Reliability analysis of defect-containing structures using partial safety factors

Liwu Wei

Paper presented at ASME 28th International Conference on Ocean, Offshore and Arctic Engineering, OMAE 2009, Honolulu, Hawaii, 31 May - 5 June 2009. Paper# OMAE 2009 - 80165.

Abstract

Some standards of structural integrity assessment such as BS 7910 and API 579-1/ASME FFS-1 recommend values of partial safety factor (PSF) applied to the deterministic engineering critical assessments of flaw-containing structures to achieve certain reliability levels. However, it is still uncertain as to whether the use of the PSFs can achieve the target reliability level specified in the codes, or excessively exceed the targets (un-conservative) or under-reach the targets (too conservative). This work was undertaken to make investigations into these issues raised from the use of PSFs through case studies involving deterministic fitness-for-service analysis incorporating PSFs and probabilistic fracture mechanics analysis. Two cases, a through-thickness crack and a surface-breaking elliptical crack in a plate subjected to tension, were considered. The results in terms of failure probability from the studied cases have shown a general trend that for each of the four PSFs recommended in BS 7910, the failure probability decreased as the assessments changed from the elastic fracture region to the plastic collapse region in the failure assessment diagram. Some over-conservatism has been found in certain situations from the use of PSFs recommended in BS 7910:2005. Cautions are given for application of the PSFs for integrity assessment of the structures and components containing flaws.

Introduction

Structural integrity is of paramount concern in many industrial sectors including oil & gas, nuclear, construction, aerospace and automobile which routinely operate a number of safety-critical structures such as offshore platforms and nuclear reactors. For the integrity assessment of defect-containing structures, the deterministic analysis according to the failure assessment procedures from standards such as BS 7910:2005 (1), API 579-1/ASME FFS-1 (2) and R6 (3) is most commonly used, in which each input quantity is treated as a definitive variable. Clearly, the deterministic analysis cannot include the effects due to the uncertainties in input data, and thus is unable to quantify the reliability level of a structure. An extension of the deterministic assessment methods to a full probabilistic fracture mechanics (P_fM) based reliability analysis would enable the reliability level of a structure to be quantified. However, this requires a complete knowledge of the relevant failure models and the distributions for each of the input quantities, and thus can be very complex and time consuming to carry out.

Partial safety factors (PSFs) have been derived in the last decade for two assessment codes, BS 7910:2005 and API 579-1/ASME FFS-1, enabling the reliability level of a defect-containing structure to be estimated by a deterministic assessment instead of invoking a complex and time-consuming full reliability analysis. The PSFs recommended in these codes are given for the key variables such as stress, toughness, yield strength corresponding to different target reliability levels (which are also termed as failure probability, P_f) and coefficient of variation (COV). However, the use of PSFs in structural integrity assessment is not widely practised. One reason for this is that it is often difficult to have sufficient statistical data to decide the statistical distribution type of a parameter. Another reason is that it is still uncertain whether the use of the PSFs would be unsafe (i.e. the actual P_f larger than the target P_f) or over-conservative (i.e. the actual P_f smaller than the target P_f).

This work was carried out to investigate the uncertainties associated with the use of PSFs through the study of a few cases (through-thickness flaws and surface-breaking flaws in a plate). This was addressed by comparing the outcome from a PFM based reliability analysis with that from a deterministic assessment incorporating PSFs.

Full structural reliability analysis

A random variable reliability problem is generally defined by the limit state function g(Z) for a structural component, where Z is a set of random physical basic variables such as loads and basic material parameters. The mathematical description of a limit state function g(Z) depends largely on the failure criterion applied to the structural component to be assessed. In this work, the Level 2A FAD (failure assessment diagram) in BS 7910:2005 was considered, and the limit state function has the following form:

β-point) on the failure surface (defined by the limit state function) using one of the built-in search algorithms (an iterative process), while calling the FORTRAN program to evaluate the limit state function. There are normally two methods to calculate the P_f from the β-point value (i.e. reliability index): first-order reliability method (FORM) and second-order reliability method (SORM). The SORM usually produces more accurate results than the FORM but requires much more numerical efforts. Usually, the probability estimate based on FORM is sufficiently accurate for many practical applications [4]. The FORM was employed in this investigation, and the relationship between P_f and β for the FORM is given by γσ) in API 579-1/ASME FFS-1 are generally higher than the corresponding values recommended in BS 7910:2005 while the reverse trend is seen in terms of both fracture toughness and flaw size. In particular, the BS 7910 PSFs on fracture toughness are very much higher than the corresponding values in API 579-1/ASME FFS-1 especially for low uncertainty level in applied stress. The net effect is likely to be that assessments conducted to the BS 7910:2005 PSFs would be much more conservative than those to the PSFs in API 579-1/ASME FFS-1.

The PSFs in BS 7910 were derived from first order second moment reliability method (FORM) to establish the most likely combination of the main random variables - fracture toughness, applied stress and flaw size - leading to failure according to the limit state Eq [1].^[7] This combination gives the so-called design values of the variables (K_mat*, σ*, a*) from which the partial safety factors for loading effects are defined as the ratio of design value to a specified characteristic value (K_mat', σ', a') and for resistance effects as the inverse of this. That is,

Φ π/2 (see Fig.1). The results are presented in Table 3. Similarly, the results of P_f obtained for the surface-breaking crack in a plate under tension are shown in Table 4.

Table 3 Results of P_f at different PSFs given in BS 7910 for a centre through-thickness crack in a plate under tension obtained from Region 1 (elastic region)

Table 4 Results of P_f at different PSFs given in BS 7910 for a surface-breaking crack in a plate under tension from Region 1 (elastic region)

Table 3 demonstrates that in Region 1 slightly conservative assessments of P_f resulted from the application of the PSF values recommended by BS 7910:2005 except at PSF4 with a COV of 0.1 where an over-conservative estimate of P_f is given by BS 7910 (10^-5) compared with that from the PFM analysis in this work (8.28x10^-7). Similar observations can be found for the results obtained for the surface-breaking crack, as shown in Table 4.

Further investigations were carried out for the centre through-thickness crack by considering a set of conditions leading to the assessments falling in the elastic-plastic region (Region 2) where π/6 < Φ ≤ π/3 and the plastic collapse region (Region 3) where Φ ≤ π/6. The COV of 0.1 was assumed in these cases. The results are shown in Table 5 for Region 2 and Table 6 for Region 3.

Table 5 Results of P_f at different PSFs given in BS 7910 for a centre through-thickness crack in a plate under tension obtained from Region 2 (elastic-plastic region)

Table 6 Results of P_f at different PSFs given in BS 7910 for a centre through-thickness crack in a plate under tension obtained from Region 3 (plastic collapse region)

The results obtained in Region 2 (Table 5) show reasonable agreement between the P_f values given in BS 7910:2005 and those from a PFM analysis except at PSF4 where considerable conservatism was incurred. The P_f value from a PFM analysis (1.22x10^-9) is about four orders of magnitude smaller than that given in the standard (10^-5).

As shown in Table 6, substantial discrepancies exist between the P_f values obtained in Region 3 from a PFM analysis and those given in BS 7910:2005. The conservatism was increased with the level of PSFs, varying from one order of magnitude at PSF1, through three orders of magnitude at PSF2 and PSF3, to four orders of magnitude at PSF4.

The variations of P_f corresponding to different PSFs in different regions of the FAD are clearly demonstrated in Fig.2. It shows a general trend that for each of the four PSFs, the failure probability decreased as the assessments changed from the elastic fracture region to the plastic collapse region. It is interesting to note that the P_f levels at nominal conditions (i.e. critical conditions corresponding to characteristic values) are 0.22 in the elastic fracture region, 0.07 in the elastic-plastic fracture region and 0.04 in the plastic collapse region.

In summary, based on the cases studied in this work, the use of the PSFs given in BS 7910:2005 in a deterministic structural integrity assessment could result in excessively conservative estimations of P_f. The extent of conservatism would depend on where the assessments lie on the FAD, with the plastic collapse region giving the most excessive conservatism. Wilson^[11] also found that the recommended PSFs in BS 7910 were excessively conservative for some conditions.

