Category Archives: Engineering

Kindness and Cholera in California

California likely leads the nation in socially activist government kindness. It also leads the nation in homelessness, chronic homelessness, and homeless veterans. The US Council on Homelessnesses estimates that, on any given day, 129,972 Californians are homeless, including 6,702 family households, and 10,836 veterans; 34,332 people are listed among “the chronic homeless”. That is, Californians with a disability who have been continuously homeless for one year or cumulatively homeless for 12 months in the past three years. No other state comes close to these numbers. The vast majority of these homeless are in the richer areas of two rich California cities: Los Angeles and San Francisco (mostly Los Angeles). Along with the homeless in these cities, there’s been a rise in 3rd world diseases: cholera, typhoid, typhus, etc. I’d like to explore the relationship between the policies of these cities and the rise of homelessness and disease. And I’d like to suggest a few cures, mostly involving sanitation.

Most of the US homeless do not live in camps or on the streets. The better off US homelessness find it is a temporary situation. They survive living in hotels or homeless shelters, or they “couch-serf,” with family or friends. They tend to take part time jobs, or collect unemployment, and they eventually find a permanent residence. For the chronic homeless things are a lot grimmer, especially in California. The chronic unemployed do not get unemployment insurance, and California’s work rules tend to mean there are no part time jobs, and there is not even a viable can and bottle return system in California, so the homeless are denied even this source of income*. There is welfare and SSI, but you have to be somewhat stable to sign up and collect. The result is that California’s chronic homeless tend to live in squalor strewn tent cities, supported by food handouts.

Californians provide generous food handouts, but there is inadequate sewage, or trash collection, and limited access to clean water. Many of the chronic homeless are drug-dependent or mentally ill, and though they might benefit from religion-based missions, Los Angeles has pushed the missions to the edges of the cities, away from the homeless. The excess food and lack of trash collection tends to breed rats and disease, and as in the middle ages, the rats help spread the diseases.

Total homelessness by state, 2018; California leads the nation. The better off among these individuals do not live on the streets, but in hotels or homeless shelters. For most, this is a short term situation. The rest, about 20%, are chronically homeless. About half of these live on the streets without adequate sewage and water. Many are drug-dependent.

The first major outbreaks of the homeless camps appeared in Los Angeles in August and September of 2017. They reappeared in 2018, and by late summer, rates were roughly double 2017’s. This year, 2019, looks like it could be a real disaster. The first case of a typhoid infected police officer showed up in May. By June there were six police officers with typhoid, and that suggests record numbers are brewing among the homeless.

To see why sanitation is an important part of the cure, it’s worth noting that typhoid is a disease of unclean hands, and a relative of botulism. It is spread by people who go to the bathroom and then handle food without washing their hands first. The homeless camps do not, by and large, have hand washing stations. and forced hygiene is prohibited. Los Angeles has set up porta-potties, with no easy hand washing. The result is typhoid epidemic that’s even affecting the police (six policemen in June!).

rate od disease spread. — R-naught, reproduction number for some diseases, CDC.

Historically, the worst outbreaks of typhoid were spread by food workers. This was the case with “typhoid Mary of the early 20th century.” My guess is that some of the police who got typhoid, got it while trying to feed the needy. If so, this fellow could become another Typhoid Mary. Ideally, you’d want shelters and washing stations where the homeless are. You’d also want to pickup the dirtier among the homeless for forced washing and an occasional night in a homeless shelter. This is considered inhumane in Los Angeles, but they do things like this in New York, or they did.

Typhus is another major disease of the California homeless camps. It is related to typhoid but spread by rodents and their fleas. Infected rodents are attracted to the homeless camps by the excess food. When the rodents die, their infected fleas jump to the nearest warm body. Sometimes that’s a person, sometimes another animal. In a nastier city, like New York, the police come by and take away old food, dead animals, and dirty clothing; in Los Angeles they don’t. They believe the homeless have significant squatters rights. California’s kindness here results in typhus.

Reproduction number and generation time for some diseases.

The last of the major diseases of the homeless camps is cholera. It’s different from the others in that it is not dependent on squalor, just poor health. Cholera is an airborne disease, spread by coughing and sneezing. In California’s camps, the crazy and sick dwell close to each other and close to healthy tourists. Cholera outbreaks are a predictable result. And they can easily spread beyond the camps to your home town, and if that happens a national plague could spread really fast.

I’d discussed R-naught as a measure of contagiousness some months ago, comparing it to the reproductive number of an atom bomb design, but there is more to understanding a disease outbreak. R-naught refers merely to the number of people that each infected person will infect before getting cured or dying. An R-naught greater than one means the disease will spread, but to understand the rate of spread you also need the generation time. That’s the average time between when the host becomes infected, and when he or she infects others. The chart above shows that, for cholera, r-naught is about 10, and the latency period is short, about 9 days. Without a serious change in California’s treatment of the homeless, each cholera case in June will result in over 100 cases in July, and well over 10,000 in August. Cholera is somewhat contained in the camps, but once an outbreak leaves the camps, we could have a pandemic. Cholera is currently 80% curable by antibiotics, so a pandemic would be deadly.

Hygiene is the normal way to prevent all these outbreaks. To stop typhoid, make bathrooms available, with washing stations, and temporary shelters, ideally these should be run by the religious groups: the Salvation Army, the Catholic Church, “Loaveser and Fishes”, etc. To prevent typhus, clean the encampments on a regular basis, removing food, clothing, feces and moving squatters. For cholera, provide healthcare and temporary shelters where people will get clean water, clean food, and a bed. Allow the homeless to work at menial jobs by relaxing worker hiring and pay requirements. A high minimum wage is a killer that nearly destroyed Detroit. Allow a business to hire the homeless to sweep the street for $2/hour or for a sandwich, but make a condition that they wash their hands, and throw out the leftovers. I suspect that a lot of the problems of Puerto Rico are caused by a too-high minimum wage by the way. There will always be poor among you, says the Bible, but there doesn’t have to be typhoid among the poor, says Dr. Robert Buxbaum.

*California has a very strict can and bottle return law where — everything is supposed to be recycled– but there are very few recycling centers, and most stores refuse to take returns. This is a problem in big government states: it’s so much easier to mandate things than to achieve them.

July 30, 2019. I ran for water commissioner in Oakland county, Michigan, 2016. If there is interest, I’ll run again. One of my big issues is clean water. Oakland could use some help in this regard.

Thermal stress failure

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Take a glass, preferably a cheap glass, and set it in a bowl of ice-cold water so that the water goes only half-way up the glass. Now pour boiling hot water into the glass. In a few seconds the glass will crack from thermal stress, the force caused by heat going from the inside of the glass outside to the bowl of cold water. This sort of failure is not mentioned in any of the engineering material books that I had in college, or had available for teaching engineering materials. To the extent that it is mentioned mentioned on the internet, e.g. here at wikipedia, the metric presented is not derived and (I think) wrong. Given this, I’d like to present a Buxbaum- derived metric for thermal stress-resistance and thermal stress failure. A key aspect: using a thinner glass does not help.

