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Determination of partial safety factors of parameters for integrity assessment of welded structures containing defects
ht.
PII:SO308-0161(96)00037-3
ELSEVIER
J. Pres.
Ves. & Piping
12 (1997)
19-25
0 1997 Elsevier Science Limited. All rights reserved Printed in Northern Ireland 030%0161/97/$17.00
Determination of partial safety factors of parameters for integrity assessment of welded structures containing defects Chen Guohua & Dai Shuho Department
of Mechanical
Engineering,
Nanjing
University
of Chemical
Technology,
Nanjing
210009, P.R. China
(Received 10 June 1996;accepted2 July 1996)
Combined with the probabilistic safety assessmentmethod and based on reliability theory and parameter sensitivity analysis,the margin of reliability degree is defined and a method of determining partial safety factors is presented. The effect and rationality of the partial safety factors of different methodsdealing with the residual stressesare analyzed in detail. Finally, the method and the results are compared with those used in PD 6493: 1991. 0 1997Elsevier ScienceLtd.
the secondary stresses in determining the partial safety factors. In the present research (based on the probabilistic safety assessment method, stochastic simulation and the parameter sensitivity analysis), a method to determine the partial safety factors is put forward with a 20% margin of reliability under conditions with or without secondary stresses.
1 INTRODUCTION
When carrying out the safety assessment of structures containing defects (especially for pressure vessels and
piping), the assessment parameters are uncertain because of a number of reasons. Up to now, the uncertainty of parameters and the partial safety factors have been used in PD 6493: 1991.’ The same idea will be adopted in JWES-2805-a revised plan on the fracture assessment of welded joints containing defects2-but it is still not considered in other guides or standards.3-6 The partial safety factors not only embody the uncertainty of parameters to some extent and ensure the reliability of assessing results, but are also convenient for engineering application. It is an important breakthrough that partial safety factors are defined to deal with the parameter uncertainty of welded structures containing defects. It is also an important part of probabilistic safety assessment methodology. The secondary stresses (such as residual and thermal stress) are among the factors that affect structural failure. Therefore, when carrying out the safety assessment of structures containing defects, it is necessary to take account of the effect of the secondary stresses if they exist in the structures,
2 THE FAILURE (FLSF)
LIMIT
STATE
FUNCTION
Based on the two-criteria failure analysis method,lA the FLSF without secondary stresses can be stated as follows:7 g(L,, K,) = (1 - 0.14,$[0.3
+ 0.7 exp( -0*65L$]
-K,=O
(1)
L = PIPda, gy)
(2)
K = K(a, P)lK,,
(3)
where, K(a, P) is the stress strength factor for an applied load P and PL is the corresponding limit load evaluated for the yield stress ITS. Substituting eqns (2) and (3) for eqn (l), the FLSF without secondary stresses can be stated as follows:
especially for welded structures without heat treat-
ment. Up to now, the revising factor p has been introduced to reflect the effect of the secondary stresses on the failure assessment results for the welded structures in two main guidance documents.‘.4 It is necessary, therefore, to consider also the effect of
g(L,K)
19
=U -O.l4[PIP&, a,>l’> X{0.3+0.7exp[-0*65(P/P,(a, -K(a, P)lK,, = 0
u,))~]} (4)
20
Chen Guohua. Dai Shuho
If secondary stresses exist, a revising factor p is introduced to calculate parameter K,: K, = K(a, P)lK,, + p
(5)
The FLSF with secondary stresses can be stated as follows: g(L, 4) = Cl - @14[PIPL(a,
0 5 L, IO.8
P = 9%
p = -5$,/9(4L, p=o
- 5) 0.8
(8) I
a,)]‘>
X (0.3 + 0.7 exp[-0.65(P/P,(a, - K(a, f'>lK~,
procedure for the treatment of secondary stresses for level 2 and level 3 assessments. It was defined as follows for level 3 assessment.’
3.3 The revising parameter and Li
u,))~]}
p recommended
by Pan
- P
= 0
(6)
Failure probability can be obtained by the Monte Carlo method after defining the FLSF and carrying out probability safety assessment.7
Pan and Li also presented a method to treat the secondary stresses through theoretical analysis, finite element computation and test validation. It can be expressed as:‘o,1’ P = *1
p = &(ll
3 APPROACHES TO DEAL REVISING FACTOR
WITH
The secondary stresses affect the structural fracture failure in a very complicated manner. When carrying out the elastic-plastic fracture assessment, many researchers have proposed analyzing methods based on different ideas. These typically include: the method used in CEGB R6 (Rev. 3)4 and PD 6493: 19911 which was recommended by Ainsworth;’ the method recommended by Qi9 for level 2 and level 3 analysis of PD 6493: 1980; and the method presented by Pan and Li for Chinese guidance.“,” 3.1 The revising factor p used in CEGB R6 (Rev. 3) The revising factor p used in R6 (Rev. 3) was obtained based on the value of $ which was derived from option 1. The value of p is independent of the material properties. For the purpose of simplicity and safety, it is defined as follows.8 L, IO.8
p = 4$,(1.05 - L,) p=o
0.8 5 L, I 1.05 L, > 1.05 I
(7)
where: $r is the maximum value of $ at a given reference stress and $ is the shift factor for different L. 3.2 The revising factor p recommended
L, 2 1.1
p=o
THE
P = *1
- lOL,)/3
L, 5 0.8 0.8 < L,< 1.1
(9) I
This idea overcomes the drawbacks in the factor p presented by Ainsworth and Qi; it can ensure the safety of the assessment and is not over-conservative under other conditions.
