principal stress as a function of the distance to the crack tip has been estimated both in the longitudinal and radial directions. A schematic representation of the expected evolution is shown in Fig. 7. Since the same trends are expected whatever the direction (longitudinal or radial), the same comments can be given. Such a diagram allows the determination of the effective distance Xeff, i.e., the size of the fracture process zone (FPZ) which extends from the crack tip up to the onset of the LEFM stress distribution (the straight line on the figure). Consequently, the effective stress can be deduced in order to calculate the stress intensity factor Kρ which is expressed as follows:
K ρ = σeff 2πXeff
(12)
Table 3 shows the values of effective stress, effective distance and stress intensity factor calculated by the volumetric method, based on the distribution of the maximum principal stresses as a function of the distance from the crack tip in the longitudinal
Fig. 6. Meshing model around the crack zone (longitudinal section). 485
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Fig. 7. A schematic distribution of the maximum principal stresses as a function of the distance from the crack tip and deducing the dimension of the plastic zone and the value of effective stress. Table 3 Effective stress, effective zone dimension, and stress intensity factor for ASME standard. Pf (MPa)
σmax (MPa)
σeff (MPa)
Xeff (m)
Xn (m)
Kρ (MPa.√m)
D (mm)
Standard
2.657 3.102 3.468 3.684 3.996
15.08 13.75 15.13 15.63 17.12
13.15 13.65 14.88 15.41 16.54
0.0021 0.005 0.004 0.0043 0.0053
0.004 0.009 0.0082 0.009 0.009
1.510 2.419 2.358 2.532 3.018
75 90 110 125 160
ASME Longitudinal Direction
direction for ASME standard. Table 4 shows also the values of effective stress, effective distance and stress intensity factor calculated by the volumetric method, based on the distribution of the maximum principal stresses as a function of the distance from the crack tip in the radial direction for ASME standard. Similarly, effective stress, effective distance and stress intensity factor calculated by the volumetric method in the longitudinal and radial direction for the other standards lead to the same trends. When comparing the effective stress values in longitudinal and radial directions for all pipes and standards, we note that the effective stress values in the radial direction are always greater than those in the longitudinal direction. This means that the crack will likely propagate in the radial direction. Therefore, the effective stress values in the radial direction are used to calculate the constraint factor L** because they cause a greater risk on the ligament of the pipes and cause the transformation of the crack from surface crack to through-wall crack and thus the plastic collapse of the pipe. 5.2. Calculation of constraint factors L** and L*** The constraint factor L** is given by the following expression:
L∗∗ = σeff / σu
(13)
Furthermore, the constraint factor L*** is also calculated by the following expression:
L∗∗∗ = σg . c / σu
(14)
where σg. c is the critical gross stress and σu is the ultimate tensile strength. The critical gross stress is defined as follows:
σg . c = Pf . D /2t
(15)
where Pf is the critical failure pressure, D is the pipe diameter, and t is the total wall thickness. The values of the critical net stress, mean stress, effective stress, and gross stress for ASME standard in the radial direction are Table 4 Effective stress, effective zone dimension, stress intensity factor for ASME standard. Pf (MPa)
σmax (MPa)
σeff(MPa)
Xeff (m)
Xn (m)
Kρ (MPa.√m)
D (mm)
Standard
2.657 3.102 3.468 3.684 3.996
40.57 60 64.5 62 50
34.8 55.5 59.5 56 33.5
0.00083 0.0006 0.00062 0.0006 0.00155
0.0018 0.00143 0.00125 0.0014 0.003
2.513 3.407 3.713 3.438 3.305
75 90 110 125 160
ASME Radial Direction
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shown in Table 5. The values of the corresponding constraint factors are reported in Table 6. Similarly, for the other standards, the same trends are obtained in terms of the values of critical net stress, mean stress, effective stress, gross stress, and the corresponding constraint factors. 6. Results and discussion The constraint factor gives a good idea of how much stress is increased due to constraints related to the triaxiality state of the stress field. Furthermore, the constraint factor depends on the pipe geometry, ligament size, pipe wall thickness and on the stress gradient. Fig. 8 presents schematically the stress-strain curve of the pipe material with and without a constraint. The symbol ‘?’ in Fig. 8 refers to the kind of critical stress and the corresponding constraint factor. When a constraint is accounted for, the level of the stress values depends on the definition of the constraint factor, as previously shown. Fig. 9 gives an illustrative example of the stress distribution around the crack tip in the radial direction for a pipe of diametr D = 125 mm and subjected to a critical failure pressure calculated by the ASME standard. It shows also the values of different critical stresses compared to ultimate tensile strength (σu = Rm) and it is observed that:
σeff > σN . c > σm > σu (Rm) > σg . c Fig. 10 shows the variation of constraint factors in terms of pipe diameter (75, 90, 110, 125, 160 mm) and standard (ASME B31G, M. ASME B31G, DNV, and CHOI) used in the calculation of critical failure pressure. We note generally that:
L > L∗ > L∗∗∗ From Fig. 10, we note that the values of the factor L*** (blue) are nearly equal or less than one. This confirms the fact that the ductile failure is not sensitive to critical global stress but rather to critical local stress. Thus, the factor L*** cannot be considered as a constraint factor and its values are taken as indicative values. As to constraint factor L** (violet), it is primarily the most realistic value of the constraint factor if we assume that the ductile failure requires a fracture process zone; thus fracture occurs when the effective stress reaches a critical value. Furthermore, L** rarely exceeds the value of 3.0, which is smaller than the theoretical value of 10, which is the constraint factor L** value for plane strain and Poisson ratio ν = 0.45 for polyethylene. Finally, regarding the constraint factor values L* (sky blue) which are always smaller than L (red), it can be said that both L and L* values refer to the average value of the nominal stress applied on the ligament. The critical stress σm represents the average value of the stress distribution on the ligament while the critical stress σN,c represents the critical net stress affecting the reduced thickness of the pipe, and this certainly explains the difference between the factors L and L*. 7. Conclusions From the previous results, the following conclusions can be drawn:
• The results of the standards ASME B31G, Modified ASMEB 31G, and CHOI in terms of critical failure pressures, critical stresses • • • •
and constraint factors are close to each other, while DNV standard results give more conservative values. Therefore, one of the three first standards is recommended. The values of constraint factor L∗ are always smaller than the values of factor L for all diameters and for all standards. The factor L∗could be taken as an appropriate constraint factor. Consequently, a suitable failure criterion for the cracked pipes can be derived so that the failure occurs when the critical stress σm reaches a value equal to ultimate tensile strength σu multiplied by constraint factor L∗ by means: σm = L∗. σu. The factor L∗∗ computed from the fracture mechanics considerations is the most realistic value of the local constraint factor because it focuses on the stress distribution at the crack tip. To make it an engineering tool for cracked pipes design, it must be simplified because its evaluation is quite complex. The deduced failure criterion based upon L∗enables the prediction of the remaining life of the studied pipes, so that maintenance and replacement works of drinking water pipes can be estimated.
Acknowledgements The authors gratefully acknowledge Aleppo University & Lille University for the facilities and logistical support throughout this Table 5 Values of critical net stress, mean stress, effective stress, gross stress for ASME standard in radial direction. Pf (MPa)
σN,
2.657 3.102 3.468 3.684 3.996
55.371 43.632 38.155 35.979 33.308
c
(MPa)
σm (MPa)
σeff (MPa)
σg. c (MPa)
σu(MPa)
D (mm)
Standard
27.817 35.290 27.755 27.187 26.367
34.8 55.5 59.5 56 33.5
14.657 17.027 19.078 20.199 21.901
25 25 25 25 25
75 90 110 125 160
ASME Radial Direction
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Table 6 Values of constraint factors for ASME standard in radial direction. Pf (MPa)
L
L*
L**
L***
a/D
D (mm)
Standard
2.657 3.102 3.468 3.684 3.996
2.284 1.800 1.574 1.484 1.374
1.147 1.455 1.145 1.121 1.087
1.435 2.289 2.454 2.310 1.382
0.605 0.702 0.787 0.833 0.904
0.067 0.056 0.045 0.040 0.031
75 90 110 125 160
ASME Radial Direction
Fig. 8. Effect of constraint factor on elastic-plastic behavior: definition of constraint factor (Rm = σu).
Fig. 9. Levels of critical net stress, mean stress, effective stress and gross stress compared to ultimate tensile strength (Rm = σu).
research.
Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Declaration of competing interests None.
