**Reliability Analysis of Defect-Containing Structures using Partial Safety Factors**

**Liwu Wei**

TWI Ltd, Granta Park

Great Abington

Cambridge, CB21 6AL, UK

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_{f}M) 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:

where K_{I} is the applied stress intensity factor, K_{mat} is fracture toughness, ρ is the plasticity interaction factor and K_{r} is the permitted fracture ratio as given in Eq [2] for the Level 2A FAD in BS 7910:2005.

where L_{r} is the plastic collapse ratio, defined as:

where σ_{ref} is the reference stress which is a function of flaw geometry and applied stresses. σ_{Y} is the yield strength or 0.2% proof strength. In this work, for all the cases investigated, L_{r} was less than the L_{rmax} value prescribed by (σ_{Y} + σ_{U})/2σ_{Y} (σ_{U} is the tensile strength).

When g(Z) is greater than 0, the assessment point is situated inside the FAD (the area enclosed by the assessment line and two axes), indicating a safe condition for the assessed body. When g(Z) is less than 0 the assessment point is located outside the FAD, leading to an unsafe condition for the assessed body. When g(Z) = 0, the assessment point is situated on the assessment line, reaching a critical (*i.e.* limit state) condition. *Figure 1* schematically illustrates the three states. It should be noted that the safe or unsafe condition is set against the criterion prescribed by Eq [2], not necessarily physically a safe or failed condition. The failure probability is thus given by

where f_{z}(z) is the multivariate density function of Z.

The implementation of the PFM based reliability analysis represented by Eq [4] was conducted in the structural reliability analysis software SYSREL-Pro 9.6^{[4]} in conjunction with a FORTRAN program of the limit state function as shown in Eq [1]. The implementation process of the probabilistic model is mathematically very complex, and the details of implementation can be referred to the manual of the software [4]. A brief description of the implementation of the model is given here. The software firstly examined the input data of the model by the input processor module, and the stochastic processor module then performed probability distribution transformations and generated the values from the distribution types of input variables to be used in the calculation of the limit state functions. Subsequently, the reliability processor module operated to search the point (called the 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

Where Φ() is the standard normal distribution function.

## Partial safety factors in deterministic assessement

Although limit state design codes with recommendations for PSFs have been in use for conventional guidance on design of steel structures to avoid failure by plastic collapse and by buckling for some years, they were not extended to fracture based structural integrity assessments until the publication of BSI document PD 6493:1991.^{[5]} BS 7910:2005 and API 579-1/ASME FFS-1 are the only major structural integrity assessment procedures to include PSFs where fracture and plastic collapse are the dominant failure modes. BS 7910:1999^{[6]} replaced PD 6493:1991 and the original PSFs were updated based on development work by Burdekin et al.^{[7]} The current version of BS 7910, namely BS 7910:2005, has not made any change to the recommendations on PSFs since then. API 579-1/ASME FFS-1 appears to have derived PSFs on a similar basis to the BS 7910 work but there are differences.^{[8]} As found in reference^{[8]}, the PSFs on applied stress (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,

Thus the failure equation with partial safety factors becomes:

where Y is the non dimensional coefficient to take account of the crack geometry.

Usually the characteristic value of a variable is the value that would typically be used in a deterministic analysis for that variable. For example, this could be an upper bound value, say 95^{th} percentile value for a loading variable and a lower bound, say mean - two standard deviation or 5^{th} percentile value for a resistance variable. However, other percentile values or mean values can be used if these are more convenient in the derivation of PSFs. Wirsching and Mansour^{[9]} assumed mean values as characteristic values for applied stress, flaw size and fracture toughness in deriving PSFs for API 579-1/ASME FFS-1. In the derivation of PSFs for BS 7910:2005, Burdekin et al^{[7]} also assumed mean values for the characteristic applied stress and flaw size but the mean minus one standard deviation value for a Weibull distribution, considered equivalent to the minimum of three fracture toughness tests, was adopted for fracture toughness. In this work, the PSFs recommended in BS 7910:2005 were applied to the deterministic assessments in terms of Eq [7]. The assessments were carried out to BS 7910:2005 using TWI software Crackwise 4.^{[10]}

## Case studies

### Overview

Two crack types in a plate subjected to tension were considered: the through-thickness crack and the surface-breaking crack. For each of the crack types, various combinations of main random variables (stress, flaw size, toughness, and yield strength) were considered to give nominal assessment points in the three main regions of the FAD. That is, in the elastic, elastic-plastic and plastic collapse regions. These would be defined approximately in terms of the angle Φ = tan^{-1}(K_{r}/L_{r}): Φ ≤ π/6 (plastic collapse or plastic collapse dominating), π/6 < Φ ≤ π/3 (elastic-plastic) and π/3 < Φ ≤ π/2 (elastic). *Fig.1* illustrates the three regions. The consideration of the calibration points in the three regions serves to reveal the varieties in reliability corresponding to the different regions associated with the same PSFs used.

