The limit load calculations for pipelines with axial complex-shaped defects
Based on the analytical model for the plastic limit state, a numerical procedure for estimating the remaining strength of a complex-shaped defect has been developed. Comparison of the calculation results obtained by the proposed procedure with the experimental data and the calculation results obtain...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-484752013-08-20T07:23:14Z The limit load calculations for pipelines with axial complex-shaped defects Bogdan, A.V. Lokhman, I.V. Ageev, S.M. Orynyak, I.V. Научно-технический раздел Based on the analytical model for the plastic limit state, a numerical procedure for estimating the remaining strength of a complex-shaped defect has been developed. Comparison of the calculation results obtained by the proposed procedure with the experimental data and the calculation results obtained by other methods showed its efficiency. С использованием аналитической модели пластического предельного состояния разработана методика численного расчета остаточной прочности объекта с дефектом сложной формы. Сравнение результатов расчета по предложенной методике с таковыми по другим методикам и экспериментальными данными свидетельствует о ее эффективности. 2009 Article The limit load calculations for pipelines with axial complex-shaped defects / A.V. Bogdan, I.V. Lokhman, S.M. Ageev, I.V. Orynyak // Проблемы прочности. — 2009. — № 1. — С. 61-68. — Бібліогр.:4 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/48475 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Bogdan, A.V. Lokhman, I.V. Ageev, S.M. Orynyak, I.V. The limit load calculations for pipelines with axial complex-shaped defects Проблемы прочности |
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Based on the analytical model for the plastic limit state, a numerical procedure for estimating the remaining strength of a complex-shaped defect has been developed. Comparison of the calculation results obtained by the proposed procedure with the experimental data and the calculation results obtained by other methods showed its efficiency. |
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Article |
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Bogdan, A.V. Lokhman, I.V. Ageev, S.M. Orynyak, I.V. |
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Bogdan, A.V. Lokhman, I.V. Ageev, S.M. Orynyak, I.V. |
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Bogdan, A.V. |
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The limit load calculations for pipelines with axial complex-shaped defects |
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The limit load calculations for pipelines with axial complex-shaped defects |
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The limit load calculations for pipelines with axial complex-shaped defects |
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The limit load calculations for pipelines with axial complex-shaped defects |
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The limit load calculations for pipelines with axial complex-shaped defects |
| title_sort |
limit load calculations for pipelines with axial complex-shaped defects |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2009 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/48475 |
| citation_txt |
The limit load calculations for pipelines with axial complex-shaped defects / A.V. Bogdan, I.V. Lokhman, S.M. Ageev, I.V. Orynyak // Проблемы прочности. — 2009. — № 1. — С. 61-68. — Бібліогр.:4 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-04T08:59:51Z |
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UDC 539.4
The Limit Load Calculations for Pipelines with Axial Complex-Shaped
Defects
A. V. B ogdan,a I. V. L o k h m an ,b S. M . Ageev,a and I. V. O ry n y ak a
a Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine,
Kiev, Ukraine
b SC Ukrtransgas, Kiev, Ukraine
Based on the analytical model fo r the plastic limit state, a numerical procedure fo r estimating the
remaining strength o f a complex-shaped defect has been developed. Comparison o f the calculation
results obtained by the proposed procedure with the experimental data and the calculation results
obtained by other methods showed its efficiency.
K e y w o r d s : defect, pipe, strength reduction factor, internal pressure, ultimate
strength, lim it load.
In tro d u c tio n . An important constituent o f the pipeline reliability assurance
is the in-line inspection, using w hich the remaining wall thickness t n o f a pipe at
its discrete points is found. Each o f the points is characterized by the axial
coordinate along the pipeline s and the angular coordinate p around the circle of
the cross-section.
