Mathematical Methods of Experimental Design in Nondestructive Testing
Konstantin Kovler and Isaak Schamban
National Building Research Institute - Faculty of Civil Engineering
Technion - Israel Institute of Technology, Haifa 32000, Israel
- International Symposium on NDT Contribution to|
the Infrastructure Safety Systems, 1999 NOV 22-26 Torres,
published by UFSM, Santa Maria, RS, Brazil
Non-destructive quality control of structural materials is based on the calibration dependence "indirect non-destructive characteristic Z - material property R" determined by both destructive and non-destructive tests. The form and character of the calibration expression depend on technological, physical and other factors (Fi) affecting both Z and R. However, the numerical values of Fi can differ: when the calibration dependence is determined in laboratory and when it is utilized in practice. Moreover, the influencing factors Fi can sometimes have an opposite effect on the material property. To improve the reliability of non-destructive testing, the calibration dependencies should be completed with the special experimental-statistical models of the regression coefficients depending on technological and physical factors. Such experimental-statistical models can be determined by mathematical methods of experimental design and used for more accurate estimation of the material properties in real testing conditions. The suggested approach is illustrated by the example of assessment of concrete strength using ultrasonic pulse velocity method.
Non-destructive quality control of structural materials is based on the dependence "indirect non-destructive characteristic Z - material property R" determined by both destructive and non-destructive tests:
|R = R (Z, bi), i = 0, 1, 2, ..., n||(1)
where bi are regression coefficients of the dependence.
The calibration dependence (1) is affected by many factors. Therefore, a sufficiently large number of specimens should be tested to establish a reliable correlation between R and Z. At the same time, unfortunately, even with a very large number of specimens the final accuracy of the non-destructive estimation of the material property remains low. For example, the strength estimation of concrete by means of ultrasonic pulse method is possible only with ±20% accuracy, and even this can be achieved only under the best conditions .
In our opinion, the most important errors, from the point of view of the final accuracy of the ND estimation, are not the errors related to the measurement of R, but those related to the unstable character of the key dependence (1). Determining calibration dependence of the type (1) only is not enough to predict the material property in real structures. It is well-known that the difference between the material property (a) estimated in laboratory samples and (b) determined in the real structure in situ, is often higher than the difference between the values of this property found by destructive and non-destructive methods. Therefore, the accuracy of the estimation is better, when the material of the tested structure is more or less identical to the material of the specimens used for the obtaining of the calibration dependencies "Z - R".
The Regression Coefficients as variable parameters
In general, the form and character of the expression (1) depend on technological, physical and other factors (Fi) affecting both non-destructive characteristic Z and material property R. However, the numerical values of the influencing factors Fi can significantly differ in two cases:
one) when the calibration dependence (1) is determined in the laboratory and
two) when the calibration dependence (1) is utilized in practice.
Moreover, the influencing factors Fi can sometimes have an opposite effect on the material property. For example, it is well known that the ultrasonic pulse velocity increases with the increase of the moisture content of the porous/capillary body, because ultrasonic waves in the water propagate faster, then in the air. Therefore, it could be expected that the strength of such material should also increase, because usually the faster is ultrasonic wave, the higher are the strength and stiffness of the material. At the same time, the wetting of porous materials is known to result in the decrease of their strength. Such a paradoxical effect triggered the authors to develop a special algorithm of obtaining correlation relationships "Z - R". The obtained models are fundamental, but should be proved and adapted for the specific testing conditions.
The presence of several "constant" coefficients bi (i = 0, 1, 2, ..., n), which in fact depend on the material composition and environment, is a common property of all the calibration dependencies. It is the use of the "constant" coefficients decreases the accuracy of non-destructive quality control, because these "constants" are isolated both from technological parameters of the manufacture process, which has to be optimized in the technologies of the 21st century, and from the real environment.
To improve the reliability of non-destructive testing, the calibration dependencies of type (1) should, in our opinion, be completed with the special experimental-statistical models of the regression coefficients depending on technological and physical factors. Such experimental-statistical models can be determined by mathematical methods of experimental design and used for more accurate estimation of the material properties in real testing conditions.
In general, the mathematical methods of experimental design help to decrease significantly the amount of the experimental work in the determination of the multi-factorial dependencies. They can be also used for optimizing mix proportions and content of admixtures, for the technical and cost analysis of the technological efficiency and for the development of new mathematical models to describe the material behavior.
Example: Assesment of concrete strength
Using the methods of experimental design, the quantitative dependencies of the regression coefficients b0 and b1 on the technological factors (water/cement ratio, water content and logarithm of concrete age) and physical factors (temperature and relative humidity) can be determined. Approbation of the suggested approach shows that the variation coefficient of the dependence "fc - V" significantly decreases.
The suggested approach can be illustrated by the example of assessment concrete strength using ultrasonic pulse velocity method. Since there is no theoretical relationship between pulse velocity and concrete strength, the calibration curve must be established empirically for given concrete.
According to the British Standard BS 1881, Part 203, the mean pulse velocity and mean strength obtained from each set of three nominally identical test specimens provide the data to construct a correlation curve. A correlation curve produced in this way related only to concretes containing the same materials, and produced, cured and tested in a similar way. The procedure recommended by RILEM is similar to that of British Standard.
