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Plastic collapse pressures for defected cylindrical vessels
Int. J. Pres. Ves. & Piping 60 (1994) 65-69 (~ 1994 ElsevierScience Limited Printed in Northern Ireland. All rights reserved 0308-0161/94/$07.00 ELSEVIER
Plastic collapse pressures for defected cylindrical vessels K. Zarrabi School o f Mechanical and Manufacturing Engineering, The University o f New South Wales, PO Box 1, Kensington, New South Wales 2033, Australia (Received 10 August 1993; accepted 24 September 1993)
Any defect in a cylindrical pressure vessel may conservatively be inscribed within a rectangular slot. Plastic collapse pressures for the cylindrical vessels with part-through rectangular slots are computed employing elastic-plastic finite element analysis. The results are represented for a wide range of dimensions of the vessels and slots using appropriate non-dimensional parameters, as well as being compared with the limited data available in the literature. In the limit when the width or length of the slot approaches zero, the part-through slot approximates a part-through longitudinal or circumferential crack, respectively. Therefore, the results of the present work may conservatively be used as part of assessing the integrity of cracked, eroded or corroded cylindrical vessels, tubes, and pipes.
plate and therefore are approximate solutions only, especially when employed to assess the integrity of cylindrical vessels. The present work is concerned with determination of the plastic collapse pressure of a cylindrical vessel with a part-through thickness slot (Fig. 1) using non-linear finite element analysis; the shell may be either thin or thick. In the limit when the width or length of the slot approaches zero, the part-through slot approximates a part-through longitudinal or circumferential crack, respectively. Therefore, the results of the present work may conservatively be used as part of assessing the integrity of cracked, eroded or corroded vessels, tubes and pipes.
INTRODUCTION Cylindrical pressure vessels, pipes and tubes may contain crack-like defects. These defects may be inherent in the material or may exist at the beginning of service, being caused by manufacturing and installation processes. Alternatively, the defects may develop during service as a result of microstructural damages. Cylindrical pressure vessels, pipes and tubes are normally subjected to a uniform internal pressure. As part of assessing the integrity of such structures under design and in-service loadings, it is necessary to determine their plastic collapse pressures. The plastic collapse pressure is the maximum internal pressure that a cylindrical vessel can sustain. Kitching and Zarrabi t'2 have obtained lower bound solutions to collapse pressures for thin shells with part-though thickness slots only. Ainsworth and Coleman 3 have proposed a lower bound solution for a cylindrical vessel with a part-through thickness circumferential crack. Miller 4 has reviewed the existing limit loads of various structures containing defects. From his review, it is clear that the majority of these limit loads are obtained by considering the local plastic collapse of the ligament beneath the crack in a
FINITE ELEMENT MODELLING In accordance with the limit theorems, it has been assumed that the material behaved in an elastic-perfectly plastic manner. The material was postulated to obey the von Mises yield criterion with the isotropic hardening rule; it was initially homogeneous and isotropic and remained so during plastic deformation. The loading was taken to be a uniform internal pressure with the 65
66
K. Zarrabi
IRi____
Fig. 1. Cylindrical shell with a part-through thickness slot.
vessel ends subjected to a uniformly distributed pressure. Strictly speaking, the problem is a threedimensional one. To cover a wide range of dimensions of the vessels and slots, several elastic-plastic finite element analyses would have to be performed. This would have been expensive, both in terms of the modelling and computer C P U times. Kitching and Zarrabi 1'2 have, however, shown that the effect of the slot width on the collapse pressure is generally small. To simplify the analysis, and as a first step, it was assumed that the slot extended around the circumference of the vessel completely so that the vessel could be analysed using axisymmetric finite element models (Fig. 2); note that this assumption leads to a conservative estimate of the plastic collapse pressure. The ANSYS 5 finite element package was used to perform the axisymmetric analyses. Because of symmetry, only half of the vessel length was modelled (Fig. 3) and, typically, a model had 163 axisymmetric isoparametric elements allowing a quadratic variation of displacement within each
t tvt ~
P
_!tt
Pi
R i
- -
Ia
C.L Fig. 3. Typical axisymmetric mesh.
element with maximum wavefront 150 and a computer run of 3000 s on a SUN 3 platform. Later as a second step a few three-dimensional finite element models were employed to investigate the effect of the slot width on the collapse pressure.
t
Ro
R E S U L T S A N D DISCUSSIONS General ~lexl/
ed area
symmetryline Fig. 2. Axisymmetric model.
