**Gage Repeatability and Reproducibility, generally known as a GR&R, is a statistical method used in process control (SPC) for measuring the variation and precision present in a measuring device and the subsequent effectiveness of the instrument to be used as a measuring tool. There are, as the name indicates, two components that comprise of the GR&R, repeatability, related to the potential of the gage itself to provide consistent results (precision) under repeated test of the same manner, and reproducibility, or the potential of the gage to offer repeated results irrespective of the operator performing the test (variation among operators).**

In Rockwell Hardness Testing, a GR&R study can be a valuable measure of how precise the tester is performing and how much variation the tester may be contributing to the overall process. However, GR&R evaluation in Rockwell are not attained without some complications, but once these are understood and accounted for, GR&R assessment offers useful process data. While several types of gages, such as calipers or micrometers, can be GR&R performance tested relatively easily and without sample variation (using a gage block of a known value to make repeated measurements on the same block and calculating the repeatability), a GR&R evaluation on a Rockwell tester poses inbuilt difficulties.

As a Rockwell test is never carried out in the same spot, a primary obstacle is the test samples variation itself. Due to this, as no material is completely uniform in hardness, a variation is introduced. This phenomenon is usually accounted for by introducing process tolerances to the mathematical analysis when carrying out the GR&R calculations.

## The Purpose

The purpose of performing a GR&R study is to determine how much of the process tolerance is being used up by the difference in a hardness testing instrument (also referred to as equipment variation or repeatability) as well as between operators (also referred to as reproducibility or appraiser variation). When the combination of these sources (repeatability and reproducibility or R&R) becomes a vital portion of the process tolerance, an operator cannot be sure whether they are measuring the hardness of a part or just generating random numbers with the Rockwell tester machine.

For statistical process control (SPC) to work efficiently, the combined variation should be less than 10% of the process tolerance (< 10% GR&R). Rockwell hardness testing machines with a GR&R between 10% and 30% may be acceptable on an interim basis, and machines with a GR&R of greater than 30% should not be used for SPC.

## Why Perform a GR&R?

A survey executed several years ago for ASTM on 30 testers in daily use, demonstrated that 90% of the instruments checked failed a direct verification even though they passed a normal indirect verification using test blocks. There is no doubt that these testers were using up most if not all of the allowable tolerance. This will greatly increase the error calculation when used for determining the uncertainty of the machine. Performing a GR&R can reveal a lot about how well a system is reading Rockwell hardness and if users need additional verification such as direct, performed.

Almost all of the testers used today have not had their basic functions (depth measuring, force, time cycle) verified since they were manufactured. Many times the shipping process alone can develop problems with these delicate instruments. After testers are shipped to the user, ASTM does not need direct verifications of the test forces and displacement measuring system. It is not uncommon to see testers in use that are more than 40 years old. That means the forces they apply during the test and the devices they use for measuring the depth have not been checked for accuracy for 40 years.

Springs are often used for applying the test forces. Springs are known to lose their force in time and after 40 years, no one is aware of what the actual force really is since only indirect verifications using standardized test blocks have to be performed periodically in order to monitor the instruments. Clever servicemen have spent a lifetime tweaking testers to “read the blocks” using the full ASTM allowed tolerances. With the unit of measurement for one regular Rockwell point equivalent to just 2 µm (almost 0.002 mm or 0.000080 inch depth), it becomes obvious that such exact measurement needs an extremely precise measuring system and an importantly controlled process.

Failure to properly prepare and then execute a Rockwell hardness test and guarantee a frequently calibrated and maintained tester can lead to compromised test data or false readings, potentially contributing to the production as well as delivery of substandard product that could have catastrophic and detrimental effects in the integrity and the performance of the goods they are used in. When it is easy to see that errors in the machine performance can rapidly translate to poor product, a GR&R can offer a revealing and measurable assessment of how well a machine is performing.

## Understanding Testers Repeatability and Reproducibility

Most users do not have quantitative knowledge of how well their testers perform. The R&R of a tester is often determined by performing a GR&R study. By doing GR&R studies periodically, it is easy to establish and then monitor the performance of an instrument. A typical GR&R study can rapidly establish the short and long-term performance of a tester including operator influence.

A long or full method GR&R study involves 10 different test blocks and three operators, each making three tests on each block. The total of 90 tests will tell users what part of their samples tolerance is going to be used by the inaccuracy of the tester. A mini or short GR&R involves 10 different test blocks with three tests on each by a single operator for a total of 30 tests. Comparisons on Rockwell testers between the two methods show negligible differences in the final result.

## The Process

Due to the earlier mentioned variation in materials, mainly on production parts, GR&R testing should be carried out on standardized test blocks in order to reduce the material variation influence as much as possible. Test blocks, by design, are manufactured to be as uniform as possible, thus making them the best material to carry out the GR&R test. In a long or full method GR&R, each of three operators should carry out a single hardness test on each of the 10 test blocks sequentially, beginning with block #1 and through block #10 to complete the first run.