For further calculations the two-dim ensional system is reduced to a
unidimensional one: the m aximum defect depth is projected onto the axial line as
specified in DNV-RP-F101 [1]. A n example o f this system o f points, which is
called a c o m p le x d e fe c t , is presented in Fig. 1. The simplest m ethod for analyzing
the strength o f a complex defect is its replacem ent by a rectangular defect whose
length is equal to that o f the complex defect l and depth is equal to the m aximum
depth o f the complex defect d . Clearly, the results obtained in this way w ould be
rather conservative. However, they are useful in the case where there is a need to
obtain, within a short period o f time, an approximate estimate o f the complex
defect strength and to get an answer to whether it is necessary at all to carry out a
more accurate calculation that takes into account the real profile o f a complex
defect. Thus, we initially consider some approaches to strength estimation for
defects o f typical shapes (rectangular, elliptic, and parabolic) and then the
calculation methods that involve the use o f a real profile o f a complex defect.
1. E stim ation o f S treng th o f a P ipe w ith Defects o f Typical Shape.
1.1. T rea tm en t o f an I s o la te d D e f e c t A c c o r d in g to D N V -R P -F 1 0 1 . One o f the
current and generally recognized techniques for estimating corrosion defects is
the international standard DNV Recom m ended Practice [1]. According to this
practice, for an isolated defect o f length l and depth d , the expression for
determining the internal failure pressure is rather simple and has the following
form:
( 1)
© A. V. BOGDAN, I. V. LOKHMAN, S. M. AGEEV, I. V. ORYNYAK, 2009
ISSN 0556-171X. Проблемы прочности, 2009, № 1 61
A. V. Bogdan, I. V. Lokhman, S. M. Ageev, and I. V. Orynyak
where
Q = V 1+ 0.31(l / 4 D t ) 2 , (2)
o u is the ultim ate strength o f the pipe steel, t is the pipe wall thickness, and D is
its outer diameter. According to [1], relationship (1) is empirical and derived on
the basis o f a great num ber o f calculations by the finite element m ethod (FEM)
and full-scale experiments. In contrast to this approach to the estimation o f the
remaining strength o f a pipe w ith single defects, we consider below an analytical
m odel which has been developed by us and is based on the limit state theory.
Axial coordinate (mm)
Fig. 1. Profile of a complex defect.
1.2. T h e A n a ly t ic a l M o d e l . The strength condition for a defected pipe can be
written as
o u t
P ^ « ( K '0 — , (3)
where a(X, r ) is the strength reduction factor, X = l/V R t, R is the inner radius,
t is the wall thickness, r = t n / t , t n is the remaining w all thickness, t n = t — d ,
o u is the ultimate strength o f the pipe material, and P is the applied internal
pressure. We shall consider the basic idea o f the a(X , r ) determination.
Consider the equilibrium equation in the radial direction:
N p + d Q x
P = ~ R + ~ d k ’ (4)
w here N ^ is the circum ferential force, cp is the circum ferential angular
coordinate, x is the axial coordinate, and Q p and Q x are the transverse forces.
For a defect-free pipe, each point o f the pipe is under identical conditions,
therefore, the transverse forces are equal to zero. Since N p = o p t , the lim it state
is a tta in ed w hen Op = o u, th e re fo re , the ad m issib le in te rn a l p ressu re
P l l ( r = 1) = O u t R or a ( r = 1) = 1
For a section with a part-through defect o f the maximum depth d , the
m aximum value o f N p j R in (4) is equal to o u ( t — d ) / R and this force cannot
resist the larger values o f the internal pressure P . Thus, the presence o f an axial
defect results in the imbalance between the circumferential stresses and the
62 ISSN 0556-171X. npodxeMbi npounocmu, 2009, N9 1
The Limit Load Calculations fo r Pipelines
internal pressure which has to be counterbalanced by an increm ent in the
transverse forces to maintain equilibrium. The transverse forces, in turn, induce
bending moments. The state o f the cylinder changes to the limit state when the
bending moments reach the critical values that m eet the chosen criterial conditions.
