Duff Abrams announced his water-cement ratio theory (or as he called it, the “water ratio”) in Lewis Institute Bulletin 1, “Design of Concrete Mixtures”. While it was the water-cement ratio relationship that gained Abrams historical acclaim, another of his innovations that is used every day in the concrete industry was also introduced in that publication – the fineness modulus (FM). While we primarily use it today to describe the fineness of sand, Abrams used it as a mix design method and addressed the combined fineness modulus of both coarse and fine aggregate. This article won’t attempt to repeat all of Abrams article, but just give you the high points. If you want to use the combined FM to develop your own mix designs, I suggest you read the entire publication.
What is the fineness modulus? The FM is the cumulative percent retained on a specified series of sieves divided by 100. Abrams defined those sieves as the #100, #48, #28, #14, #8, #4, 3/8”, ¾” and 1-1/2” sieves. When using a larger maximum aggregate size, double the sieve size, i.e. 3”, 6”, 12”. Over time those sieves have changed to the #100, #50, #30, #16, #8, #4, 3/8”, ¾” and 1-1/2” sieves, but I don’t know why the switch. The original sieves had particle sizes that doubled in diameter for each sieve. The reason the sieves don’t appear to be doubling is based on how the sieves are labeled. For those that aren’t familiar with the U.S. sieve sizes, the #100 sieve has 100 openings per inch. This doesn’t mean that the particle diameter is 1/100th of an inch because the diameter of the wires used in the mesh has to be considered. The actual opening is really 0.0059 inches and not 0.01 inches. The #50 sieve has an opening of 0.0117 inches, or pretty close to double the #100 sieve. While Abrams’ sieve sizes may have truly been more accurate, the modern sieves have replaced his series. Below are examples of the FM calculations for a coarse and fine aggregate.
Aggregate Cumulative Percent Retained
|Sieve||Coarse Agg||Fine Agg|
There are several items of note here. First, the coarse aggregate includes values for the #8, #16, #39, #50 and #100 sieves, even if they are not shown on the reported grading. If the values aren’t shown, they can either be assumed as 100 (if material is retained on the #4 sieve, it will also be retained on the #100 sieve), assumed to be the value reported retained on the #200 sieve (if that is shown) or assumed to be an interpolation between the #200 sieve and the smallest reported sieve. Of course, assumptions are difficult to make if you are a computer program, so if you are comparing your results to a computer program you need to know the basis for the programs calculations.
The second factor of note is that the 1” and ½” sieves are not reported. If you are working with an ASTM C-33 #57 stone, the specifications for that stone include the 1” and ½” sieves and not the ¾” and 3/8” sieves. Often the ¾” and 3/8” sieves aren’t even reported when the testing is based on ASTM specifications. This is unfortunate. Always request the full complement of sieves whenever you are having a sieve analysis performed. If you don’t have the correct sieves, you can either interpolate the values or just not calculate the coarse aggregate FM.
The third factor of note is that I have not reported any metric equivalents, which I am usually good about doing. The reason for this is that, as far as I have been able to find, there is no standardly defined metric equivalent of the fineness modulus. Some people use the hard metric equivalent sieves – 37.5mm, 19.0mm, 9.5mm, 4.75mm, 2.36mm, 1.18mm, 0.60mm, 0.30mm and 0.15mm sieves. Others use soft conversions – 40mm, 20mm, 10mm, 5mm, 2.5mm, 1.25mm, 0.625mm, 0.312mm, 0.16mm. Some use a combination of the two. Quite a while back I posted a request for information about metric equivalents of the FM on the Aggregate Research website. The following message summarizes the best responses to that: http://www.aggregateresearch.com/forum/viewmessage.aspx?MID=5788. (For those of you that are not aware of it, the Aggregate Research website is a great one. While there isn’t much discussion in the forums anymore, they have a backlog of terrific message threads that were often contributed to by the top people in the industry. I suggest you check it out at www.aggregateresearch.com. )
Using the FM to proportion mixes. Now that we have beaten the discussion about calculating the fineness modulus to death, let’s talk about proportioning mixes like Abrams did. Table 3 in “Design of Concrete Mixtures” contains recommended combined FMs for mixes containing various cement quantities. Unfortunately Abrams defined cement quantities by loose volume. In other words, how many shovels full of cement combined with how many shovels full of combined aggregate? I imagine Abrams was a little more scientific, using fixed volume buckets, but it is still hard to relate to the weights that we use today. However, you can still look at the broad patterns in the table. For example, as the cement content goes up, the fineness modulus (and quantity of coarse aggregate) goes up. As the maximum aggregate size increases the coarse aggregate content increases. (Remember that a pea gravel mix needs more sand than a 1”, 25mm, stone mix needs.
Abrams also showed in Table 4 that, as the combined fineness modulus goes up the water demand goes down. However, he also showed that as cement content goes up, the amount of water per sack of cement goes down. This is probably comparable to reports that for a reasonable cement content, changing the cement content does not appreciably affect water demand.
Figure 3 shows that as you add coarse aggregate to a mix (up to a certain point) the strength increases. However, if you add too much coarse aggregate the strength decreases. (You will see more about this in my discussion about the Coarseness Factor chart in a few weeks.)
After you determine your desired combined FM, it is simple to calculate the proportions of aggregate necessary to reach that FM. Fortunately Abrams only used 1 coarse aggregate and 1 fine aggregate in his examples so you can use 2 simultaneous equations:
X = % stone
Y = % sand
X + Y = 1.00
Target FM = (FMx * X) + (FMy * Y)
If you resolve the equation it becomes:
Target FM = (FMx * X) + (FMy * (1.00 – X))
Target FM = (FMx * X) + (FMy * 1) – (FMy * X)
Target FM = X * (FMx – FMy) + (FMy)
(Target FM – FMy) = X * (FMx – FMy)
(Target FM – FMy) / (FMx – FMy) = X
X = (Target FM – FMy) / (FMx – FMy)
Y = 1 – X
If we use the example for a Target FM = 5.50, FMx = 7.25 and FMy = 2.7 then then X resolves to .615 (61.5%) and Y = 1 – X or .385 (38.5%).
Now that you have been dragged through your worst memories of high school algebra, be thankful we have spreadsheets and specialized computer software to do the heavy lifting for us.
If you want to know more about Abrams’ “Designing Concrete Mixtures” you can get a copy of it on the Portland Cement Association’s DVD, “Concrete Research Library”. http://members.cement.org/EBiz50/ProductCatalog/Product.aspx?ID=81 It only costs $100 and contains over 1,200 technical papers, including some of the original research by Abrams, Walker and a host of other notable people from concrete’s past. I highly recommend it.
If you made it this far I congratulate you. You are either a masochist or a real concrete geek (my kind of person). In the coming weeks we will discuss more mix design techniques based on combined grading, such as the Coarseness Factor Chart, % Mortar, 8-18, Mix Suitability Factor and others. I look forward to hearing your comments.
Until next time,