# theoretically the strongest concrete

From one book on masonry I get a mix for concrete as such:
1.6 cement :: 3.2 sand :: 6 gravel
From another masonry book I get this for concrete mix:
5 cement :: 12.5 sand :: 20 aggregate or gravel
For mortar I get this as a mix ratio:
1 portland cement :: 1 lime :: 4 to 6 parts sand
What I am wondering is like a checkerboard or chess board where the white squares are each one particle of portlandcement and the black squares are each one grain of sand. And that the strongest and ideal mix to make both Concrete and to make Mortar is simple this ratio:
1 portland cement to 1 of sand
So that each sand particle is connected to other sand particles by the portland cement between them. Trouble is getting a mix so that it is perfectly mixed as to result in a 3-dimensional perfect packing such as a chessboard is a perfect 2 dimensional packing.
Has anyone done experimental tests to see at what ratio of portlandcement to sand yields the strongest most durable concrete??
I would hazard to guess that if uniformly mixed that the 1 to 1 mix is the strongest.
Is there any proof to my above assertion?
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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What you are asking isn't as easy as it sounds. Concrete is not like a checkerboard, where all the squares are the same size and shape. Concrete is a combination of rock, sand, cement and water, all of varying sizes and shapes (or amorphous).
Think of it this way - in woodworking the strongest glued joint is the thinnest one. Concrete is sort of the same way. Water surrounds the cement particles to make paste, and you want the least amount of water. Paste surrounds sand particles to make mortar, and you want the least amount of paste. Mortar surrounds the rock particles to make concrete, and you want the least amount of mortar. You could theoretically calculate all this for spheres, or using computer simulations, but the National Institute of Standards and Technology uses super-computers to do it, and is just now, after about 10 years, getting to the point where they can do it. Actually, their program doesn't optimize the concrete, it just tries to predict what the specified concrete will do.
If you are familiar with the subject, you might comment on my question on "Predicting spherical packing of multiple particle sizes" from last week.
Jay Shilstone
On Mon, 15 Sep 2003 11:11:26 -0500, Archimedes Plutonium

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Mon, 15 Sep 2003 16:10:09 -0500 James Shilstone wrote:

It should be easy to research and find the answer. One easily can get a uniform grade of sand and see if a 1 :: 1 mix is stronger than say a 1:: 2 mix and various other mixes.

What is intriguing about the strongest concrete is that it is sort of like another Kepler Packing Problem where the spaces between the balls is filled with portland cement.
So, in the Kepler Packing Problem in 3-d, each sphere is surrounded by how many gaps?
Perhaps the strongest concrete is a direct result of the Kepler Packing Problem.

I proved the Kepler Packing Problem as that of "kissing points".
So if you are going to pose a different problem with variable particle sizes, this new problem is also solvable once you factor in the adjustment of "kissing points". For example, say you have 60 balls of which 1/3 are one size, and 1/3 another and the last 1/3 another, then once you factor in the new kissing points to determine the smallest packing, well, you are on the way to a generalized formula for variable packing sizes.
As for Concrete and the maximum strength, I believe it is somewhere near a
1 :: 1
mix, where 1 is portland cement and the 1 part sand is of uniform size and where the mix is so well mixed.
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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Karl Hallowell wrote:
whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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Part of the problem also revolves around the fact that the more water that is added to the concrete, the lower the strength. We do not have a binary (marbles and B-Bs), or a ternary mix (marbles, B-Bs and cement), but a quaternary (4 part) blend of marbles, B-Bs, cement and water. Studies have shown that we can reduce water by getting a good blend of large and small particles (marbles and B-Bs). If we just have 1 size particle, the water demand goes up and strength goes down. Some studies have shown that reducing the maximum aggregate size will increase strength, but we still need a variety of particle sizes to minimize water demand.
I am a concrete technologist, not a mathematician. If you know of a computer program or web model that will predict voids in a Kepler Packing system using a variety of sphere sizes, I would like to know where.
Thanks, Jay Shilstone
On Tue, 16 Sep 2003 01:29:19 -0500, Archimedes Plutonium

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James Shilstone wrote:

I consider the water as temporary.