The above considerations are presented in more detail in [2], where the following
expressions for a LL were found:
1 + 2 X2 x o ( l - r o)
a LL = ----------2------------- (5a)
1 + 2X2 (1 - x 0 ) V }
for a rectangular defect,
1 + X2 (1 — (1 — x 0 )^ 2 3/32)(1 - x 0 )4/3
a LL ~ , , ) 2 n w / o (5b)1+ X2 (1— X 0 )4/3
for a semi-elliptical defect, and
1+ X2(1 — (1 —x 0 ) 8 9 ( 1 —x 0 ) ^ ^
a LL 1 , 12/1 x (5c)
1+ X (1—x 0 )
for a parabolic defect.
As was noted above, the replacem ent o f a complex defect by a rectangular
one leads to rather conservative estimates. Therefore, below we consider the
approaches which take into account the real shape o f a complex defect and yield a
more accurate estimate o f the strength.
2. S treng th C alcu lation fo r C om plex Defects.
2.1. T he E ffe c t iv e A r e a f o r a C o m p le x D e fe c t. One o f the simplest methods
o f strength calculation for a pipeline w ith a complex defect involving the use of
its real profile is the m ethod based on the concept o f the effective area.
In [3], the following expression for strength evaluation for a complex defect
is proposed:
f = T _ % A0
(6)
where A is the effective area, A 0 is the total area across the wall thickness in the
longitudinal direction, M is the Folias factor, and o is the yield strength o f the
pipeline material. W ith this approach, the multiplier (1 — A/A0 ) can be interpreted
as the ratio o f the average remaining wall thickness t na to the original defect-free
wall thickness t o f the defected area. D ifferent zones o f the complex defect are
exam ined sequentially and the minimum wall thickness is chosen. The subjective
nature o f this m odel is obvious.
Another currently available m ethod o f strength analysis for a complex defect
is the method proposed in DNV-RP-F101 [1]. The basic idea o f this method
consists in splitting a complex defect into simple ones with their strength
calculation using the expression described in item 1.1.
ISSN 0556-171X. npoöxeMbi npounocmu, 2009, N 1 63
A. V. Bogdan, I. V. Lokhman, S. M. Ageev, and I. V. Orynyak
As an alternative to the existing traditional approaches, the researchers o f the
Pisarenko Institute o f Problems o f Strength developed a numerical procedure for
assessing complex shaped defects, which rests upon clear-cut physical models.
2.2. N u m e r ic a l P r o c e d u r e f o r D e te r m in in g th e S tre n g th R e d u c tio n F a c to r f o r
a C o m p le x -S h a p e d D e fe c t. Let us introduce the notion o f the dimensionless
internal pressure p = ( p R ) / (o u t) , where o u is the ultimate strength o f the
pipeline material. We m ark N points on the profile o f a complex defect at the
interval d x = t , where t is the wall thickness. Each point is characterized by the
remaining strength o t = ( t n i/ t )o u , where t ni is the remaining wall thickness at
the point num bered i.
The basis o f the critical state m odel is that the increments in the applied
m om ent to the left o f the point under consideration (towards smaller point
numbers) are positive and those to its right, are negative. In addition, we assume
that when moving to the left, we have negative increments d x = —t. These
assumptions were made for the purpose o f similarity o f further computations.
We assign a numerical value to the dimensionless internal pressure p.
Keeping in m ind the above considerations, we search for points where the
strength condition m ay be not satisfied. Before formulating the procedure of
searching, let us find the value o f the increment in the transverse force on the
segment d x (the subscript i on the remaining strength o is omitted):
d Q x N p o ut o t o u t v- ^
1 T = p — R = — — R = ~ T l p — o ] , (7)
where o = o /o u is the dimensionless remaining strength.
It is clear that according to the assumption o f a positive increm ent in the
bending m om ent to the left o f the point with the num ber k, it is necessary that the
condition o f the transverse force Q x decrease be fulfilled. Thus
p — o k ^ 0 (8)
because d x < 0. Condition (8) is a necessary condition to choose a point of
possible failure.
