Risk assessment based on performantial criterion for inspection of offshore structures in presence of large cracks

ABSTRACT When performing risk analysis, it is often uneasy to find the link between limit state and consequences. This paper focuses on efficiency based limit states in case of large cracks on offshore structures. Randomness and uncertainties on loading as well as on crack measurement and detection are introduced.


Introduction
Nowadays reassessment of structures is a great challenge for public or private owners. In offshore field, this subject appeared in the 1990s when the platforms that were installed in the 1970s reached their initial life time of about 20 years. Most of them were steel framed structures made with welded tubular components and called Jacket platforms: they were about 2000 to be reassessed in the 1990s all over the word in various sites such as the gulf of Mexico, the gulf of Guinea and the North Sea. This challenge is still of first importance with more and more sophisticated methods such as Risk Based Inspection Rouhan et al., 2004). Owners and certification companies must provide decision aid tools to extend the life-time, with or without repairing, or to remove the structure. In this last case, the removing is a delicate and dangerous operation which includes important risks as detailed by (Kroon, 2004). We focus here on the other cases where the operator requires data up-dating (modification of the structure, metocean data, inspection results) and decision aid-tools able to include hazards and the new requirements of regulation. Deterministic approaches are not generally efficient and considered as too conservative; in fact, during reassessment it is common to compute fatigue lifetime of joints of several months when no crack occured since 30 years. That leads to state the importance of reliability and risk analysis which offers since 20 years the basic theoretical concepts to approach this complexity (Wenche et al., 2000).
In addition, they are a great amount of these structures includes an important reserve of integrity due to redundancy at design stage and and they need of simplification of building process (using of similar components to prevent human errors). Thus they are not optimized from reliability point of view and local damages may be tolerated. That's the case for through cracks, obtained when the crack depth reaches the steel thickness. Usual reliability studies (Madsen, 1997), are based on a local limit state which considers the geometry of the crack (depth or length) as critical values. The joint typology is introduced only in an implicit way when damage is computed using the Paris law. As critical values are very fair (steel thickness), these criteria are very conservative and don't allow to analyze the stress distribution in presence of cracks. One reason is that the scale effect needed to include local mechanical quantities in global structural analysis.
We suggest here a complete approach based on hybrid physical-analytical response surfaces in the aim to: -analyze the effect of through crack on the structural analysis , -model wave loading in storm conditions (Schoefs, 1996).
What distinguishes physical response surfaces from analytical response surfaces is the use of deterministic mechanical models. The selected structural model is based on a cracked joint finite element. In terms of a probabilistic analysis, this approach leads to an increase in dimension of probabilistic space which acts on reliability level if no precaution is taken (correlation, conditioning).
After the reminder of key points for risk analysis in section 2, the paper focuses in section 3 on response surface methodology and their building. It allows to provide in section 4 a parametric reliability analysis for a tripod structure: parameters are the mean crack length and its position. Performantial limit state functions are selected and based on the displacement at the top of the structure and strain energy. The paper ends in section 5 with a parametric risk based inspection of this structures when including the reliability of No Destructive Techniques: parameters introduce costs of repair, failure and inspection.

Basic key points of risk estimation
Risk estimation aims to evaluate risk "R", defined for a given event "e" in Equation [1]: where p(e) is the probability associated to event "e" and C(e) the vector of consequences due to this event.
In case of offshore platforms analysis, p(e) is estimated both for exploitation (oil pipes, electrical and hydraulic systems, fire protection...) and for structure on which the present paper focuses on. Here, assessment of p(e) comes from a reliability study according to a performantial state function. Components of vector C(e) are generally loss of human life, financial loss of exploitation, environmental consequences in terms of biological loss (wildlife, flora) and economical (cleaning, tourist picture)... When possible, vector of consequences is reduced to only one component, sum of all homogenized components, for example under a monetary form. Otherwise, a multicriterion analysis is necessary.

Concept of physical response surface and its application
R.S.M. has been largely developed during the last decade and response function models, including non-linear ones, are now tractable especially for the specific needs of structural reliability analysis (Labeyrie et al., 1995). This methodology is basically a formal representation based on geometrical ideas in link with a surface building in the probabilistic space. It leads to investigate properties of a physical process versus a system approach. That means the required response (e.g. mechanical behaviour) to be considered as the output of a system, which varies in response to the changing levels of several input variables (stimuli) called basic variables ( Figure 1).