Before gong on to the general case of thermal stress failure, lets consider the glass, and try to compute the magnitude of the thermal stress. The glass is being torn apart and that suggests that quite a lot of stress is being generated by a ∆T of 100°C temeprarture gradient.

To calcule the thermal stress, consider the thermal expansivity of the material, α. Glass — normal cheap glass — has a thermal expansivity α = 8.5 x10^-6 meters/meter °C (or 8.5 x10^-6 foot/foot °C). For every degree Centigrade a meter of glass is heated, it will expand 8.5×10^-6 meters, and for every degree it is cooled, it will shrink 8.5 x10^-6 meters. If you consider the circumference of the glass to be L (measured in meters), then
∆L/L = α ∆T.

where ∆L is the change in length due to heating, and ∆L/L is sometimes called the “strain.”. Now, lets call the amount of stress caused by this expansion σ, sigma, measured in psi or GPa. It is proportional to the strain, ∆L/L, and to the elasticity constant, E (also called Young’s elastic constant).

σ = E ∆L/L.

For glass, Young’s elasticity constant, E = 75 GPa. Since strain was equal to α ∆T, we find that

σ =Eα ∆T

Thus, for glass and a ∆T of 100 °C, σ =100°C x 75 GPa x 8.5 x10^-6 /°C = 0.064 GPa = 64MPa. This is about 640 atm, or 9500 psi.

As it happens, the ultimate tensile strength of ordinary glass is only about 40 MPa = σ_u. This, the maximum force per area you can put on glass before it breaks, is less than the thermal stress. You can expect a break here, and wherever σ_u < Eα∆T. I thus create a characteristic temperature difference for thermal stress failure:

The Buxbaum failure temperature, ß = σ_u/Eα

If ∆T of more than ß is applied to any material, you can expect a thermal stress failure.

The Wikipedia article referenced above provides a ratio for thermal resistance. The usits are perhaps heat load per unit area and time. How you would use this ratio I don’t quite know, it includes k, the thermal conductivity and ν, the Poisson ratio. Including the thermal conductivity here only makes sense, to me, if you think you’ll have a defined thermal load, a defined amount of heat transfer per unit area and time. I don’t think this is a normal way to look at things. As for including the Poisson ratio, this too seems misunderstanding. The assumption is that a high Poisson ratio decreases the effect of thermal stress. The thought behind this, as I understand it, is that heating one side of a curved (the inside for example) will decrease the thickness of that side, reducing the effective stress. This is a mistake, I think; heating never decreases the thickness of any part being heated, but only increases the thickness. The heated part will expand in all directions. Thus, I think my ratio is the correct one. Please find following a list of failure temperatures for various common materials.

Stress strain properties of engineering materials including thermal expansion, ultimate stress, MPa, and Youngs elastic modulus, GPa.

You will notice that most materials are a lot more resistant to thermal stress than glass is and some are quite a lot less resistant. Based on the above, we can expect that ice will fracture at a temperature difference as small as 1°C. Similarly, cast iron will crack with relatively little effort, while steel is a lot more durable (I hope that so-called cast iron skillets are really steel skillets). Pyrex is a form of glass that is more resistant to thermal breakage; that’s mainly because for pyrex, α is a lot smaller than for ordinary, cheap glass. I find it interesting that diamond is the material most resistant to thermal failure, followed by invar, a low -expansion steel, and ordinary rubber.

Robert E. Buxbaum, July 3, 2019. I should note that, for several of these materials, those with very high thermal conductivities, you’d want to use a very thick sample of materials to produce a temperature difference of 100*C.

Making The City of New Orleans profitable

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The City of New Orleans is the name of the only passenger train between Chicago and New Orleans. It’s also the name of a wonderful song by Steve Goodman, 1971. Hear it, sung by Arlo Guthrie with scenes from a modern ride.

“Riding on the City of New Orleans
Illinois Central Monday morning rail
Fifteen cars and fifteen restless riders
Three conductors and twenty-five sacks of mail
All along the southbound odyssey
The train pulls out at Kankakee
Rolls along past houses, farms and fields
Passin’ trains that have no names
Freight yards full of old black men
And the graveyards of the rusted automobiles…”

Every weekday, this train leaves Chicago at 9:00 PM and gets into New Orleans twenty hours later, at 5:00 PM. It’s a 925 mile trip at a 45 mph average: slow and money-losing, propped up by US taxes. Like much of US passenger rail, it “has the disappearing railroad blues.” It’s a train service that would embarrass the Bulgarians: One train a day?! 45 mph average speed!? It’s little wonder is that there are few riders, and that they are rail-enthusiasts: “the sons of Pullman porters, and the sons of engineers, Ride[ing] their father’s magic carpets made of steel.” The wonder, to me was that there was ever fifteen cars for these, “15 restless riders”.

A sack of mail being picked up on the fly.

I would be happy to see more trips and a faster speed, at an average speed of at least 60 mph. This would require 85 mph or higher between stops, but it would save on salaries, and it would bring in some new customers. But even if these higher speeds cost nothing extra, in net, you’d still need something more to make the trip profitable; a lot more if the goal is to add another train. Air-traffic will always be faster, and the automobile, more convenient. I find a clue to profitability in the fifteen cars of the song and in the sacks of mail.

Unless I’m mistaken, mail traffic was at least as profitable as passenger traffic, and those “twenty-five sacks of mail” were either very large, or just the number on-loaded at Kankakee. Passenger trains like ‘the city of New Orleans’ were the main mail carriers till the late 1970s, a situation that ended when union disputes made it unprofitable. Still, I suspect that mail might be profitable again if we used passenger trains only for fast mail — priority and first class — and if we had real fast mail again. We currently use trucks and freight trans for virtually all US mail, we do not have a direct distribution system. The result is that US mail is vastly slower than it had been. First class mail used to arrive in a day or two, like UPS now. But these days the post office claims 2 to 4 business days for “priority mail,” and ebay guarantees priority delivery time “within eight business days”. That’s two weeks in normal language. Surely there is room for a faster version. It costs $7.35 for a priority envelope and $12.80 for a priority package (medium box, fixed price). That’s hardly less than UPS charges.

Last day of rail post service New York to Washington, DC. .June 30, 1977.

Passenger trains could speed our slow mail a lot, if it were used for this, even with these slow speeds. The City of New Orleans makes this trip in less than a day, with connections available to major cities across the US. If priority mail went north-south in under one day, people would use it more, and that could make the whole operation profitable. Trains are far cheaper than trucks when you are dealing with large volumes; there are fewer drivers per weight, and less energy use per weight. Still there are logistical issues to making this work, and you want to move away from having many post men handling individual sacks, I think. There are logistical advantages to on-loading and off-loading much larger packages and to the use of a system of standard sizes on a moving conveyor.

