TWI Bulletin, November - December 2003
In part two of this feature on the practical aspects of radiography this work describes the validation and refinement of a simple model of radiography, originated by Pollitt in 1962. ( Part 1)
George Georgiou is an independent NDT consultant and director of Jacobi Consulting Ltd. He is currently under contract to TWI as an NDT consultant and is carrying out research in a number of different areas. In particular he has been directly involved in developing techniques for Computed Radiography in National and European projects. He has had many years experience in mathematical modelling of NDT methods such as radiography, ultrasound and thermography. During the 1990s he was a BSI representative on a number of European committees involved in the development of CEN standards for all the major NDT methods carrying out bespoke pieces of research for the various committees. During much of the 1990s he was section manager of the NDT section at TWI.
Charles Schneider is a Principal Project Leader in the NDT Technology Section at TWI. He has worked in the fields of inspection reliability and theoretical modelling since 1985, holding posts within the Central Electricity Generating Board and its successor companies. Since joining TWI in 1997, his work has included inspection qualification, and the application of statistical techniques to both inspection reliability and fatigue testing.
As Charles Schneider and George Georgiou explain, defects are treated as smooth, parallel-sided slots. It builds on similar work, which was presented at NDT'99, on thick-section welds (thickness range 50-114mm). The latest experimental data were collected from 12 realistic planar defects in welds of thicknesses 10-50mm. They were radiographed under various exposure conditions, giving 278 defect/radiograph combinations (supplementing the 308 defect/radiograph combinations already available from TWI's earlier work on thick-section welds). Each radiograph was evaluated 'blind' by two radiographers. The specimens were then sectioned to determine defect size, orientation and gape.
The experimental data show variations in detectability that are strongly correlated with theoretical predictions. We have refined Pollitt's original theory, improving the observed correlation with the experimental data. In all cases, the refined theory is either accurate or pessimistic. The broader range of the available experimental data has also enabled us to improve the accuracy of our previously published statistical models for the reliability of radiography. TWI is already using such models to support the continued operation of its clients' plant.
The story so far...
During 1995-1999, TWI performed several detailed studies on the radiography of large planar defects in thick-section welds. The work considered a number of issues, such as the capability of 1950s and 1960s radiography, the use of statistical models to predict defect detectability and the effect of human factors on defect detectability. The thicknesses studied were generally in the range 50-114mm. The main application of the work was to quantify the capability of the construction radiography performed on the welds of the Magnox steel reactor pressure vessels, but the work also had generic value in providing a better understanding of radiographic capability. In particular, TWI developed models for the radiographic detectability of these defects, based on earlier publications by Pollitt and Halmshaw. The experimental results confirmed the value of these models in predicting the radiographic detectability of planar defects in thick-section welds.
This paper is based on an extension of the above work to investigate the detectability of planar defects in thinner section welds (thickness range 10-50mm). This new work, funded by the UK nuclear licensees, has specific applications in the nuclear power industry, but it is again believed to have generic value in improving the understanding of radiographic capability. The paper compares the experimental data with our earlier results and shows how the new thin-wall data has enabled us to improve our previously published models. A separate paper published in the last issue of the Bulletin provided a more detailed account of the practical aspects of the work.
Experimental programme
Three new butt-welded test specimens were manufactured, which contained 12 planar longitudinal welding defects with through-wall extents up to about 3mm. The defects included hydrogen-induced cracking, lack of sidewall fusion and lack of root fusion. Care was taken to ensure that the defects introduced had a realistic metallurgical morphology. The specimens were radiographed under various exposure conditions, including angled shots to simulate different weld preparation angles, and the use of spacer plates to simulate thicker specimens. The thickness range evaluated was 10mm to 50mm, and both X-ray and gamma radiography was used. This gave a total of 278 defect/radiograph combinations, each of which was evaluated 'blind' by two interpreters (supplementing the 308 defect/radiograph combinations already available from TWI's earlier work on thick-section welds).
TWI sectioned the specimens to determine each defect's size, orientation and gape. The performance of the radiography was then analysed as a function of these defect parameters and of radiographic parameters, such as source type and source-to-film distance.
As in our previous work, the experimental results show variations in detectability with parameters such as penetrated thickness, gape and orientation, which are broadly in line with what is expected on simple theoretical arguments ( eg decreasing detectability with increasing wall thickness). However, it is difficult to display the raw data in a form that enables a more quantitative comparison with theory. This is because of the large number of interrelated parameters that vary from defect to defect or exposure to exposure, making it difficult to isolate the effects of individual parameters.