Figure 4 shows a typical computed internal pressure versus radial displacement of point A, where point A is marked in Fig. 2. Referring to
Plastic collapse pressures for defected cylindrical vessels 50 [ !
67
Pc = 43.75 MPa
40
30 e~
N
20
10
I
h
I
{
I
0.05
0.1
0.15
0.2
0.25
Radial displacement of point A (mm) Fig. 4. Typical internal p r e s s u r e versus radial d i s p l a c e m e n t .
Fig. 4, the plastic collapse pressure is defined as the intersection of the tangent to the plastic region and the ordinate axis representing the internal pressure. To represent the results in general form, three non-dimensional parameters were employed, viz: p,
Pc =--; Po
a p=--" R~'
h £2=-n
where Pc is the collapse pressure of the vessel containing a part-through slot; P o = V ~ / 2 try In Ro/Ri is the collapse pressure of the defect-free vessel; ov is the yield strength of the material; R is the mean radius of the vessel; H is the thickness of the vessel; and the other dimensions
are as defined in Figs 1 and 2. These parameters have been used by Kitching and Zarrabi ',2 to represent their lower bound results for thin shells containing part-through slots, but those authors have used P0 = oyR/H. To investigate the validity of the above non-dimensional parameters for the present study, elastic-plastic finite element analyses of a thin and a thick shell with various sizes of the slots were performed. The computed P* is plotted against p for various if2 in Fig. 5. It is apparent from Fig. 5 that, within the range of the acceptable engineering approximation, the above non-dimensional parameters are valid for a wide range of dimensions of the cylindrical vessels and slots.
1 0.8
p.
0.6 -
I
I
~
0.4 i
1
(
0.2'
L
0
0
0.2
L 0.4
0.6
0.8
1
1.2
1.4
L 1.6
1.8
Q h/H-O.gO (Thick)
P
h/H-O.50 (Thick)
h/H-0.10 (Thick)
h/H-0.90 (Thin)
x
h/H-O.50 (Thin)
h/H-0.10 (Thin)
Fig. 5. P * versus p for various Q =
h/H
b a s e d on thin and thick vessels.
68
K. Zarrabi
~.~
0,~
p*
"
0.1
Q = 0,7
~
~
Q = 0.5
~ = 0.3
0.4
0.2
Q=0.1
0
0
I
I
I
0.5
1
1.5 P
Fig. 6. P* versus p for various if2.
Axisymmetric models
The computed results from the axisymmetric finite element models are plotted in Fig. 6 using various symbols. To represent the c o m p u t e d collapse pressures analytically, they have been fitted to a bivariate polynomial of the Chebyshev type. 6 That is, it is assumed that:
e*(p, •) =
i
j
aiJ,(p)
(Q)
where T~(p) is the Chebyshev polynomial of the first kind of degree 3 in the argument p and so is T/(f~) in the argument Q. The prime on the two summation signs, following standard convention, indicates that the first term in each sum is halved. The coefficients a 0 are obtained by minimising a measure of error, E R R , defined as: ERR = ~-(P*(p,
£2) - p , ) 2
The fitted curves are depicted in Fig. 6 as solid lines. As might be expected, P* decreases with an increase of p and a decrease of f2. The computed collapse pressures for /9 = 1.28 and g2 = 0.50 are c o m p a r e d with the experimental collapse and burst pressures obtained by Kitching and Zarrabi 2 in Table 1. Table 1 shows that the non-dimensional collapse pressures c o m p u t e d using the axisymmetric finite element model were generally lower than the experimental collapse and burst pressures. The exception is when or= 15.66 ° for which the computed P* = 0 . 4 6 is slightly higher than the experimental collapse pressure P* = 0.43 but it is still lower than the non-dimensional burst pressure of 0.83. The above comparison indicates that the computed collapse pressures can conservatively be used for design and integrity type of analyses.