How would a revised mail service work? I’d suggest using a version of intermodal logistics. Currently this route consists of 20 stops including the first and last, Chicago and New Orleans. This suggests an average distance between stops of 49 Miles. Until the mid 70s, , mail would be dropped off and picked up at every stop, with hand sorting onboard and some additional on-off done on-the-fly using sacks and hooks, see picture above. For a modern version, I would suggest the same number of passenger stops, but fewer mail pick ups and drop offs, perhaps only 1/3 as many. These would be larger weight, a ton or more, with no hand sorting. I’d suggest mail drop offs and pick ups every 155 miles or so, and only of intermodal containers or pods: ten to 40 foot lengths. These containers plus their contents would weigh between 2,500 and 25,000 pounds each. They would travel on flatcars at the rear of the passenger cars, and contain first class and priority mail only. Otherwise, what are you getting for the extra cost?

The city of New Orleans would still leave Chicago with six passenger cars, but now these would be followed by eight to ten flatcars holding six or more containers. They’d drop off one of the containers at a stop around the 150 mile mark, likely Champaign Urbana, and pick up five or so more (they’d now have ten). Champaign Urbana is a major east-west intermodal stop, by the way. I’d suggest the use of six or more heavy forklifts to speed the process. At the next mail-stop, Centralia, two containers might come off and four or more might come on. Centralia is near St. Louis, itself a major rail hub for trains going west. See map below. The next mail stop might be Memphis. Though it’s not shown as such, Memphis is a major east-west rail hub; it’s a hub for freight. A stripped down mail-stop version of passenger train mail like this seems quite do-able — to me at least. It could be quite profitable, too.

Amtrak Passenger rail map. The city of New Orleans is the dark blue line going north-south in the middle of the map.

Intermodal, flat-bed trucks would take the mail to sorting locations, and from there to distribution points. To speed things, the containers might hold pre-sorted sacks of mail. Intermodal trucks might also carry some full containers east and west e.g. from Centralia to St. Louis, and some full flatcars could be switched on and off too. Full cars could be switched at the end, in New Orleans for travel east and west, or in the middle. There is a line about “Changing cars in Memphis Tennessee.” I imagine this relates to full carloads of mail joining or leaving the train in Memphis. Some of these full intermodal containers could take priority mail east and west. One day mail to Atlanta, and Houston would be nice. California in two days. That could be a money maker.

At this point, I would like to mention “super-fast” rail. The top speeds of these TGV’s “Transports of Grande Vitess” are in the range of 160 mph (265 km/hr) but the average speeds are lower because of curves and the need to stop. The average speeds are roughly 125 mph on the major routes in Europe, but they require special rails and rail beds. My sense is that this sort of special-use improvement is not worth the cost for US rail traffic. While 60 -90 mph can be handled on the same rails that carry freight, the need for dedicated track comes with a doubling of land and maintenance costs. And what do you have when you have it? The bullet rail is still less than half as fast as air travel. At an average speed of 125 mph, the trip between Chicago and New Orleans would take seven hours. For business travelers, this is not an attractive alternative to a two hour flight, and it is not well suited for intermodal mail. The fuel costs are unlikely to be lower than air travel, and there is no easy way to put mail on or off a TGV. Mail en-route would slow the 125 mph speed further, and the use of intermodal containers would dramatically increase the drag and fuel cost. Air travel has less drag because air density is lower at high altitude.

Meanwhile, at 60 mph average speeds, train travel can be quite profitable. Energy use is 1/4 as high at 60 mph average as at 120 mph. An increase of average speed to 60 mph would barely raise the energy use compared to TGV, but it would shorten the trip by five hours. The new, 15 hour version of “The City of New Orleans” would not be competitive for business travel, but it would be attractive for tourists, and certainly for mail. Having fewer hours of conductor/ engineer time would save personnel costs, and the extra ridership should allow the price to stay as it is, $135 one-way. A tourist might easily spend $135 for this overnight trip: leaving Chicago after dinner and arriving at noon the next day. This is far nicer than arriving at 5:00 PM, “when the day is done.”

Robert Buxbaum, June 21, 2019. One summer during graduate school, I worked in the mail room of a bank, stamping envelopes and sorting them by zip code into rubber-band tied bundles. The system I propose here is a larger-scale version of that, with pre-sorted mail bags replacing the rubber bands, and intermodal containers replacing the sacks we put them in.

How to avoid wet basements

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My house is surrounded my mulch — it absorbs enough rainwater that I rarely have to water.

Generally speaking water gets to your basement from rain, and the basic way you avoid wet basements is by providing some more attractive spot for the rainwater to go to. There are two main options here: divert the water to a lake or mulch-filled spot at least 8 feet away from your home, or divert it to a well-operated street or storm drain. My personal preference is a combination of both.

At right I show a picture of my home taken on a particularly nice day in the spring. Out front is a mulch-filled garden and some grass. On the side, not shown is a driveway. Most of the rain that hits our lawn and gardens is retained in 4 inches of mulch, and waters the plants. Four inches of mulch-covered ground will hold at least four inches of rainwater. Most of the rain that hits the house is diverted to downspouts and flows down the driveway to the street. Keeping some rainwater in the mulch means you don’t have to pay so much to water the trees and shrubs. The tree at the center here is an apple tree. I like fruit trees like this, they really suck up water, and I like the apples. We also have blueberries and roses, and a decorative pear (I like pears too, but they are messy).

In my opinion, you want some slope even in the lawn area, so excess rainwater will run to the sewers and not form a yard-lake, but that’s a professional preferences; it’s not always practical and some prefer a brief (vernal ) lake. A vernal lake is one that forms only in the spring. If you’ve got one, you may want to fill it with mulch or add trees that are more water tolerant than the apple, e.g. swamp oak or red cedar. Trees remove excess water via transpiration (enhanced evaporation). Red Cedars grow “knees” allowing them to survive with their roots completely submerged.

For many homes, the trick to avoiding a flooded basement is to get the water away from your home and to the street or a retention area.

When it comes to rain that falls on your hose, one option is to send it to a vernal lake, the other option is to sent it to the street. If neither is working, and you find water in your basement, your first step is to try to figure out where your rainwater goes and how it got there. Follow the water when it’s raining or right after and see where it goes. Very often, you’ll discover that your downspouts or your driveway drain into unfortunate spots: spots that drain to your basement. To the extent possible, don’t let downspout water congregate in a porous spot near your house. One simple correction is to add extenders on the downspouts so that the water goes further away, and not right next to your wall. At left, I show a simple, cheap extender. It’s for sale in most hardware stores. Plastic or concrete downspout pans work too, and provide a good, first line of defense agains a flood basement. I use several to get water draining down my driveway and away from the house.

Sometimes, despite your best efforts, your driveway or patio slopes to your house. If this is the case, and if you are not quite ready to replace your driveway or patio, you might want to calk around your house where it meets the driveway or patio. If the slope isn’t too great, this will keep rainwater out for a while — perhaps long enough for it to dry off, or for most of the rainwater to go elsewhere. When my driveway was put in, I made sure that it sloped away from the house, but then the ground settled, and now it doesn’t quite. I’ve put in caulk and a dirt-dam at the edge of the house. It keeps the water out long enough that it (mostly) drains to the street or evaporates.