4 PROBABILISTIC PROCEDURE
SAFETY
ASSESSMENT
According to the FLSF established in Section 2, the uncertainties of four parameters are considered. They are applied load P, fracture toughness KI,, flow stresses (T and defect size a. The failure probability is calculated by a Monte Carlo simulation. The value of p in eqn (5) is treated in three different ways, as described in Section 3. The probability safety assessment procedure is briefly conducted as follows: (1) define the distribution type of each random variable; produce samples of them; (2) calculate L, from eqn (2); (3) calculate K, from eqn (3) or (5); (4) substitute K, and L, for FLSF (4) and (6), checking if the value of g (L,, K,) is greater than zero; (5) Repeat (l)-(4) N times. Let the total times that g (L,, K,) is above zero be LL. The failure probability Pf is then (1 - LJN).
5 THE SENSITIVITY ANALYSIS ASSESSMENT PARAMETERS
OF
by Qi
Addressing the irrationality in PD 6493: 1980 to the treatment of residual stresses, Qi put forward a new
The purpose of sensitivity analysis is to quantitatively analyze the effect of each parameter on the failure probability of a structure containing defects, identify
Safety factors
of parameters
for integrity assessment
the main parameter which affects the structural reliability, and propose a reliable basis for determining the partial safety factors of each assessment parameter.
Table
2. The sensitivity in analyzing toughness without consideration
Fracture toughness changing times (decrease)
Reliability degree
5.1 Choice of engineering case
The engineering case used in this paper has been selected from Ref. 3 and involves semi-elliptical surface cracks in a plate used in a nuclear power station. The plate thickness is ‘40 mm, the mean value of membrane stress (T, is lOOMPa, and the mean value of secondary stress (T,, is 180 MPa. The half-length of the crack is 18 mm and the mean value of depth is 9mm. The mean value of fracture toughness and yield strength is 50 fi MPa and 163 MPa at given operation. According to Ref. 12 and PD 6493: 1991 Appendix A,’ the variance coefficient of each assessment parameter is determined as follows. (1) Material fracture toughness is fitted for a log-normal distribution with a coefficient of variation of O-07. (2) Material yield strength fly is fitted for a normal distribution with a coefficient of variance of 0.05. (3) The variance coefficient of membrane stress and secondary stress are defined as 0.09, and both of them are fitted for a normal distribution. (4) The defect depth size is defined as a normal distribution with a coefficient of variance 0.03. 5.2 Computing and analyzing results without consideration of factor p
The computed reliability degree is 0.92903 without considering the effect of secondary stress. The sensitivity analysis results for each parameter are shown in Tables 1 to 4. 5.3 Computing and analyzing results with factor p considered
The computed reliability degree is shown in Table 5 with the secondary stress factor p considered. The sensitivity analysis results of each parameter are shown in Tables 6 to 9. Table
1. The sensitivity in analyzing results of applied load without consideration of factor p
1.1 1.2
Margin of reliability degree (%I
0.813 07
12.4818
0.64133
30.968
OF
It can be seen from Table 5 that the reliability degree value is at its largest with the factor p in R6, and the reliability degree value is at its smallest with the factor p of Qi. The reliability degree value with the factor of Pan and Li lies between them. The reliability degree value with R6 is only slightly larger than the value of Pan and Li, but they differ greatly from the value of Qi. It can be shown that the factor p given by Qi is too conservative. On the other hand, the computing results in Table 5 can reflect to some extent that the factor in R6 is unsafe under some conditions. Qi’s factor p overcomes those unsafe conditions, but introduces a too conservative factor under other conditions. Pan and Li’s factor p not only overcomes the unsafe conditions of R6’s factor p, but also does not introduce too conservative factors such as Qi’s factor p. Therefore, Pan and Li’s factor p is much more rational. It is in accordance with the results in Refs 10 and 11. In fact, Pan and Li’s factor p has been adopted by the safety assessment standard for pressure vessels and piping containing defects in service (3rd draft),13 whose background is the Eighth National Five-Year Science Research Program. It can be seen from Tables l-4 and Tables 6-9 that the reliability degree values clearly change with Qi’s factor p. It makes all parameters very sensitive, but there is little clear difference between R6’s factor and Pan and Li’s factor p for the sensitivity of each parameter. Therefore, in the present research, we only
Table
3. The sensitivity in analyzing results of defect size without consideration of factor p
Reliability degree
Margin of reliability degree (%)*
Defect size changing times (increase)
1.1
0.713 3 0.41063
23.22 55.8
1.2
* See section 7.1.
results of fracture of factor p
6 THE RATIONALITY ANALYSIS DIFFERENT FACTORS
Applied load changing times (increase) I.2
21
of welded structures
1.3 1.4
Reliability degree 0.84400 0.793 50 0.73983
Margin of reliability degree (“/I 9.1526 14.588 20.265
22
Chen Guohua, Dai Shuho
Table 4. The sensitivity in analyzing results of yield strength without considerationof factor p
Yield strength changing times (decrease)
Reliability degree
Margin of reliability degree (%)
1.1 1.2 1.3
0.855 73 0.74180 0.588 07
7.8899 20.1532 36.7006
consider the effect of R6’s and Pan and Li’s factors on the structural reliability.