Table 1. Comparison of computed and experimental non-dimensional collapse and burst pressures
tr Axisymmetric FE P* Experimental a P* Experimental a non-dimensional burst pressures aFrom Ref. 2.
1-3° 0.46 0-56 0.71
3-9° 15-7° 0 . 4 6 0-46 0-52 0.43 0 . 6 7 0.83
Plastic collapse pressures for defected cylindrical vessels
69
Table 2. Comparison of the P* computed in the present study with those of Ref. 3
t2 Axisymmetric FE P* P* from Ref. 3
0-10 0-27 0.23
0-30 0.67 0.64
0.50 0.93 1.00
0-70 0-90 1-00 1.00 1-00 1.00
The computed collapse p r e s s u r e s for a c o m p l e t e c i r c u m f e r e n t i a l c r a c k are c o m p a r e d with the c o r r e s p o n d i n g l o w e r b o u n d p r e s s u r e s o b t a i n e d by A i n s w o r t h a n d C o l e m a n 3 in T a b l e 2 which indicates a g o o d a g r e e m e n t b e t w e e n the two sets o f results. H o w e v e r , w h e n f2 = 0-5 it a p p e a r s that the l o w e r b o u n d p r e s s u r e c a l c u l a t e d in R e f . 3 is slightly optimistic.
Fig. 7. Typical three-dimensional model.
i 0,8 P%
0.6 0.4
0.2
0 0
20
40
60
80
100
120
140
160
180
c~ ( d e g ) Fig. 8. P* versus o: for p = 0.5 and f2 = h/H = 0.5.
Three-dimensional models
REFERENCES
T o o b s e r v e the effect o f t h e slot w i d t h o n the collapse p r e s s u r e , several three-dimensional finite e l e m e n t m o d e l s h a v e b e e n p e r f o r m e d . F o r the sake o f b r e v i t y , t h e m e s h o f o n e o f the m o d e l s is s h o w n in Fig. 7 only. T h e d i m e n s i o n s o f the vessels a n d slots are c h o s e n in such a way that p = 0.5 a n d f2 = 0.5 a n d cr v a r i e d f r o m z e r o to 180 °. T h e c o m p u t e d P * v e r s u s ce is s h o w n in Fig. 8 which i n d i c a t e s a v a r i a t i o n o f 2 5 % for P * as the slot w i d t h varies f r o m z e r o to 180 ° . A l t h o u g h for a g e n e r a l c o n c l u s i o n o t h e r v a l u e s o f p a n d f2 m u s t be c o n s i d e r e d , f r o m the a b o v e results it s e e m s t h a t the a x i s y m m e t r i c results m a y b e c o n s e r v a t i v e by as m u c h as 2 5 % d e p e n d i n g o n the slot's width.
1. Kitching, R. & Zarrabi, K., Lower bound to limit pressure for cylindrical shell with part-through slot. Int. J. Mech. Sci., 23 (1981) 31-48. 2. Kitching, R. & Zarrabi, K., Limit and burst pressures for cylindrical shells with part-through slots. Int. J. Pres. Ves. & Piping, 10 (1982) 235-70. 3. Ainsworth, R. A. & Coleman, M. C., Example of an application of an assessment procedure for defects in plant operating in the creep range. Fatigue Fract. Engg Mater. Struct., 10 (1987) 129-40. 4. Miller, A. G., Review of limit loads of structures containing defects. Int. J. Pres. Ves. & Piping, 32 (1988) 197-227. 5. ANSYS Revision 5.0, Swanson Analysis Systems, Inc., Houston, PA, 1992. 6. Cox, M. G. & Hayes, J. G., Curve fitting: A guide and suite of algorithms for the non-specialist user. Report NAC26, National Physical Laboratory, Oxford, 1974.