A drain valve. Use this to keep other people’s sewer water out of your basement.

There is one more source of wet basement water, one that hits the houses in my area once a year or so. In our area of Oakland county, Michigan, we have combined storm and sanitary sewers. Every so often, after a big rain, other people’s rainwater and sanitary sewage will come up through the basement drains. This is really a 3rd world sewer system, but we have it this way because when it was put in, in the 1900s, it was first world. One option if you have this is to put in a one-way drain valve. There are various options, and I suggest a relatively cheap one. The one shown at right costs about $15 at Ace hardware. It will keep out enough water, long enough to protect the important things in your home. The other option, cheaper and far more hill-billy, is to stuff rags over your basement drains, and put a brick over the rags. I’ll let you guess what I have in my basement.

Robert Buxbaum, June 13, 2019

How tall could you make a skyscraper?

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Built in 1931, the highest usable floor space of the Empire State building is 1250 feet (381m) above the ground. In 1973, that record was beaten by the World Trade Center building 1, 1,368 feet (417 m, building 2 was eight feet shorter). The Willis Tower followed 1974, and by 2004, the tallest building was the Taipei Tower, 1471 feet. Building heights had grown by 221 feet since 1931, and then the Burj Khalifa in Dubai, 2,426 ft ( 739.44m):. This is over 1000 feet taller than the new freedom tower, and nearly as much taller than the previous record holder. With the Saudi’s beginning work on a building even taller, it’s worthwhile asking how tall you could go, if your only limitations were ego and materials’ strength.

Burj Khalifa, the world’s tallest building, Concrete + glass structure. Dubai tourism image.

Having written about how long you could make a (steel) suspension bridge, the maximum height of a skyscraper seems like a logical next step. At first glance this would seem like a ridiculously easy calculation based on the math used to calculate the maximum length of a suspension bridge. As with the bridge, we’d make the structure from the strongest normal material: T1, low carbon, vanadium steel, and we’d determine the height by balancing this material’s yield strength, 100,000 psi (pounds per square inch), against its density, .2833 pounds per cubic inch.

If you balance these numbers, you calculate a height: 353,000 inches, 5.57 miles, but this is the maximum only for a certain structure, a wide flag-pole of T1 steel in the absent of wind. A more realistic height assumes a building where half the volume is empty space, used for living and otherwise, where 40% of the interior space contains vertical columns of T1 steel, and where there’s a significant amount of dead-weight from floors, windows, people, furniture, etc. Assume the dead weight is the equivalent of filling 10% of the volume with T1 steel that provides no structural support. The resulting building has an average density = (1/2 x 0.2833 pound/in³), and the average strength= (0.4 x 100,000 pound/in²). Dividing these numbers we get a maximum height, but only for a cylindrical building with no safety margin, and no allowance for wind.

H’_max-cylinder = 0.4 x 100,000 pound/in²/ (.5 x 0.2833 pound/in³) = 282,400 inches = 23,532 ft = 4.46 miles.

This is more than ten times the Burj Khalifa, but it likely underestimates the maximum for a steel building, or even a concrete building because a cylinder is not the optimum shape for maximum height. If the towers were constructed conical or pyramidal, the height could be much greater: three times greater because the volume of a cone and thus its weight is 1/3 that of a cylinder for the same base and height. Using the same materials and assumptions,

The tallest building of Europe is the Shard; it’s a cone. The Eiffel tower, built in the 1800s, is taller.

H’_max-cone = 3 H’_max-cylinder = 13.37 miles.

A cone is a better shape for a very tall tower, and it is the shape chosen for “the shard”, the second tallest building in Europe, but it’s not the ideal shape. The ideal, as we’ll see, is something like the Eiffel tower.

Before speaking about this shape, I’d like to speak about building materials. At the heights we’re discussing, it becomes fairly ridiculous to talk about a steel and glass building. Tall steel buildings have serious vibration problems. Even at heights far before they are destroyed by wind and vibration , the people at the top will begin to feel quite sea-sick. Because of this, the tallest buildings have been constructed out of concrete and glass. Concrete is not practical for bridges since concrete is poor in tension, but concrete can be quite strong in compression, as I discussed here. And concrete is fire resistant, sound-deadening, and vibration dampening. It is also far cheaper than steel when you consider the ease of construction. The Trump Tower in New York and Chicago was the first major building here to be made this way. It, and it’s brother building in Chicago were considered aesthetic marvels until Trump became president. Since then, everything he’s done is ridiculed. Like the Trump tower, the Burj Khalifa is concrete and glass, and I’ll assume this construction from here on.

let’s choose to build out of high-silica, low aggregate, UHPC-3, the strongest concrete in normal construction use. It has a compressive strength of 135 MPa (about 19,500 psi). and a density of 2400 kg/m³ or about 0.0866 lb/in³. Its cost is around $600/m³ today (2019); this is about 4 times the cost of normal highway concrete, but it provides about 8 times the compressive strength. As with the steel building above, I will assume that, at every floor, half of the volume is living space; that 40% is support structure, UHPC-3, and that the other 10% is other dead weight, plumbing, glass, stairs, furniture, and people. Calculating in SI units,

H’_{max-cylinder-concrete} = .4 x 135,000,000 Pa/(.5 x 2400 kg/m³ x 9.8 m/s²) = 4591 m = 2.85 miles.

The factor 9.8 m/s² is necessary when using SI units to account for the acceleration of gravity; it converts convert kg-weights to Newtons. Pascals, by the way, are Newtons divided by square meters, as in this joke. We get the same answer with less difficulty using inches.

H’_{max-cylinder-concrete} = .4 x 19,500 psi/(.5 x.0866 lb/in³) = 180,138″ = 15,012 ft = 2.84 miles

These maximum heights are not as great as for a steel construction, but there are a few advantages; the price per square foot is generally less. Also, you have fewer problems with noise, sway, and fire: all very important for a large building. The maximum height for a conical concrete building is three times that of a cylindrical building of the same design:

H’_{max–cone-concrete} = 3 x H’_{max-cylinder-concrete} = 3 x 2.84 miles = 8.53 miles.

Mount Everest, picture from the Encyclopedia Britannica, a stone cone, 5.5 miles high.

That this is a reasonable number can be seen from the height of Mount Everest. Everest is rough cone , 5.498 miles high. This is not much less than what we calculate above. To reach this height with a building that withstands winds, you have to make the base quite wide, as with Everest. In the absence of wind the base of the cone could be much narrower, but the maximum height would be the same, 8.53 miles, but a cone is not the optimal shape for a very tall building.