7 DETERMINATION SAFETY FACTOR PARAMETER
OF THE PARTIAL OF EACH ASSESSMENT
7.2 The determination of the partial safety factor
When carrying out safety assessment of structures containing defects by the probabilistic safety assessment method, it is first necessary to obtain the parameters and the stochastic distribution law through a lot of preparation. Secondly, it is also necessary to carry out a lot of simulation and calculation by computer. It is not convenient for engineering application. Because of this, it is necessary to set up a simple and direct method which not only reflects the uncertainties in assessing parameters but also is convenient for engineering application. Furthermore, these methods can rationally change the uncertainty aspects into certain problems. A powerful method which is presented by the authors to solve the problem is to make use of the sensitivity analysis results to determine the partial safety factor based on probabilistic safety assessment. 7.1 Definition of the margin of the reliability degree
The margin (y) of the reliability degree can be stated as a comparatively changing ratio of the reliability degree (R,) calculated with an assessing parameter unchanged to the reliability degree (R’) computed with the assessing parameter which changes amount by a given time (increase or decrease). It can be expressed as follows: RI--R0
Y=
Table
I I ~
x 100%
After defining the margin of the reliability degree, the partial safety factor for each assessing parameter can be determined according to the results of sensitivity analysis. It can be found from the results of the sensitivity analysis of applied load (shown in Tables 1 and 6) that the partial safety factor can be taken as 1.1 for the applied load. It can be found from the results of the sensitivity analysis of material fracture toughness (shown in Tables 2 and 7) that the partial safety factor can be taken as 1.1 for material fracture toughness. It can be found from the results of the sensitivity analysis of defect size (shown in Tables 3 and 8) that the partial safety factor can be taken as 1.4 for the defect size. It can be found from the results of the sensitivity analysis for yield strength (shown in Tables 4 and 9) that the partial safety factor can be taken as 1.3 for the yield strength. The partial safety factors for each assessing parameter are shown in Table 10. The determination method for the partial safety factors described above was based on the reliability analysis of structures containing defects, on the sensitivity analysis of assessing parameters with or without the consideration of secondary stresses, and also with the effect on the structural reliability of different revising factors for the secondary stresses.
(10)
RI
5. The computed results of reliability different revising factors
where: y is the margin of reliability degree; R0 is the reliability degree calculated with the assessing parameter unchanged; and R’ is the reliability degree calculated with the assessing parameter changed. In fact, the value of y reflects the sensitivity of the assessing parameters to the reliability and it embodies the effect on the reliability of structures containing defects when the assessing parameters change. Therefore, it is rational to determine the partial safety factor by changing multiples of each parameter at a given value of y, and its basis is reliability analysis. In order to ensure the assessing result with high reliability, the value of y is taken as 20% and the changing multiple of each assessing parameter is taken as its partial safety factor.
degree
under
Method of factor p
Method in CEGB R6
Method of Qi
Method of Pan
Assessment results (reliability)
0.806 13
0.517 87
0.788 07
8 COMPARISON METHOD AND PD6493: 1991
BETWEEN THE PRESENT THE METHOD IN
8.1 Determination of the partial safety factors in PD6493: 1991
According to Ref. 14, the failure criterion is that the actual defect size acr is less than or equal to the critical
Safety factors
of parameters
for integrity
assessment
23
of welded structures
Table 6. The sensitivity of analyzing resultsof applied load with factor p considered Factor p used in R6
Applied load changing times (decrease)
Reliability degree
Margin of reliability degree (%)
Reliability degree
Margin of reliability degree (%)
O-965 60 0.99677
19.78 23.648
O-88463 0.98987
70.82 91.14
1-l 1.2
defect size a,, when determining the partial safety factors in PD6493: 1991. The FLSF is: z = acr - a, With the safety factor becomes:
yO considered, the FLSF (12)
In the meantime, the FLSF was built up based on CEGB R6 (Rev. 2):15
The relationship curves between the safety factor and failure probability were obtained by sensitivity studies under different conditions, and the partial
safety factors were obtained
Changing times (increase)
1.1
Table
Changing times (decrease)
1.1 1.2 1.3 1.4
of analyzing
It can be determined from Section 8.1 that there are some drawbacks to the method in PD6493: 1991: (1) failure criterion was based on linear elastic fracture mechanics; (2) no effect on the partial safety factors of the secondary stresses was considered; (3) the fracture toughness was treated as the minimum one of the three test values or equivalent, not taken as the normal or log-normal distribution. In order to overcome those drawbacks, some new ideas are introduced in the present research as follows: (1) the elastic-plastic
results of fracture
Margin of reliability degree (%)
0.92447 0.97377 0.99037
1.2 1.3
23-36 26.68
14.68 20.80 22.85
8. The sensitivity
fracture
mechanics failure
analysis was adopted in the FLSF;
Factor p usedin R6 Reliability degree
0.97217 0.99830
Margin of reliability degree (%)
8.2 Comparison between the two methods to determine the partial safety factors
as Table 11.
