By Claude Goguen, P.E., LEED AP
In the last issue of Precast Inc., we talked about the concept of variability and how a good quality control program should include measures to ensure consistency in your product. Now think of how you view your suppliers. You want them to be reliable. You want to expect a level of service and quality of product, and they must meet or exceed those expectations. In other words, you want to trust your suppliers. Your customers are expecting the same of you: consistent, high-quality products made to their exact specifications.
Let’s focus on that word “consistency.” To achieve consistency, we must minimize variability, and to do that, we need to obtain consistent testing results. The first step in tightening those results is to be aware of the disparity in the first place. We will look at some ways to measure and track variability that will enable you to better detect any issues.
Averages don’t tell us everything
When you think of concrete quality, you’ll likely conjure up an image of a concrete cylinder. After all, the compression test is one of the most important in the QC process. Let’s say we break a cylinder, and the strength we get is 4,850 psi. The next cylinder breaks at 4,970 psi. Now that we have two values, we can compute the average, which in this case is 4,910 psi.
But that really doesn’t give us a lot to go on. Averages don’t tell everything we need to know. Case in point: The average depth of the Mississippi River is 11 in. That average doesn’t help the guy running a barge with a 9-ft draft. He’s more concerned about the depth where he’ll actually be traveling.
This is where we introduce frequency distributions and standard deviations. These will provide a measurement to evaluate the consistency of your test results. Standard deviations are a broad measure of quality in your production operation and will provide a clear sign if changes are necessary. There are two basic types of variation:
- Common variation – Inherent in every process describing why nothing is ever exactly the same.
- Special variation – Something happens that is not usually part of the process.
The use of standard deviations allows identification of special variations, enabling them to be controlled without overreacting to common variability. In the case of strengths, for example, by continually monitoring your 28-day strength test results and evaluating the standard deviations with each mix, you can determine whether you can reduce costs or improve strengths, or need to redesign a mix altogether. Cost savings may result from reducing the amount of cement or by eliminating product rejections caused by low strengths. Both are dependent on the degree of control you have over your mixes.
There will always be some variation in the strengths you achieve. It’s practically impossible to get the same compressive strength test after test. What you’re primarily concerned with is always meeting or surpassing the specified 28-day compressive strength, which we will refer to as ƒc. In order to do that, you will have to work with a mix designed to deliver a higher concrete strength. This higher strength is called the required average compressive strength, and we will refer to as ƒcr. You must choose an ƒcr that will ensure all your products achieve the specified strength (or greater). Obviously you cannot do this without knowing exactly how much your strengths vary.
Therefore, the standard deviation is needed to determine ƒcr. As you reduce your degree of variation (through the controls we discussed in Part 1 of this series), the less the ƒcr needs to be, and that translates to a cost savings for the company.
Formulas and definitions
Normally, you should have at least 30 consecutive tests or two groups of consecutive tests totaling 30 tests or more. With this number of tests, a result is the average of any two cylinder breaks from the same sample. Later on, we will discuss how to proceed if only 15 to 29 consecutive tests are available, or if no data are available to establish a standard deviation.
Figure 1 illustrates several fundamental statistical concepts.In this abbreviated example, six consecutive 28-day strength test results have been graphed for one particular mix. The heavy horizontal line is the average strength of all six tests. In statistical language, this is identified as X (mean or average). Figure the average by adding all the test values and dividing by the number of test results, N:
But if you have a 4,000 psi specification requirement, three of these tests did not meet the required strength. They might represent product cast on three different days or even at different times during the same day.
It is useful to have a single value represent the spread of these numbers above and below the average. If we take the difference between each individual test and the average, without regard to whether they are above or below average, then add them up and divide by the total number of differences, N, the result is the mean deviation.
The mean deviation is one measure of variability. The smaller the value of N – and the smaller the variance of each test from the average, X – the better your group of numbers. But you should also know that one or two large differences in a group of tests will have a significant impact on the mean deviation. Those large differences are disturbing, because they indicate that some individual tests are not really indicative of the whole group. Perhaps someone forgot to handle a cylinder properly and left it to dry out in the sun all day. Obviously, the strength result of that cylinder would stand out – it would be an “outlier.”
A standard deviation seeks to emphasize the impact of very low or very high test values, or outliers. To do this, we apply a simple mathematical concept: Square the differences before adding them together, then divide by (N-1), one less than the total number of samples. The square root of this value is known as the standard deviation. This measure of variability, the standard deviation, is normally written as the letter s. The mathematical formula for standard deviation is:
where s = standard deviation in psi
Σ = summation of
n = N = number of tests
Xi = each individual test in psi
X = average strength in psi
Expanding this formula, the standard deviation of the tests given in the previous example would be:
The equations in this article can be calculated with a typical hand-held business or scientific calculator, and the answers will magically appear. But for the sake of explanation, here are the calculations by hand:
Is this considered good or bad? Here are some threshold values published by AC 318-11:
Besides being an indicator of your quality control, the standard deviation can determine ƒcr , the required average strength. As required by ACI 318-11, two formulas are used when at least 30 tests are available. You must choose the larger ƒcr from the following calculations for specified concrete strengths of 5,000 psi or less:
Continue with the example above, where the standard deviation was calculated as 626 psi. If you are looking for a specified 28-day strength, ƒc , or 4,000 psi, your required strength (ƒcr) should be the larger result of the two formulas:
Thus, in this scenario you should aim at producing 4,959 psi concrete, or more practically 5,000 psi concrete, in order to ensure that all of your breaks will be at least the specified 4,000 psi.
Now let’s look at two sets of consecutive test records that total at least 30 tests (on concrete made with similar materials, QC procedures and operating conditions). The standard deviation is the statistical average of the values calculated from each test record. The following formula provides the statistical average deviation, s, of the values calculated from each test record:
When the number of tests is between 15 and 30, multiply the standard deviation calculated by the above formula with the appropriate number from the ACI 318 table in Figure 2.
Finally, if you do not have sufficient test records for calculating a standard deviation, you will need to work up a mix for a new specification. Once again, ACI has a table for that. Determine the required average strength from ACI 318, Table 184.108.40.206 (see Figure 3), select mix proportions that will achieve the required average strength, and make trial mixes and/or several field tests.
As you work diligently to reduce variation in your test results, your standard deviation result will decrease as shown in Figure 4. The standard deviation of the green curve is 100 psi. The standard deviation of the blue curve is 45 psi. This means in order to hit a specified flexural strength of 580 psi, the mix design associated with the green curve will have to be designed for 810 psi simply because of its large variability. The mix design associated with the blue curve, however, can be designed for 680 psi – 16% less – because of its lower variability.
Knowing how much variation you have in your test results can be quite an eye-opener. It can be the first step toward taking control of certain factors to tighten up your tolerances. While standard deviation calculations can be a great tool, it’s not the only way to control variability. You can simply plot results on graph paper to give you a baseline. Use whatever works for you and your resources at hand.
A good quality control system encompasses three parts: Say what you do, do what you say, and prove it. Anyone can have a shelf of manuals and procedures laying out their QC program. Those who rise above will work on the second part by creating a quality culture that always strives to enhance consistency. Being aware of your variability is Step 1. Take the time to plot your results and see how you are currently faring. This will set a baseline to measure improvement. Document these measures and record them. Your customers will appreciate your efforts.
Claude Goguen, P.E., LEED AP, is NPCA’s director of Technical Services.