I will now calculate the optimal shape for a tall building in the absence of wind. I will start at the top, but I will aim for high rent space. I thus choose to make the top section 31 feet on a side, 1,000 ft², or 100 m². As before, I’ll make 50% of this area living space. Thus, each apartment provides 500 ft² of living space. My reason for choosing this size is the sense that this is the smallest apartment you could sell for a high premium price. Assuming no wind, I can make this part of the building a rectangular cylinder, 2.84 miles tall, but this is just the upper tower. Below this, the building must widen at every floor to withstand the weight of the tower and the floors above. The necessary area increases for every increase in height as follows:

dA/dΗ = 1/σ dW/dH.

Here, A is the cross-sectional area of the building (square inches), H is height (inches), σ is the strength of the building material per area of building (0.4 x 19,500 as above), and dW/dH is the weight of building per inch of height. dW/dH equals A x (.5 x.0866 lb/in³), and

dA/dΗ = 1/ ( .4 x 19,500 psi) x A x (.5 x.0866 lb/in³).

dA/A = 5.55 x 10^-6 dH,

∫dA/A = ∫5.55 x 10^-6 dH,

ln (A_base/A_top) = 5.55 x 10^-6 ∆H,

Here, (A_base/A_top) = A_{base sq feet} /1000, and ∆H is the height of the curvy part of the tower, the part between the ground and the 2.84 mile-tall, rectangular tower at the top.

Since there is no real limit to how big the base can be, there is hardly a limit to how tall the tower can be. Still, aesthetics place a limit, even in the absence of wind. It can be shown from the last equation above that stability requires that the area of the curved part of the tower has to double for every 1.98 miles of height: 1.98 miles = ln(2) /5.55 x 10^-6 inches, but the rate of area expansion also keeps getting bigger as the tower gets heavier. I’m going to speculate that, because of artistic ego, no builder will want a tower that slants more than 45° at the ground level (the Eiffel tower slants at 51°). For the building above, it can be shown that this occurs when:

dA/dH = 4√A_base. But since

dA/dH = A 5.55 x 10^-6 , we find that, at the base,

5.55 x 10^-6 √A_base = 4.

At the base, the length of a building side is L_base = √A_base= 4 /5.55 x 10^-6 inches = 60060 ft = 11.4 miles. Artistic ego thus limits the area of the building to slightly over 11 miles wide of 129.4 square miles. This is about the area of Detroit. From the above, we calculate the additional height of the tower as

∆H = ln (A_base/A_top)/ 5.55 x 10^-6 inches = 15.1/ 5.55 x 10^-6 inches = 2,720,400 inches = 226,700 feet = 42.94 miles.

H_max-concrete = 2.84 miles + ∆H = 45.78 miles. This is eight times the height of Everest, and while air pressure is pretty low at this altitude, it’s not so low that wind could be ignored. One of these days, I plan to show how you redo this calculation without the need for calculus, but with the inclusion of wind. I did the former here, for a bridge, and treated wind here. Anyone wishing to do this calculation for a basic maximum wind speed (100 mph?) will get a mention here.

From the above, it’s clear that our present buildings are nowhere near the maximum achievable, even for construction with normal materials. We should be able to make buildings several times the height of Everest. Such Buildings are worthy of Nimrod (Gen 10:10, etc.) for several reasons. Not only because of the lack of a safety factor, but because the height far exceeds that of the highest mountain. Also, as with Nimrod’s construction, there is a likely social problem. Let’s assume that floors are 16.5 feet apart (1 rod). The first 1.98 miles of tower will have 634 floors with each being about the size of Detroit. Lets then assume the population per floor will be about 1 million; the population of Detroit was about 2 million in 1950 (it’s 0.65 million today, a result of bad government). At this density, the first 1.98 miles will have a population of 634 million, about double that of the United States, and the rest of the tower will have the same population because the tower area contracts by half every 1.98 miles, and 1/2 + 1/4 + 1/8 + 1/16 … = 1.

Nimrod examining the tower, Peter Breugel

We thus expect the tower to hold 1.28 Billion people. With a population this size, the tower will develop different cultures, and will begin to speak different languages. They may well go to war too — a real problem in a confined space. I assume there is a moral in there somewhere, like that too much unity is not good. For what it’s worth, I even doubt the sanity of having a single government for 1.28 billion, even when spread out (e.g. China).

Robert Buxbaum, June 3, 2019.

How long could you make a suspension bridge?

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The above is one of the engineering questions that puzzled me as a student engineer at Brooklyn Technical High School and at Cooper Union in New York. The Brooklyn Bridge stood as a wonder of late 1800s engineering, and it had recently been eclipsed by the Verrazano bridge, a pure suspension bridge. At the time it was the longest and heaviest in the world. How long could a bridge be made, and why did Brooklyn bridge have those catenary cables, when the Verrazano didn’t? (Sometimes I’d imagine a Chinese engineer being asked the top question, and answering “Certainly, but How Long is my cousin.”)

I found the above problem unsolvable with the basic calculus at my disposal. because it was clear that both the angle of the main cable and its tension varied significantly along the length of the cable. Eventually I solved this problem using a big dose of geometry and vectors, as I’ll show.

Vector diagram of forces on the cable at the center-left of the bridge.

Consider the above vector diagram (above) of forces on a section of the main cable near the center of the bridge. At the right, the center of the bridge, the cable is horizontal, and has a significant tension. Let’s call that T°. Away from the center of the bridge, there is a vertical cable supporting a fraction of roadway. Lets call the force on this point w. It equals the weight of this section of cable and this section of roadway. Because of this weight, the main cable bends upward to the left and carries more tension than T°. The tangent (slope) of the upward curve will equal w/T°, and the new tension will be the vector sum along the new slope. From geometry, T= √(w² +T°²).

Vector diagram of forces on the cable further from the center of the bridge.

As we continue from the center, there are more and more verticals, each supporting approximately the same weight, w. From geometry, if w weight is added at each vertical, the change in slope is always w/T° as shown. When you reach the towers, the weight of the bridge must equal 2T Sin Θ, where Θ is the angle of the bridge cable at the tower and T is the tension in the cable at the tower.

The limit to the weight of a bridge, and thus its length, is the maximum tension in the main cable, T, and the maximum angle, that at the towers. Θ. I assumed that the maximum bridge would be made of T1 bridge steel, the strongest material I could think of, with a tensile strength of 100,000 psi, and I imagined a maximum angle at the towers of 30°. Since there are two towers and sin 30° = 1/2, it becomes clear that, with this 30° angle cable, the tension at the tower must equal the total weight of the bridge. Interesting.

Now, to find the length of the bridge, note that the weight of the bridge is proportional to its length times the density and cross section of the metal. I imagined a bridge where the half of the weight was in the main cable, and the rest was in the roadway, cars and verticals. If the main cable is made of T1 “bridge steel”, the density of the cable is 0.2833 lb/in3, and the density of the bridge is twice this. If the bridge cable is at its yield strength, 100,000 psi, at the towers, it must be that each square inch of cable supports 50,000 pounds of cable and 50,000 lbs of cars, roadway and verticals. The maximum length (with no allowance for wind or a safety factor) is thus

L(max) = 100,000 psi / 2 x 0.2833 pounds/in3 = 176,500 inches = 14,700 feet = 2.79 miles.