7. The sensitivity
Reliability degree
(11)
z = YOacr- a,
Table
Factor p of Pan
Factor p of Qi
of analyzing
Factor p usedin R6 Reliability degree
Margin of reliability degree (%)
0.863 0.903 97 0.93217 0.95383
7.055 12.14 15.635 18.32
toughness
with factor p considered
Factor p of Qi Reliability degree 0.72087 0.85650 0.92957
Margin of reliability degree (%) 39.20 65.39 79.50
Factor p of Pan Reliability degree 0.91350 0.96833 0.98847
Margin of reliability degree (%) 15.92 22.87 25.43
results of defect size with factor p considered
Factor p of Qi Reliability degree 0.55177 0.58403 0.60957 0.63087
Margin of reliability degree (%) 6.55 12.78 17.71 21.82
Factor p of Pan and Li Reliability degree 0.83097 o%J447 0+%987 0.90817
Margin of reliability degree (%) 5.44 9.69 12.92 15.24
24
Chen Guohua, Table
9. The sensitivity
Changing times (increase)
results of yield strength
Factor p usedin R6
1.1 1.2 1.3
(2) two
of analyzing
conditions
Margin of reliability degree (%)
Reliability degree
Margin of reliability degree (%)
Reliability degree
Margin of reliability degree (%)
0882 77 0.92467 0.94593
9.51 14.7 17.34
0.72963 0.85103 0.9113
40.89 64.33 75.07
0.8988 0.94363 0.96557
14.05 19.74 22.52
with
or without
secondary
9 CONCLUSIONS
on the reliability theory, combined with the probabilistic safety assessing method, it was concluded that Pan and Li’s method is much more rational, with the uncertainty of assessing parameters and the effect of three typical revising factors on the margin of reliability degree considered. The partial safety factors of each parameter were determined with a 20% margin of reliability through a sensitivity analysis of structural reliability. The uncertainties of each parameter can be reflected to a certain extent, and the uncertain problems are changed into certain problems. It is also convenient for engineering application.
It is shown from the analysis that the partial safety factors determined with the secondary stresses considered ensure that assessing results are safer than those without the secondary stresses considered. Compared with the determination method in PD 6493: 1991, some drawbacks are overcome, and the partial safety factors in the present
paper
are determined
on the basis of both
sensitivity analysis and the margin of reliability.
(5) Although
the uncertainties of parameters are represented by partial safety factors to a certain
Table
10. The partial
Table
11. The partial
safety factors in PD6493: 1991’*‘5
Assessingparameters Load (COVS%) Toughness (minimum value of three test value) Defect size (standard error 2-5 mm)
Normal Severe 1.1
1.4
1 1.1
1.2 1.4
extent, it is recommended that the reliability of
(1) Based
(4)
Factor p of Pan and Li
Reliability degree
variable fitted for a distribution (including fracture toughness and yield stresses); (4) the concept of the margin of reliability degree was introduced.
(3)
with factor p considered
Factor p of Qi
stresses are considered in detail; (3) each assessingparameter is taken as a random
(2)
Dai Shuho
safety factors for each parameter
Assessingparameters
Partial safety factors
Applied loads Fracture toughness Defect size Yield strength
1.1 1.2 1.4 1.3
important structures or components is assessed using detailed probabilistic safety assessment methodology. ACKNOWLEDGMENT
The authors would like to express their appreciation to the National Natural Science Foundation of China for supporting the research. REFERENCES 1. Guidance on the method for assessing the acceptability of flaws in fusion welded structures. Published document,BSI-PD 6493, 1991. 2. JapaneseSociety of Welding, A revision plan on the fracture assessmentof the welding joint containing defects, 1991-1994.(In Japanese.) 3. Bergman, M. et al., A Procedure for Safety Assessment of Components with Cracks, SA/FoU-REPORT 91/01, The SwedishPlant Inspectorate, 1991. 4. Milne, I., Ainsworth, R. A., Dowling, A. R. and Stewart, A. T., Assessmentof the integrity of structures containing defects. CEGB Report /H/RG-Rev. 3, 1986. 5. Bloom, J. M. and Malik, S. N., Procedure for the assessment of the integrity of nuclear pressurevessels and piping containing defects. NP-2431,1982. 6. China Pressure Vessel Defect Assessment,(CVDA), CSNE, Society of Pressure Vessel Technology and CSChE, Society of Chemical Mechanical Engineering, 1984. 7. Chen, G. H. and Dai, S. H., Study on the reliability assessment methodology for pressurevesselscontaining defects. International Journal of Pressure Vessels and Piping, 1996,69, 273-277. 8. Ainsworth, R. A., The treatment of thermal and residual stress in fracture assessments.Engineering Fracture Mechanics, 1986,24(l), 65-76. 9. Qi, D. M., Recommendationson the treatment of
Safety factors
of parameters
for integrity assessment of welded structures
residual stress in PD6493 for the assessment of the significance of weld defects. Engineering Fracture Mechanics, 1992, 41(2), 257-270. 10. Pan, H. L., Fracture mechanics analysis and engineering fracture assessment procedure for welding structure in the presence of secondary stresses. Ph.D. thesis, East China University of Science Technology, 1995. 11. Pan, H. L. and Li, P. L., The recommendable procedure for treating residual stress in the fracture assessment diagram. CPVT, 1996, 13(l), 9-16. 12. Dai Shu-Ho and Wang Ming-O, Reliability Analysis in Engineering Applications, Van Nostrand Reinhold, New
25
York, 1992, pp. 69-72. 13. The safety assessment standard for pressure vessels containing defects in service, 3rd draft, 1995, p. 18. (In Chinese.) 14. Plane, C. A., Cowling, M. L., Nwegbu, V. K. and Burdekin, F. M., The determination of safety factors for defect assessment using reliability analysis methods. Third International Symposium on Integrity of Offshore Structures. September, 1987, pp. 395-420. 15. The assessment standards of pressure vessels containing defects abroad. Translated by Jin Shu-Feng. Labor Press House, Beijing, 1982, pp. 277-316. (In Chinese.)