The following calibration dependencies, for example, are used in the practice of ultrasonic control of concrete strength : fc = a0 + a1V+ a2V2 or fc = b0 Vb1 - in the Slovak Standard STN 73 1371, fc = b0 eb1V or fc = a0 + a1V - in the Russian Standard GOST 17624. In these dependencies fc is the mean value of compressive strength of tested specimens, MPa; a0, a1 a2, b0, b1, b2 are the regression coefficients, V is the measured longitudinal ultrasonic pulse velocity, e is the base of the natural logarithm.
According to the Russian Standard GOST 17624, the relation between the pulse velocity and the strength of the concrete should be established for the given concrete before testing a structure. The mathematical form recommended by this standard for the dependence "fc - V" is either linear or exponential. The linear form fc = a0 + a1V is recommended, when the strength range is narrow. For wider strength range an exponential function is used:
The experimental data show that mostly the type and amount of coarse aggregate influence the dependence "fc - V". If so, for ordinary concretes, in which coarse aggregate has usually strength higher than that of the concrete, the amount of coarse aggregate is determined by the volume of the mortar matrix, or, in other words, by two technological parameters: water/cement (or cement/water) ratio and water content.
As far as physical factors are regarded, the main influencing factors are the temperature (T) and moisture (M). At the same time, some decrease in pulse velocity can be neglected, if the concrete temperature is kept in the range of 5 - 30°C. As was noted earlier, the moisture influences concrete strength and pulse velocity by opposite way. In addition, the more porous is concrete macrostructure, the greater is expected difference between the strengths of dry and wet material.
In the present paper an attempt was made to evaluate the regression coefficients b0 and b1 of the relationship (2), depending on some technological and physical factors Fi,j.
The quantitative relationships for the regression coefficients b0 and b1 were determined with the mathematical methods of experimental design. The experimental plan was so called complete 24 factorial design, with 4 factors varied: cement/water ratio (F1), water content (F2), logarithm of the concrete age (F3) and moisture of the material (F4). Each factor was varied in 2 levels, maximum (+1) and minimum (-1). The influencing factors and their interval of variation are shown in Table 1. Such an approach is a particular case of multifactorial multilevel designs, in which the functional relationship between the response of interest and the levels of the k quantitative experimental variables may be graduated by a general polynomial of degree d in the levels of the variables. In the simplest case of two-level design, we code the levels in standardized units so that the 2 values taken by each of the variables are = +1 and -1 and suppose also that the second degree graduating polynomial fitted by the method of least squares is
||Levels of variation
||Interval of variation
|Water/binder ratio (c/w)
|Water content (w), kg/m3
|Age of concrete (t), days
|Moisture content (M), %
|Table 1:The conditions of the experimental design|
The cement used was ASTM Type I, ordinary Portland cement, grade 300. The coarse aggregate was crushed limestone with 19-mm maximum size and the fine aggregate was quartz sand with fineness modulus of 1.52. Superplasticizer of naphthalene formaldehyde sulfonate type was used in the amount necessary for achieving a slump of S4 - S5. The specimens were cast in the form of prisms with the size of 280×70×70 mm. The specimens were tested on compressive strength by both destructive and non-destructive (ultrasonic pulse velocity) methods at 3 and 28 days age. The ultrasonic measurements were performed with the apparatus equipped by the transducers having a natural frequency of 54 kHz, the path length at direct transmission was 100 mm.
As a result of the destructive and non-destructive measurements, the values of the coefficients b0 and b1 were calculated as
The final dependencies of the regression coefficients were obtained in the form:
b0 = 4.77 + 0.101 F1 + 0.181 F2 + 1.245 F3 + 0.172 F4 - 0.622 F1F2 - 0.260 F1F3 + 0.388 F1F4 + 0.550 F2F3 + 0.067 F2F4 + 0.220 F3F4,
b1 = 0.408 + 0.074 F1 - 0.013 F2 - 0.024 F3 - 0.028 F4 + 0.036 F1F2 - 0.001 F1F3 - 0.022 F1F4 - 0.024 F2F3 + 0.008 F2F4 - 0.014 F3F4,
where F1 = (c/w-2.0)/0.6, F2 = (w-180)/30, F3 = (lg t-0.977)/0.485, F4 = (M-5.5)/2.5.
To prove the obtained dependencies, the new series of concrete was cast as 100-mm cubes and cylinders with 100-mm diameter and 52-mm height. The concrete composition (kg per cubic meter) was 300:210:620:1200 (cement:water:sand:gravel). The specimens were tested at the age of 7, 14, 28, 56 and 90 days. The results are shown in Fig. 1.
It can be seen that the calculated values of concrete strength are close to the experimental ones, but the older is concrete, the higher is the difference, because the age of part of the specimens tested in 56 and 90 days after casting exceeded the range accepted at the experimental design (3 - 28 days). If to remain within the range of concrete ages not exceeding 28 days, the regression line of predicted strength becomes significantly closer to the ideal correlation having the slope of 1. The slope of this regression line is 0.95, i. e. the difference between the predicted strength and that of the destructive method is only by 5%.
To improve the reliability of non-destructive testing, the calibration dependencies should be completed with the special experimental-statistical models of the regression coefficients depending on technological and physical factors. Such experimental-statistical models can be determined by mathematical methods of experimental design and used for more accurate estimation of the material properties in real testing conditions. Approbation of the suggested approach shows that the accuracy of the non-destructive estimation is improved.
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