Modulus Of Rigidity Experiment

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Arleen Jerdee

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Aug 3, 2024, 4:48:20 PM8/3/24

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This study treats the measurement uncertainties that we can find in the stiffness modulus of the bituminous test. We present all the sensors installed on rigidity modulus measurement chains and also their uncertainty ranges. Several parameters influence the rigidity module's value, such as the parameters related to experimental conditions, and others are rather connected to the equipment's specification, which are the speed, the loading level, the temperature, the tested sample dimension, and the data acquisition, etc. All these factors have a great influence on the value of the modulus of rigidity. To qualify the uncertainty factors, we used two approaches: the first one is made by following the method described by the GUM (Guide to the expression of uncertainty in measurement), the second approach based on the numerical simulation of the Monte Carlo. The two results are then compared for an interval of confidence of 95%. The paper also shows the employment of the basic methods of statistical analysis, such as the Comparing of two variances. Essential concepts in measurement uncertainty have been compiled and the determination of the stiffness module parameters are discussed. It has been demonstrated that the biggest source of error in the stiffness modulus measuring process is the repeatability has a contribution of around 45.23%.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Results should not be presented as a single number but similarly to a continuous random variable so as an interval named the confidence interval. Thus, with the specified probability that is assigned to this interval, it can be assumed that the resulting measurement value is contained within this interval, providing that the measurement and the accuracy analysis have been properly conducted. Calculating measurement uncertainty is related to such notions as tolerance and required measurability. Tolerance is a denominate number that represents the difference between the maximal and minimal values of a certain property. It determines the interval, in which the real values of the specified property for the individual exemplars of the produced items should be contained.

There are requirements for test laboratories to evaluate and report the uncertainty associated with their test results. Such requirements may be demanded by a customer who wishes to know the bounds within which the reported result may be reasonably assumed to lie, or the laboratory itself may wish to understand which aspects of the test procedure have the greatest effect on results so that this may be monitored more closely or improved.

Bituminous mixtures are highly heterogeneous material, which is one of the reasons for high measurement uncertainty when subjected to tests. The results of such tests are often unreliable, which may lead to making bad professional judgments. They can be avoided by carrying out reliable analyses of measurement uncertainty adequate for the research methods used and conducted before the actual research is done. This paper presents the calculation of measurement uncertainty using as an example the determination of the stiffness Modulus of the asphalt mixture, which, in turn, was accomplished using the indirect tension method.

The parameters that can modify the results of these rigidity modulus tests are too many to count such as the parameters related to the test conditions and others, rather, related to the specification of the equipment which is the speed and the level of loading, the temperature, the size of the sample tested and the data acquisition, etc. All these factors have a great influence on the value of the modulus of rigidity.

To qualify the uncertainty factors, we used two approaches: Approach 1 is made by following the approach described by the GUM (Guide to the expression of uncertainty in measurement), the approach 2 by the numerical simulation of the Monte Carlo. The two results are then compared for an interval of confidence of 95%.

For the realization of the tests, a hydraulic press of capacity 50 kN is used. The main Characteristics are the following ones:

Defining the various test parameters is made using specific software.

The control of the press is made using the software. This program allows you to define the different parameters and create the desired solicitation signals: The control of the press and the thermal enclosure wall as well as all the necessary elements for the programming of an experiment.

The machine is tracked metrologically by control charts obtained from the tests on a working etalon (Etalon Ring).

The deformation measuring system consisting of two linear sensors (LVDT: Linear Variable Differential Transformers), capable of measuring the transient diametral deformation of the specimen during the application of a charge pulse with an accuracy better than 1m.

For the standard uncertainty of the resolution, Calibration, Drift and Sensitivity of the sensors, the Load rise time, and the Temperature, their evaluation of standard uncertainty is based on scientific judgment using all of the relevant information available.

The pulse duration of the applied load shall be 124 milliseconds, followed by a rest period of 3 seconds. The magnitude of the charge is necessary to produce a transient increase of 4 microns in the horizontal diameter of the specimen Table 1.

During the test, 10 pulses of conditioning charge are first applied to allow the press to calibrate the magnitude of the charge and the duration. Then 5 more charge pulses are applied for each direction, the variation of the diameter deformation is measured horizontal and applied load. The modulus of rigidity is calculated using the measurement of the 5 charging pulses according to the following formula [1] Table 2:(1)

The factors that can influence the value of the modulus of rigidity of the asphalt are mainly: temperature, loading speed, and stress level, frequency or loading time, sample size tested, and repeatability of the test process. All of these factors are of paramount importance to the value of the Modulus.

The expanded uncertainty corresponds to a quantity defining an interval, around the result of measurement; it can be expected to include a large fraction of the distribution of values that could be reasonably attributed to the measurand

where K is the coverage factor, we can take as an coverage factor K = 2 to have a level of confidence from 95% The final result expressed with two significant figures, by using the rule of rounding of Gauss. If all sources of uncertainty are taken into consideration, then there will be greater uncertainty [10].

To determine the most influential parameters, the contribution analysis is used to identify the input parameters that contribute the most to the uncertainty of the final results. They are characterized by uncertainty and high sensitivity. However, a parameter with excessive sensitivity and small uncertainty, or vice versa, can be a significant contributor.

According to the analysis of this histogram, we find that repeatability has a contribution of around 45.23% represents the highest percentage index, compared to other factors (force sensor, LVDT displacement sensors, and the thickness of the specimen), this is explained by the values of the uncertainties of these factors which are negligible Figure 4.

Monte Carlo Simulation (MCS) is the second method to estimate the uncertainty of measurements, where evaluating the measurement uncertainty by the MCS method can be carried out by establishments of the model equation for the measurand in the function of the individual parameters of influence, then selecting the significant sources of uncertainty, identification of the probability density functions corresponding to each source of uncertainty selected, and selecting the number Monte Carlo trials, and finally calculating the M results by applying the equation that was defined for the measurand [11] Figure 5.

The probability densities of each input quantity are defined in Table 4 of the GUM method. The standard deviations of each variable are obtained by taking the standard uncertainties calculated by the GUM approach.

When all the distributions of the input quantities have been defined, we can then generate realizations of each input quantity by drawing in their probability density function, it is necessary to have a sufficiently powerful random number generator, the calculation is carried out in a specific software (MC-Ed) It allows to define a Modulus for each Modulus size (load, strain, specimen thickness, load surface factor, etc.) and then assembles them into a Meta-Modulus. Finally, he chooses the numerical value of each term of the model, its associated uncertainty, and the probability function.

We note that the difference between the two confidence intervals obtained is very small and that the interval calculated by numerical simulation of Monte-Carlo is included in that of the propagation of the uncertainties. Moreover, these results are quite in agreement with each other and we can consider that the results given by the numerical simulation of Monte Carlo validate the law of propagation of the uncertainties.

We also applied the comparison criterion described in the document (BIPM JCGM 10). Indeed, it is advisable to apply the GUM method by adding terms of higher degree if necessary and to give a confidence interval of 95% for the output variable. Then, to validate the GUM method, it is necessary to apply the numerical simulation of Monte Carlo in order to provide a standard uncertainty and a confidence interval of 95%. It must then be determined whether the confidence intervals obtained by the Monte Carlo method and the law of propagation of the uncertainties are in agreement with a degree of approximation calculated as follows Table 5.