PII:SO308-0161(96)00037-3
ELSEVIER
J. Pres.
Ves. & Piping
12 (1997)
19-25
0 1997 Elsevier Science Limited. All rights reserved Printed in Northern Ireland 030%0161/97/$17.00
Determination of partial safety factors of parameters for integrity assessment of welded structures containing defects Chen Guohua & Dai Shuho Department
of Mechanical
Engineering,
Nanjing
University
of Chemical
Technology,
Nanjing
210009, P.R. China
(Received 10 June 1996;accepted2 July 1996)
Combined with the probabilistic safety assessmentmethod and based on reliability theory and parameter sensitivity analysis,the margin of reliability degree is defined and a method of determining partial safety factors is presented. The effect and rationality of the partial safety factors of different methodsdealing with the residual stressesare analyzed in detail. Finally, the method and the results are compared with those used in PD 6493: 1991. 0 1997Elsevier ScienceLtd.
the secondary stresses in determining the partial safety factors. In the present research (based on the probabilistic safety assessment method, stochastic simulation and the parameter sensitivity analysis), a method to determine the partial safety factors is put forward with a 20% margin of reliability under conditions with or without secondary stresses.
1 INTRODUCTION
When carrying out the safety assessment of structures containing defects (especially for pressure vessels and
piping), the assessment parameters are uncertain because of a number of reasons. Up to now, the uncertainty of parameters and the partial safety factors have been used in PD 6493: 1991.’ The same idea will be adopted in JWES-2805-a revised plan on the fracture assessment of welded joints containing defects2-but it is still not considered in other guides or standards.3-6 The partial safety factors not only embody the uncertainty of parameters to some extent and ensure the reliability of assessing results, but are also convenient for engineering application. It is an important breakthrough that partial safety factors are defined to deal with the parameter uncertainty of welded structures containing defects. It is also an important part of probabilistic safety assessment methodology. The secondary stresses (such as residual and thermal stress) are among the factors that affect structural failure. Therefore, when carrying out the safety assessment of structures containing defects, it is necessary to take account of the effect of the secondary stresses if they exist in the structures,
2 THE FAILURE (FLSF)
LIMIT
STATE
FUNCTION
Based on the two-criteria failure analysis method,lA the FLSF without secondary stresses can be stated as follows:7 g(L,, K,) = (1 - 0.14,$[0.3
+ 0.7 exp( -0*65L$]
-K,=O
(1)
L = PIPda, gy)
(2)
K = K(a, P)lK,,
(3)
where, K(a, P) is the stress strength factor for an applied load P and PL is the corresponding limit load evaluated for the yield stress ITS. Substituting eqns (2) and (3) for eqn (l), the FLSF without secondary stresses can be stated as follows:
especially for welded structures without heat treat-
ment. Up to now, the revising factor p has been introduced to reflect the effect of the secondary stresses on the failure assessment results for the welded structures in two main guidance documents.‘.4 It is necessary, therefore, to consider also the effect of
g(L,K)
19
=U -O.l4[PIP&, a,>l’> X{0.3+0.7exp[-0*65(P/P,(a, -K(a, P)lK,, = 0
u,))~]} (4)
20
Chen Guohua. Dai Shuho
If secondary stresses exist, a revising factor p is introduced to calculate parameter K,: K, = K(a, P)lK,, + p
(5)
The FLSF with secondary stresses can be stated as follows: g(L, 4) = Cl - @14[PIPL(a,
0 5 L, IO.8
P = 9%
p = -5$,/9(4L, p=o
- 5) 0.8
(8) I
a,)]‘>
X (0.3 + 0.7 exp[-0.65(P/P,(a, - K(a, f'>lK~,
procedure for the treatment of secondary stresses for level 2 and level 3 assessments. It was defined as follows for level 3 assessment.’
3.3 The revising parameter and Li
u,))~]}
p recommended
by Pan
- P
= 0
(6)
Failure probability can be obtained by the Monte Carlo method after defining the FLSF and carrying out probability safety assessment.7
Pan and Li also presented a method to treat the secondary stresses through theoretical analysis, finite element computation and test validation. It can be expressed as:‘o,1’ P = *1
p = &(ll
3 APPROACHES TO DEAL REVISING FACTOR
WITH
The secondary stresses affect the structural fracture failure in a very complicated manner. When carrying out the elastic-plastic fracture assessment, many researchers have proposed analyzing methods based on different ideas. These typically include: the method used in CEGB R6 (Rev. 3)4 and PD 6493: 19911 which was recommended by Ainsworth;’ the method recommended by Qi9 for level 2 and level 3 analysis of PD 6493: 1980; and the method presented by Pan and Li for Chinese guidance.“,” 3.1 The revising factor p used in CEGB R6 (Rev. 3) The revising factor p used in R6 (Rev. 3) was obtained based on the value of $ which was derived from option 1. The value of p is independent of the material properties. For the purpose of simplicity and safety, it is defined as follows.8 L, IO.8
p = 4$,(1.05 - L,) p=o
0.8 5 L, I 1.05 L, > 1.05 I
(7)
where: $r is the maximum value of $ at a given reference stress and $ is the shift factor for different L. 3.2 The revising factor p recommended
L, 2 1.1
p=o
THE
P = *1
- lOL,)/3
L, 5 0.8 0.8 < L,< 1.1
(9) I
This idea overcomes the drawbacks in the factor p presented by Ainsworth and Qi; it can ensure the safety of the assessment and is not over-conservative under other conditions.
