The paper introduces a probabilistic rationale with the associated software for modelling the uncertainties interaction arising from the above mentioned sources. Emphasis is placed on the influence of defects (crack) size distribution on the structural reliability. A probabilistic sensitivity analysis is outlined and applied to the case of a storage tank failure.
among which the techniques of image reconstruction and failure process simulation are at the core of this approach.
The know how which emerged from the techniques and procedures related to the concept of operational reliability, obviously concerns any field of industry, being already the object of international regulations such as ISO 9000 or CEI (Comité Electrotechnique International) standards.
The basic task of industrial inspection is to detect and locate defects during manufacturing and operation. It plays an ever increasing role in early, fast and reliable detection of the potential of failure, in order to maintain productivity and operating availability of plants. It is a major factor of capital protection by excluding failures that penalise nowadays with about 4 % the GNP, of any developed country, besides the whole range of negative consequences on human and ecological environment.
The traditional one, aims to guarantee a "safe" operational life under prescribed operational circumstances. This is the "safe-life" approach, which has been developed continuously since Wöhler experiments on fatigue of railway axes until the early 50th. This approach has been developed for safe-life prediction under progressive cumulative damage induced by fatigue, corrosion, creep, irradiation, wear or even natural ageing. Safety factors are applied to experimentally derived endurance to failure in order to state the safe operational life. The adherence to this approach, widely used nowadays in conventional engineering implies the retirement or replacement of involved components and structures as the safe life has been reached. The product built-in reliability is based only on the inspection performed during fabrication, in order to assure "defect-free" quality before operation. This approach which one could state as "Elysian"seeks for perfection in fabrication. Its basic message is "no defects!".
Unfortunately, in real life it can not be maintained. To many practitioners it seems dangerous not to consider the possibility of overlooking initial defects or defects generated during operation. Many catastrophic failures, especially in aerospace field have highlighted the inconsistency of the safe-life procedure.
As a result, a new design, fabrication and maintenance philosophy has emerged in the high-tech industry which put emphasise on safe operation even in the presence of defects, which may exist from the beginning or, develop by cumulative damage during operation. This new engineering philosophy is focused on the "damage-tolerance" (up to a certain extent) within a specified time intervals bounded by in-service inspections. Under this rationale, inspections intervals dictate the repair or retirement of damaged components or the whole structure. The reliability of the product may be thus significantly enhanced at reasonable costs.
The damage-tolerant approach while recognising that the risk of failures can not be completely excluded can contribute to minimise the failure risk down to tolerable levels (between 10-4 to 10-6 probability of failure) by combining the knowledge and reasoning of designer, production and quality assurance people, beginning from manufacture and in-service inspection until the component retirement.
It follows that inspection reliability coupled with explicite failure risk quantification play a central role in a successful application of damage-tolerant philosophy.
The reliability of inspection techniques may be understood and quantified in probabilistic terms. Broadly speaking the inspection reliability is defined as the probability of not overlooking an existing defect (probability of detection, POD) and correct sizing the defect. Whatever simple this definition may appear, it encompasses many complex issues ranging from the specification of the nature of defects to influencing factors related with the inspection instrumentation, product nature, the involved human factor and the available expertise for inspection data processing and assessing.
Figure 1 illustrates the interrelationship between key factors involved in the concept of inspection reliability. A detailed discussion in these issues falls outside the scope of this overall view. Pertinent information can be found elsewhere /1/. However, while some traditional notions in fig. 1 may seem self-explanatory for inspection practitioners other, wide or closely related with the trends of NDT-reliability improvements, deserve to be outlined.
Fig. 1: Inspection Reliability
One major drive for improvement is given by the increasing capacity of automated data acquisition and processing by the currently available computer techniques. Computer modelling procedures associated with NDT offer new opportunities for optimal test parameter selection, together with the quantification of procedure limits.
With the new developed modelling algorithms, practically the whole NDT testing situation can be covered. The NDT modelling systems developed at IzfP-Saarbrücken under the logo of CAI (Computer Aided Inspection) are a combination of modelling the geometry and material response of the inspected component together with the testing techniques and its scanning parameters.
Fig. 2 Modelling a NDT system
In ultrasonic (US) inspection, the result of the modelling is a high frequency data field which may be used as input data field for further processing. As an example, when using ALOK (Amplitude - Time - Locus - Curves, /2/) procedure the data compression and logarithmic behaviour of the amplifier have to be simulated by data pre-processing. Using SAFT (Syntehtic Aperture Focusing Technique), the high-frequency ultrasonic data (A-scans) fields can be used directly.
In a detection process a "sensitivity analysis" i.e. test simulation with parameter variation may be easily performed and compared with the traditional procedure of implementing "real artificial" flaw into a real component, a singular circumstance among an infinity of possibilities. After the selection of testing situations and defect parameters, the modelling module calculates the expected signals at each intended probe position. These signals are then used within the testing procedure like ALOK or SAFT (Synthetic Aperture Focusing Technique) to get an image of the defect /3/. The evaluation of the image leads to a statement about the reliability of defect detection.
Within NDT modelling systems one essential module has been developed in order to predict three-dimensional images of the interior of the material. This module consists of two parts:
Although consideration of failure risk in probabilistic terms is often unappealing to engineering and inspection community, it represents a firm trend for future developments. Probabilistic approach enables a much closer representation of reality than a currently accepted deterministic analysis which assumes that inputs are known exactly or can be represented by conservative upper and lower bounds /6/.