Figure 1. System modelling using response surfaces
System has to be modelled by some mathematical function of random variables X i involved in the system and characterised by statistical information θ i (moments, free or parametric distribution functions...). For reliability analysis, the investigated response is either strength, load variables or limit state function. The surrogates to original models are classically analytical functions (mainly polynomial expression) whose parameters are estimated from experimental plans. It is to notice that these polynomial or derived functions are used as simplest readily available smoothing curves, without any appeal to their theoretical properties as asymptotic approximations to the true response function (Bouyssy et al., 1994;Muzeau et al., 1993). Such a "blind approach" allows non-intrusive reliability analysis with the use of standard finite element sofware, but it remains very questionable and limits have been already proved and illustrated (Labeyrie et al. 1995;Leira et al. 2003).
This context leads to state the importance for defining building criteria, according to the specific attempt on reliability analysis. Three major criteria, which may conflict, have been proved to be relevant by Labeyrie and Schoefs (1996): -physical meaning, in the sense that approximation functions are based as far as possible on some understanding of the underlying mechanism of the physical process, -distribution effects; the selection of basic variables and model fitting should reflect how the whole distribution is well transferred through the response surface (the so-called distribution effects) especially for non-linear transfer as some energetic transfers (loading of wind or wave), -goodness of fit measurement; the metrics of the Sobolev space can be suggested to introduce some mean least squares error on the partial derivatives as a measure of the fitting goodness (Schoefs, 1996), -computational tractability; an increase of the complexity level in stochastic modelling doesn't ensure necessarily more realistic results and is always a time consuming option when computing.
As an alternative to usual regression techniques for the fitting of model to a data base, we suggest here a building of response surfaces which respects the previous criteria. We propose to combine the stochastic modeling of basic field and deterministic transfer function. This method called physical response surface, based Basic variables X i Response Transfer response function f Î Î on sensitivity and uncertainty studies, allows to rank the input variables and to target the feature efforts of probabilistic modeling for the dominant random variables. This approach is self-consistent too in case of extrapolation of model once probabilistic support of basic variables stay between the physical limit bounds of the model. The illustration presented below focuses on the energetic transfer for the computing of severe wave loading (storm conditions) acting on cracked Jacket offshore structures, composed with welded steel tubular components (see Figure 2). Two response surfaces are suggested: the first one concerns the loading and the second one the components of stiffness matrix in case of crack presence.
From the resulting response surface, reliability studies based on performantial criterion (displacement or strain energy) can then be performed. The response function takes an physical analytical form in simple cases  but in complex real cases the using of Monte-Carlo simulation and finite-element model are needed to perform directly reliability computations or to fit an analytic response surface.

Building of selected response surfaces
Response surface of wave loading in severe storm conditions is presented in (Schoefs, 1996). The source of environmental loading resides in kinematics field of water particles near the structural component. Knowing this kinematics field, Morison equations give the distributed load on such a wet component as the summation of a drag (Equation [2]) and an inertial (Equation [3]) component (Morison et al., 1950). Equations [2] and [3] give the corresponding response surfaces.
[ ] Environment: fluid-structure interaction 5VTWEVWTG L VJTQWIJ ETCEM where a, b, c et d are stochastic fields that depend of the set of environmental variables: extreme wave height and the wave number k. In the following, they are computed using the first order Stokes model where "n" equals 1 (Airy waves). Coefficients C X , C D , C M , C' X allows to introduce fluid-structure interaction. The factor θ mg is the increase of section diameter D due to marine growth and ρ is the density of fluid. Vectors Inspection of welding joint of offshore structures are generally practiced in harsh conditions; hazards on crack length "a" as well as the capability to detect a crack (see section 5) must be taken into account. The uncertainty on measurement leads to an uncertainty in mechanical properties (geometrical) and thus to a random mechanical behavior. This randomness on geometry leads to high computation costs when using the common analytical response surface methodology associated with a finite element software (Lemaire, 1997). In fact, the well-known cost of re-meshing is then of first importance. Here we select a mixed response surface "physicalanalytical". An analytical approximation of stiffness matrix component is then deduced from simplified physical model called Cracked Joint Finite Element (CJFE). It is presented on Figure 3 in case of a crack with a plan of symmetry (here (x, y); then two parameters "e y " and "k z " are introduced.
We model the large through crack in considering the resulting loss of stiffness (two flexion springs with stiffness k y and k z ) and neutral fiber displacement in the cracked section of the beam (two eccentricities e y and e y ). Expression of these response surfaces whatever the joint typology (T-joint, K-joint...) are available in (Rguig, 2005).