4 PROBABILISTIC PROCEDURE
SAFETY
ASSESSMENT
According to the FLSF established in Section 2, the uncertainties of four parameters are considered. They are applied load P, fracture toughness KI,, flow stresses (T and defect size a. The failure probability is calculated by a Monte Carlo simulation. The value of p in eqn (5) is treated in three different ways, as described in Section 3. The probability safety assessment procedure is briefly conducted as follows: (1) define the distribution type of each random variable; produce samples of them; (2) calculate L, from eqn (2); (3) calculate K, from eqn (3) or (5); (4) substitute K, and L, for FLSF (4) and (6), checking if the value of g (L,, K,) is greater than zero; (5) Repeat (l)-(4) N times. Let the total times that g (L,, K,) is above zero be LL. The failure probability Pf is then (1 - LJN).
5 THE SENSITIVITY ANALYSIS ASSESSMENT PARAMETERS
OF
by Qi
Addressing the irrationality in PD 6493: 1980 to the treatment of residual stresses, Qi put forward a new
The purpose of sensitivity analysis is to quantitatively analyze the effect of each parameter on the failure probability of a structure containing defects, identify
Safety factors
of parameters
for integrity assessment
the main parameter which affects the structural reliability, and propose a reliable basis for determining the partial safety factors of each assessment parameter.
Table
2. The sensitivity in analyzing toughness without consideration
Fracture toughness changing times (decrease)
Reliability degree
5.1 Choice of engineering case
The engineering case used in this paper has been selected from Ref. 3 and involves semi-elliptical surface cracks in a plate used in a nuclear power station. The plate thickness is ‘40 mm, the mean value of membrane stress (T, is lOOMPa, and the mean value of secondary stress (T,, is 180 MPa. The half-length of the crack is 18 mm and the mean value of depth is 9mm. The mean value of fracture toughness and yield strength is 50 fi MPa and 163 MPa at given operation. According to Ref. 12 and PD 6493: 1991 Appendix A,’ the variance coefficient of each assessment parameter is determined as follows. (1) Material fracture toughness is fitted for a log-normal distribution with a coefficient of variation of O-07. (2) Material yield strength fly is fitted for a normal distribution with a coefficient of variance of 0.05. (3) The variance coefficient of membrane stress and secondary stress are defined as 0.09, and both of them are fitted for a normal distribution. (4) The defect depth size is defined as a normal distribution with a coefficient of variance 0.03. 5.2 Computing and analyzing results without consideration of factor p
The computed reliability degree is 0.92903 without considering the effect of secondary stress. The sensitivity analysis results for each parameter are shown in Tables 1 to 4. 5.3 Computing and analyzing results with factor p considered
The computed reliability degree is shown in Table 5 with the secondary stress factor p considered. The sensitivity analysis results of each parameter are shown in Tables 6 to 9. Table
1. The sensitivity in analyzing results of applied load without consideration of factor p
1.1 1.2
Margin of reliability degree (%I
0.813 07
12.4818
0.64133
30.968
OF
It can be seen from Table 5 that the reliability degree value is at its largest with the factor p in R6, and the reliability degree value is at its smallest with the factor p of Qi. The reliability degree value with the factor of Pan and Li lies between them. The reliability degree value with R6 is only slightly larger than the value of Pan and Li, but they differ greatly from the value of Qi. It can be shown that the factor p given by Qi is too conservative. On the other hand, the computing results in Table 5 can reflect to some extent that the factor in R6 is unsafe under some conditions. Qi’s factor p overcomes those unsafe conditions, but introduces a too conservative factor under other conditions. Pan and Li’s factor p not only overcomes the unsafe conditions of R6’s factor p, but also does not introduce too conservative factors such as Qi’s factor p. Therefore, Pan and Li’s factor p is much more rational. It is in accordance with the results in Refs 10 and 11. In fact, Pan and Li’s factor p has been adopted by the safety assessment standard for pressure vessels and piping containing defects in service (3rd draft),13 whose background is the Eighth National Five-Year Science Research Program. It can be seen from Tables l-4 and Tables 6-9 that the reliability degree values clearly change with Qi’s factor p. It makes all parameters very sensitive, but there is little clear difference between R6’s factor and Pan and Li’s factor p for the sensitivity of each parameter. Therefore, in the present research, we only
Table
3. The sensitivity in analyzing results of defect size without consideration of factor p
Reliability degree
Margin of reliability degree (%)*
Defect size changing times (increase)
1.1
0.713 3 0.41063
23.22 55.8
1.2
* See section 7.1.
results of fracture of factor p
6 THE RATIONALITY ANALYSIS DIFFERENT FACTORS
Applied load changing times (increase) I.2
21
of welded structures
1.3 1.4
Reliability degree 0.84400 0.793 50 0.73983
Margin of reliability degree (“/I 9.1526 14.588 20.265
22
Chen Guohua, Dai Shuho
Table 4. The sensitivity in analyzing results of yield strength without considerationof factor p
Yield strength changing times (decrease)
Reliability degree
Margin of reliability degree (%)
1.1 1.2 1.3
0.855 73 0.74180 0.588 07
7.8899 20.1532 36.7006
consider the effect of R6’s and Pan and Li’s factors on the structural reliability.