The random variables were assumed to have the same distribution types and coefficient of variation (COV, ie the ratio of mean to standard deviation) as adopted in the work^{[7]} used in the derivation of the PSFs in BS 7910.

### Data input

The input data for the PFM based reliability analysis of the through-thickness crack and surface-breaking crack in a plate subjected to tension are presented in *Tables 1 and 2*, respectively. Four random variables, flaw length of a through-thickness crack or flaw height of a surface-breaking crack (2a), toughness (K_{mat}), yield strength (σy) and primary stress (P_{m}), were considered. As in the work of Burdekin et al^{[7]}, the distribution types of the four random variables (a, K_{mat}, σy and P_{m}) were assumed to be Normal, Weibull, Lognormal and Normal, respectively. These distributions were established based on the statistical analysis of a large amount of test/measurement data. Two levels of COV, 0.1 and 0.2, were considered for flaw height, toughness and primary stress, while considering only one level of COV (0.1) for yield strength. The characteristic value of the toughness was taken as the mean minus one standard deviation (*i.e.* 1SD), while the mean - 2SD was assumed for the yield strength.

**Table 1 Input data for the through-wall crack in a plate subject to tension**

Variable | Distribution type | Mean | COV, % | Characteristic or nominal value |
---|---|---|---|---|

FAD | BS 7910:2005 Level 2A | |||

Component types | Centre-cracked plate in tension | |||

Thickness, B, mm | Fixed | 50 | Nominal | |

Plate width, mm | Fixed | 1000 | Nominal | |

Tensile strength, MPa | Fixed | 420 | Nominal | |

Flaw length, a, mm | Normal | 50 | 10, 20 | Mean |

Toughness, K_{mat}, Nmm^{-3/2} |
Weibull | 800, 5000, 10000 | 10, 20 | Mean - 1SD* |

Yield strength, MPa | Lognormal | 350 | 10 | Mean - 2SD |

Primary stress (normalised), P_{m}/Nominal |
Normal | 1 | 10, 20 | Nominal value |

Note:

* SD stands for standard deviation

**Table 2 Input data for the surface-breaking crack in a plate subject to tension**

Variable | Distribution type | Mean | COV, % | Characteristic or nominal value |
---|---|---|---|---|

FAD | BS 7910:2005 Level 2A | |||

Component types | Surface semi-elliptical plate in tension | |||

Thickness, B, mm | Fixed | 50 | Nominal | |

Plate width, mm | Fixed | 1000 | Nominal | |

Flaw height, a, mm | Normal | 10 | 10, 20 | Mean |

Flaw aspect ratio, c/a | Fixed | 2 | Nominal | |

Toughness, K_{mat}, Nmm^{-3/2} |
Weibull | 800 | 10, 20 | Mean - 1SD |

Yield strength, MPa | Lognormal | 350 | 10 | Mean - 2SD |

Tensile strength, MPa** | Fixed | 420 | Nominal | |

Primary stress (normalised), P_{m}/Nominal |
Normal | 1 | 10, 20 | Nominal value |

For the through-thickness crack (see *Table 1*), three mean values of toughness (800, 5000 and 10000Nmm^{-3/2}) were adopted to render nominal assessment points situating in three regions of the FAD, *i.e.* elastic, elastic-plastic and plastic collapse region (see *Fig.1* for the definition of the three regions).

For the surface-breaking crack (see *Table 2*), only one mean level of toughness (800Nmm^{-3/2}) was considered, corresponding to the elastic region.

A set of nominal mean primary stresses were defined in such a way that the critical conditions in terms of Level 2A were reached after the inclusion of various PSFs (see Eq [7]). The PSFs corresponding to the four different P_{f} levels given in BS 7910:2005 were investigated. For simplicity, the notations below are used in this report:

PSF1 denotes Level 1 of PSFs corresponding to a P_{f} of 2.3x10^{-1} (*i.e.* a reliability index β_{r} of 0.739). PSF2 denotes Level 2 of PSFs corresponding to a P_{f} of 1x10^{-3} (*i.e.* a reliability index β_{r} of 3.09). PSF3 denotes Level 3 of PSFs corresponding to a P_{f} of 7x10^{-5} (*i.e.* a reliability index β_{r} of 3.8). PSF4 denotes Level 4 of PSFs corresponding to a P_{f} of 1x10^{-5} (*i.e.* a reliability index β_{r} of 4.27).

