- Email: [email protected]
Hybrid formulation solution for stress analysis of curved pipes with welded bending joints
Engineering Fracture Mechanics 77 (2010) 2992–2999
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Hybrid formulation solution for stress analysis of curved pipes with welded bending joints Luisa R. Madureira a,*, Francisco Q. Melo b a b
Department of Mechanical Engineering, Faculty of Engineering of the University of Porto, Portugal Department of Mechanical Engineering, University of Aveiro, Portugal
a r t i c l e
i n f o
Article history: Received 7 August 2009 Received in revised form 2 April 2010 Accepted 12 April 2010 Available online 14 April 2010 Keywords: Hybrid formulation Curved pipes Stress intensity factors Fourier series
a b s t r a c t A method is presented for evaluating the stress intensity factor of part-through cracks in a thin pipe elbow. A hybrid formulation solution is used to evaluate the stress field close to the crack area. The stress field values are then inputted into a previously developed method published in the literature to evaluate the stress intensity factor in cylindrical shells. Results from cylindrical shells with part-through cracks are extended to double-curvature pipe configurations that contain the same kind of flaw. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Curved pipes play an important role in piping engineering, and an accurate evaluation of their stress field is important for critical applications. The process of manufacturing curved pipes generally involves plastic forming and/or welding methods to join pipe parts when the pipe cannot be manufactured as a whole piece. In both processes, high loads or imperfect welding can create part-through cracks caused by lack of material adhesion or local ruptures. The stress intensity factor or SIF, in abbreviated form, is used in fracture mechanics to predict the stress state close to the crack tip and can be determined by numerical or experimental methods. The former method has largely been adopted by project engineers, as experimental techniques are generally expensive and demand very precise methods for finding the displacement field close to the defect zone. Numerical methods in process engineering are frequently identified with finite element techniques. The structure is modeled with shell finite elements, while the crack front is modeled with singular finite elements. Research in this area has seen a considerable number of contributions that provide more accurate and advanced techniques for determining the SIF of piping accessories that contain part-through cracks. In fact, these partial defects draw more attention than complete cracks since thorough inspection procedures can be used to detect them at an early stage. The possibility of crack generation and propagation in pipe elbows subjected to bending loads has been studied using fracture mechanics methods. Viswanath et al. [1] obtained stress intensity factors in elbows subjected to uniform in-plane bending. In this study, a part-through crack along the longitudinal axis at the crown pipe was studied. The crack opening mode was found to be due to transverse section ovalization, a serious stress mode that induces rapid crack growth in this kind of piping accessories. The authors used a finite element commercial code to perform the analysis. Yahiaoui et al. [2] analyzed the influence of crack size on limit loads also using finite element analysis. Oliveira et al. [3] studied the stress intensity factors of surface elliptical cracks existing in pipe elbows using high order 8-node shell elements. To simulate the remaining ligament compliance at the crack zone, they used the line-spring model, a method that has performed very well for this class of three-dimensional structural problems. * Corresponding author. Fax: +351 22 5081445. E-mail addresses: [email protected] (L.R. Madureira), [email protected] (F.Q. Melo). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.04.009
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
2993
Nomenclature a ai(x) bi(x) C d/ E h KI L M0 O(h) r R SIF t U(r, u) u u v w
a m f
r rx X
maximum crack depth ovalization function warping function elastic pipe properties matrix bending angle Young’s modulus pipe thickness nominal stress intensity factor transverse section center arch bending moment ovalization displacement pipe transverse section radius pipe curvature radius stress intensity factor equivalent thickness total internal energy displacement field longitudinal shell displacement circumferential shell displacement transverse shell displacement ellipse axis ratio poisson0 s ratio position point in the crack profile where the SIF is measured vector of internal stresses hoop stress the integral domain
However, because finite element modeling for fracture mechanics problems requires highly refined meshes and heavy computational work, an alternative method is proposed. The process combines shape factors available in the literature with a hybrid solution formulation where unknown functions, ovalization and warping, are used together with Fourier trigonometric expansions [4–6]. These shape factors are used in the evaluation of SIF parameters for a generalized crack existing in a doubly curved surface shell. Recently Marie et al. [7] published the stress intensity factors and J calculation for cracked elbows where values for KI and rref are tabled for several crack geometries. Most previous studies use a crack front that is oriented along the longitudinal (or axial) pipe elbow line arch. This study focuses on cracks along the circumference line of the welded flange-pipe joint. As the shell structure is based on a semimembrane deformation model the bending stress along the longitudinal direction is not considered, but instead the traction stress due to the longitudinal direction force is. The evaluation of the stress intensity factor is based on linear elastic fracture mechanics. The weld joining the flange to the pipe toroidal surface is assumed as a T-joint and for this kind of problem a stress concentration factor of 2 is normally used [8] and will multiply the nominal stresses calculated in this work. Also, following the work of Carpinteri et al. [9] for part-through cracks in cylindrical shells, stress intensity factors are evaluated for different crack geometries, once the curved pipe radius/thickness ratio considered allows this approach. 2. Methodology In a curved pipe with end flanges, the flange-pipe interface is a likely location for defects, as sketched in Fig. 1. Here, the goal is to analyze the structural integrity of a pipe where a crack similar to the one displayed above is detected in a piping part. The role of the stress intensity factor (SIF) is to quantify the elastic stress field in the close neighborhood of the crack front. For this purpose, this work follows that of Carpinteri et al. [9], in which thin and thick walled straight pipes that have a part-through crack were subjected to bending loads. Due to the presence of the defect, the pipe integrity is defined by the maximum SIF along the crack profile [9]. The displacement field is given by [6,10–13]:
wðx; hÞ ¼
v ðx; hÞ ¼ uðx; hÞ ¼
5 X i¼2 5 X i¼2 5 X i¼2
ai ðxÞ cosðihÞ
ai ðxÞ sinðihÞ i
bi ðxÞ cosðihÞ
ð1Þ
2994
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
M
x= -L
Crack location
x= 0 Crack location
θ
C
C x= +L M
Section C-C Fig. 1. Part-through crack in a welded flange joint of a curved pipe.