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4.5 4 3.5 3
L
2.5
L*
2
L**
1.5
L***
1 0.5 0
75 75 75 75 90 90 90 90 110 110 110 110 125 125 125 125 160 160 160 160 A M D C A M D C A M D C A M D C A M D C
Fig. 10. Variations of constraint factor values in terms of diameter and standard used to calculate the critical pressure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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E. Melan, ZurPlastizitat des raumlichenKontinuums, Ingenieur-Archiv 8 (1938) 116–126. D. Broek, Elementary Engineering Fracture Mechanics, MartinusNijhoff Publishers, The Hague, 1982. A.R. Dowling, G. Towley, The effect of defect on structural failures: a two-criteria approach, Int. J. Press. Vessel. Pip. 32 (1975) 77–107. A.G. Miller, Review of limit loads of structures containing defects, Int. J. Press. Vessel. Pip. 32 (1988) 197–327. G. Yagawa, Y. Kitajima, H. Ueda, Three-dimensional fully plastic solutions for semi-elliptical surface cracks, Int. J. Press. Vessel. Pip. 53 (3) (1993) 457–510. Y. Lei, J-integral and limit load analysis of semi-elliptical surface cracks in plates under combined tension and bending, Int. J. Press. Vessel. Pip. 81 (1) (2004) 43–56. I. Sattari-Far, P. Dillstrom, Local limit load solutions for surface cracks in plates and cylinders using finite element analysis, Int. J. Press. Vessel. Pip. 81 (1) (2004) 57–66. M. Staat, V.D. Khoi, Limit loads of circumferentially flawed pipes and cylindrical vessels under internal pressure, Int. J. Press. Vessel. Pip. 83 (3) (2006) 188–196. M. Staat, Local and global collapse pressure of longitudinally flawed pipes and cylindrical vessels, Int. J. Press. Vessel. Pip. 82 (3) (2005) 17–25. G. Pluvinage, General approaches of pipeline defect assessment, in: Pluvinage, Elwany (Eds.), Safety, Reliability and Risks Associated with Water, Oil and Gas Pipelines, 2008, pp. 1–22. A.S.M.E. B31G, Manual for Determining the Remaining Strength of Corroded Pipelines, (Revision of ASME B31G-2009), New York, USA, http://files.asme.org/ Catalog/Codes/PrintBook/33501, (2012). DNV-RP-F101, Recommended Practice, Corroded Pipelines, (1999) (Reprint with corrections 2001, Det DetNorskeVeritas). J.B. Choi, B.K. Goo, J.C. Kim, Y.J. Kim, W.S. Kim, Development of limit load solutions for corroded gas pipelines, Int. J. Press. Vessel. Pip. 80 (2) (2003) 121–128. O. Bouledroua, M. Hadj Meliani, Z. Azari, A. Sour, N. Merah, G. Pluvinage, Effect of sandblasting on tensile properties, hardness and fracture resistance of a line pipe steel used in Algeria for oil transport, J. Fail Anal. Preven. 17 (5) (04 August 2017) 890–904. G. Pluvinage, O. Bouledroua, M. Hadj Meliani, K. Suleiman Rami, Corrosion defect analysis using domain failure assessment diagram, Int. J. Pres. Ves. Pip. August 165 (2018) 126–134. M. Soudani, O. Bouledroua, M. Hadj Meliani, K. El-miloudi, B.G.N. Muthanna, A. Khelil, et al., Corrosion inspection and recommendation on the internal wall degradation caused rupture of 6″ gas line pipe, J. Bio. Tribo. Corros. 4 (2018) 28. M. Hadj Meliani, O. Bouledroua, A. Alhussein, Y.G. Matvienko, G. Pluvinage, Some aspects sandblasting and hydrogen on the material master failure curve in engineering fracture mechanics, An. Mecanica Fractura 32 (2015). M. Hadj Meliani, M. Benarous, Z. Azari, G. Pluvinage, Constraint parameter for a longitudinal surface notch in a pipe submitted to internal pressure, Key Eng. Mater. 399 (2009) 3–11. M. Hadj Meliani, Z. Azari, M. Al-Qadhi, N. Merah, G. Pluvinage, A two-parameter approach to assessing notch fracture behaviour in clay/epoxy nanocomposites, Compos. Part B 80 (2015) 126–133. A. Elazzizi, M. Hadj Meliani, A. Khelil, G. Pluvinage, Y.G. Matvienko, The master failure curve of pipe steels and crack paths in connection with hydrogen embrittlement, Int. J. Hydrog. Energy 40 (5) (2015) 2295–2302. M. Soudani, M. Hadj Meliani, K. El-Miloudi, O. Bouledroua, C. Fares, M.A. Benghalia, Efficiency of green inhibitors against hydrogen embrittlement on mechanical properties of pipe steel API 5L X52 in hydrochloric acid medium, J. Bio. Tribo-Corrosion 4 (3) (2018) 36. J. Capelle, J. Gilgert, I. Dmytrakh, G. Pluvinage, Sensitivity of pipelines with steel API X52 to hydrogen embrittlement, Int. J. Hydrog. Energy 33 (24) (2008) 7630. M. Hadj Meliani, Z. Azari, G. Matvienko Yu, G. Pluvinage, The effect of hydrogen on the master failure curve of APL 5L gas pipe steels, Procedia Eng. 10 (2011) 942–947. M. Soudani, M. Hadj Meliani, K. El-miloudi, Z. Azari, A.A. Sorour, N. Merah, et al., Reduction of hydrogen embrittlement of API 5l X65 steel pipe using a green inhibitor, Int. J. Hydrog. Energy 43 (24) (14 June 2018) 11150–11159.
