The paper makes a research into the uncertainty in the flaw determination and constructs the statistical relationship between the measured flaw sizes by NDT and the true flaw sizes. According to the statistical relationship, the distribution of true flaw sizes is obtained at a given measured flaw size. Furthermore, a quantitative method for determining the true flaw size is also successfully developed by making full use of the measured flaw sizes.
Keywords: Inspection, flaw size, distribution, quantitative method, reliability
When the integrity of the engineering structure with flaws is assessed, the flaw size in the structures must firstly be inspected accurately. But owing to the effect of random and fuzzy factors, non-destructive inspections, such as ultrasonic testing, carried out in an industrial environment to reveal hidden flaws in structures,' are imperfect'. There are obvious errors between the measured and true flaw sizes, and the measured sizes of the identical flaw are different because of different inspectors. The reliability of ultrasonic inspection depends on several factors. The most important among these are, e.g. the type, size and orientation of the defect, crack morphology, location of the crack, type of material inspected, geometrical restrictions(inaccessibility) in scanning, human factors, inspection procedure and inspection equipment, and environmental conditions in inspection work.
The paper makes a research into the uncertainty in the flaw determination and constructs the mathematical relation between the measured flaw sizes by NDT and the true flaw sizes. The distribution of true flaw sizes is obtained at a given measured flaw size. Furthermore, A quantitative method for determining the true flaw size is also successfully developed by making full use of the measured flaw sizes from different inspectors.
2. THE PROBABILITY MODEL FOR FLAW SIZING
In analyses of a large number of non-destructive size data, a linear relation between the logarithm of the measured flaw size and the logarithm of the true flaw size with normally distributed deviations has proved satisfactory. Furthermore, the use of lognormal distribution is mathematically convenient. and the parameters of the corresponding regression model can also be interpreted physically.
In the lognormal model , the true flaw size, a, is related to the measured flaw size, a', in the following way[2-4]:
Where e is a normally distributed random error term with zero mean and variance s2, and are the regression parameters.
According to this model, the expected value of the logarithm of the true flaw size is
Which means that the median of the true flaw size is equal to exp( lo + l1 ln a'). The variance of the logarithm of the true flaw size is assumed to be independent of the flaw size:
The model definition given in eqns(1)-(3) corresponds to the fact that the true flaw size is a log-normally distributed random variable.
The parameters and s2 can be determined by estimating the parameters of the corresponding linear regression model from data consisting of known flaw sizes and corresponding measurement results. Before performing a linear regression, the data must be reduced to a set of n pairs, ( ai', ai), where ai' is the measured flaw size for the ith flaw and ai' is the true flaw size for the ith flaw. Given the n pairs of ( ai', ai) data points to be fit by the regression analysis, the estimates and for lo and l1 can be obtained respectively. The formulas for and are
Where and are given by
Then the regression equation can be written as
The distributions of the estimates and are:
The covariance of the and are
Therefore the mean and variance of are:
The estimate 2 for the variance s2 can be written as
Assuming n = n-2 then
3. THE TRUE FLAW SIZE DISTRIBUTION
4. THE EXPRESSION METHOD OF THE FLAW SIZE
The ultrasonic inspection results of the stainless steel component is given in Table 1. The true length of the crack can be obtained by anatomizing specimen.
|True length ai
|Inspection length ai
|Table 1: The ultrasonic inspection results of the stainless steel component(mm)|
According to the linear regression analysis method, the regression equation of crack true length ai and inspection length ai' can be written as
and the standard deviation estimator for s is 0.2728。
If the identical crack with the same weld is inspected ultrasonically m times independently, the crack length are respectively, according to the method mentioned above, the distribution of the true crack length can be expressed as
Then the crack length aR with reliability R is
Likewise, the one-side confidence upper limit of the crack length aRU with confidence g and reliability R can be obtained by
In this paper, the uncertainty in the flaw determination was studied and the model for flaw sizing on the basis of statistical relationship between the measured and true flaw size was constructed. The model discussed is based on the log-normal model. Using the model, it is possible to take into account the prior information of the flaw size and combine it with the measurement results.
Through the model, the distribution of true flaw sizes is obtained at a given measured flaw size Furthermore, A quantitative method for determining the true flaw size is also successfully developed by making full use of the measured flaw sizes from different inspectors. According to the method, the flaw size with confidence g and reliability R can be obtained easily. One example is given to illustrate the effectiveness of the model and the method.
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