If it turns out to be true that the best or optimum mix of concrete is a ratio of that corresponding to the Kepler Packing Problem where the density is
pi/3sqrt2 or approx 74%
Then that would entail a concrete mix of approx 1 :: 4 ratio for superior concrete.
Jay, you say you are a concrete expert. I wonder if you also have studied asphalt such as road blacktop.
Where instead of the glue as portlandcement, the glue then becomes tar.
Now, I wonder if the best or optimum asphalt blacktop is also a ratio of 1 :: 4 for 1 unit of tar to 4 units of aggregate.
It would seem to me that the Theory of glues for the optimum product that there should be no difference in ratio for the blacktop and for the concrete. And that the superior concrete should follow the same Kepler Packing theory as does the superior blacktop.
But, however, if the optimum concrete and blacktop follows the 3-d chessboard pattern and not the Kepler Packing pattern would indicate the the ratio is that of 1 :: 1 of 1 unit of portlandcement to 1 unit of aggregate, likewise blacktop of tar and aggregate.
I suppose I can conduct this Experiment but the blacktop would get awful messy.
If the Experiment turns out that Kepler Packing is the optimum blacktop and the optimum concrete mix, then the question arises as to whether this is a Coincidence or whether the Kepler Packing is the geometrical equivalent of the optimum material product of strength and endurance. I suppose physics would say that the density and the most dense object is tied to strength and durability.
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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On Wed, 17 Sep 2003 00:58:55 -0500, Archimedes Plutonium

The water isn't temporary, except that after 60 years all the cement that can combine with the water has done so. Water remains in the pores of the concrete for decades. Even in the desert it can exist inside the concrete for years. The product of the water - cement reaction is actually smaller than the individual components, so even when the water reacts, a space remains. However, the more water initially in the mixture, the lower the concrete strength at any given time.

I am not an asphalt expert, but I know a bit about it. There was some work done by Fuller and Thompson back in the 1950's that resulted in the .45 power chart, which is used in asphalt. A graph is drawn where the X-axis is the aggregate particle size taken to the .45 power (with a specific set of sieves used for determining the aggregate gradation). The Y axis is the percent of aggregate that passes each sieve. A line is drawn from the origin of the graph (0,0) up to 100% passing the largest nominal size of the aggregate (say 3/4"). The "optimal grading" for asphalt falls near this line. Some studies have shown that if the grading falls directly on the line (which can be done in the lab but not in real life during production) the aggregate can have so few voids there is insufficient room for the asphaltic binder. I don't know if Fuller and Thompson ever compared their work to Kepler's.
There is a guy named Joe Dewar in the UK who has done a lot of work with designing concrete based on aggregate voids. I won't post his email address in an open newsgroup, but you can probably find him on the web. He has a computer program called MixSim for concrete mix designs. (It competes with our seeMIX program.)
Jay Shilstone
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Wed, 17 Sep 2003 10:05:05 -0500 James Shilstone wrote:

Thanks James. I am having a difficult time of envisioning that .45 power and how it relates to the quantity of tar versus aggregate. Whether it relates at all to the amount of tar used.
Just from looking at blacktop, I suspect the ratio of mix is that of 1 :: 1. Where one part is tar and the other part is aggregate. Am I correct or wrong, James?

Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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On Wed, 17 Sep 2003 11:49:25 -0500, Archimedes Plutonium

I'm not certain, but I think that asphalt emulsions usually range about 4-6% by volume. If the numbers were much higher, the asphalt would be prohibitively expensive. I found this web address that gives some general info: http://www.streetprint.com/spec_support/300-105-0004.htm
In asphalt they actually want some permeable voids that provide a cushioning effect for the applied load.
Jay Shilstone
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James Shilstone wrote:

Hello Jay, it appears that I cannot make a nice easy cross-comparison with concrete portland cement and that of blacktop asphalt.
I suppose the binder of portland cement is 100% of the portland cement and not a small fraction of 4-6%, whereas the binder in asphalt is that tar binder which is a mere 4-6%. I was hoping that there would be a nice and easy comparison but it appears not so.
Jay, has anyone worked out the cost difference because the cost in energy of oil to heat up limestone in order to produce portland cement compared to the cost of getting tar of 4-6%. It seems that the binder of blacktop is so much cheaper because of the quantity needed.
However, there is a constant and that constant is the fact of packing aggregate as per Kepler Packing and as per a 3-d chessboard. And after packing them in a Kepler Packing is to fill the voids with either portlandcement or with asphalt-tar and then compare the strength of each.
So we still have as constants the geometric configuration of the aggregate.
And it may turn out to be the case that concrete is strongest when it has a 3-d chessboard configuration with a 1 :: 1 ratio and that blacktop is the strongest with a Kepler Packing configuration with a 1 :: 4 ratio.
P.S. perhaps the reason tar is not comparable to cement is that tar is a glue whereas cement becomes an integral part of the concrete and not a glue.
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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On Thu, 18 Sep 2003 10:42:08 -0500, Archimedes Plutonium

The binder for portland cement concrete is the HYDRATED cement (composed of cement plus water). We also call cement + water (and something called "entrained air" - which is microscopic air bubbles) "paste". The paste fraction of concrete typically takes up about 30% of the volume of the concrete. Of that 30%, about 21% of the total concrete is composed of water and entrained air.
The binder in asphalt as I said is about 4-6%, but there is also air between the particles. The voids make up about another 4-5% of the asphaltic concrete.
It is hard to compare asphaltic concrete to portland cement concrete due to the nature of the binder. The asphaltic emulsion has a greater viscosity than the portland cement paste. Also, the cement reacts with the water and rapidly (2-5 hours) forms a spiny product that holds the concrete together. With asphaltic concrete you don't have a reaction, but you do have a greater quantity of very fine particles, primarily to create surface area to hold on to the asphaltic emulsion.

One of the biggest cost differences between asphalt and concrete is that you can put traffic on the asphalt almost immediately, but the concrete needs to cure and harden, sometimes for as much as a month. I'm not certain, but I think that on a per ton or per cubic yard basis, concrete is usually cheaper than asphalt.

Remember that in asphalt not all the voids are filled. This is part of the asphalt design.
Do you mind telling me what your particular interest in this is? Is it an academic question or are you trying to address a specific application? If you are responding to an academic problem, you can probably assume away many of the variables. If you are responding to a real-life problem, I would be happy to refer you to sources that could provide you with a lot better education in asphalt than I can.
Jay Shilstone

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Thu, 18 Sep 2003 12:04:26 -0500 James Shilstone wrote:

Luckly some roadwork nearby of blacktop is going on and was able to observe that the underlayer is a pure tar and then the upper layers seems to be a mix.
Jay, I am doubtful of this 4-6% figure as binder. Because looking at asphalt in that it is black in color indicates that there had to be much more tar constituents than a mere 4-6%. Maybe the binder is a small fraction of the tar-constituents.
So I am wondering if in a cubic centimeter of blacktop that the tar elements comprise much more than 4-6%. Perhaps close to 30% with the 70% of sand and gravel.
Just the appearance of fresh blacktop suggests more than a mere 4-6%.
Perhaps the 4-6% is a tiny fraction of the tar elements.

I suppose if one heats the tar hot enough that the diffusion of the tar throughout the sand and gravel is much easier than the diffusion of portlandcement in the concrete matrix. So I am guessing that it is far easier to insure uniformity of mix with tar than with portland cement.

Well I was asking more on the lines of theory than on the lines of present day commerce. Something of theory that a cubic meter of portlandcement requires the heat from petroleum of say (I am guessing) 4 barrels of oil. Whereas the oil required to make a cubic meter of tar requires 3 barrels of oil and where the tar of a cubic meter can surface one kilometer of highway yet a cubic meter of portland cement could only pave 1/100 of a kilometer of highway.

