The first question most people ask about concrete is, “How strong is it?” Yes, there are other important characteristics about the concrete, but strength is usually the most important one. In fact, most “hidden concrete”, such as footings, columns, beams and slabs, really is not affected by durability concerns. The primary function of the concrete is to provide support. If we don’t need to worry about maximum water/cementitious ratios for durability concerns, there are other ways to design mixtures just for strength. When concrete has a prior performance history the primary way we determine the cementitious content of concrete is through the use of statistical overdesign techniques, such as are in ACI documents.
“Statistics” – a word that strikes fear into the hearts of many a 9 th grade student. But fear not, dear readers. With today’s calculators, spreadsheets and other computer software we no longer have to struggle through calculating averages, standard deviations and coefficient of variations. However, we do need to understand what these things are and what they mean. Many of the images below are taken from ACI 214, “Guide to Evaluation of Strength Test Results of Concrete”, which is the basis for the strength overdesign calculations found in ACI 318, the Building Code.
First we need to understand some concepts:
f’c – Design strength – This is the strength the engineers specifies for the concrete in the structure. Many people think that this means that all concrete tests must achieve this strength or be considered failures, but that isn’t so. You can actually have some tests that fall below f’c and still be considered to comply with the Building Code. More on this later.
f’cr – Required average compressive strength – If we were to batch 10 loads of concrete, each with the same weights of materials from the same mix design, then later tested the hardened concrete at 28 days, we would get 10 different test results. Minor changes in materials, batching and mixing, casting of the cylinders (or cubes), curing and testing would result in variations in the concrete test results. Some would be higher and some would be lower. For this reason we don’t design concrete mixes just to achieve Design Strength. We increase the target average strength based on the spread, or standard deviation of the historical test results. This higher strength is the “required average compressive strength”.
They say a picture is worth 1,000 words, so let me show you some pictures from ACI 214.
This is a frequency distribution. Don’t be afraid. It can’t hurt you. It is just a picture. Do you see the dots lined up in columns in the picture? Each dot represents a test. The tests are grouped together into “buckets” or ranges, such as “all the tests between 22 and 24 MPa (which translates to 3190 psi to 3480 psi). If you look closely there are 10 dots in that column. That means 10 of our tests fell between 22 and 24 MPa. 9 tests fell between 20 and 22 MPa and 9 tests fell between 24 and 26 MPa, and so on. When you look at the dots, you can see that most of the tests fell in the middle, with fewer and fewer tests occurring at higher and lower strengths. When you draw a curve that “sort of” fits the outline of the dots could get something that looks sort of like a bell. Do you remember back in school when everyone did really badly on a test and almost no one passed? The teacher would say, “I’m grading this test on the bell curve.” and everyone would give a sigh of relief. This is the bell curve she was talking about.
The main point is this. When you cast a bunch of concrete batches and test them, some tests will be higher and other tests will be lower. There in the middle is the “average”. The width of that spread of tests is related to the “standard deviation”. The higher the standard deviation the wider the spread. The lower the standard deviation the tighter the spread as shown below.
So let’s say we want to design our concrete for an f’c of 20 MPa. Look at the chart below.
This is where things start to get a little tricky. We now need to discuss something called “probability”. We already know that we want to achieve a minimum strength, f’c. We also know that the average strength has to be somewhere above the minimum strength, otherwise ½ of our tests will be below the specified strength. We also know that a group of concrete tests will be spread out over a range and that the standard deviation is a measure of how wide that range will be. The bell curve describes that spread. The higher the standard deviation the wider the curve.
Look at the graph above. In this example we have a design strength, f’c, of 20 MPa. Do you see how the curves are all to the right of the average? However, each of the curves also has a small section to the left, or below, the design strength. The thing you can’t really tell is that no matter how narrow the curve and how far to the right we move it, there will always be a small part of the curve that goes below design strength. Sure the odds of us getting a test below design strength is approaching the odds of us winning the lottery, but there is still a chance we will get a test below design strength.
This is where ACI 214, 301 and 318 come in. ACI specifies that we design concrete mixes so that 2 conditions must be met:
- There is a 1 in 100 chance that a single test will fall below specified strength (this varies slightly for design strengths above 5000 psi)
- There is a 1 in 100 chance that the average of any 3 consecutive tests will fall below specified strength
What this means is that ACI expects that some of the concrete tests will fall below specified strength. ACI also allows this and says that just because a single test is slightly below specified strength, it doesn’t mean that the test is a failing test.
I have some good news and some bad news. The good news is that at this point I think you have had enough statistics so we are going to stop talking about it. The bad news is that we are going to switch to algebra.
ACI 318 provides two sets of equations to tell us how much we need to overdesign our concrete. This first set is for concrete at or below 5000 psi (or 35 MPa). The second set is for concrete designed over 5000 psi.
Again, don’t be frightened. The f’cr is simply the average strength required to produce the design strength, f’c, based on a standard deviation, “s”. If you are working in metric units the equations look like this:
For example, let’s say we have a design strength, f’c, of 3000 psi and a standard deviation, s, of 500 psi and that the standard deviation is based on 30 tests (the recommended number of tests, but more on this in our next installment). Our target strength will be the greater of the following 2 equations:
- 3000 psi + (1.34 x 500 psi) = 3000 psi + 670 psi = 3670 psi
- 3000 psi + (2.33 x 500 psi) -500 psi = 3000 psi + 1165 psi – 500 psi = 3665 psi
Therefore our target strength is 3670 psi (the larger of the two values).
Determining cementitious content
The problem with using ACI overdesign equations is that they just tell us what our target strength should be and not how much cementitious material we need to use in our mixture. For that we need a Rosetta Stone to translate between strength and cementitious content. The one I use the most is the “cementitious efficiency”.
As I stated above, the standard deviation is based on the spread of 30 test results. From those same test results we can also calculate an average. Let’s say the average strength was 3900 psi. Now we need to know the amount of cementitious material used in the mix design we previously tested. Let’s say we used 400 lbs of cement and 100 lbs of fly ash, for a total cementitious content of 500 lbs. Now we resort to algebra again (but just a simple equation):
- Cement efficiency = average strength / total quantity of cementitious material
- 3900 psi / 500 lbs of cementitious = 7.8 psi / lb
I need to present more discussion on cementitious efficiency at a later date, but for now we will just use this value. You will notice it is expressed as psi/lb, which is similar to miles/gallon for gasoline in your car. If you are working in metric units I recommend you use MPa/(100 kg of cementitious). Your numbers will wind up being similar, but not the same.
To continue our example above, if we need to produce an average strength of 3670 psi and our cementitious efficiency is 7.8 psi/lb, we use the following equation:
- Qty cementitious = Target strength / cementitious efficiency
- 3670 psi / 7.8 psi/lb = 471 lbs of cementitious material
There you have it! We need 471 lbs of combined cementitious material for this target strength at the calculated standard deviation. In summary our steps are:
- Select the design strength, f’c
- Calculate the historical standard deviation, s
- Calculate the minimum recommended strength, f’cr
- Calculate the cementitious efficiency from the historical average strength
- Divide the minimum recommended strength by the cementitious efficiency
The whole process isn’t that hard, but there are some caveats. Unfortunately this post has gotten rather long, so I think I need to save the remainder for next time. For complete information on what is specified in ACI documents, you will need to purchase copies at www.concrete.org .
Until then, I hope this will suffice to get you started. If you have any questions or comments just let me know.
Until next time,