Plastic collapse pressures for defected cylindrical vessels K. Zarrabi School o f Mechanical and Manufacturing Engineering, The University o f New South Wales, PO Box 1, Kensington, New South Wales 2033, Australia (Received 10 August 1993; accepted 24 September 1993)
Any defect in a cylindrical pressure vessel may conservatively be inscribed within a rectangular slot. Plastic collapse pressures for the cylindrical vessels with part-through rectangular slots are computed employing elastic-plastic finite element analysis. The results are represented for a wide range of dimensions of the vessels and slots using appropriate non-dimensional parameters, as well as being compared with the limited data available in the literature. In the limit when the width or length of the slot approaches zero, the part-through slot approximates a part-through longitudinal or circumferential crack, respectively. Therefore, the results of the present work may conservatively be used as part of assessing the integrity of cracked, eroded or corroded cylindrical vessels, tubes, and pipes.
plate and therefore are approximate solutions only, especially when employed to assess the integrity of cylindrical vessels. The present work is concerned with determination of the plastic collapse pressure of a cylindrical vessel with a part-through thickness slot (Fig. 1) using non-linear finite element analysis; the shell may be either thin or thick. In the limit when the width or length of the slot approaches zero, the part-through slot approximates a part-through longitudinal or circumferential crack, respectively. Therefore, the results of the present work may conservatively be used as part of assessing the integrity of cracked, eroded or corroded vessels, tubes and pipes.
INTRODUCTION Cylindrical pressure vessels, pipes and tubes may contain crack-like defects. These defects may be inherent in the material or may exist at the beginning of service, being caused by manufacturing and installation processes. Alternatively, the defects may develop during service as a result of microstructural damages. Cylindrical pressure vessels, pipes and tubes are normally subjected to a uniform internal pressure. As part of assessing the integrity of such structures under design and in-service loadings, it is necessary to determine their plastic collapse pressures. The plastic collapse pressure is the maximum internal pressure that a cylindrical vessel can sustain. Kitching and Zarrabi t'2 have obtained lower bound solutions to collapse pressures for thin shells with part-though thickness slots only. Ainsworth and Coleman 3 have proposed a lower bound solution for a cylindrical vessel with a part-through thickness circumferential crack. Miller 4 has reviewed the existing limit loads of various structures containing defects. From his review, it is clear that the majority of these limit loads are obtained by considering the local plastic collapse of the ligament beneath the crack in a
FINITE ELEMENT MODELLING In accordance with the limit theorems, it has been assumed that the material behaved in an elastic-perfectly plastic manner. The material was postulated to obey the von Mises yield criterion with the isotropic hardening rule; it was initially homogeneous and isotropic and remained so during plastic deformation. The loading was taken to be a uniform internal pressure with the 65
66
K. Zarrabi
IRi____
Fig. 1. Cylindrical shell with a part-through thickness slot.
vessel ends subjected to a uniformly distributed pressure. Strictly speaking, the problem is a threedimensional one. To cover a wide range of dimensions of the vessels and slots, several elastic-plastic finite element analyses would have to be performed. This would have been expensive, both in terms of the modelling and computer C P U times. Kitching and Zarrabi 1'2 have, however, shown that the effect of the slot width on the collapse pressure is generally small. To simplify the analysis, and as a first step, it was assumed that the slot extended around the circumference of the vessel completely so that the vessel could be analysed using axisymmetric finite element models (Fig. 2); note that this assumption leads to a conservative estimate of the plastic collapse pressure. The ANSYS 5 finite element package was used to perform the axisymmetric analyses. Because of symmetry, only half of the vessel length was modelled (Fig. 3) and, typically, a model had 163 axisymmetric isoparametric elements allowing a quadratic variation of displacement within each
t tvt ~
P
_!tt
Pi
R i
- -
Ia
C.L Fig. 3. Typical axisymmetric mesh.
element with maximum wavefront 150 and a computer run of 3000 s on a SUN 3 platform. Later as a second step a few three-dimensional finite element models were employed to investigate the effect of the slot width on the collapse pressure.
t
Ro
R E S U L T S A N D DISCUSSIONS General ~lexl/
ed area
symmetryline Fig. 2. Axisymmetric model.