This was more than three times the length of the Verrazano bridge, whose main span is ‎4,260 ft. I attributed the difference to safety factors, wind, price, etc. I then set out to calculate the height of the towers, and the only rational approach I could think of involved calculus. Fortunately, I could integrate for the curve now that I knew the slope changed linearly with distance from the center. That is for every length between verticals, the slope changes by the same amount, w/T°. This was to say that d²y/dx² = w/T° and the curve this described was a parabola.

Rather than solving with heavy calculus, I noticed that the slope, dy/dx increases in proportion to x, and since the slope at the end, at L/2, was that of a 30° triangle, 1/√3, it was clear to me that

dy/dx = (x/(L/2))/√3

where x is the distance from the center of the bridge, and L is the length of the bridge, 14,700 ft. dy/dx = 2x/L√3.

We find that:
H = ∫dy = ∫ 2x/L√3 dx = L/4√3 = 2122 ft,

where H is the height of the towers. Calculated this way, the towers were quite tall, higher than that of any building then standing, but not impossibly high (the Dubai tower is higher). It was fairly clear that you didn’t want a tower much higher than this, though, suggesting that you didn’t want to go any higher than a 30° angle for the main cable.

I decided that suspension bridges had some advantages over other designs in that they avoid the problem of beam “buckling.’ Further, they readjust their shape somewhat to accommodate heavy point loads. Arch and truss bridges don’t do this, quite. Since the towers were quite a lot taller than any building then in existence, I came to I decide that this length, 2.79 miles, was about as long as you could make the main span of a bridge.

I later came to discover materials with a higher strength per weight (titanium, fiber glass, aramid, carbon fiber…) and came to think you could go longer, but the calculation is the same, and any practical bridge would be shorter, if only because of the need for a safety factor. I also came to recalculate the height of the towers without calculus, and got an answer that was shorter, for some versions, a hundred feet shorter, as shown here. In terms of wind, I note that you could make the bridge so heavy that you don’t have to worry about wind except for resonance effects. Those are the effects are significant, but were not my concern at the moment.

The Brooklyn Bridge showing its main cable suspension structure and its catenaries.

Now to discuss catenaries, the diagonal wires that support many modern bridges and that, on the Brooklyn bridge, provide support at the ends of the spans only. Since the catenaries support some weight of the Brooklyn bridge, they decrease the need for very thick cables and very high towers. The benefit goes down as the catenary angle goes to the horizontal, though as the lower the angle the longer the catenary, and the lower the fraction of the force goes into lift. I suspect this is why Roebling used catenaries only near the Brooklyn bridge towers, for angles no more than about 45°. I was very proud of all this when I thought it through and explained it to a friend. It still gives me joy to explain it here.

Robert Buxbaum, May 16, 2019. I’ve wondered about adding vibration dampers to very long bridges to decrease resonance problems. It seems like a good idea. Though I have never gone so far as to do calculations along these lines, I note that several of the world’s tallest buildings were made of concrete, not steel, because concrete provides natural vibration damping.

A hydrogen permeation tester

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Over the years I’ve done a fair amount of research on hydrogen permeation in metals — this is the process of the gas dissolving in the metal and diffusing to the other side. I’ve described some of that, but never the devices that measure the permeation rate. Besides, my company, REB Research, sells permeation testing devices, though they are not listed on our site. We recently shipped one designed to test hydrogen permeation through plastics for use in light weight hydrogen tanks, for operation at temperatures from -40°C to 85°C. Shortly thereafter we got another order for a permeation tester. With all the orders, I thought I’d describe the device a bit — this is the device for low permeation materials. We have a similar, but less complex design for high permeation rate material.

Shown below is the central part of the device. It is a small volume that can be connected to a high vacuum, or disconnected by a valve. There is an accurate pressure sensor, accurate to 0.01 Torr, and so configured that you do not get H₂ + O₂ reactions (something that would severely throw off results). There is also a chamber for holding a membrane so one side is help in vacuum, in connection to the gauge, and the other is exposed to hydrogen, or other gas at pressures up to 100 psig (∆P =115 psia). I’d tested to 200 psig, but currently feel like sticking to 100 psig or less. This device gives amazingly fast readings for plastics with permeabilities as low as 0.01 Barrer.

REB Research hydrogen permeation tester cell with valve and pressure sensor.

To control the temperature in this range of interest, the core device shown in the picture is put inside an environmental chamber, set up as shown below, with he control box outside the chamber. I include a nitrogen flush device as a safety measure so that any hydrogen that leaks from the high pressure chamber will not build up to reach explosive limits within the environmental chamber. If this device is used to measure permeation of a non-flammable gas, you won’t need to flush the environmental chamber.

I suggest one set up the vacuum pump right next to the entrance of the chamber; in the case of the chamber provided, that’s on the left as shown with the hydrogen tank and a nitrogen tank to the left of the pump. I’ve decided to provide a pressure sensor for the N₂ (nitrogen) and a solenoidal shutoff valve for the H₂ (hydrogen) line. These work together as a safety feature for long experiments. Their purpose is to automatically turn off the hydrogen if the nitrogen runs out. The nitrogen flush part of this process is a small gauge copper line that goes from the sensor into the environmental chamber with a small, N₂ flow bleed valve at the end. I suggest setting the N₂ pressure to 25-35 psig. This should give a good inert flow into the environmental chamber. You’ll want a nitrogen flush, even for short experiments, and most experiments will be short. You may not need an automatic N₂ sensor, but you’ll be able to do this visually.

Basic setup for REB permeation tester and environmental chamber

I shipped the permeation cell comes with some test, rubbery plastic. I’d recommend the customer leave it in for now, so he/she can use it for some basic testing. For actual experiments, you replace mutest plastic with the sample you want to check. Connect the permeation cell as shown above, using VCR gaskets (included), and connect the far end to the multi-temperature vacuum hose, provided. Do this outside of the chamber first, as a preliminary test to see if everything is working.

For a first test live the connections to the high pressure top section unconnected. The pressure then will be 1 atm, and the chamber will be full of air. eave the top, Connect the power to the vacuum pressure gauge reader and connect the gauge reader to the gauge head. Open the valve and turn on the pump. If there are no leaks the pressure should fall precipitously, and you should see little to no vapor coming out the out port on the vacuum pump. If there is vapor, you’ve got a leak, and you should find it; perhaps you didn’t tighten a VCR connection, or you didn’t do a good job with the vacuum hose. When things are going well, you should see the pressure drop to the single-digit, milliTorr range. If you close the valve, you’ll see the pressure rise in the gauge. This is mostly water and air degassing from the plastic sample. After 30 minutes, the rate of degassing should slow and you should be able to measure the rate of gas permeation in the polymer. With my test plastic, it took a minute or so for the pressure to rise by 10 milliTorr after I closed the valve.