Let us assume that on passing sequentially the complex defect profile points,
the point satisfying condition (8) is found. Then we take k = 0 for this point and
consider that the points w ith negative numbers are to the left o f this point and
those with positive numbers to the right o f it. Also assume that the transverse
force Q x = 0 at this point. The aim o f the calculation is to find the increm ent in
the bending m om ent and to compare it w ith its critical value. So we move to the
left (in the direction o f the negative numbers o f the points). The transverse forces
can be found from the following expressions:
o a t _
Q —1 = Q 0 — — ( p — o 0 )d x ,
...................................................... (9)
o a t _
Q —i = Q —i+ i — — ( p — o —i+ i)dx ,
64 ISSN 0556-171X. npodxeMbi npounocmu, 2009, N9 1
The Limit Load Calculations fo r Pipelines
or in dimensionless quantities:
q _ i = q 0 + ( p - a q),
( 10)
q - i = q - i + i + ( p - O - i + 0
_ a A
where Q - i = q - i
R
Let us find the increments in the bending moments:
a a t t 2
M 0 = Q o t + ~ R T ( P - a o ^ ,
a a t t 2
M - i = M - i + i + Q - i + i + — ( p - a - i + i ) — ,
( 11)
or using ( 10) and (11) in dimensionless quantities
- P - a 0
mo = — ~—
where M - i = m_
a a t
- - , - , ( P - a - i+1)
m - i = m -i+ 1 + q - i+ 1 + --------2-------
3
( 12)
R
a a t 2
The condition o f strength has the form M _ < —- — or in a dimensionless
form
_ R
^ - t ■ (13)
_ _ R
We keep on moving to the left until m _ > 0 and m _ < — . I f strength
condition (13) is not fulfilled, we go to the right (towards positive subscripts).
Otherwise, we search for the next point where the condition o f fracture possibility
(8) holds true.
We move to the right in a similar way, w ith the only difference that on the
right, according to the m odel assumption, m t should be negative. I f on the right
the strength condition (13) is not fulfilled either, fracture occurs. I f no point
where fracture occurs is found for a given dimensionless internal pressure, we
increase the value o f the dimensionless internal pressure and repeat the above
steps o f the numerical procedure. I f such a point has been found, we decrease the
value o f the internal pressure. The search technique can be similar, e.g., to that of
ISSN 0556-171X. npoöxeMbi npounocmu, 2009, N9 1 65
A. V. Bogdan, I. V. Lokhman, S. M. Ageev, and I. V. Orynyak
dividing a segment in two. In this case, the dimensionless internal pressure will be
determined with a prescribed accuracy. It is obvious that the obtained value of the
dimensionless internal pressure p will be equal to the strength reduction factor a.
The validity o f the developed algorithm was checked on two model examples,
wherein the geometrical parameters o f defects were set in conventional units
related to a single step d x = t = 1
E x a m p le 1. Consider a rectangular defect w ith the following parameters:
length L = 5, R t = 20, and t n / t = 0.3. Calculations by formula (5a) gives
a = 0.5545 and by the algorithm constructed a = 0.556.
E x a m p le 2 . Consider two rectangular defects o f length 2L and depth d in a
pipe, the distance between the defects being equal to 2k (Fig. 2).
Fig. 2. Stressed state for two symmetrical defects.
The value o f the strength reduction factor a is num erically found from a
system o f equations where the first equation can be obtained from expression (5a)
assuming that the defect h a lf length is equal to L + z (z is the displacement of
the point, where Q x = 0 w ith respect to the defect center) and the second
equation is obtained from the conditions o f equality o f the areas o f figures Fj
and F-2 :
a = 1 + 2
( L + z ) 2
R t
r (1- r ) 1 + 2
( L + z ) 2
R t
(1—x)
(14)
( l — z ) ( a — x ) = k(1 - a ) .
Consider two rectangular defects (Fig. 2) with the following parameters:
L = 9, R t = 80, and t = 1, i.e. X = 1.0125 and t n / t = 0.3. We calculate the strength
o f such a defective segment by the developed semi-analytical procedure for two
defects w ith the use o f the proposed universal numerical algorithm and a
technique described in DNV-RP-F101 [1].