Figure 4 shows a typical computed internal pressure versus radial displacement of point A, where point A is marked in Fig. 2. Referring to
Plastic collapse pressures for defected cylindrical vessels 50 [ !
67
Pc = 43.75 MPa
40
30 e~
N
20
10
I
h
I
{
I
0.05
0.1
0.15
0.2
0.25
Radial displacement of point A (mm) Fig. 4. Typical internal p r e s s u r e versus radial d i s p l a c e m e n t .
Fig. 4, the plastic collapse pressure is defined as the intersection of the tangent to the plastic region and the ordinate axis representing the internal pressure. To represent the results in general form, three non-dimensional parameters were employed, viz: p,
Pc =--; Po
a p=--" R~'
h £2=-n
where Pc is the collapse pressure of the vessel containing a part-through slot; P o = V ~ / 2 try In Ro/Ri is the collapse pressure of the defect-free vessel; ov is the yield strength of the material; R is the mean radius of the vessel; H is the thickness of the vessel; and the other dimensions
are as defined in Figs 1 and 2. These parameters have been used by Kitching and Zarrabi ',2 to represent their lower bound results for thin shells containing part-through slots, but those authors have used P0 = oyR/H. To investigate the validity of the above non-dimensional parameters for the present study, elastic-plastic finite element analyses of a thin and a thick shell with various sizes of the slots were performed. The computed P* is plotted against p for various if2 in Fig. 5. It is apparent from Fig. 5 that, within the range of the acceptable engineering approximation, the above non-dimensional parameters are valid for a wide range of dimensions of the cylindrical vessels and slots.
1 0.8
p.
0.6 -
I
I
~
0.4 i
1
(
0.2'
L
0
0
0.2
L 0.4
0.6
0.8
1
1.2
1.4
L 1.6
1.8
Q h/H-O.gO (Thick)
P
h/H-O.50 (Thick)
h/H-0.10 (Thick)
h/H-0.90 (Thin)
x
h/H-O.50 (Thin)
h/H-0.10 (Thin)
Fig. 5. P * versus p for various Q =
h/H
b a s e d on thin and thick vessels.
68
K. Zarrabi
~.~
0,~
p*
"
0.1
Q = 0,7
~
~
Q = 0.5
~ = 0.3
0.4
0.2
Q=0.1
0
0
I
I
I
0.5
1
1.5 P
Fig. 6. P* versus p for various if2.
Axisymmetric models
The computed results from the axisymmetric finite element models are plotted in Fig. 6 using various symbols. To represent the c o m p u t e d collapse pressures analytically, they have been fitted to a bivariate polynomial of the Chebyshev type. 6 That is, it is assumed that:
e*(p, •) =
i
j
aiJ,(p)
(Q)
where T~(p) is the Chebyshev polynomial of the first kind of degree 3 in the argument p and so is T/(f~) in the argument Q. The prime on the two summation signs, following standard convention, indicates that the first term in each sum is halved. The coefficients a 0 are obtained by minimising a measure of error, E R R , defined as: ERR = ~-(P*(p,
£2) - p , ) 2
The fitted curves are depicted in Fig. 6 as solid lines. As might be expected, P* decreases with an increase of p and a decrease of f2. The computed collapse pressures for /9 = 1.28 and g2 = 0.50 are c o m p a r e d with the experimental collapse and burst pressures obtained by Kitching and Zarrabi 2 in Table 1. Table 1 shows that the non-dimensional collapse pressures c o m p u t e d using the axisymmetric finite element model were generally lower than the experimental collapse and burst pressures. The exception is when or= 15.66 ° for which the computed P* = 0 . 4 6 is slightly higher than the experimental collapse pressure P* = 0.43 but it is still lower than the non-dimensional burst pressure of 0.83. The above comparison indicates that the computed collapse pressures can conservatively be used for design and integrity type of analyses.