The physical modelling of fracture process is based on Fracture Mechanics (FM) principles, a new field of technology that aims to quantify the conditions under which an existing crack in a structural element extends to failure under operational loading. According to FM principles, applied to brittle materials under static loading, failure occurs when a compounded parameter (the stress intensity factor, SIF), K, function of applied stress ( and crack size a, attains a critical value Kc - the fracture toughness, a material characteristic. Hence, fracture occurs if:
This basic FM criterion, pertinent for predominatly elastic circumstance before fracture, derives from the first principle of thermodynamics. Accordingly, the initiation of unstable crack extension occurs when the strain energy release rate equals the energy that is necessary for the creation of new free surfaces along the front crack under extension.
While Kc as material characteristic is determined by well established laboratory technique, K is assessed by analytical or numerical solutions according to the geometry of the structural element in interaction with the crack size and loading. Generally, K assumes a functional form:
where Y (.) is a geometrical correction factor. For a through thickness crack in a large body, Y = 1.
Under static loading, the scatter in the crack size (and location) dominates the random pattern of the fracture process. Under repeated loading, with random variation of stresses from cycle-to-cycle, there is an interaction effect with statistical scatter of the crack size.
As concerns the scatter of the crack size it has an objective "intrinsic" distribution as result of manufacturing processes and in service operational circumstances. This can be described by a random variable with the probability density (PD), fa (a). However, scatter of the observed crack size s, is influenced by the uncertainty which stems from NDT technique (process, instrumentation, human factor). One can define a probability of detection (POD), in terms of repartition function PD and its associated PD, fD (s). It follows that PD of the size of existing defects fa(a) can be inferred from the PD derived from NDT, fs (s) and the POD, PD:
Many research programs are now under development for deriving information on POD related to the physical principles underlying the NDT procedure.
As concerns the intrinsic PD of the crack size, a favoured description has been given by Marshal  in terms of exponential distribution:
see Eq. (4)
where aDm is the mean value of the crack size in the population of defects.
Under static loading circumstances (= const) and a crack population of sufficiently small crack size in comparison with the size of the structural element, (Y=1), the PD of SIF, fK (K), follows from conjunction of Eqs (2) and (4). A Rayleigh distribution results /10/: see Eq. (5)
where Kmam is SIF related to the mean crack size, a m . The simplest probabilistic fracture mechanics interpretation that retains, however, the trend pattern may be put forward by considering in the failure criterion, Eq (1) a deterministic value of the fracture toughness K c =K o . Under the assumption that, at least, one crack exists in the considered structural element, the probability of the event described by Eq (1), i.e. of failure, is: see Eq. (6)
and combined with Eq (5) an explicit relationship results /10/ by Eq. (7).
Equation (7) enables to outline the philosophy of probabilistic risk assessment by "sensitivity analysis". Figure 4 illustrates the probability of fracture of a structural component under homogeneous traction loading , the statistical distribution of the through thickness crack size being described by an exponential rule, Eq (4), with a m = 6,25 mm, a value recommended by Marshal for nuclear power reactor pressure vessels, as result of PISC - round robin investigation. This exemplification implies also that at least one crack exists in the component.
According to the outlined parametric analysis it is obvious that a range exists where a small stress increase may lead to an important increase of failure risk. The benefit from increasing material fracture toughness is clearly illustrate in Fig. 4 by the decrease in failure probability.
If the component contains n - cracks of size variability described by f a (a), Eq (4), and accounting on P f , Eq. (7), the probability of fracture associated with the presence of a single crack, then from the consideration of all routes of failure, the probability of fracture is given by Eq. (8).
Probabilistic fracture mechanics procedures have been developed for elastic - plastic circumstance as is the case of current structural steels. Failure Assessment Diagram (FAD) or R6-Procedure  is at the base of a FADSIM-Computer Code developed for this purpose. This approach of failure risk has been applied to a post-factum analysis of a large storage vessel failure. By modelling the crack size at fracture initiation by a normal distribution (mean value of crack size inferred from measurements on fractured fragments) and the fracture toughness uncertainty according to a 3-parameters Weibull distribution (Ko = 150 MPa m Ka = 200 MPa m and b = 4) a probability of fracture of nearly 50 %, has been evinced demonstrating the imminence of fracture. A sensitivity probabilistic analysis have outlined the deleterious effect of high stress concentration at the failed welded joint, combined with the excessive crack depth as a result of stress corrosion crack extension. The lack of in-service NDT monitoring prevented to acknowledge in due time, the high potential of failure.
The probability risk assessment philosophy has been adopted in many high-tech industries like nuclear (e.g. Nuclear Regulatory Commission - NRC, Regulatory Guide - 1987 - for PWR vessels integrity under pressurised thermal shocks) or aerospace (US Air Force ASIP and ENSIP Programs) industry /6/, /7/.
As outlined in Fig. 5, the base approach of mechanical failure risk having probabilistic fracture mechanics as key ingredient, is a new synthesises with contributions from many engineering fields. Loading statistics (random-loading pattern), NDT reliability and material strength statistical scatter are the necessary input information for modelling the potential of a failure. This general rationale has distinct features when related with ductile/brittle fracture, or cumulative damage due to fatigue, corrosion, wear, creep or irradiation.
As a result, damage-tolerant philosophy incorporates probabilistic approach in a natural way. The failure risk is quantified in terms of failure probability per year of operation. Decisions on repairs or retirements made on the basis of probabilistic assessment of failure risk represent now a major advance in comparison with "yes-no" statement underlying the traditional approach to safety and the associated "yes-no" acceptance criteria in NDT.
Finally, decisions on the acceptable risk-level must be made. Such an undertaking that is strongly influenced by the nature of the product and its operational circumstances is a matter, involving technical, economical, social and on global issues political decision, as well.
Fig 5: Multidisciplinary approach with probabilistic fracture mechanics
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