The following pipe deformations used to find the solution according to the semi-membrane strain field are given by differential equations [6,10–13]:
@u 1 þ ðw cos h v sin hÞ @x R ¼0
ex ¼ eh
1 @u @ v usin h þ þ r @h @x ! R 1 @2w @v kh ¼ 2 r @h2 @h
ð2Þ
cxh ¼
The solution is derived by minimizing the total elastic energy U involved in the pipe distortion process with a set of unknown functions [14]:
Uðr; uÞ ¼
1 2
Z
rT C1 r dX þ
X
Z
rT u d X
ð3Þ
X
C is the pipe elastic properties matrix. The first integral refers to the internal elastic deformation energy, and the second is the deformation work performed by the unknown vector r of the internal stresses against the admissible displacement field u. An external entity M0, the bending moment, performs additional work at a bending angle d/. The system of external forces performs deformation work (called the external work) upon the displacement functions ai and bi. The total energy of the structure U is the sum of this external work with the corresponding elastic deformation. This value should be zero if the displacement functions are exact, but for the displacement field approximations used here (Fourier expansions), U has a residual value. This residual allows for the optimization of expressions to minimize U. The variation of the total energy U is found as follows:
dU ¼
Z
drT C1 r dX þ
X
Z X
drT u dX þ
Z
rT du dX
ð4Þ
X
where the integral domain X is [0, L] [0, 2p] and L represents the length of the transverse section of the center arch. The total variation in U is the result of separate contributions from variations in the stress field r and in the displacement u. Equilibrium is achieved by minimizing the function U separately for each of the unknowns:
8 @U @U @U > ¼ 0; ¼ 0; ¼0 < @N @Nxhi @mhi xi i2 i2 i2 0 > @U @U 0 @U @U : @a dai þ @a ¼ 0; dbi þ @b ¼0 0 dbi 0 dai @bi i i2
i
i
i2
The following system of differential equations can be solved by evaluation of eigenvalues and eigenvectors [4,5]:
9pD 5prA prA prS 0 3 0 a b M0 þ a p Sb þ 2 4 2 4 r3 4R 4R 3 4Rr 16R 32R2 prS 00 5prA 64pD prA 2prA prS 0 prA prS 0 0 a a3 ¼ þ b b þ þ a þ p Sb þ 3 5 3 9 r3 3R 6R 2 3R 6R 4 9R2 5R2
prS
a002 ¼
prA
þ 2
ð5Þ
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
! 152 pD 5prA prS 0 3prA prS 0 0 a4 þ þ b3 pSb4 þ b 2 3 16 r 8R 8R 8R 8R 5 32R 32R prS 00 4prA 13prA 576pD 3prA prS 0 0 a a5 ¼ þ b pSb5 a þ þ þ 3 5 25 r3 5R 10R 4 15R2 25R2 pS 4pS prS prS 2prA prS 0 a prAb002 ¼ b3 þ þ 2 b2 2 b4 þ pSa02 þ 2R r 3R 6R 3 4R 4R pS 9pS 2prS pS prS prðA SÞ 0 5prA prS 0 b3 þ b4 2 b5 a2 þ pSa03 þ a prAb003 ¼ b2 þ 2 2R r 2R 4R 8R 8R 4 4R 4R prS pS 16pS 2prS prA prS 0 prS 3prA 0 b4 þ pS4 a0 þ a þ prAb004 ¼ 2 b2 b3 þ a 2R r 3R 6R 3 10R 5R 5 4R 4R2 prS pS 25pS prS 3prA prS 0 2 b5 a þ p S5 a 0 prAb005 ¼ 2 b3 b4 þ 2R r 8R 8R 4 4R 4R
prS
a004 ¼
5prA
a þ 2 2
17prA
2995
þ
ð6Þ
2.1. Determination of the nominal stress field in a curved pipe under bending To assess the solution method discussed here, stress analysis is carried out for a curved pipe with rigid end flanges. The material properties and dimensions are: Transverse section radius, r – 50 mm. Wall thickness, h – 2 mm. Mean curvature radius, R – 200 mm. Pipe angle, / – 90°. Transverse section center arch, L – 157.0796 mm (=R/). Material – stainless steel, type 304, Cr18Ni8: E = 210 GPa,
m ¼ 0:3.