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Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Analytical and numerical investigation using limit analysis on the ductile failure of pipes containing surface cracks
T
⁎
M. Mouwakeha, , S. Masria, M. Hadj Melianib,d, R.K. Suleimanc, G. Pluvinaged, M. Nait-Abdelazize a
Department of Applied Mechanics, Faculty of Mech. Eng., Aleppo University, Aleppo, Syria LPTPM, Hassiba Benbouali University of Chlef, Chlef 02000, Algeria c Center of Research Excellence in Corrosion (CoRE-C), King Fahd University of Petroleum & Minerals (KFUPM), Dhahran 31261, Saudi Arabia d University of Lorraine, LEM3 Laboratory, Ecole d'Ingénieur de Metz, France e University of Lille, Unité Mécanique de Lille, Av. Paul Langevin, 59650 Villeneuve d'Ascq, France b
A R T IC LE I N F O
ABS TRA CT
Keywords: Limit analysis Constraint factor Surface crack Finite element analysis Polyethylene pipe
Using cracked pipes design standards and finite element analysis, the limit load analysis for pipes containing surface cracks was determined. The study was performed on five pipes of different diameters with a constant crack length and depth. The crack geometry is a semi-elliptical surface crack. The cracked pipes are subjected to internal pressures which are obtained from formulas of cracked pipes design standards. Due to the ductile behavior of polyethylene pipes, the failure occurs when the critical stress reaches a value equal to the ultimate tensile strength multiplied by a constraint factor. In this work, the constraint factor was calculated and its evolution with the pipe diameter was analyzed. Three different definitions of a constraint factor based on global or local approaches were also compared, so that a new failure criterion can be obtained. The new failure criterion makes the prediction of the pipe residual life possible which, in turn, facilitates a systematic approach to maintenance and replacement of pipes.
1. Introduction The first appearance of the theory of limit analysis was in the late 1930s of the last century [1]. The theory constitutes a branch of the plasticity theory that is related to elastic/perfectly plastic behavior. In the mid-1950s, a large number of analytical solutions were firstly found to calculate the ultimate load of beams and shells, leading to more realistic values of the capability to resist plastic collapse. This led progressively to the design of concepts based on a limit state that is replacing those based on allowable stress. The introduction of Linear Elastic Fracture Mechanics in the mid-1950s [2] led designers to consider principally the risk of brittle fracture governed by the global stress in apparent opposition to the theory of plastic collapse governed by the nominal stress. In the early1970s, Dowling and Towley [3] showed that the theories of limit analysis and fracture mechanics can be combined to form a novel approach, i.e., the two criteria approach which is the basis for the method of integrity-failure diagrams. The newly developed idea in the method of two criteria is to consider an interaction between the two mechanisms so that the ultimate load Pult is less than that given by limit analysis Pl.a and higher than that given by elastic fracture mechanics Pf.m: Pf.m < Pult < Pl.a. Numerous works have been conducted to obtain limit load solutions in pipes containing surface cracks subjected only to internal pressure, or to combined loads (internal pressure and bending) [4–10]. Since pipes generally fail in a ductile manner due to the
⁎
Corresponding author at: LPTPM, Hassiba BenBouali University of Chlef, P.O.Box. 151, Hay Salem, 02000 Chlef, Algeria. E-mail address: moufi[email protected] (M. Mouwakeh).
https://doi.org/10.1016/j.engfailanal.2019.06.007 Received 14 March 2019; Received in revised form 19 May 2019; Accepted 4 June 2019 Available online 05 June 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.
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σN,c σm σeff σg.c. E L Pf σyy
Nomenclature a l D t P σu (Rm)
crack depth crack length pipe diameter pipe thickness internal pressure in pipe ultimate tensile strength
critical net stress mean or average stress effective stress critical gross stress elastic Young modulus constraint factor critical failure pressure opening stress
behaviour of the constitutive material, failure prediction tools are mostly based on the limit analysis. This failure criterion assumes that failure occurs when critical net stress σNc reaches ultimate tensile strength σu. We note that ductile failure is sensitive to critical net stress σNc(load divided by the ligament cross section) whatever brittle failure is sensitive to gross stress σgc (load divided by the entire section). The above-mentioned criterion needs to be modified to take into account constraints due to loading mode and geometry effects in the following manner:
σN,c = L. σu
(1)
where L is called the constraint factor. Design standards for pipes with defects such as ASME B31G, Modified ASME B31G [11], DNV RF101 [12] and CHOI formulas [13] are based on limit analysis and allow the calculation of the critical failure pressure for corroded pipes. They also incorporate a safety factor which defines a lower bound of the experimental results [14–16]. The basic question is the adopted value of the constraint factor and its closeness to values generally obtained. In addition, it is interesting to capture the evolution of constraint factor with ligament size, defect and pipe geometries as follows:
a a L = L⎛ , ⎞ ⎝D t ⎠
(2)
where D is the pipe diameter, t is the thickness, and a is the defect depth. In this work, the determination of the critical failure pressure using the above-mentioned standards has firstly been conducted for five cases by varying both D and t parameters. The geometry of the defect was remained unchanged. After calculating the critical failure pressures Pf, values of the critical net stresses and the corresponding constraint factors L can be consequently calculated. Indeed, according to Eq. (1), the constraint factor can be expressed as follows [17–19]:
L = σN,c/σu
(3)
Moreover, a second way to obtain the constraint factor was investigated. Using a finite element method (FEM), the calculation of stress distribution along the ligament was achieved, which allows us to obtain the average values of maximal principal stresses σm. Another calculation of the constraint factor deduced from FEM analysis can be therefore obtained as follows:
L∗ = σm/σu
(4)
Finally, the volumetric method [10] was also used to offer an alternative way to compute the constraint factor [20–24]. This method requires one to determine, from the variation of stress distribution away from crack tip, the fracture process zone (FPZ) using the volumetric method which is a local fracture criterion. It assumed that the fracture process requires a certain volume which is assumed to be a cylinder in which the diameter of the cylinder is the effective distance. The physical meaning of this fracture process volume is “the high stress region” where the necessary fracture energy release rate is stored. Inside the fracture zone, the effective stress σeff, which is the average value of the stress distribution, acts as a local fracture stress. Consequently, a third definition of the constraint factor can be assumed:
L∗∗ = σm/σu
(5)
This work aims to compare and discuss these different constraint factors in order to propose a new failure criterion through the designing of pressurized pipes, taking into account the following remarks:
• Limit analysis formulas (ASME B31G, Modified ASME B31G, DNV RF101, and CHOI) consider the defect as a lack of material. They neglect the stress concentration induced by the defect, • Similar to limit analysis formulas, they refer to the nominal stress, i.e., the average stress in the ligament surface under defect, • The volumetric method takes into account the stress concentration induced by the defect, i.e., the stress concentration relative to the gross stress to separate from the stress increase due to the reduction of the ligament surface, • This stress concentration is evaluated by the constraint factor, • The comparison between the methods makes it possible to estimate the error using a much more conservative limit analysis. This
is why limit analysis is used to establish a relative value. This paper is organized as follows: in Section 2, we will present the material and geometries before describing the calculation of critical failure pressures and stresses in Section 3 according to the standards previously introduced and the corresponding constraint factors L expressed by Eq. (3). Sections 4 and 5 are dedicated to the calculation of the additional constraint factors L* and L**. In Section 6, the results obtained through the different methods are discussed and followed by the concluding remarks. 481
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2. Material, pipes and crack geometry The studied material is a high-density polyethylene whose main properties are: σu = 25 MPa, ρ = 960 kg/m3, E = 450 MPa, υ = 0.45, where σu, ρ, E, and υ are the ultimate tensile stress, mass density, Young's modulus and Poisson's ratio, respectively. In this study, five pipes with different wall thicknesses are chosen: (D = 75 mm, t = 6.8 mm), (D = 90 mm, t = 8.2 mm), (D = 110 mm, t = 10 mm), (D = 125 mm, t = 11.4 mm), and (D = 160 mm, t = 14.6 mm). The crack geometry is assumed to be a semi-elliptical surface crack as shown in Fig. 1, where l = 2c is the crack length and a is the crack depth. The crack dimensions remain constant for all pipes. The crack dimensions are the following: l = 2c = 100 mm, a = 5 mm. 3. Calculation of constraint factor L using design standards 3.1. Critical failure pressure The critical failure pressure is calculated for different pipes using the formula given by the various standards. 3.1.1. ASME B31G standard This semi-empirical approach was developed in the late 1960s and early 1970s and is generally used to calculate the residual strength of the pressurized defected pipes [11]. Based on a wide and comprehensive series of experiments conducted on different sections of defected pipes, the failure pressure was formulated, taking into account the following two conditions: 1. The maximum value of the circumferential stress should not exceed the yield stress, i.e., σθθ < σy. 2. A relatively short defect can be represented in the form of a parabola while a long one is represented in the form of a semielliptical shape. According to this standard, critical failure pressure is given by the following relationship:
Pf =
⎡ 2(1.1σy ) t ⎢ 1 − ⎢ D ⎢1 − ⎣
( )( ) ⎤⎥ ⎥ ( ) ( ) ⎥⎦ 2
3
a
a
2
3
t
t
M
where
M=
l 2 D L 2 D 1 + 0.8 ⎛ ⎞ ⎛ ⎞ for 0.8 ⎛ ⎞ ⎛ ⎞ ≤ 4 ⎝D⎠ ⎝ t ⎠ ⎝D⎠ ⎝ t ⎠
(6)
where D, a, t, M, σy and l are the outer diameter, crack depth, wall thickness, bulging factor, yield stress and longitudinal corrosion defect length, respectively. 3.1.2. Modified ASMEB 31G standard The modified ASME B31G standard incorporates the new bulging factor M, so that the critical failure pressure Pf is given by the following relationship:
Pf =
M= For:
( ) ⎤⎥
a ⎡ 2(1.1σy ) t ⎢ 1 − 0.85 t ⎢ D (a t) ⎢ 1 − 0.85 M ⎣
⎥ ⎥ ⎦
(7)
l 2 D l 4 D 2 1 + 0.6275 ⎛ ⎞ ⎛ ⎞ − 0.003375 ⎛ ⎞ ⎛ ⎞ ⎝D⎠ ⎝ t ⎠ ⎝D⎠ ⎝ t ⎠
l 2 D D t
( ) ( ) ≤ 50
Fig. 1. Pipe and crack geometry. 482
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3.1.3. DNV RP-F101 standard The DNV RP-E101 standard is the first general and comprehensive standard for the calculation of pipes containing surface cracks subject to solely internal pressure loads, or combined loads consisting of internal pressure and bending loads [12]. According to this standard, the critical failure pressure is given by the following relationship:
Pf =
For: Q =
2(σU) × D−t
t ⎡ 1 − a/t ⎤ ⎢ ⎣ 1 − (a/t)/Q ⎥ ⎦
1 + 0.31
( ) where D, a t, and σu are identical to the terms previously introduced, and Q is the bulging factor. 1 Dt
(8)
2
,
3.1.4. CHOI standard Combining limit analysis and finite element analysis of a pipe containing a surface crack [13], the equation of critical failure pressure is derived and given by the following relationship: 2
Pf = 0.9 ×
2(σU) × t ⎡ ⎛ L ⎞ ⎛ L ⎞⎤ ⎥ ⎢C0 + C1 Rt + C2 Di ⎝ ⎠ ⎝ Rt ⎠ ⎦ ⎣ ⎜
⎟
⎜
⎟
(9)
where σU, Di, t, and R are the ultimate tensile strength, inside diameter, wall thickness and average pipe radius, respectively and C0, C1, and C2 are given in terms of a/t. Fig. 2 shows the variation of critical failure pressure as a function of a/D ratio. We note that the critical failure pressure decreases linearly with the increase of a/D ratio or with decreasing the diameter. As shown in Fig. 2, the standards ASME 31G, modified ASME 31G, and CHOI exhibit the same trends, i.e., the critical failure pressure decreases linearly versus a/D ratio, while it remains constant for DNV standard.
3.2. Calculation of critical net stress Critical net stress is calculated from the critical failure pressure values for different diameters using the formula:
σN , c = Pf . D/2t∗
(10)
where Pf is the critical failure pressure and t∗ is the thickness of the ligament and equals to: (11)
t∗ = t − a
Similarly, the variation of the critical net stress versus a/D ratio is shown in Fig. 3. We note that the critical net stress increases upon increasing the a/D ratio.
3.3. Calculation of constraint factor L The constraint factor L is calculated according to the different standards by using Eq. (3). Table 1 below shows the obtained values. Since the constraint factor is obtained by dividing the critical net stress by a constant (the ultimate tensile stress σu=25 MPa), the evolution of this parameter as a function of a/D ratio is the same to that shown in Fig. 3. While the values of constraint factor obtained for ASME, Modified ASME and Choi methods exhibit the same trends with a relative difference of about 15%, the DNV standard is most conservative, the value of the constraint factor being approximately two times higher than that issued for the other standards when considering the maximum a/D ratio.
C ri tical Pressure Pf (MPa)
6
5
DNV
4
ASME
3 2
Modified ASME
1
Choi
0 0,03
0,04
0,05 a/D
0,06
0,07
Fig. 2. Variation of critical failure pressure versus a/D ratio. 483
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120 DNV
100
(Mpa)
60
n,c
80
40
ASME
Modified ASME
20 0 0,03
Choi 0,04
0,05 a/D
0,06
0,07
Fig. 3. Variation of critical net stress versus a/D ratio. Table 1 Values of constraint factors for all standards. D (mm)
L ASME
L M.ASME
L DNV
L Choi
75 90 110 125 160
2.284 1.8 1.574 1.484 1.374
1.804 1.56 1.442 1.393 1.328
4.151 2.818 2.2 1.96 1.673
1.568 1.508 1.499 1.468 1.381
4. Calculation of the constraint factor L* using fem 4.1. Calculation of the mean of maximum stresses acting on the ligament The finite element software ABAQUS® was used for the computations reported in this study. The material was assumed to be elastic/perfectly plastic obeying the Von-Mises flow criterion. Critical failure pressure values which were previously computed have been applied on the internal surface of the pipes. In terms of boundary conditions, the ends of the pipe are fixed. FE calculations were performed for different diameters to obtain the equivalent Von-Mises and maximum principal stresses. The meshing of the 3D finite element model consists of hexahedral elements. A longitudinal section of the pipe illustrating the mesh around the crack zone is given in Fig. 4. An example of the maximum principal stress distribution is shown in Fig. 5. Mean maximum stress σm, which is the average of the maximum principal stresses acting over the ligament, can be deduced as a function of the critical failure pressure.
4.2. Calculation of the constraint factor L* The corresponding constraint factor L* deduced from the mean maximum stress was obtained by using Eq. (4). Table 2 shows the constraint factor values L* according to mean maximum stress values obtained from FE computation.