Table 1. Comparison of computed and experimental non-dimensional collapse and burst pressures
tr Axisymmetric FE P* Experimental a P* Experimental a non-dimensional burst pressures aFrom Ref. 2.
1-3° 0.46 0-56 0.71
3-9° 15-7° 0 . 4 6 0-46 0-52 0.43 0 . 6 7 0.83
Plastic collapse pressures for defected cylindrical vessels
69
Table 2. Comparison of the P* computed in the present study with those of Ref. 3
t2 Axisymmetric FE P* P* from Ref. 3
0-10 0-27 0.23
0-30 0.67 0.64
0.50 0.93 1.00
0-70 0-90 1-00 1.00 1-00 1.00
The computed collapse p r e s s u r e s for a c o m p l e t e c i r c u m f e r e n t i a l c r a c k are c o m p a r e d with the c o r r e s p o n d i n g l o w e r b o u n d p r e s s u r e s o b t a i n e d by A i n s w o r t h a n d C o l e m a n 3 in T a b l e 2 which indicates a g o o d a g r e e m e n t b e t w e e n the two sets o f results. H o w e v e r , w h e n f2 = 0-5 it a p p e a r s that the l o w e r b o u n d p r e s s u r e c a l c u l a t e d in R e f . 3 is slightly optimistic.
Fig. 7. Typical three-dimensional model.
i 0,8 P%
0.6 0.4
0.2
0 0
20
40
60
80
100
120
140
160
180
c~ ( d e g ) Fig. 8. P* versus o: for p = 0.5 and f2 = h/H = 0.5.
Three-dimensional models
REFERENCES
T o o b s e r v e the effect o f t h e slot w i d t h o n the collapse p r e s s u r e , several three-dimensional finite e l e m e n t m o d e l s h a v e b e e n p e r f o r m e d . F o r the sake o f b r e v i t y , t h e m e s h o f o n e o f the m o d e l s is s h o w n in Fig. 7 only. T h e d i m e n s i o n s o f the vessels a n d slots are c h o s e n in such a way that p = 0.5 a n d f2 = 0.5 a n d cr v a r i e d f r o m z e r o to 180 °. T h e c o m p u t e d P * v e r s u s ce is s h o w n in Fig. 8 which i n d i c a t e s a v a r i a t i o n o f 2 5 % for P * as the slot w i d t h varies f r o m z e r o to 180 ° . A l t h o u g h for a g e n e r a l c o n c l u s i o n o t h e r v a l u e s o f p a n d f2 m u s t be c o n s i d e r e d , f r o m the a b o v e results it s e e m s t h a t the a x i s y m m e t r i c results m a y b e c o n s e r v a t i v e by as m u c h as 2 5 % d e p e n d i n g o n the slot's width.
1. Kitching, R. & Zarrabi, K., Lower bound to limit pressure for cylindrical shell with part-through slot. Int. J. Mech. Sci., 23 (1981) 31-48. 2. Kitching, R. & Zarrabi, K., Limit and burst pressures for cylindrical shells with part-through slots. Int. J. Pres. Ves. & Piping, 10 (1982) 235-70. 3. Ainsworth, R. A. & Coleman, M. C., Example of an application of an assessment procedure for defects in plant operating in the creep range. Fatigue Fract. Engg Mater. Struct., 10 (1987) 129-40. 4. Miller, A. G., Review of limit loads of structures containing defects. Int. J. Pres. Ves. & Piping, 32 (1988) 197-227. 5. ANSYS Revision 5.0, Swanson Analysis Systems, Inc., Houston, PA, 1992. 6. Cox, M. G. & Hayes, J. G., Curve fitting: A guide and suite of algorithms for the non-specialist user. Report NAC26, National Physical Laboratory, Oxford, 1974.