An in-plane uniform bending moment M0 loads the pipe, where symmetry conditions were considered. In the calculations, a specified rotation angle of d/ = 0.002 rad at a flanged end was prescribed. This corresponds to a value M = 35332456 N mm. Boundary conditions are imposed:
bi ð0Þ ¼ 0; bi ðLÞ ¼ 0 a0i ð0Þ ¼ 0; ai ðLÞ ¼ 0
ð7Þ
The values obtained for the ovalization parameters at x = 0 are:a2 = 103.42 d/ mm, a3 = -31.03d/ mm, a4 = 8.15da mm, a5 = 3.89d/ mm. (d/ = 0.002 rad). The pipe ovalization displacement is given by the trigonometric expansion, for x = 0:
OðhÞ ¼ ð103:42 cos 2h 31:03 cos 3h 8:15 cos 4h þ 3:89 cos 5hÞdu mm The hoop stress at section x = 0 or x = L, as a consequence of the bending moment, is given by (8):
"
rx
# X 5 5 1X 1 0 ¼ A r cos hW ðxÞ þ ai ðxÞ cos h cos ih þ sin ihsinh þ bi ðxÞ cos ih MPa R i¼2 i i¼2 00
ð8Þ
Two more examples are studied in order to focus on the effect of different radius or thickness for this kind of pipe. EXAMPLE 2: h = 5, r = 50 (in mm). EXAMPLE 3: h = 2, r = 40 (in mm). Evaluations lead to the stress distributions for x = L that are shown in Fig. 2 where at the extrados the values obtained for the nominal stresses are 130, 180 and 293 Mpa for Examples 1, 2 and 3 respectively. 2.2. Structural evaluation of a curved pipe with a flaw in a welded joint subjected to a bending moment The method is applied to a curved pipe with cracked rigid end flanges. Fig. 3 shows a crack resulting from the lack of adhesion that occurs in the extrados curved pipe. The combined pipe and welded joint thickness are 5 mm for Example 1; therefore, the equivalent pipe thickness is assumed to be 5 mm with a part-through crack as shown in Fig. 4 [9]. The following parameters are necessary for evaluating the stress intensity factor (SIF) associated with the crack geometry: a = maximum crack depth ael = minor half axis of ellipse (the crack has an elliptical profile) n = a/t (non-dimensional parameter; measured at the largest crack depth)
2996
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
Fig. 2. Longitudinal membrane stress, rx at x = L from uniform bending load.
C Flaw location
Crack location
Welded joint 2
3
Flange
θ C Section C-C Fig. 3. Elbow with end flanges: welding detail and flaw location (in mm).
Y
r t
J P B CN L A β
0.1h 0.5h
h ζ
bel
a ael X
Fig. 4. Parameterization of a semi-elliptical part-through crack existing in a pipe [9].
a = ael/bel (ellipse principal axes ratio) s = ael/a (offset of ellipse major axis related to the tangent to the external pipe circumference; usually s = 1). An auxiliary coordinate f, refers to the position point in the crack profile where the stress intensity factor is measured; coordinate f starts at the intersection point of the crack profile with the pipe’s external circumference; f is used as a normalized ratio fh (see Fig. 4): t (equivalent thickness = pipe + weld) = 5 mm.
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
2997
Semi-elliptical crack: a = 3 mm (a = ael); s = 1; a/t 0.6; R/t: assumed 10, the maximum available in the graphical results of Carpinteri et al. [9] (equivalent R 52.5 mm; internal pipe radius).
a = 0.6 (ellipse axis ratio). f = 0.6 Material: stainless steel, type 304, Cr18Ni8. Given these parameters, the non-dimensional stress intensity factor, in reference to the nominal value, can be estimated:
K I ¼ rM
pffiffiffiffiffiffi pa
ð9Þ
This case study has a = 3 mm; the nominal value for KI is:
K I ¼ rM
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pa ¼ rM p 0:003 ¼ 0:097 rM N m1=2
ð10Þ
Then, these values will be multiplied by 2 due to stress concentration effects as explained previously. 3. Discussion Fig. 5 shows the non-dimensional SIF for a defect in a straight pipe and with the geometric parameters described above. Again, using analysis from Carpinteri et al. [9], the non-dimensional SIF is K I;M ffi 1.08 as can be read in Fig. 5. The effective value of the stress intensity factor is:
K IM ¼ K LM rM
pffiffiffiffiffiffi pa ¼ 1:08 0:097 rM
ð11Þ
Following Example 1 as in Section 2.1, a curved pipe where the shell surface close to the crack zone is approximately cylindrical is considered. The pipe is subjected to a bending moment that results from a prescribed in-plane flange rotation of 0.002 rad, as described in Fig. 2. The maximum bending stress at the extrados position in the flange weld was rM ¼ 130 MPa. The corresponding value for the stress intensity factor from Eq. (11) is 13.63 MPa m1/2, but then multiplied by the factor 2 Eq. (12) now gives
2 0:097 1:08 130N
pffiffiffiffiffi m ¼ 2 13:63 ¼ 27:26
ð12Þ 1/2
The critical SIF value for this kind of stainless steel is KIC = 50 MPa m [8] and the results show that for Example 1 this value is not reached by far. Considering now different crack geometries, with circular, elliptic or straight front and for the Examples mentioned, various SIFs were evaluated and are summarized in Tables 1 and 2. Considering for instance a circular front with a crack depth of 2.4 mm for Example 3, looking again in Fig. 5 where for a = 1 the factor is 0.8, the stress intensity factor is (multiplied by the factor 2 of stress concentration effects in the presence of a crack)
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi p 0:0024 0:8 293N m ¼ 40:7
Fig. 5. Dimensionless stress intensity factor for a part-through crack with geometry parameters from [9] r/t = 10.