7 DETERMINATION SAFETY FACTOR PARAMETER
OF THE PARTIAL OF EACH ASSESSMENT
7.2 The determination of the partial safety factor
When carrying out safety assessment of structures containing defects by the probabilistic safety assessment method, it is first necessary to obtain the parameters and the stochastic distribution law through a lot of preparation. Secondly, it is also necessary to carry out a lot of simulation and calculation by computer. It is not convenient for engineering application. Because of this, it is necessary to set up a simple and direct method which not only reflects the uncertainties in assessing parameters but also is convenient for engineering application. Furthermore, these methods can rationally change the uncertainty aspects into certain problems. A powerful method which is presented by the authors to solve the problem is to make use of the sensitivity analysis results to determine the partial safety factor based on probabilistic safety assessment. 7.1 Definition of the margin of the reliability degree
The margin (y) of the reliability degree can be stated as a comparatively changing ratio of the reliability degree (R,) calculated with an assessing parameter unchanged to the reliability degree (R’) computed with the assessing parameter which changes amount by a given time (increase or decrease). It can be expressed as follows: RI--R0
Y=
Table
I I ~
x 100%
After defining the margin of the reliability degree, the partial safety factor for each assessing parameter can be determined according to the results of sensitivity analysis. It can be found from the results of the sensitivity analysis of applied load (shown in Tables 1 and 6) that the partial safety factor can be taken as 1.1 for the applied load. It can be found from the results of the sensitivity analysis of material fracture toughness (shown in Tables 2 and 7) that the partial safety factor can be taken as 1.1 for material fracture toughness. It can be found from the results of the sensitivity analysis of defect size (shown in Tables 3 and 8) that the partial safety factor can be taken as 1.4 for the defect size. It can be found from the results of the sensitivity analysis for yield strength (shown in Tables 4 and 9) that the partial safety factor can be taken as 1.3 for the yield strength. The partial safety factors for each assessing parameter are shown in Table 10. The determination method for the partial safety factors described above was based on the reliability analysis of structures containing defects, on the sensitivity analysis of assessing parameters with or without the consideration of secondary stresses, and also with the effect on the structural reliability of different revising factors for the secondary stresses.
(10)
RI
5. The computed results of reliability different revising factors
where: y is the margin of reliability degree; R0 is the reliability degree calculated with the assessing parameter unchanged; and R’ is the reliability degree calculated with the assessing parameter changed. In fact, the value of y reflects the sensitivity of the assessing parameters to the reliability and it embodies the effect on the reliability of structures containing defects when the assessing parameters change. Therefore, it is rational to determine the partial safety factor by changing multiples of each parameter at a given value of y, and its basis is reliability analysis. In order to ensure the assessing result with high reliability, the value of y is taken as 20% and the changing multiple of each assessing parameter is taken as its partial safety factor.
degree
under
Method of factor p
Method in CEGB R6
Method of Qi
Method of Pan
Assessment results (reliability)
0.806 13
0.517 87
0.788 07
8 COMPARISON METHOD AND PD6493: 1991
BETWEEN THE PRESENT THE METHOD IN
8.1 Determination of the partial safety factors in PD6493: 1991
According to Ref. 14, the failure criterion is that the actual defect size acr is less than or equal to the critical
Safety factors
of parameters
for integrity
assessment
23
of welded structures
Table 6. The sensitivity of analyzing resultsof applied load with factor p considered Factor p used in R6
Applied load changing times (decrease)
Reliability degree
Margin of reliability degree (%)
Reliability degree
Margin of reliability degree (%)
O-965 60 0.99677
19.78 23.648
O-88463 0.98987
70.82 91.14
1-l 1.2
defect size a,, when determining the partial safety factors in PD6493: 1991. The FLSF is: z = acr - a, With the safety factor becomes:
yO considered, the FLSF (12)
In the meantime, the FLSF was built up based on CEGB R6 (Rev. 2):15
The relationship curves between the safety factor and failure probability were obtained by sensitivity studies under different conditions, and the partial
safety factors were obtained
Changing times (increase)
1.1
Table
Changing times (decrease)
1.1 1.2 1.3 1.4
of analyzing
It can be determined from Section 8.1 that there are some drawbacks to the method in PD6493: 1991: (1) failure criterion was based on linear elastic fracture mechanics; (2) no effect on the partial safety factors of the secondary stresses was considered; (3) the fracture toughness was treated as the minimum one of the three test values or equivalent, not taken as the normal or log-normal distribution. In order to overcome those drawbacks, some new ideas are introduced in the present research as follows: (1) the elastic-plastic
results of fracture
Margin of reliability degree (%)
0.92447 0.97377 0.99037
1.2 1.3
23-36 26.68
14.68 20.80 22.85
8. The sensitivity
fracture
mechanics failure
analysis was adopted in the FLSF;
Factor p usedin R6 Reliability degree
0.97217 0.99830
Margin of reliability degree (%)
8.2 Comparison between the two methods to determine the partial safety factors
as Table 11.