Sensitivity analysis
Suggested response surfaces lead to increase the dimension of the probabilistic space. In fact, space of basic random variables has a dimension of 8 {H, k, C X , C D , C M , C' X , θ mg , a} where H and k are correlated. Space of intermediate design variables, transferred through the CJFE has a dimension of 11 {H, k, C X , C D , C M , C' X , θ mg , e y , e y , k y , k z } where H and k firstly and the set (e y , e y , k y , k z ) secondly are correlated. From reliability point of view, the consequence is an increase of the probability of failure. Sensitivity studies are dedicated to this question by performing Monte-Carlo simulations or perturbation techniques. A sensitiveness ratio is defined in  and several sensitivity studies are available for the CJFE model alone  or in presence of extreme wave loading in the case of a wet cracked vertical component . They shown that, in the CJFE model, only stiffness plays a significant role. As a consequence, probabilistic space has a dimension of 9 even 8 in the case of low through crack size: in fact, k y does not act any more in this case.

Description of the structure
A tripod structure is selected for illustration ( Figure 4). Some of them are used for burning non-exploited gaz of oil fields. They are generally unmanned and placed far away from coasts. An analysis only based on economical considerations can then be accepted. A beam finite element model of the studied platform is represented too on Figure 4; the water depth is about 40 m and the structures comprises 14 tubular components. More details are available in (Rguig, 2005). In view to reduce computing costs, no call to an external FE software is realized. All structural analysis, CJFE model programming, pre/post-processing and computing of stochastic field of loading are made with Matlab software.

Description of random variables
Probability density functions, conditioning between variables, coefficients of variations (c.o.v) and truncatures of samples to keep simulated events in physical supports are presented in Table 1. They refer to severe environmental conditions (storm) in the Frigg field, near Shetland Islands in North Sea (Doucet et al., 1987). Simulations are performed with Modified Latin Hypercube Sampling: it consists in selection of events in the middle of iso-likely intervals. Where H s ,T stat , θ, denote respectively the significant wave height, the sea state duration and the wave heading. The hypothesis of exponential distribution for crack length "a" comes from statistical analysis of inspection results in North Sea (Moan et al., 1997) and is extended here to the case of large cracks only; the upper bound for truncature is deduced from geometrical considerations (tube perimeter).

Computation of probability of failure
Reliability estimation is carried out by Monte-Carlo simulations. Two limit states are considered. The first one is based on the displacement U 6 at the top of the structure (node P6), the critical value (here 0.041 m) being motivated by the maximal curve of the gas pipe attached to this vertical component. The second one considers the strain energy of the structure E s , and aims to reflect the structural integrity. In this case, the critical value of the strain energy is more complex to define; it can be associated to a percentage of the strain energy of the non-cracked structure  or to another critical quantity. We suggest this last way. We present on Figure 5 the realizations of the couples (U 6 , E s ) that have been obtained under random loading on the non-cracked tripod structure. The fitting of this behavior (curve on the Figure 5) gives an average relationship between U 6 and E s ; a critical value E c of 2.99 10 4 N.m is then deduced. It corresponds to the critical displacement U c of 0.041 m. This value is deduced from the yield stress in the gaz pipe due to a deflection of the gaz pipe that is fixed on the structure.

Figure 5. Realization of couples (U 6 , E s ); non-cracked tripod structure loaded with random waves
The limit states take the usual form of safety margins: For a though crack of half-open angle ϕ located on node P5 and which axis of symmetry makes an angle ψ with the wave heading P1P2 (Figure 4), results of failure probability computation are plotted on Figure 6 for several configurations. Note that we assume for this study the wave heading to be deterministic because of storm conditions that are studied here.

displacement (up) and strain (down) safety margins
We note first that axis 0° is an axis of symmetry for these curves: it can be explained by the symmetry in mechanical behavior, wave heading being deterministic and in the plane of symmetry of the structure and CJFE model being symmetric and bi-lateral. Then, despite of the selection of equivalent bounds in the sense of Figure 5, values of probability of failure obtained with the two limit state functions are very different: there is a factor of 5.25 for an half open angle of 20°( from 0.0754 to 0.397 respectively for the strain and displacement safety margins). Moreover, their evolution with ϕ and ψ are different too. It underlines the need of improved limit state when performing risk analysis where absolute values can be looked for. In the following, displacement criterion only will be addressed.