ð13Þ
2998
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
Table 1 Stress intensity factors for part-through cracks (60%). Sample (mm)
Weld (mm)
r max at h = p (MPa)
Crack depth (mm)
SIF straight front
SIF eliptic front a = 0.6
SIF circular front
r = 50, h = 5 r = 50, h = 2 r = 40, h = 2
Weld = 5 Weld = 3 Weld = 2
180 130 293
a=6 a=3 a = 2.4
80.0 45.4 91.6
43.4 27.2 55.0
39.6 20.2 40.7
Table 2 Stress intensity factors for part-through cracks (30%). Sample (mm)
Weld (mm)
r max at h = p (MPa)
Crack depth (mm)
SIF straight front
SIF eliptic front a = 0.6
SIF circular front
r = 50, h = 5 r = 50, h = 2 r = 40, h = 2
Weld = 5 Weld = 3 Weld = 2
180 130 293
a=3 a = 1.5 a = 1.2
52.4 27.1 54.4
Not available 18.4 37.1
29.3 14.6 30.0
Fig. 6. Stress intensity factors for part-through cracks in flat plates under tension [15].
A method used by Gonçalves and Castro [15] to calculate the SIF using the line-spring model with finite elements is shown in Fig. 6. and can be compared with this present solution assuming flat plate-like properties at the crack zone following their approach. In their example, SIFs of semi-elliptical part-through cracks in flat plates subjected to pure tension or bending were studied. The crack properties are illustrated in Fig. 6, where a/c = 0.6 corresponds to a in Fig. 5 and angle / corresponds to point A. The factor Q from their work is given by the following equation:
Q ¼ 1 þ 1:464ða=cÞ1:65
ð14Þ
Then, when a/c = 0.6, the factor Q is 1.63. The SIF calculated by Gonçalves and Castro [15] applied to flat plates is
pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi K Iplate = Q ¼ 1:35= 1:63 ¼ 1:057
ð15Þ
It can be noted that this value is close to the result of 1.08 obtained by Carpinteri et al. [9] using cylindrical shells, (5), so these results show that the method here proposed is suitable for thin shells with relatively large curvatures. 4. Conclusions It was shown that SIF results were quite similar either using cylindrical shell or flat plate approaches for the same crack geometry, and so the method can be applied to a doubly curved shell with a part-through crack, provided that the shell has two distinct principal curvatures, where one is much smaller than the other which is an advantage in addressing this complex problem of curved pipes under bending. Considering the examples analyzed with different pipe radius and thickness, the main influence in the results obtained is when the radius is shorter because the nominal stresses are much higher, but then a smaller ratio radius/equivalent thickness for the same relative crack depth increases the stress intensity factors for pipes with the same radius because the
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
2999
dominant factors are the nominal stress and the value of the crack depth as can be seen in Eq. (11). So the combination of the three factors in these calculations is of leading importance in the analysis of piping structures with shallow cracks. References [1] Viswanatha N, Bhate N, Kushwaha HS. Stress-intensity factors for elbows with axial surface crack at crown. Trans. SMiRT 14th international conference in structural mechanics in reactor technology, Lyon, France; 17–22 August 1997. paper G06-5. [2] Yahiaoui K, Moffat DG, Moreton DN. Pipe elbows with cracks part 1: a parametric study of the influence of crack size on limit loads due to pressure and opening bending. J Strain Anal 2000;35(1):35–46. [3] Oliveira CM, Melo F, Castro PT. The elastic analysis of arbitrary thin shells having part-through cracks, using the integrated line spring and the semiloof shell elements. Engng Fract Mech 1991;39(6):1027–35. [4] Madureira L, Melo F. Stress analysis of curved pipes with hybrid formulation. Int J Pres Vess Pip 2004;81(3):243–9. [5] Madureira L, Melo F. A hybrid formulation in the stress analysis of curved pipes. Engng Comput 2000;17(8):970–80. [6] Millard A, Roche R. Elementary solutions for the propagation of ovalization along straight pipes and elbows. Int J Pres Vess Pip 1984;16(1):101–29. [7] Marie S, Chapulio S, Kayser Y, Lacire MH, Drubay B, Barthelet B, et al. French RSE-M and RCC-MR code appendices for flaw analysis: presentation of the fracture parameters calculation — Part IV: cracked elbows. Int J Pres Vess Pip 2007;84(10–11):659–86. [8] Avallone E, Baumeister T. Mark’s standard handbook for mechanical engineers. McGraw-Hill; 1997 [chapter 5]. [9] Carpinteri A, Brighenti R, Spagnoli A. Part-through cracks in pipes under cyclic bending. Nucl Engng Des 1998;185:1–10. [10] Flügge W. Thin elastic shells. 2nd ed. Berlin: Springer; 1973. [11] Hose D, Kitching R. Behavior of GRP smooth pipe bends with tangent pipes under flexure or pressure loads: a comparison of analyses by conventional and finite element techniques. Int J Mech Sci 1993;35(7):549–75. [12] Kitching R. Smooth and mitred pipe bends. In: Gill SS, editor. The stress analysis of pressure vessels & pressure vessels components. Pergamon Press; 1970 [chapter 7]. [13] Findlay GE, Spence J. Flexibility of smooth circular curved tubes with flanged end constraints. Int J Pres Vess Pip 1979;7:13–29. [14] Zinekiewicz OC, Taylor R. The finite element method. 5th ed. Butterworth-Heinemann; 2000. [15] Gonçalves JPM, Castro PT. Application of the line spring model to some complex geometries and comparison with three-dimensional results. Int J Pres Vess Pip 1999;76(8):551–60.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Hybrid formulation solution for stress analysis of curved pipes with welded bending joints Luisa R. Madureira a,*, Francisco Q. Melo b a b
Department of Mechanical Engineering, Faculty of Engineering of the University of Porto, Portugal Department of Mechanical Engineering, University of Aveiro, Portugal
a r t i c l e
i n f o
Article history: Received 7 August 2009 Received in revised form 2 April 2010 Accepted 12 April 2010 Available online 14 April 2010 Keywords: Hybrid formulation Curved pipes Stress intensity factors Fourier series