### Results and discussion

The first cases were investigated concerning the centre through-thickness crack in a plate subjected to tension. Four different levels of PSF (PSF1 to PSF4) and a COV of 0.1 and 0.2 were applied. A low toughness of 800 Nmm^{-3/2} was chosen so that the assessment region belonged to the elastic region (Region 1) where π/3 < 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)**

| PSF | ||||||||
---|---|---|---|---|---|---|---|---|---|

COV = 0.1 | COV = 0.2 | ||||||||

| PSF1 | PSF2 | PSF3 | PSF4 | PSF1 | PSF2 | PSF3 | PSF4 | |

Stress, σ | 1.05 | 1.2 | 1.25 | 1.3 | 1.1 | 1.25 | 1.35 | 1.4 | |

Flaw size, a | 1 | 1.4 | 1.5 | 1.7 | 1.05 | 1.45 | 1.55 | 1.8 | |

Fracture toughness, K_{mat} |
1 | 1.3 | 1.5 | 1.7 | 1 | 1.8 | 2.6 | 3.2 | |

Yield strength | 1 | 1.05 | 1.1 | 1.2 | 1 | 1.05 | 1.1 | 1.2 | |

P_{f} |
BS 7910 | 2.3x10^{-1} |
10^{-3} |
7x10^{-5} |
10^{-5} |
2.3x10^{-1} |
10^{-3} |
7x10^{-5} |
10^{-5} |

This work | 1.385x10^{-1} |
1.94x10^{-4} |
1.36x10^{-5} |
8.28x10^{-7} |
1.28x10^{-1} |
1.01x10^{-3} |
6.35x10^{-5} |
9.79x10^{-6} |

**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)**

| PSF | ||||||||
---|---|---|---|---|---|---|---|---|---|

COV = 0.1 | COV = 0.2 | ||||||||

| PSF1 | PSF2 | PSF3 | PSF4 | PSF1 | PSF2 | PSF3 | PSF4 | |

Stress, σ | 1.05 | 1.2 | 1.25 | 1.3 | 1.1 | 1.25 | 1.35 | 1.4 | |

Flaw size, a | 1 | 1.4 | 1.5 | 1.7 | 1.05 | 1.45 | 1.55 | 1.8 | |

Fracture toughness, K_{mat} |
1 | 1.3 | 1.5 | 1.7 | 1 | 1.8 | 2.6 | 3.2 | |

Yield strength | 1 | 1.05 | 1.1 | 1.2 | 1 | 1.05 | 1.1 | 1.2 | |

P_{f} |
BS 7910 | 2.3x10^{-1} |
10^{-3} |
7x10^{-5} |
10^{-5} |
2.3x10^{-1} |
10^{-3} |
7x10^{-5} |
10^{-5} |

This work | 1.215x10^{-1} |
1.45x10^{-4} |
9.64x10^{-6} |
5.17x10^{-7} |
1.23x10^{-1} |
9.02x10^{-4} |
5.56x10^{-5} |
7.93x10^{-6} |

*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)**

| COV = 0.1 | ||||
---|---|---|---|---|---|

| PSF1 | PSF2 | PSF3 | PSF4 | |

Stress, σ | 1.05 | 1.2 | 1.25 | 1.3 | |

Flaw size, a | 1 | 1.4 | 1.5 | 1.7 | |

Fracture toughness, K_{mat} |
1 | 1.3 | 1.5 | 1.7 | |

Yield strength | 1 | 1.05 | 1.1 | 1.2 | |

P_{f} |
BS 7910 | 2.3x10^{-1} |
10^{-3} |
7x10^{-5} |
10^{-5} |

This work | 3.01x10^{-2} |
2.85x10^{-5} |
2.89x10^{-6} |
1.22x10^{-9} |

**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)**

| COV = 0.1 | ||||
---|---|---|---|---|---|

| PSF1 | PSF2 | PSF3 | PSF4 | |

Stress, σ | 1.05 | 1.2 | 1.25 | 1.3 | |

Flaw size, a | 1 | 1.4 | 1.5 | 1.7 | |

Fracture toughness, K_{mat} |
1 | 1.3 | 1.5 | 1.7 | |

Yield strength | 1 | 1.05 | 1.1 | 1.2 | |

P_{f} |
BS 7910 | 2.3x10^{-1} |
10^{-3} |
7x10^{-5} |
10^{-5} |

This work | 1.545x10^{-2} |
2.357x10^{-6} |
3.928x10^{-8} |
2.907x10^{-9} |

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:

- 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 in the plastic collapse region. - 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 region to the plastic collapse region. - 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

- BS 7910:2005 Incorporating Amendment No.1: 'Guide on methods for assessing the acceptability of flaws in metallic structures', British Standards Institution, 2005.
- API 579-1/ASME FFS-1 2007 Fitness-for-service.
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- BS 7910:1999 Incorporating Amendment No.1: 'Guide on methods for assessing the acceptability of flaws in metallic structures', British Standards Institution, 2000.
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*Engineering Failure Analysis*14, 2007. - Wirsching P H and Mansour A E: 'Incorporation of structural reliability methods into fitness-for-service procedures'. The Materials Properties Council Inc., 1998.
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