Definition
Due to harsh conditions of inspection for divers, inspection results are often modeled in a probabilistic way: in fact wave shaking, current velocity, turbidity and weight of equipment introduce uncertainties and remove on site conditions from test in laboratory. We used here a bayesian definition of inspection results in terms of detection . It is based on the probabilities of 4 events: Let us denote by "X", the event "crack presence" which takes value 0 in case of absence and 1 in case of presence. We show that the probability associated to these events can be expressed as a function of the Probability of Detection (PoD = P(d(X) = 1|X = 1)), the probability of false alarm (PFA = P(d(X) = 1|X= 1)) and the probability of crack presence (γ = P(X = 1)). Equations [6] and [7] give expressions for P(E 2 ) and P(E 3 ). Note that "γ" is a stochastic process indexed with time and that depends on initial conditions of building, thermal and mechanical fatigue process and incidents: most of these effects are still not understood and modelled in offshore field. We consider the structure at a given time and here "γ" is fixed at value 0.1 which corresponds to an estimation of the probability to observe large cracks assumed to be through cracks. Sensitivity studies are available in . The parametric surface P(E 2 ) is drawn on Figure 7.
ROC curves (receiver operating characteristic) characterized the performance of the technic and human chain "NDT tool -acquisition and treatment -diverinspector" and link the PoD and the PFA for a given crack size. To estimate the probabilities [6] and [7], given a ROC curve, a projection of this curve on the surface P(E i ) is realized and presented on Figure 7 for P(E 2 ) and two ROC curves.
Three ROC curves are plotted one on Figure 8 for  The expectations of the probability for events E 2 and E 3 are computed for the three ROC curves plotted on Figure 8 and presented in Table 1 with γ equals 0.

Risk functions and results
In terms of risk estimation, based on economic criteria only, the over-cost ( where P f|E 3 is the probability computed in 4.3, and knowing the inspection and repair policy: -no detection leads to no repair and consequently to a risk of failure, -a detection leads to a repair. In this last case, the failure is not considered because the risk of failure is of second order: only the over-cost (Equation [9]) due to a bad decision is taken into account. Let us used the dimensionless parameters Evolution of E(C) nd for ROC 1 Evolution of E(C) d for ROC 1 Evolution of E(C) nd for ROC 2 Evolution of E(C) d for ROC 2 Evolution of E(C) nd for ROC 3 Evolution of E(C) d for ROC 3 (c 1 ,c 2 ) when using the three CND techniques presented in Figure 8. An half-open angle of 20° and a probability of crack presence "γ" are selected for illustrations. In case of longer cracks, these risks increase but the trends and ranking of techniques are similar. The risk decreases clearly when increasing NDT performance, from ROC curve 3 to 1. The over-cost in case of detection is lower than the risk in case of no detection. It appears that the evolution with parameters c 1 and c 2 are strongly non linear and lead to the conclusion that this analysis must be placed in a global cost study. In fact, at given level of c 2 and cost of repair, if NDT 1 is significantly more expensive than NDT 2 or 3, this one can be less attractive in terms of risk. A more complete study allows to introduce combination of NDT performance is available in .

Conclusions
This paper presents the several steps of a risk estimation in the case of cracked joint of offshore structures submitted to extreme wave loading: -the response surface modeling of wave loading and mechanical behaviour, -the selection of random variables, -the reliability estimation, -the hazards on length measurements and detection of cracks.
A safety margin based on the displacement of a specific node is compared to another one based on strain energy. It is shown that the corresponding level of probability of failure are different despite the selected critical values are linked. Only the limit state based on displacement is used for the risk analysis. Three techniques are then compared in terms of mean performance and are introduced in the risk analysis. A cost parametric study allows to select the best one when looking the costs of detection and non-detection.