a b s t r a c t A method is presented for evaluating the stress intensity factor of part-through cracks in a thin pipe elbow. A hybrid formulation solution is used to evaluate the stress field close to the crack area. The stress field values are then inputted into a previously developed method published in the literature to evaluate the stress intensity factor in cylindrical shells. Results from cylindrical shells with part-through cracks are extended to double-curvature pipe configurations that contain the same kind of flaw. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Curved pipes play an important role in piping engineering, and an accurate evaluation of their stress field is important for critical applications. The process of manufacturing curved pipes generally involves plastic forming and/or welding methods to join pipe parts when the pipe cannot be manufactured as a whole piece. In both processes, high loads or imperfect welding can create part-through cracks caused by lack of material adhesion or local ruptures. The stress intensity factor or SIF, in abbreviated form, is used in fracture mechanics to predict the stress state close to the crack tip and can be determined by numerical or experimental methods. The former method has largely been adopted by project engineers, as experimental techniques are generally expensive and demand very precise methods for finding the displacement field close to the defect zone. Numerical methods in process engineering are frequently identified with finite element techniques. The structure is modeled with shell finite elements, while the crack front is modeled with singular finite elements. Research in this area has seen a considerable number of contributions that provide more accurate and advanced techniques for determining the SIF of piping accessories that contain part-through cracks. In fact, these partial defects draw more attention than complete cracks since thorough inspection procedures can be used to detect them at an early stage. The possibility of crack generation and propagation in pipe elbows subjected to bending loads has been studied using fracture mechanics methods. Viswanath et al. [1] obtained stress intensity factors in elbows subjected to uniform in-plane bending. In this study, a part-through crack along the longitudinal axis at the crown pipe was studied. The crack opening mode was found to be due to transverse section ovalization, a serious stress mode that induces rapid crack growth in this kind of piping accessories. The authors used a finite element commercial code to perform the analysis. Yahiaoui et al. [2] analyzed the influence of crack size on limit loads also using finite element analysis. Oliveira et al. [3] studied the stress intensity factors of surface elliptical cracks existing in pipe elbows using high order 8-node shell elements. To simulate the remaining ligament compliance at the crack zone, they used the line-spring model, a method that has performed very well for this class of three-dimensional structural problems. * Corresponding author. Fax: +351 22 5081445. E-mail addresses: [email protected] (L.R. Madureira), [email protected] (F.Q. Melo). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.04.009
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
2993
Nomenclature a ai(x) bi(x) C d/ E h KI L M0 O(h) r R SIF t U(r, u) u u v w
a m f
r rx X
maximum crack depth ovalization function warping function elastic pipe properties matrix bending angle Young’s modulus pipe thickness nominal stress intensity factor transverse section center arch bending moment ovalization displacement pipe transverse section radius pipe curvature radius stress intensity factor equivalent thickness total internal energy displacement field longitudinal shell displacement circumferential shell displacement transverse shell displacement ellipse axis ratio poisson0 s ratio position point in the crack profile where the SIF is measured vector of internal stresses hoop stress the integral domain
However, because finite element modeling for fracture mechanics problems requires highly refined meshes and heavy computational work, an alternative method is proposed. The process combines shape factors available in the literature with a hybrid solution formulation where unknown functions, ovalization and warping, are used together with Fourier trigonometric expansions [4–6]. These shape factors are used in the evaluation of SIF parameters for a generalized crack existing in a doubly curved surface shell. Recently Marie et al. [7] published the stress intensity factors and J calculation for cracked elbows where values for KI and rref are tabled for several crack geometries. Most previous studies use a crack front that is oriented along the longitudinal (or axial) pipe elbow line arch. This study focuses on cracks along the circumference line of the welded flange-pipe joint. As the shell structure is based on a semimembrane deformation model the bending stress along the longitudinal direction is not considered, but instead the traction stress due to the longitudinal direction force is. The evaluation of the stress intensity factor is based on linear elastic fracture mechanics. The weld joining the flange to the pipe toroidal surface is assumed as a T-joint and for this kind of problem a stress concentration factor of 2 is normally used [8] and will multiply the nominal stresses calculated in this work. Also, following the work of Carpinteri et al. [9] for part-through cracks in cylindrical shells, stress intensity factors are evaluated for different crack geometries, once the curved pipe radius/thickness ratio considered allows this approach. 2. Methodology In a curved pipe with end flanges, the flange-pipe interface is a likely location for defects, as sketched in Fig. 1. Here, the goal is to analyze the structural integrity of a pipe where a crack similar to the one displayed above is detected in a piping part. The role of the stress intensity factor (SIF) is to quantify the elastic stress field in the close neighborhood of the crack front. For this purpose, this work follows that of Carpinteri et al. [9], in which thin and thick walled straight pipes that have a part-through crack were subjected to bending loads. Due to the presence of the defect, the pipe integrity is defined by the maximum SIF along the crack profile [9]. The displacement field is given by [6,10–13]:
wðx; hÞ ¼
v ðx; hÞ ¼ uðx; hÞ ¼
5 X i¼2 5 X i¼2 5 X i¼2
ai ðxÞ cosðihÞ
ai ðxÞ sinðihÞ i
bi ðxÞ cosðihÞ
ð1Þ
2994
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
M
x= -L
Crack location
x= 0 Crack location
θ
C
C x= +L M
Section C-C Fig. 1. Part-through crack in a welded flange joint of a curved pipe.