Figure 3 presents the dependences o f the strength reduction factor a on the
dimensionless distance between the defects k /L obtained by the method based on
relations (14), with the use o f the numerical procedure, and the procedure
66 ISSN 0556-171X. npodxeMbi npounocmu, 2009, N9 1
The Limit Load Calculations fo r Pipelines
described in the DNV-RP-F101 [1] (dashed line). The num erical and analytical
solutions (14) coincide and are presented by a solid line.
Examples 1 and 2 indicate that the numerical algorithm for determining the
strength reduction factor is correct and devoid o f the drawbacks associated with
simplification or schematization, and it is appropriate for strength calculation for
a pipe w ith an arbitrary strength distribution at each point. Further we consider
examples o f strength calculation for real complex-shaped defects using different
computation methods.
__ .---- -----
\\
/
/ y
Fig. 3. Strength reduction factor a versus dimensionless distance between the defects k/L: solid
line - numerical procedure, dashed line - DNV-RP-F101 [1].
2.3. D e te r m in a tio n o f S tren g th f o r R e a l D e fe c ts . Consider determination of
the internal failure pressure for real defects.
E x a m p le 1. (DNV-RP-F101 [1], B.3.2). The input parameters o f a model
defect: D = 611 mm, t = 8.2 mm, and f u = 571 N/mm . The profile geometry for
a complex shaped defect is presented in Table 1.
T a b l e 1
The Profile Geometry for a Complex Shaped Defect
Point number Length (mm) Depth (mm) Point number Length (mm) Depth (mm)
1 0 0 7 173.4 1.8
2 28.9 1.0 8 202.3 2.8
3 57.8 1.1 9 231.2 2.8
4 86.7 1.1 10 260.1 1.6
5 115.6 1.1 11 289.0 0
6 144.5 1.3
Strength calculations for such a defect are perform ed using computer
programs based on the algorithm from DNV-RP-F101 [1] and the numerical
procedure developed by us.
ISSN 0556-171X. npoöxeMbi npounocmu, 2009, N 1 67
A. V. Bogdan, I. V. Lokhman, S. M. Ageev, and I. V. Orynyak
Results o f calculations:
- failure pressure calculated according to the DNV: P f = 13.22 N/mm ;
2
- failure pressure calculated by our numerical procedure: P f =13.34 N/mm .
E x a m p le 2 ([4]). The input parameters o f a real defect (its profile geometry is
shown in Fig. 4): D = 457.7 mm, t = 6.04 mm, and f u = 428.5 N /m m 2.
Axial coordinate (mm)
Fig. 4. Profile of a complex-shaped defect [3].
Results o f strength calculation:
- failure pressure calculated according to the DNV: P f = 8.51 N/mm ;
- failure pressure calculated using the effective area: P f = 8.70 N/mm ;
- failure pressure calculated by the num erical procedure from item 2 .2 :
P f = 8.71 N /m m 2;
- experimental failure pressure: P f = 10.13 N/mm .
The results obtained demonstrate that the numerical procedure for calculating
the failure pressure developed by us is correct, physically justified and enables
obtaining more accurate estimates o f the failure pressure in a pipe as compared
with the known calculation procedures.
1. D N V R e c o m m e n d e d P r a c t ic e - D N V -R P -F 1 0 1 - C o r r o d e d P ip e l in e s , Det
Norske Veritas, Norw ay (2004).
2. I. V. Orynyak, “Leak and break models o f ductile fracture o f pressurized
pipe with axial defects,” in: Proc. IPC2006 6th Int. Pipeline Conf. (Sept.
25-29), Calgary, Canada (2006).
3. J. F. K iefner and P. H. Vieth, “PC program speeds new criterion for
evaluating corroded pipe,” O il & G a s J ., 88 , No. 34, 91-93 (1990).
4. R. D. Souza, A. C. Benjamin, R. D. Vieira, et al., “Part 4: Rupture tests of
pipeline segments containing real long corrosion defects,” E xp . T ech ., 31,
No. 1, 46-51 (2007).
Received 11. 06. 2008
68 ISSN 0556-171X. npodxeMbi npounocmu, 2009, N 1
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