However, it is worth pointing out that due to the very complex nature in deriving PSFs, the above observations of over-conservatism should not be treated as universally true. It could be also true that an over-optimistic (i.e. non-conservative) estimation of P_f is generated when using the PSFs given in BS 7910 for a structure under certain working conditions. This was demonstrated in the work^[12] which showed a non-conservative case when using the PSFs from PD 6493:1991 which are slightly different from those recommended in BS 7910.

Thus, substantially more studies of cases covering a variety of structures containing different types of flaw are needed in order to draw solid conclusions regarding the PSFs recommended in codes or to establish a new set of more realistic PSFs.

Due to the uncertainty of P_f associated with the use of PSFs given in BS 7910:2005 as demonstrated in the case studies, alternatively, it would be more justifiable and reliable to carry out a PFM based reliability analysis to assess the reliability of a structure, particularly for one of paramount structural integrity concern, as recommended in^[12].

Conclusions

The following conclusions can be drawn from this work:

1. The cases investigated (a through-thickness crack and a surface-breaking crack in a plate subjected to tension) demonstrated that conservatism in terms of P_f could result from the use of BS 7910 recommendations of PSFs.The extent of conservatism is associated with the region of the FAD in which the assessment lies, and a target P_f value from BS 7910 could be four orders of magnitude higher than that from a PFM based reliability analysis inthe plastic collapse region.
   
   
2. The P_f results from the cases studied have also shown a general trend that for each of the four PSFs recommended in BS 7910, the failure probability decreased as the assessments changed from the elastic fracture regionto the plastic collapse region.
   
   
3. More investigations are needed to fully identify the uncertainties relating to the P_f levels corresponding to different PSFs recommended in BS 7910 or to establish a new set of more realistic PSFs.

Acknowledgments

This work was funded by a TWI exploratory research programme. Thanks are also due to Drs Isabel Hadley and Henryk Pisarski of TWI for helpful discussions and comments on this work. The help from Dr Amin Muhammed of Shell Global Solutions, particularly in using the STRUREL software for probabilistic analysis involved, is also acknowledged.

References

1. BS 7910:2005 Incorporating Amendment No.1: 'Guide on methods for assessing the acceptability of flaws in metallic structures', British Standards Institution, 2005.
2. API 579-1/ASME FFS-1 2007 Fitness-for-service.
3. R6 Revision 4: 'Assessment of the integrity of structures containing defects', British Energy, 2007.
4. RCP: STRUREL Professional Version 6, 2007.
5. BSI PD 6493:1991: 'Guidance on methods for assessing the acceptability of flaws in fusion welded structures', British Standards Institution, 1991.
6. BS 7910:1999 Incorporating Amendment No.1: 'Guide on methods for assessing the acceptability of flaws in metallic structures', British Standards Institution, 2000.
7. Burdekin F M, Hamour W, Pisarski H G and Muhammed A: 'Derivation of partial safety factors for BS 7910:1999', Paper S640/006/99, IMechE Seminar, Flaw assessment in pressure equipment and welded structures PD 6493to BS 7910. London, 8 June, 1999.
8. Muhammed, A: 'Background to the derivation of partial safety factors for BS 7910 and API 579', Engineering Failure Analysis 14, 2007.
9. Wirsching P H and Mansour A E: 'Incorporation of structural reliability methods into fitness-for-service procedures'. The Materials Properties Council Inc., 1998.
10. TWI: 'Crackwise 4 - BS 7910 fracture/fatigue assessment procedures', 2007.
11. Wilson, R: 'A comparison of the simplified probabilistic method in R6 with the partial safety factor approach', Engineering Failure Analysis 14, 2007.
12. Pisarski, H G: 'Comparison of deterministic and probabilistic CTOD flaw assessment procedures', Paper OMAE 98-2354, Proc. The ASME 17th International Conference on offshore mechanics and arctic engineering,1998.