The following pipe deformations used to find the solution according to the semi-membrane strain field are given by differential equations [6,10–13]:
@u 1 þ ðw cos h v sin hÞ @x R ¼0
ex ¼ eh
1 @u @ v usin h þ þ r @h @x ! R 1 @2w @v kh ¼ 2 r @h2 @h
ð2Þ
cxh ¼
The solution is derived by minimizing the total elastic energy U involved in the pipe distortion process with a set of unknown functions [14]:
Uðr; uÞ ¼
1 2
Z
rT C1 r dX þ
X
Z
rT u d X
ð3Þ
X
C is the pipe elastic properties matrix. The first integral refers to the internal elastic deformation energy, and the second is the deformation work performed by the unknown vector r of the internal stresses against the admissible displacement field u. An external entity M0, the bending moment, performs additional work at a bending angle d/. The system of external forces performs deformation work (called the external work) upon the displacement functions ai and bi. The total energy of the structure U is the sum of this external work with the corresponding elastic deformation. This value should be zero if the displacement functions are exact, but for the displacement field approximations used here (Fourier expansions), U has a residual value. This residual allows for the optimization of expressions to minimize U. The variation of the total energy U is found as follows:
dU ¼
Z
drT C1 r dX þ
X
Z X
drT u dX þ
Z
rT du dX
ð4Þ
X
where the integral domain X is [0, L] [0, 2p] and L represents the length of the transverse section of the center arch. The total variation in U is the result of separate contributions from variations in the stress field r and in the displacement u. Equilibrium is achieved by minimizing the function U separately for each of the unknowns:
8 @U @U @U > ¼ 0; ¼ 0; ¼0 < @N @Nxhi @mhi xi i2 i2 i2 0 > @U @U 0 @U @U : @a dai þ @a ¼ 0; dbi þ @b ¼0 0 dbi 0 dai @bi i i2
i
i
i2
The following system of differential equations can be solved by evaluation of eigenvalues and eigenvectors [4,5]:
9pD 5prA prA prS 0 3 0 a b M0 þ a p Sb þ 2 4 2 4 r3 4R 4R 3 4Rr 16R 32R2 prS 00 5prA 64pD prA 2prA prS 0 prA prS 0 0 a a3 ¼ þ b b þ þ a þ p Sb þ 3 5 3 9 r3 3R 6R 2 3R 6R 4 9R2 5R2
prS
a002 ¼
prA
þ 2
ð5Þ
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
! 152 pD 5prA prS 0 3prA prS 0 0 a4 þ þ b3 pSb4 þ b 2 3 16 r 8R 8R 8R 8R 5 32R 32R prS 00 4prA 13prA 576pD 3prA prS 0 0 a a5 ¼ þ b pSb5 a þ þ þ 3 5 25 r3 5R 10R 4 15R2 25R2 pS 4pS prS prS 2prA prS 0 a prAb002 ¼ b3 þ þ 2 b2 2 b4 þ pSa02 þ 2R r 3R 6R 3 4R 4R pS 9pS 2prS pS prS prðA SÞ 0 5prA prS 0 b3 þ b4 2 b5 a2 þ pSa03 þ a prAb003 ¼ b2 þ 2 2R r 2R 4R 8R 8R 4 4R 4R prS pS 16pS 2prS prA prS 0 prS 3prA 0 b4 þ pS4 a0 þ a þ prAb004 ¼ 2 b2 b3 þ a 2R r 3R 6R 3 10R 5R 5 4R 4R2 prS pS 25pS prS 3prA prS 0 2 b5 a þ p S5 a 0 prAb005 ¼ 2 b3 b4 þ 2R r 8R 8R 4 4R 4R
prS
a004 ¼
5prA
a þ 2 2
17prA
2995
þ
ð6Þ
2.1. Determination of the nominal stress field in a curved pipe under bending To assess the solution method discussed here, stress analysis is carried out for a curved pipe with rigid end flanges. The material properties and dimensions are: Transverse section radius, r – 50 mm. Wall thickness, h – 2 mm. Mean curvature radius, R – 200 mm. Pipe angle, / – 90°. Transverse section center arch, L – 157.0796 mm (=R/). Material – stainless steel, type 304, Cr18Ni8: E = 210 GPa,
m ¼ 0:3.