If you like, you can now repeat this preliminary experiment with hydrogen connect the hydrogen line to one of the two ports on the top of the permeation cell and connect the other port to the rest of the copper tubing. Attach the H₂ bleed restrictor (provided) at the end of this tubing. Now turn on the H₂ pressure to some reasonable value — 45 psig, say. With 45 psi (3 barg upstream) you will have a ∆P of 60 psia or 4 atm across the membrane; vacuum equals -15 psig. Repeat the experiment above; pump everything down, close the valve and note that the pressure rises faster. The restrictor allows you to maintain a H₂ pressure with a small, cleansing flow of gas through the cell.

If you like to do these experiments with a computer record, this might be a good time to connect your computer to the vacuum reader/ controller, and to the thermocouple, and to the N₂ pressure sensor.

Here’s how I calculate the permeability of the test polymer from the time it takes for a pressure rise assuming air as the permeating gas. The volume of the vacuumed out area after the valve is 32 cc; there is an open area in the cell of 13.0 cm²and, as it happens, the thickness of the test plastic is 2 mm. To calculate the permeation rate, measure the time to rise 10 millitorr. Next calculate the millitorr per hour: that’s 360 divided by the time to rise ten milliTorr. To calculate ncc/day, multiply the millitorr/hour by 24 and by the volume of the chamber, 32 cc, and divide by 760,000, the number of milliTorr in an atmosphere. I found that, for air permeation at ∆P = one atm, I was getting 1 minute per milliTorr, which translates to about 0.5 ncc/day of permeation through my test polymer sheet. To find the specific permeability in cc.mm/m².day.atm, I multiply this last number by the thickness of the plastic (2 mm in this case), divide by the area, 0.0013 m², and divide by ∆P, 1 atm, for this first test. Calculated this way, I got an air permeance of 771 cc.mm/m².day.atm.

The complete setup for permeation testing.

Now repeat the experiment with hydrogen and your own plastic. Disconnect the cell from both the vacuum line and from the hydrogen in line. Open the cell; take out my test plastic and replace it with your own sample, 1.87” diameter, or so. Replace the gasket, or reuse it. Center the top on the bottom and retighten the bolts. I used 25 Nt-m of torque, but part of that was using a very soft rubbery plastic. You might want to use a little more — perhaps 40-50 Nt-m. Seal everything up. Check that it is leak tight, and you are good to go.

The experimental method is the same as before and the only signficant change when working with hydrogen, besides the need for a nitrogen flush, is that you should multiply the time to reach 10 milliTorr by the square-root of seven, 2.646. Alternatively, you can multiply the calculated permeability by 0.378. The pressure sensor provided measures heat transfer and hydrogen is a better heat transfer material than nitrogen by a factor of √7. The vacuum gauge is thus more sensitive to H₂ than to N₂. When the gauge says that a pressure change of 10 milliTorr has occurred, in actuality, it’s only 3.78 milliTorr. The pressure gauge reads 3.78 milliTorr oh hydrogen as 10 milliTorr.

You can speed experiments by a factor of ten, by testing the time to rise 1 millitorr instead of ten. At these low pressures, the gauge I provided reads in hundredths of a milliTorr. Alternately, for higher permeation plastics (or metals) you want to test the time to rise 100 milliTorr or more, otherwise the experiment is over too fast. Even at a ten millTorr change, this device gives good accuracy in under 1 hour with even the most permeation-resistant polymers.

Dr. Robert E. Buxbaum, March 27, 2019; If you’d like one of these, just ask. Here’s a link to our web site, REB Research,

Why concrete cracks and why sealing is worthwhile

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The oil tanker Palo Alto is one of several major ships made with concrete hulls.

Modern concrete is a wonderful construction material. Major buildings are constructed of it, and major dams, and even some ships. But under the wrong circumstances, concrete has a surprising tendency to crack and fail. I thought I’d explain why that happens and what you can do about it. Concrete does not have to crack easily; ancient concrete didn’t and military or ship concrete doesn’t today. A lot of the fault lies in the use of cheap concrete — concrete with lots of filler — and with the cheap way that concrete is laid. First off, the major components of modern concrete are pretty uniform: sand and rock, Portland cement powder (made from cooked limestone, mostly), water, air, and sometimes ash. The cement component is what holds it all together — cements it together as it were — but it is not the majority of even the strongest concretes. The formula of cement has changed too, but the cement is not generally the problem. It doesn’t necessarily stick well to the rock or sand component of concrete (It sticks far better to itself) but it sticks well enough that spoliation, isn’t usually a problem by itself.

What causes problem is that the strength of concrete is strongly affected (decreased) by having lots of sand, aggregate and water. The concrete used in sidewalks is as cheap as possible, with lots of sand and aggregate. Highway and wall concrete has less sand and aggregate, and is stronger. Military and ship concrete has little sand, and is quite a lot stronger. The lowest grade, used in sidewalks, is M5, a term that refers to its compressive strength: 5 Mega Pascals. Pascals are European (Standard International) units of pressure and of strength. One Pascal is one Newton per square meter (Here’ a joke about Pascal units). In US (English) units, 5 MPa is 50 atm or 750 psi.

Ratios for concrete mixes of different strength; the numbers I use are double these because these numbers don’t include water; that’s my “1”.

The ratio of dry ingredients in various concretes is shown at right. For M5, and including water, the ratio is 1 2 10 20. That is to say there is one part water, two parts cement, 10 parts sand, and 20 parts stone-aggregate (all these by weight). Added to this is 2-3% air, by volume, or nearly as much air as water. At least these are the target ratios; it sometimes happens that extra air and water are added to a concrete mix by greedy or rushed contractors. It’s sometimes done to save money, but more often because the job ran late. The more the mixer turns the more air gets added. If it turns too long there is extra air. It the job runs late, workers will have to add extra water too because the concrete starts hardening. I you see workers hosing down wet concrete as it comes from the truck, this is why. As you might expect, extra air and water decrease the strength of the product. M-10 and M-20 concrete have less sand, stone, and water as a proportion to cement. The result is 10 MPa or 20 MPa strength respectively.

A good on-site inspector is needed to keep the crew from adding too much water. Some water is needed for the polymerization (setting) of the concrete. The rest is excess, and when it evaporates, it leaves voids that are similar to the voids created by having air mix in. It is not uncommon to find 6% voids, in commercial concrete. This is to say that, after the water evaporates, the concrete contains about as much void as cement by volume. To get a sense of how much void space is in the normal concrete outside your house, go outside to a piece of old concrete (10 years old at least) on a hot, dry day, and pour out a cup of water. You will hear a hiss as the water absorbs, and you will see bubbles come out as the water goes in. It used to be common for cities to send inspectors to measuring the void content of the wet (and dry) concrete by a technique called “pycnometry” (that’s Greek for density measurement). I’ve not seen a local city do this in years, but don’t know why. An industrial pycnometer is shown below.

Pyncnometer used for concrete. I don't see these in use much any more.