7. The sensitivity
Reliability degree
(11)
z = YOacr- a,
Table
Factor p of Pan
Factor p of Qi
of analyzing
Factor p usedin R6 Reliability degree
Margin of reliability degree (%)
0.863 0.903 97 0.93217 0.95383
7.055 12.14 15.635 18.32
toughness
with factor p considered
Factor p of Qi Reliability degree 0.72087 0.85650 0.92957
Margin of reliability degree (%) 39.20 65.39 79.50
Factor p of Pan Reliability degree 0.91350 0.96833 0.98847
Margin of reliability degree (%) 15.92 22.87 25.43
results of defect size with factor p considered
Factor p of Qi Reliability degree 0.55177 0.58403 0.60957 0.63087
Margin of reliability degree (%) 6.55 12.78 17.71 21.82
Factor p of Pan and Li Reliability degree 0.83097 o%J447 0+%987 0.90817
Margin of reliability degree (%) 5.44 9.69 12.92 15.24
24
Chen Guohua, Table
9. The sensitivity
Changing times (increase)
results of yield strength
Factor p usedin R6
1.1 1.2 1.3
(2) two
of analyzing
conditions
Margin of reliability degree (%)
Reliability degree
Margin of reliability degree (%)
Reliability degree
Margin of reliability degree (%)
0882 77 0.92467 0.94593
9.51 14.7 17.34
0.72963 0.85103 0.9113
40.89 64.33 75.07
0.8988 0.94363 0.96557
14.05 19.74 22.52
with
or without
secondary
9 CONCLUSIONS
on the reliability theory, combined with the probabilistic safety assessing method, it was concluded that Pan and Li’s method is much more rational, with the uncertainty of assessing parameters and the effect of three typical revising factors on the margin of reliability degree considered. The partial safety factors of each parameter were determined with a 20% margin of reliability through a sensitivity analysis of structural reliability. The uncertainties of each parameter can be reflected to a certain extent, and the uncertain problems are changed into certain problems. It is also convenient for engineering application.
It is shown from the analysis that the partial safety factors determined with the secondary stresses considered ensure that assessing results are safer than those without the secondary stresses considered. Compared with the determination method in PD 6493: 1991, some drawbacks are overcome, and the partial safety factors in the present
paper
are determined
on the basis of both
sensitivity analysis and the margin of reliability.
(5) Although
the uncertainties of parameters are represented by partial safety factors to a certain
Table
10. The partial
Table
11. The partial
safety factors in PD6493: 1991’*‘5
Assessingparameters Load (COVS%) Toughness (minimum value of three test value) Defect size (standard error 2-5 mm)
Normal Severe 1.1
1.4
1 1.1
1.2 1.4
extent, it is recommended that the reliability of
(1) Based
(4)
Factor p of Pan and Li
Reliability degree
variable fitted for a distribution (including fracture toughness and yield stresses); (4) the concept of the margin of reliability degree was introduced.
(3)
with factor p considered
Factor p of Qi
stresses are considered in detail; (3) each assessingparameter is taken as a random
(2)
Dai Shuho
safety factors for each parameter
Assessingparameters
Partial safety factors
Applied loads Fracture toughness Defect size Yield strength
1.1 1.2 1.4 1.3
important structures or components is assessed using detailed probabilistic safety assessment methodology. ACKNOWLEDGMENT
The authors would like to express their appreciation to the National Natural Science Foundation of China for supporting the research. REFERENCES 1. Guidance on the method for assessing the acceptability of flaws in fusion welded structures. Published document,BSI-PD 6493, 1991. 2. JapaneseSociety of Welding, A revision plan on the fracture assessmentof the welding joint containing defects, 1991-1994.(In Japanese.) 3. Bergman, M. et al., A Procedure for Safety Assessment of Components with Cracks, SA/FoU-REPORT 91/01, The SwedishPlant Inspectorate, 1991. 4. Milne, I., Ainsworth, R. A., Dowling, A. R. and Stewart, A. T., Assessmentof the integrity of structures containing defects. CEGB Report /H/RG-Rev. 3, 1986. 5. Bloom, J. M. and Malik, S. N., Procedure for the assessment of the integrity of nuclear pressurevessels and piping containing defects. NP-2431,1982. 6. China Pressure Vessel Defect Assessment,(CVDA), CSNE, Society of Pressure Vessel Technology and CSChE, Society of Chemical Mechanical Engineering, 1984. 7. Chen, G. H. and Dai, S. H., Study on the reliability assessment methodology for pressurevesselscontaining defects. International Journal of Pressure Vessels and Piping, 1996,69, 273-277. 8. Ainsworth, R. A., The treatment of thermal and residual stress in fracture assessments.Engineering Fracture Mechanics, 1986,24(l), 65-76. 9. Qi, D. M., Recommendationson the treatment of
Safety factors
of parameters
for integrity assessment of welded structures
residual stress in PD6493 for the assessment of the significance of weld defects. Engineering Fracture Mechanics, 1992, 41(2), 257-270. 10. Pan, H. L., Fracture mechanics analysis and engineering fracture assessment procedure for welding structure in the presence of secondary stresses. Ph.D. thesis, East China University of Science Technology, 1995. 11. Pan, H. L. and Li, P. L., The recommendable procedure for treating residual stress in the fracture assessment diagram. CPVT, 1996, 13(l), 9-16. 12. Dai Shu-Ho and Wang Ming-O, Reliability Analysis in Engineering Applications, Van Nostrand Reinhold, New
25
York, 1992, pp. 69-72. 13. The safety assessment standard for pressure vessels containing defects in service, 3rd draft, 1995, p. 18. (In Chinese.) 14. Plane, C. A., Cowling, M. L., Nwegbu, V. K. and Burdekin, F. M., The determination of safety factors for defect assessment using reliability analysis methods. Third International Symposium on Integrity of Offshore Structures. September, 1987, pp. 395-420. 15. The assessment standards of pressure vessels containing defects abroad. Translated by Jin Shu-Feng. Labor Press House, Beijing, 1982, pp. 277-316. (In Chinese.)