Fig. 4. Mesh model used for the pipe (longitudinal section). 484
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Fig. 5. Maximum principal stress distribution around crack zone (longitudinal section). Table 2 Values of constraint factors L* for all standards. D (mm)
L٭ASME
L٭M.ASME
L٭DNV
L٭Choi
75 90 110 125 160
1.148 1.456 1.145 1.122 1.088
1.012 1.308 1.068 1.062 1.026
1.836 2.076 1.467 1.386 1.277
0.914 1.275 1.102 1.121 1.111
5. Determination of constraint factor L** using volumetric method 5.1. Calculation of the effective stress σeff Similarly, by using FEM and the ABAQUS program and applying the same boundary conditions used in the above modeling, we have calculated the stress distribution around the crack tip in both longitudinal and radial directions. A mesh refinement, shown in Fig. 6, has been performed in the region close to the crack tip in order to capture as accurately as possible the stress gradient and consequently the stress distribution in the vicinity of the crack tip. The evolution of the maximum principal stress as a function of the distance to the crack tip has been estimated both in the longitudinal and radial directions. A schematic representation of the expected evolution is shown in Fig. 7. Since the same trends are expected whatever the direction (longitudinal or radial), the same comments can be given. Such a diagram allows the determination of the effective distance Xeff, i.e., the size of the fracture process zone (FPZ) which extends from the crack tip up to the onset of the LEFM stress distribution (the straight line on the figure). Consequently, the effective stress can be deduced in order to calculate the stress intensity factor Kρ which is expressed as follows:
K ρ = σeff 2πXeff
(12)
Table 3 shows the values of effective stress, effective distance and stress intensity factor calculated by the volumetric method, based on the distribution of the maximum principal stresses as a function of the distance from the crack tip in the longitudinal
Fig. 6. Meshing model around the crack zone (longitudinal section). 485
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Fig. 7. A schematic distribution of the maximum principal stresses as a function of the distance from the crack tip and deducing the dimension of the plastic zone and the value of effective stress. Table 3 Effective stress, effective zone dimension, and stress intensity factor for ASME standard. Pf (MPa)
σmax (MPa)
σeff (MPa)
Xeff (m)
Xn (m)
Kρ (MPa.√m)
D (mm)
Standard
2.657 3.102 3.468 3.684 3.996
15.08 13.75 15.13 15.63 17.12
13.15 13.65 14.88 15.41 16.54
0.0021 0.005 0.004 0.0043 0.0053
0.004 0.009 0.0082 0.009 0.009
1.510 2.419 2.358 2.532 3.018
75 90 110 125 160
ASME Longitudinal Direction
direction for ASME standard. Table 4 shows also the values of effective stress, effective distance and stress intensity factor calculated by the volumetric method, based on the distribution of the maximum principal stresses as a function of the distance from the crack tip in the radial direction for ASME standard. Similarly, effective stress, effective distance and stress intensity factor calculated by the volumetric method in the longitudinal and radial direction for the other standards lead to the same trends. When comparing the effective stress values in longitudinal and radial directions for all pipes and standards, we note that the effective stress values in the radial direction are always greater than those in the longitudinal direction. This means that the crack will likely propagate in the radial direction. Therefore, the effective stress values in the radial direction are used to calculate the constraint factor L** because they cause a greater risk on the ligament of the pipes and cause the transformation of the crack from surface crack to through-wall crack and thus the plastic collapse of the pipe. 5.2. Calculation of constraint factors L** and L*** The constraint factor L** is given by the following expression:
L∗∗ = σeff / σu
(13)
Furthermore, the constraint factor L*** is also calculated by the following expression:
L∗∗∗ = σg . c / σu
(14)
where σg. c is the critical gross stress and σu is the ultimate tensile strength. The critical gross stress is defined as follows:
σg . c = Pf . D /2t
(15)
where Pf is the critical failure pressure, D is the pipe diameter, and t is the total wall thickness. The values of the critical net stress, mean stress, effective stress, and gross stress for ASME standard in the radial direction are Table 4 Effective stress, effective zone dimension, stress intensity factor for ASME standard. Pf (MPa)
σmax (MPa)
σeff(MPa)
Xeff (m)
Xn (m)
Kρ (MPa.√m)
D (mm)
Standard
2.657 3.102 3.468 3.684 3.996
40.57 60 64.5 62 50
34.8 55.5 59.5 56 33.5
0.00083 0.0006 0.00062 0.0006 0.00155
0.0018 0.00143 0.00125 0.0014 0.003
2.513 3.407 3.713 3.438 3.305
75 90 110 125 160
ASME Radial Direction