Modelling
The Pollitt model
As before, we compared the experimental observations with the simple deterministic model of radiography described by Pollitt, which treats planar defects as smooth, parallel-sided slots. The new thin-wall data show similar variations in detectability to those reported previously. The results are again clearly correlated with the theoretical predictions, albeit not quite as strongly as for the thick-wall data.
We investigated the effect of the same incremental refinements of the original model as before. These refinements are based mainly on ideas attributed to Halmshaw. The refinements include (a) consideration of the geometrical enlargement of the radiographic image due to beam divergence and shot angle, and (b) a more rigorous treatment of radiographic unsharpness. Each of the incremental refinements improved the correlation of theory with experiment (once the thick-wall and thin-wall wall data were combined). Following these refinements, the deterministic model is either accurate or conservative in predicting each one of the 556 new experimental results.
Theoretical index of detectability
TWI has developed a 'theoretical index' of detectability, which quantifies how easily a planar defect can be detected according to the Pollitt model. The theoretical index is positive if the Pollitt model predicts detection and negative if the Pollitt model predicts non-detection. For the data from the thick-section welds, we reported at NDT '99 an approximate relationship between the probability of defect detection p and the theoretical index I theory , given by:
The constant coefficients A and B of this relationship were estimated from the experimental data, by the maximum likelihood method, using a type of regression theory known as 'logistic' regression. The function on the left-hand side of equation (1) is called the 'logit' function.
Figure 1 illustrates both sets of experimental results, together with the linear relationship that was fitted to the thick-section data. Figure 1 has been produced by grouping the data according to the theoretical index I theory and plotting an estimate of the logit function given by
is the proportion of detections within each group. The error bars indicate approximate two-sided 80% confidence limits. Figure 1 confirms that the defects are detected with high reliability (~90%) whenever the improved Pollitt model predicts detection ( ie I theory > 0). But, while the thick-section data in Figure 1 follows an approximately linear trend, this is clearly not the case for the thin-wall data (which encompasses a much wider range of values of the index). A quadratic trend clearly gives a better fit to the thin-wall data; this has also been confirmed by a statistical test of non-linearity (at the 5% significance level).
In this study, we also investigated whether functions other than the logit function (which is the most commonly used) would yield a better fit to the experimental data. Of these models, the 'quadratic gompit' model gave the best fit. The model is given by:
The coefficients A, B and C are again estimated by the maximum likelihood method, using a well-established algorithm.
Figure 2 shows the same data as Figure 1 after transforming the vertical axis to show the 'gompit' function
in place of the logit function
Note that we have not shown a trend line for the thick-section data in Figure 2, because the linear trend line plotted in Figure 1 (originally identified in our earlier work on thick-section welds) would no longer appear as a straight line after transforming the vertical axis from the logit function to the gompit function.
The goodness-of-fit of the model illustrated in Figure 2 was assessed using three different statistical tests (each at the 5% significance level). All three tests confirm that the model fits the experimental data well.
One drawback to the quadratic models above is that, for certain values of the index I theory, it will start to predict a probability of detection (POD) that is a decreasing function of I theory, instead of the increasing function of I theory that is intuitively correct. (This is because a quadratic function always has a turning point). Thus, the model illustrated in Figure 2 yields a minimum in the POD (of ~9%) at an index of about -6.4. This fortunately lies outside the range of all the experimental data except for one case of non-detection (which lies at an index of -6.5). This means that the fit to the experimental data will be almost identical, if we replace the unrealistic (decreasing) part of the quadratic model by a constant equal to the minimum value of the quadratic, ie
Even after this improvement in the realism of the model, it still seems rather implausible that the POD never falls below 9%, regardless of how small the detectability index is. We therefore recommend that the model be applied with caution for values of the index less than -6.4. On the positive side, it is difficult to envisage an application where an overestimate of the POD as 9% (rather than some lower value) would have any serious safety consequences. We suspect that the main risk would be that a safety assessment might integrate over a range of defects having different indices, including some very low indices, and that an overestimate of the POD for the low indices might have a knock-on effect on the integrated POD. Even in such a case, we suspect that the effect on the integrated POD would be relatively modest. Nevertheless, we believe there is some scope for future investigation of more complex non-linear models that are strictly increasing functions of the index.
Conclusions
The models described in this paper can be used to estimate how easily a large planar defect can be detected by radiography, from knowledge of its size, orientation and other relevant parameters. In principle, the probability of detection and an associated confidence interval can be estimated. The models apply to defects of a similar type to those we have examined experimentally; caution needs to be exercised in extrapolating the results to other defects.