An in-plane uniform bending moment M0 loads the pipe, where symmetry conditions were considered. In the calculations, a specified rotation angle of d/ = 0.002 rad at a flanged end was prescribed. This corresponds to a value M = 35332456 N mm. Boundary conditions are imposed:
bi ð0Þ ¼ 0; bi ðLÞ ¼ 0 a0i ð0Þ ¼ 0; ai ðLÞ ¼ 0
ð7Þ
The values obtained for the ovalization parameters at x = 0 are:a2 = 103.42 d/ mm, a3 = -31.03d/ mm, a4 = 8.15da mm, a5 = 3.89d/ mm. (d/ = 0.002 rad). The pipe ovalization displacement is given by the trigonometric expansion, for x = 0:
OðhÞ ¼ ð103:42 cos 2h 31:03 cos 3h 8:15 cos 4h þ 3:89 cos 5hÞdu mm The hoop stress at section x = 0 or x = L, as a consequence of the bending moment, is given by (8):
"
rx
# X 5 5 1X 1 0 ¼ A r cos hW ðxÞ þ ai ðxÞ cos h cos ih þ sin ihsinh þ bi ðxÞ cos ih MPa R i¼2 i i¼2 00
ð8Þ
Two more examples are studied in order to focus on the effect of different radius or thickness for this kind of pipe. EXAMPLE 2: h = 5, r = 50 (in mm). EXAMPLE 3: h = 2, r = 40 (in mm). Evaluations lead to the stress distributions for x = L that are shown in Fig. 2 where at the extrados the values obtained for the nominal stresses are 130, 180 and 293 Mpa for Examples 1, 2 and 3 respectively. 2.2. Structural evaluation of a curved pipe with a flaw in a welded joint subjected to a bending moment The method is applied to a curved pipe with cracked rigid end flanges. Fig. 3 shows a crack resulting from the lack of adhesion that occurs in the extrados curved pipe. The combined pipe and welded joint thickness are 5 mm for Example 1; therefore, the equivalent pipe thickness is assumed to be 5 mm with a part-through crack as shown in Fig. 4 [9]. The following parameters are necessary for evaluating the stress intensity factor (SIF) associated with the crack geometry: a = maximum crack depth ael = minor half axis of ellipse (the crack has an elliptical profile) n = a/t (non-dimensional parameter; measured at the largest crack depth)
2996
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
Fig. 2. Longitudinal membrane stress, rx at x = L from uniform bending load.
C Flaw location
Crack location
Welded joint 2
3
Flange
θ C Section C-C Fig. 3. Elbow with end flanges: welding detail and flaw location (in mm).
Y
r t
J P B CN L A β
0.1h 0.5h
h ζ
bel
a ael X
Fig. 4. Parameterization of a semi-elliptical part-through crack existing in a pipe [9].
a = ael/bel (ellipse principal axes ratio) s = ael/a (offset of ellipse major axis related to the tangent to the external pipe circumference; usually s = 1). An auxiliary coordinate f, refers to the position point in the crack profile where the stress intensity factor is measured; coordinate f starts at the intersection point of the crack profile with the pipe’s external circumference; f is used as a normalized ratio fh (see Fig. 4): t (equivalent thickness = pipe + weld) = 5 mm.
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
2997
Semi-elliptical crack: a = 3 mm (a = ael); s = 1; a/t 0.6; R/t: assumed 10, the maximum available in the graphical results of Carpinteri et al. [9] (equivalent R 52.5 mm; internal pipe radius).
a = 0.6 (ellipse axis ratio). f = 0.6 Material: stainless steel, type 304, Cr18Ni8. Given these parameters, the non-dimensional stress intensity factor, in reference to the nominal value, can be estimated:
K I ¼ rM
pffiffiffiffiffiffi pa
ð9Þ
This case study has a = 3 mm; the nominal value for KI is:
K I ¼ rM
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pa ¼ rM p 0:003 ¼ 0:097 rM N m1=2
ð10Þ
Then, these values will be multiplied by 2 due to stress concentration effects as explained previously. 3. Discussion Fig. 5 shows the non-dimensional SIF for a defect in a straight pipe and with the geometric parameters described above. Again, using analysis from Carpinteri et al. [9], the non-dimensional SIF is K I;M ffi 1.08 as can be read in Fig. 5. The effective value of the stress intensity factor is:
K IM ¼ K LM rM
pffiffiffiffiffiffi pa ¼ 1:08 0:097 rM
ð11Þ
Following Example 1 as in Section 2.1, a curved pipe where the shell surface close to the crack zone is approximately cylindrical is considered. The pipe is subjected to a bending moment that results from a prescribed in-plane flange rotation of 0.002 rad, as described in Fig. 2. The maximum bending stress at the extrados position in the flange weld was rM ¼ 130 MPa. The corresponding value for the stress intensity factor from Eq. (11) is 13.63 MPa m1/2, but then multiplied by the factor 2 Eq. (12) now gives
2 0:097 1:08 130N
pffiffiffiffiffi m ¼ 2 13:63 ¼ 27:26
ð12Þ 1/2
The critical SIF value for this kind of stainless steel is KIC = 50 MPa m [8] and the results show that for Example 1 this value is not reached by far. Considering now different crack geometries, with circular, elliptic or straight front and for the Examples mentioned, various SIFs were evaluated and are summarized in Tables 1 and 2. Considering for instance a circular front with a crack depth of 2.4 mm for Example 3, looking again in Fig. 5 where for a = 1 the factor is 0.8, the stress intensity factor is (multiplied by the factor 2 of stress concentration effects in the presence of a crack)
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi p 0:0024 0:8 293N m ¼ 40:7
Fig. 5. Dimensionless stress intensity factor for a part-through crack with geometry parameters from [9] r/t = 10.