To compensate for the random non-uniformity inherent in a test block, the operator must assure that each set of indents on each block is repeated in as close proximity as possible to each other (radially or grouped). The process is repeated for three complete runs for a total of 90 tests, nine on each test block.

## The Calculation

The GR&R calculation for Rockwell testers is fundamentally the comparison of the combination of machine and operator variation with the process tolerance. If the difference is low or the process tolerance wide by comparison, then the % GR&R will also be low. On the other hand, if the variation is high or the process tolerance narrow by comparison, the % GR&R will be high. In the analysis, the ranges of readings for each operator are calculated and then the average range is produced for each. Furthermore, the average test value is also determined for each operator. This data is utilized for generating the full GR&R result.

## Tolerance

The process tolerance aspect of the calculation is comparatively simple: it is plugged in directly from the engineering specification for part's hardness (for instance, a part calling for a hardness of 42 to 48 HRC would have a total tolerance of 6 points). Note that the calculation of GR&R is only relevant in the context of process tolerance, comparing machine and operator variation to a test block tolerance, for example, is not meaningful, as it says nothing about the machine’s appropriateness for measuring real parts. Test block tolerances are for assuring the accuracy of a machine, not its repeatability.

## Variation

The calculations for variation can appear fairly vague, but all that they do is convert average range values and operator differences into an approximation for six sigma (six times the standard deviation for all the data). Six sigma is referred to as the statistical description for a machine's total variation. Assuming the machine is varying in a normal manner, six sigma states that more than 99&337; of all tests done on a given block (or set of ten blocks in the case of GR&R) will fall within this region. It is also, in a sense, the uncertainty of the machine at that hardness level, meaning that for a given reading; the actual hardness value could be up to plus or minus three sigma away.

## The Result

With well-trained operators and/or automatic machines yielding negligible operator (appraiser) variation, it is easiest to understand the final calculation for % GR&R in the context of just the machine (equipment) variation. This calculation just divides the six sigma approximation (average range for all tests multiplied by a statistical constant) by the process tolerance and multiplies by 100. A 10% or less GR&R then requires that six sigma (total machine variation) be 10% or less of the process tolerance. The graph in Figure 2 shows typical GR&R results from different types of Rockwell testers, including analog and digital deadweight as well as closed loop. As expected, the analog instruments produce the poorest GR&R performance while a closed loop tester with greatly regulated force application yields the best performance.

Some Rockwell testers are available with features that contribute to high performing GR&R and a few manufacturers guarantee that every tester produced will go through a GR&R evaluation and will not be shipped unless it completes this test with a GR&R performance better than or equal to a stated percentage. Proof of the study is usually provided in a GR&R certificate issued with each tester. Some contributing factors to high GR&R performance include:

- Closed-loop load control: load cell provides means of feedback via the load cell and indenter (friction and wear from mechanical parts are compensated for within the loop). Closed loop control offers the most accurate form of force application
- Attaching the penetrator to the load cell so that errors from friction are prevented
- Designing so that the depth measuring scale and the force measuring device are directly in line with the indenter in a single axis arrangement
- Elimination of elevating screws (source of non-recoverable deflection)
- Employing digital readout as opposed to analog and increasing resolution to 0.01

GR&R is a useful overall measure of a Rockwell tester's performance but not a replacement for the suggested regular block/indirect verification process. Daily indirect performance verification of the testing instrument is vital; the scales being used should be verified with the help of standardized test blocks or coupons. If possible, it is suggested that the system is verified with each scale change and at each shift start-up. Blocks should be selected that are in the approximate range of the material being tested and used only on the calibrated side.

Five total readings should be made in the verification process; it is necessary for the measured values to fall within the tolerance stated on the block and the block certificate. If verification fails, the machine should be removed from operation until the suitable adjustments or repairs are made. Maintenance and authorized verification of the instrument are imperative to the continued smooth operation and assurance that the system meets the precision necessities of a Rockwell test. ASTM recommends annual maintenance and verification of a Rockwell tester and more frequent verification under extreme or heavy use conditions. Verification should be carried out by an accredited verifying agency and the report should follow and reference ASTM E18 Rockwell test method.

GR&R is indeed a useful and informative tool in examining the performance level of a Rockwell hardness tester. With hardness testing utilized as a vital and informative process in materials testing, quality control and acceptance and performance of materials, users depend on the data generated in order to verify structural integrity, heat treatment, and quality of components for determining if a material has the properties required for guaranteeing that the materials utilized in the things used daily contribute to a well-engineered, safe and efficient world. Proper technique, procedure and strict adherence to standards in addition to guaranteeing a well-performing instrument will immensely contribute to the usefulness and accuracy of Rockwell testing.

This information has been sourced, reviewed and adapted from materials provided by Buehler.

For more information on this source, please visit Buehler.