Pycnometer used for concrete. I don’t see these in use much any more.

One of the main reason that concrete fails has to do with differential expansion, thermal stress, a concept I dealt with some years ago when figuring out how cold it had to be to freeze the balls off of a brass monkey. As an example of the temperature change to destroy M5, consider that the thermal expansion of cement is roughly 1 x 10^-5/ °F or 1.8 x10^-5/°C. This is to say that a 1 meter slab of cement that is heated or cooled by 100°F will expand or shrink by 10^-3 m respectively; 100 x 1×10^-5 = 10^-3. This is a fairly large thermal expansion coefficient, as these things go. It would not cause stress-failure except that sand and rock have a smaller thermal expansion coefficients, about 0.6×10^-5 — barely more than half the value for cement. Consider now what happens to concrete that s poured in the summer when it is 80°F out, and where the concrete heats up 100°F on setting (cement setting releases heat). Now lets come back in winter when it’s 0°F. This is a total of 100°F of temperature change. The differential expansion is 0.4 x 10^-5/°F x 100°F = 4 x10^-4 meter/meter = 4 x10^-4 inch/inch.

The force created by this differential expansion is the elastic modulus of the cement times the relative change in expansion. The elastic modulus for typical cement is 20 GPa or, in English units, 3 million psi. This is to say that, if you had a column of cement (not concrete), one psi of force would compress it by 1/3,000,000. The differential expansion we calculated, cement vs sand and stone is 4×10^-4 ; this much expansion times the elastic modulus, 3,000,000 = 1200 psi. Now look at the strength of the M-5 cement; it’s only 750 psi. When M-5 concrete is exposed to these conditions it will not survive. M-10 will fail on its own, from the temperature change, without any help needed from heavy traffic. You’d really like to see cities check the concrete, but I’ve seen little evidence that they do.

Water makes things worse, and not only because it creates voids when it evaporates. Water also messes up the polymerization reaction of the cement. Basic, fast setting cement is mostly Ca₃SiO₅

2Ca₃SiO₅ + 6 H₂O –> 3Ca0SiO₂•H₂O +3Ca(OH)₂•H₂O.

The former of these, 3Ca0SiO₂•H₂O, forms something of a polymer. Monomer units of SiO₄ are linked directly or by partially hydrated CaO linkages. Add too much water and the polymeric linkages are weakened or do not form at all. Over time the Ca(OH)₂ can drain away or react with CO₂ in the air to form chalk.

concrete strength versus-curing time. Slow curing of damp concrete helps; fast dry hurts. Carbonate formation adds little or no strength. Jehan Elsamni 2011.

Portland limestone cement strength versus curing time. Slow curing and damp helps; fast dry hurts. Carbonate formation adds little or no strength. Jehan Elsamni 2011.

Ca(OH)₂ + CO₂ → CaCO₃ + H₂O

Sorry to say, the chalk adds little or no strength, as the graph at right shows. Concrete made with too much water isn’t very strong at all, and it gets no stronger when dried in air. Hardening goes on for some weeks after pouring, and this is the reason you don’t drive on 1 too 2 day old concrete. Driving on weak concrete can cause cracks that would not form if you waited.

You might think to make better concrete by pouring concrete in the cold, but pouring in the cold makes things worse. Cold poured cement will expand the summer and the cement will detach from the sand and stone. Ideally, pouring should be in spring or fall, when the temperature is moderate, 40-60°F. Any crack that develops grows by a mechanism called Rayleigh crack growth, described here. Basically, once a crack starts, it concentrates the fracture forces, and any wiggling of the concrete makes the crack grow faster.

Based on the above, I’ve come to suspect that putting on a surface coat can (could) help strengthen old concrete, even long after it’s hardened. Mostly this would happen by filling in voids and cracks, but also by extending the polymer chains. I imagine it would be especially helpful to apply the surface coat somewhat watery on a dry day in the summer. In that case, I imagine that Ca₃SiO₅ and Ca(OH)₂ from the surface coat will penetrate and fill the pores of the concrete below — the sales pores that hiss when you pour water on them. I imagine this would fill cracks and voids, and extend existing CaOSiO₂•H₂O chains. The coat should add strength, and should be attractive as well. At least that was my thought.

I should note that, while Portland cement is mostly Ca₃SiO₅, there is also a fair amount (25%) of Ca₂SiO₄. This component reacts with water to form the same calcium-silicate polymer as above, but does so at a slower rate using less water per gram. My hope was that this component would be the main one to diffuse into deep pores of the concrete, reacting there to strengthen the concrete long after surface drying had occurred.

Trump tower: 664′, concrete and glass. What grade of concrete would you use?

As it happened, I had a chance to test my ideas this summer and also about 3 years ago. The city inspector came by to say the concrete flags outside my house were rough, and thus needed replacing, and that I was to pay or do it myself. Not that I understand the need for smooth concrete, quite, but that’s our fair city. I applied for a building permit to apply a surface coat, and applied it watery. I used “Quickrete” brand concrete patch, and so far it’s sticking OK. Pock-holes in the old concrete have been filled in, and so far surface is smooth. We’ll have to see if my patch lasts 10-20 years like fresh cement. Otherwise, no matter how strong the concrete becomes underneath, the city will be upset, and I’ll have to fix it. I’ve noticed that there is already some crumbling at the sides of flags, something I attribute to the extra water. It’s not a problem yet, but hope this is not the beginning of something worse. If I’m wrong here, and the whole seal-coat flakes off, I’ll be stuck replacing the flags, or continuing to re-coat just to preserve my reputation. But that’s the cost of experimentation. I tried something new, and am blogging about it in the hope that you and I benefit. “Education is what you get when you don’t get what you want.” (It’s one of my wise sayings). At the worst, I’ll have spent 90 lb of patching cement to get an education. And, I’m happy to say that some of the relatively new concrete flags that the city put in are already cracked. I attribute this to: too much sand, air, water or air (they don’t look like they have much rock): Poor oversight.

Dr. Robert E. Buxbaum. March 5, 2019. As an aside, the 664 foot Trump Tower, NY is virtually the only skyscraper in the city to be built of concrete and glass. The others are mostly steel and glass. Concrete and glass is supposed to be stiffer and quieter. The engineer overseeing the project was Barbara Res, the first woman to oversee a major, NY building project. Thought question: if you built the Trump Tower, which quality of concrete would you use, and why.

Great waves, small circles, and the spread of ideas.

Why the earth is magnetic with the north pole heading south.

REB Research Blog

Random thoughts about hydrogen, engineering, business and life by Dr. Robert E. Buxbaum

Category Archives: Engineering

Kindness and Cholera in California

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Thermal stress failure

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Making The City of New Orleans profitable

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How to avoid wet basements

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How tall could you make a skyscraper?

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How long could you make a suspension bridge?

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A hydrogen permeation tester

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Why concrete cracks and why sealing is worthwhile

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Great waves, small circles, and the spread of ideas.

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Why the earth is magnetic with the north pole heading south.

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