ð13Þ
2998
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
Table 1 Stress intensity factors for part-through cracks (60%). Sample (mm)
Weld (mm)
r max at h = p (MPa)
Crack depth (mm)
SIF straight front
SIF eliptic front a = 0.6
SIF circular front
r = 50, h = 5 r = 50, h = 2 r = 40, h = 2
Weld = 5 Weld = 3 Weld = 2
180 130 293
a=6 a=3 a = 2.4
80.0 45.4 91.6
43.4 27.2 55.0
39.6 20.2 40.7
Table 2 Stress intensity factors for part-through cracks (30%). Sample (mm)
Weld (mm)
r max at h = p (MPa)
Crack depth (mm)
SIF straight front
SIF eliptic front a = 0.6
SIF circular front
r = 50, h = 5 r = 50, h = 2 r = 40, h = 2
Weld = 5 Weld = 3 Weld = 2
180 130 293
a=3 a = 1.5 a = 1.2
52.4 27.1 54.4
Not available 18.4 37.1
29.3 14.6 30.0
Fig. 6. Stress intensity factors for part-through cracks in flat plates under tension [15].
A method used by Gonçalves and Castro [15] to calculate the SIF using the line-spring model with finite elements is shown in Fig. 6. and can be compared with this present solution assuming flat plate-like properties at the crack zone following their approach. In their example, SIFs of semi-elliptical part-through cracks in flat plates subjected to pure tension or bending were studied. The crack properties are illustrated in Fig. 6, where a/c = 0.6 corresponds to a in Fig. 5 and angle / corresponds to point A. The factor Q from their work is given by the following equation:
Q ¼ 1 þ 1:464ða=cÞ1:65
ð14Þ
Then, when a/c = 0.6, the factor Q is 1.63. The SIF calculated by Gonçalves and Castro [15] applied to flat plates is
pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi K Iplate = Q ¼ 1:35= 1:63 ¼ 1:057
ð15Þ
It can be noted that this value is close to the result of 1.08 obtained by Carpinteri et al. [9] using cylindrical shells, (5), so these results show that the method here proposed is suitable for thin shells with relatively large curvatures. 4. Conclusions It was shown that SIF results were quite similar either using cylindrical shell or flat plate approaches for the same crack geometry, and so the method can be applied to a doubly curved shell with a part-through crack, provided that the shell has two distinct principal curvatures, where one is much smaller than the other which is an advantage in addressing this complex problem of curved pipes under bending. Considering the examples analyzed with different pipe radius and thickness, the main influence in the results obtained is when the radius is shorter because the nominal stresses are much higher, but then a smaller ratio radius/equivalent thickness for the same relative crack depth increases the stress intensity factors for pipes with the same radius because the
L.R. Madureira, F.Q. Melo / Engineering Fracture Mechanics 77 (2010) 2992–2999
2999
dominant factors are the nominal stress and the value of the crack depth as can be seen in Eq. (11). So the combination of the three factors in these calculations is of leading importance in the analysis of piping structures with shallow cracks. References [1] Viswanatha N, Bhate N, Kushwaha HS. Stress-intensity factors for elbows with axial surface crack at crown. Trans. SMiRT 14th international conference in structural mechanics in reactor technology, Lyon, France; 17–22 August 1997. paper G06-5. [2] Yahiaoui K, Moffat DG, Moreton DN. Pipe elbows with cracks part 1: a parametric study of the influence of crack size on limit loads due to pressure and opening bending. J Strain Anal 2000;35(1):35–46. [3] Oliveira CM, Melo F, Castro PT. The elastic analysis of arbitrary thin shells having part-through cracks, using the integrated line spring and the semiloof shell elements. Engng Fract Mech 1991;39(6):1027–35. [4] Madureira L, Melo F. Stress analysis of curved pipes with hybrid formulation. Int J Pres Vess Pip 2004;81(3):243–9. [5] Madureira L, Melo F. A hybrid formulation in the stress analysis of curved pipes. Engng Comput 2000;17(8):970–80. [6] Millard A, Roche R. Elementary solutions for the propagation of ovalization along straight pipes and elbows. Int J Pres Vess Pip 1984;16(1):101–29. [7] Marie S, Chapulio S, Kayser Y, Lacire MH, Drubay B, Barthelet B, et al. French RSE-M and RCC-MR code appendices for flaw analysis: presentation of the fracture parameters calculation — Part IV: cracked elbows. Int J Pres Vess Pip 2007;84(10–11):659–86. [8] Avallone E, Baumeister T. Mark’s standard handbook for mechanical engineers. McGraw-Hill; 1997 [chapter 5]. [9] Carpinteri A, Brighenti R, Spagnoli A. Part-through cracks in pipes under cyclic bending. Nucl Engng Des 1998;185:1–10. [10] Flügge W. Thin elastic shells. 2nd ed. Berlin: Springer; 1973. [11] Hose D, Kitching R. Behavior of GRP smooth pipe bends with tangent pipes under flexure or pressure loads: a comparison of analyses by conventional and finite element techniques. Int J Mech Sci 1993;35(7):549–75. [12] Kitching R. Smooth and mitred pipe bends. In: Gill SS, editor. The stress analysis of pressure vessels & pressure vessels components. Pergamon Press; 1970 [chapter 7]. [13] Findlay GE, Spence J. Flexibility of smooth circular curved tubes with flanged end constraints. Int J Pres Vess Pip 1979;7:13–29. [14] Zinekiewicz OC, Taylor R. The finite element method. 5th ed. Butterworth-Heinemann; 2000. [15] Gonçalves JPM, Castro PT. Application of the line spring model to some complex geometries and comparison with three-dimensional results. Int J Pres Vess Pip 1999;76(8):551–60.