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A Quantitative Method for Determining the Flaw Size in the Structure

Ding Keqin
(National Center of Boiler & Pressure Vessel Inspection & Research , Beijing 100013, P. R.China)
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ABSTRACT

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

1. INTRODUCTION

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[1]. 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]:

 (1)

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

 (2)

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:

 (3)

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

 (4)

 (5)

Where and are given by

 (6)
 (7)

Then the regression equation can be written as

 (8)

The distributions of the estimates and are[2]:

 (9)
 (10)

The covariance of the and are

 (11)

Therefore the mean and variance of are:

 (12)

 (13)
Then
 (14)
Where
 (15)

The estimate 2 for the variance s2 can be written as

 (16)
Assuming n = n-2 then
 (17)

3. THE TRUE FLAW SIZE DISTRIBUTION

If the identical flaw is inspected m times independently, the flaw sizes are respectively. According to eq.(1), the true flaw size can be written as

 (18)
Assuming
 (19)

Owing to ln ai i = 1,2,...,m independent hence ln a is normally distributed its mean and variance are:

 (20)
 (21)
Where
 (22)
Therefore
 (23)

We can prove that ln a is the distribution of the true flaw size[2]. According to eq.(23), the variance of ln a has contracted m multiple i.e. the precision of the flaw size determination can be greatly improved by m times independent inspection. The estimated mean of ln a is given by

 (24)
 (25)
 (26)
Where
 (27)
Therefore
 (28)

The variance estimate of ln a is and satisfying the following way:

 (29)

4. THE EXPRESSION METHOD OF THE FLAW SIZE

If the identical flaw is inspected m times independently, the flaw sizes are respectively. As mentioned above, the distribution of the true flaw size can be expressed by equation (28), therefore the flaw aRU size with confidence g and reliability R can be expressed as[2]

 (30)

In which can be calculated by the following way[2]

 (31)
 (32)

5. EXAMPLE

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[5].

 No.i 1 2 3 4 5 6 7 8 9 10 11 12 True length ai 7.6 2.9 9.0 1.6 9.2 2.1 6.6 1.9 1.8 1.8 1.3 2.9 Inspection length ai 9.0 1.5 10.0 1.0 8.0 1.5 6.0 1.5 1.5 2.0 1.0 1.0 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

 (33)

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

 (34)

Then the crack length aR with reliability R is

 (35)
 (36)

Likewise, the one-side confidence upper limit of the crack length aRU with confidence g and reliability R can be obtained by

 (37)
 (38)

6. CONCLUSION

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.

REFERENCES

1. Berens,A. P., NDE reliability data analysis. In Metals Handbook, 9th edn, Vol.17, ASM Int., 1989,pp.689-701.
2. Ding Keqin, Study on fuzzy R6 failure assessment method for the pressure vessel in service. Postdoctoral research report, Beijing University of Aeronautics & Astronautics, 1999.10. (in Chinese)
3. Fu Huiming,Ding Keqin and Liu Dengdi, The fuzzy expression method of crack size. Journal of Mechanical Strength, 2000, 22(1) (in Chinese)
4. Fu Huiming Liu dengdi, Fuzzy theory of nondestructive testing and its application. 1999, 14(1), 225-231.(in Chinese)
5. Kaisa Simoda, Urho Pulkkinen, Models for nondestructive inspection data. Reliability Engineering & System Safety. 1998,60: 1-12.

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