I can accept that but I cannot seem to accept the idea that 4-6% of volume to hold together all that gravel and sand.

I find much satisfaction and enjoyment in tackling theoretical issues. I go from one problem to another and often come back to a problem. Some how I landed on concrete, blacktop and Kepler Packing in the last few days. I am building a concrete block garage and that has focused my attention.

Perhaps the Kepler Packing Problem is entwined in this concrete and blacktop in the idea that KPP is the minimum amount of binder to make a solid mass. If one can imagine a KPP with its 74% touching balls and its 26% holes or gaps in between, and if those 26% holes were filled with tar or portland cement then the entire material is one piece solid and very strong.
But ordinary concrete and ordinary blacktop are never uniform balls that touch at kissing points but at all sorts of angles and flat pieces with air pockets and no binder.
If one could take BBs and pack them Keplerian with a density of 74% and fill the remaining 26% with tar or with portlandcement then they would achieve a solid that is very strong. Such Keplerian Packed concrete or Keplerian Packed blacktop would be a minimum solid because the Kepler Packing is a maximum density.
Experiment: to make a Kepler Packing of BBs and fill the gaps with tar and do the same with a concrete of Kepler Packing. I wonder what the characteristics of the two would be. And of course neither one of these materials have ever been created. And it should be easier to create the KPP of blacktop for I do not see how to infuse a KPP with portland cement.
Anyone know if portland cement can be infused into a Kepler Packing of BBs??
A Checkerboard Packing of concrete and blacktop would be easy in that we get tiny cubes of steel and fill the gaps with tar or with portlandcement.
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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...

... Way out! You would be lucky to get one m^3 of concrete to cover more than a few cm of road!
Suppose you had 1 m^3 of tar and a piece of road 1km long, 10m wide then how thick a layer of tar could you spread on it? Answer: 1/(1000 * 10) m = 0.1 mm - I've seen thicker paint!
If you were laying a concrete foundation for a road, then aim for several tonnes / m^2.
--
Jeremy Boden

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On Sat, 20 Sep 2003 12:47:17 +0100, Jeremy Boden

////
Some people are *so* critical..... Any street that is not more than 1 meter wide could be paved for 1/100 km with a cubic meter of concrete, couldn't it? A good four inch thickness even.... :-)
But....if a cu meter of concrete weighs 2 or 3 tonnes, and you suggest several (=3?) tonnes per sq meter, you are proposing to pour a meter depth? Naaa! You meant per linear meter of road that's say 7 meters wide, giving a road bed 14 cm deep = 5 in plus....
(Remember what they say about arguing with fools?)
Brian W
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On Fri, 19 Sep 2003 02:31:03 -0500, Archimedes Plutonium

The 4-6% value I gave you was based on actual data from an asphalt extraction of asphaltic concrete. I just had the data on 1 mix, however. You might want to go to www.asphaltinstitute.org for information straight from the horse's mouth.

Personally I would think it would be the other way around, since the asphaltic emulsion has a higher viscosity than portland cement paste.

I think people have done that, but I don't know who.

It wouldn't do to infuse the Keplerian Packing with just portland cement. You need to infuse it with cement and water paste. That should be very easy to do, depending on the ratio of water to cement. More water means lower viscosities, but also lower strength of the paste.
Jay Shilstone
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James Shilstone wrote:

Hi Jay, I am having a hard time of believing this 4-6% tar binding in typical road asphalt. Hard believing because it is so black that it must have more than 6% tar. And due to the theoretical notion that the densest of packing whether they be oranges in a grocery store or baseballs or sand grains that are somewhat round that the densest packing is a Kepler Packing and so the percentage of volume of gaps to glue together those sand grains would be 26%. Now I realize that a concrete mix of portlandcement and sand that many of those gaps in the sand will have no portlandcement.
What I am saying is the the Minimum Strongest Perfect concrete of sand that is round would be of a 74% sand and 26% portland cement filling each of those gaps of the Kepler Packing.
So I am having a difficult time of envisioning highway asphalt as being only a mere 6% by volume of a tar binder. Granted that the gravel used in asphalt and concrete is not round shaped but having many flat surfaces where one flat surface rests upon another relatively flat face surface and so you would need less tar than the Kepler Packing amount of 26%. But I cannot envision that the flat surfaces would reduce the 26% down to a mere 6%.
Perhaps there is something about the term "binder" of a 6% binder that I am missing in that the tar consists of a binder plus some other tar ingredients.
P.S. I am having trouble in getting posts to the Internet in that they appear on my ISP but not on any other ISP. Suggesting that someone wrote a virus program that censors my posts and perhaps other posts.
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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Subject: censorship of Internet posts Re: blacktop ratio tied to concrete ratio?? Re: Kepler Packing on concrete mixes Re: theoretically the strongest concrete Date: Sun, 21 Sep 2003 11:27:53 -0500 From:
Reply-To: NOdtgEMAIL Organization: whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies Newsgroups: sci.materials, sci.engr, sci.math References: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13
James Shilstone wrote:

Hi Jay, I am having a hard time of believing this 4-6% tar binding in typical road asphalt. Hard believing because it is so black that it must have more than 6% tar. And due to the theoretical notion that the densest of packing whether they be oranges in a grocery store or baseballs or sand grains that are somewhat round that the densest packing is a Kepler Packing and so the percentage of volume of gaps to glue together those sand grains would be 26%. Now I realize that a concrete mix of portlandcement and sand that many of those gaps in the sand will have no portlandcement.
What I am saying is the the Minimum Strongest Perfect concrete of sand that is round would be of a 74% sand and 26% portland cement filling each of those gaps of the Kepler Packing.
So I am having a difficult time of envisioning highway asphalt as being only a mere 6% by volume of a tar binder. Granted that the gravel used in asphalt and concrete is not round shaped but having many flat surfaces where one flat surface rests upon another relatively flat face surface and so you would need less tar than the Kepler Packing amount of 26%. But I cannot envision that the flat surfaces would reduce the 26% down to a mere 6%.
Perhaps there is something about the term "binder" of a 6% binder that I am missing in that the tar consists of a binder plus some other tar ingredients.
P.S. I am having trouble in getting posts to the Internet in that they appear on my ISP but not on any other ISP. Suggesting that someone wrote a virus program that censors my posts and perhaps other posts.
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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Subject: Re: Kepler Packing on concrete mixes Re: theoretically the strongest concrete Date: Sun, 21 Sep 2003 11:43:20 -0500 From:
Reply-To: NOdtgEMAIL Organization: whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies Newsgroups: sci.materials, sci.engr, sci.math References: 1 , 2 , 3 , 4 , 5 , 6 , 7
Erez Volach wrote:

I wonder if sand grains are mostly flat faced and not roundish? The Kepler Packing deals with spheres and it is the densest packing with 26% void gaps.
Does anyone know what the usual or average sand roundness or flatness is?
I suppose one can get gravel that is mostly roundish. I know the gravel of quartzite to make a concrete mix is mostly angular with alot of flatlike surfaces. Perhaps sand is much like this quartzite in having many flat surfaces. So that when one mixes a concrete mix of sand and aggregate gravel that many of the flat surfaces touch and leaving little if any room for portland cement to glue together those flat surfaces. And that the Kepler Packing of spheres with its density of 74% and 26% gaps for a glue. That in a sand and or gravel mix of concrete or asphalt that the 26% gaps need not have a glue because they are mostly flat surfaces touching one another. So I suppose the 26% can be drastically reduced in these mixes because of "flatness"
E.V. , I am having trouble posting in that I suspect someone wrote a virus to prevent these posts of mine from appearing on all outside ISP, except for my own ISP of dtgnet. So am double posting.
Archimedes Plutonium, a snipped-for-privacy@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
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really. I hadn't realized taht AP used the same presentation as JSH, continually restarting the item as a new group. it's a good question, though, just like the realtion to Avagadro's #.
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