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shown in Table 5. The values of the corresponding constraint factors are reported in Table 6. Similarly, for the other standards, the same trends are obtained in terms of the values of critical net stress, mean stress, effective stress, gross stress, and the corresponding constraint factors. 6. Results and discussion The constraint factor gives a good idea of how much stress is increased due to constraints related to the triaxiality state of the stress field. Furthermore, the constraint factor depends on the pipe geometry, ligament size, pipe wall thickness and on the stress gradient. Fig. 8 presents schematically the stress-strain curve of the pipe material with and without a constraint. The symbol ‘?’ in Fig. 8 refers to the kind of critical stress and the corresponding constraint factor. When a constraint is accounted for, the level of the stress values depends on the definition of the constraint factor, as previously shown. Fig. 9 gives an illustrative example of the stress distribution around the crack tip in the radial direction for a pipe of diametr D = 125 mm and subjected to a critical failure pressure calculated by the ASME standard. It shows also the values of different critical stresses compared to ultimate tensile strength (σu = Rm) and it is observed that:
σeff > σN . c > σm > σu (Rm) > σg . c Fig. 10 shows the variation of constraint factors in terms of pipe diameter (75, 90, 110, 125, 160 mm) and standard (ASME B31G, M. ASME B31G, DNV, and CHOI) used in the calculation of critical failure pressure. We note generally that:
L > L∗ > L∗∗∗ From Fig. 10, we note that the values of the factor L*** (blue) are nearly equal or less than one. This confirms the fact that the ductile failure is not sensitive to critical global stress but rather to critical local stress. Thus, the factor L*** cannot be considered as a constraint factor and its values are taken as indicative values. As to constraint factor L** (violet), it is primarily the most realistic value of the constraint factor if we assume that the ductile failure requires a fracture process zone; thus fracture occurs when the effective stress reaches a critical value. Furthermore, L** rarely exceeds the value of 3.0, which is smaller than the theoretical value of 10, which is the constraint factor L** value for plane strain and Poisson ratio ν = 0.45 for polyethylene. Finally, regarding the constraint factor values L* (sky blue) which are always smaller than L (red), it can be said that both L and L* values refer to the average value of the nominal stress applied on the ligament. The critical stress σm represents the average value of the stress distribution on the ligament while the critical stress σN,c represents the critical net stress affecting the reduced thickness of the pipe, and this certainly explains the difference between the factors L and L*. 7. Conclusions From the previous results, the following conclusions can be drawn:
• The results of the standards ASME B31G, Modified ASMEB 31G, and CHOI in terms of critical failure pressures, critical stresses • • • •
and constraint factors are close to each other, while DNV standard results give more conservative values. Therefore, one of the three first standards is recommended. The values of constraint factor L∗ are always smaller than the values of factor L for all diameters and for all standards. The factor L∗could be taken as an appropriate constraint factor. Consequently, a suitable failure criterion for the cracked pipes can be derived so that the failure occurs when the critical stress σm reaches a value equal to ultimate tensile strength σu multiplied by constraint factor L∗ by means: σm = L∗. σu. The factor L∗∗ computed from the fracture mechanics considerations is the most realistic value of the local constraint factor because it focuses on the stress distribution at the crack tip. To make it an engineering tool for cracked pipes design, it must be simplified because its evaluation is quite complex. The deduced failure criterion based upon L∗enables the prediction of the remaining life of the studied pipes, so that maintenance and replacement works of drinking water pipes can be estimated.
Acknowledgements The authors gratefully acknowledge Aleppo University & Lille University for the facilities and logistical support throughout this Table 5 Values of critical net stress, mean stress, effective stress, gross stress for ASME standard in radial direction. Pf (MPa)
σN,
2.657 3.102 3.468 3.684 3.996
55.371 43.632 38.155 35.979 33.308
c
(MPa)
σm (MPa)
σeff (MPa)
σg. c (MPa)
σu(MPa)
D (mm)
Standard
27.817 35.290 27.755 27.187 26.367
34.8 55.5 59.5 56 33.5
14.657 17.027 19.078 20.199 21.901
25 25 25 25 25
75 90 110 125 160
ASME Radial Direction
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Table 6 Values of constraint factors for ASME standard in radial direction. Pf (MPa)
L
L*
L**
L***
a/D
D (mm)
Standard
2.657 3.102 3.468 3.684 3.996
2.284 1.800 1.574 1.484 1.374
1.147 1.455 1.145 1.121 1.087
1.435 2.289 2.454 2.310 1.382
0.605 0.702 0.787 0.833 0.904
0.067 0.056 0.045 0.040 0.031
75 90 110 125 160
ASME Radial Direction
Fig. 8. Effect of constraint factor on elastic-plastic behavior: definition of constraint factor (Rm = σu).
Fig. 9. Levels of critical net stress, mean stress, effective stress and gross stress compared to ultimate tensile strength (Rm = σu).
research.
Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Declaration of competing interests None.
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4.5 4 3.5 3
L
2.5
L*
2
L**
1.5
L***
1 0.5 0
75 75 75 75 90 90 90 90 110 110 110 110 125 125 125 125 160 160 160 160 A M D C A M D C A M D C A M D C A M D C
Fig. 10. Variations of constraint factor values in terms of diameter and standard used to calculate the critical pressure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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