Simple Supports

Why Mexico City Gets Such Strong Earthquakes

September 13, 2017by briancpotter1 Comment

Mexico City is known for being incredibly susceptible to earthquakes. The recent 8.1 earthquake off Mexico’s coast was able to damage the city, despite happening 600 miles away. And the 1985 Michoacan earthquake did absolutely catastrophic damage to the city, despite occurring more than 200 miles away.

Mexico Earthquake 1985 — Damage from the 1985 Mexico City earthquake

Why does Mexico City experience such strong earthquakes, when it’s not near any major fault line?

The reason is due to the ground the city is built on. Mexico City sits on an ancient lake bed, a thin layer of extremely soft soil on top of a layer of bedrock. Soft soils such as these can amplify seismic waves, which makes anything built on top of them especially vulnerable.

Screen Shot 2017-09-12 at 10.23.28 PM — Seismic waves moving from rock to soil

Earthquakes travel through the ground in the form of seismic waves. These waves travel very fast through stiff, solid materials like bedrock, and very slow through soft soils. Because the energy flux of the waves remains constant as they move between mediums, the amplitude increases as the waves slow down moving from stiff rock to soft soil.

Screen Shot 2017-09-12 at 10.19.59 PM — 1-D model of soil amplification effects

When seismic waves encounter a discontinuity in the material they’re moving through, such as hitting a layer of soil on top of a layer of rock, they’re simultaneously reflected and refracted, like light bouncing off the surface of water. The refracted waves will then be reflected back and forth between the upper and lower surface of the soil. When the waves in the top layer have a higher amplitude (due to a lower velocity) than the waves in the bottom layer, they’ll be amplified due to resonance with the arriving waves. This is known as “soil amplification”, and it significantly increases the ground motion (and thus the seismic forces) at certain seismic wave frequencies.

Screen Shot 2017-09-12 at 10.25.48 PM — Amplification at certain frequencies

This mechanism becomes especially catastrophic if the amplified frequencies in the soil are also resonating (i.e.: matching the periods) of the buildings sitting on it. In Mexico City, the amplified periods were approximately 2-2.5 seconds, which occurs with buildings around 6-15 stories tall. Buildings at this height sustained FAR more damage than both shorter and taller buildings.

This process of soil amplification is well known, and US building codes reflect it. Every building built is assigned a site class, which is essentially a measure of how soft and crummy your soil is – the softer your soil, the larger the seismic forces you have to design for. Plenty of places in the US have soils like this: parts of San Francisco, for example, or the entire city of Charleston (which has such terrible soil that anything not built on piles is slowly sinking).

But the building code provisions are a simplification of an already-simplified model. They don’t factor in the height of the soft soil layer (a critical variable in determining the amplification factor), or the fact that certain building periods will be especially vulnerable. They also don’t take into account that certain properties of soft soils, such as their tendency towards inelastic deformation), can also serve to reduce seismic forces. So it wouldn’t surprise me if these provisions change as we experience more earthquakes, and our knowledge of them increases.

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Sources:

Seismic Site Effects – Wikipedia

International Handbook of Earthquake & Engineering Seismology

Earthquake Design Practice for Buildings

ASCE 7-10 Minimum Design Loads for Buildings

Amplification of Seismic Body Waves by Low-Velocity Surface Layers

FEMA Flood Maps

September 1, 2017September 1, 2017by briancpotterLeave a Comment

Screen Shot 2017-08-31 at 11.50.51 PM — FEMA Flood Zones for downtown Houston.

In the aftermath of Harvey, a great deal of attention is being paid to flood-prone areas of the country – specifically, the FEMA maps that show a location’s annual flood risk. These maps are important – beyond showing flooding risk, they determine whether federally-funded mortgages will require flood insurance or not.

It’s also useful to know if you’re building a chemical plant that might explode if it gets flooded.

Screen Shot 2017-09-01 at 12.37.49 AM — FEMA map showing the Arkema Chemical Plant which exploded. It’s located squarely within the flood zone.

(Spoiler alert: pretty much anywhere near a river or the coast is at significant risk of flooding.)

If you’re interested in knowing whether your house is in a flood zone (and you should be), FEMA has an online tool for viewing the flood map of an address here. You can also download a layer for viewing on google earth here.

Why Buildings Shouldn’t Last 1000 Years

August 30, 2017August 30, 2017by briancpotterLeave a Comment

One of the many criticisms leveled (badum-psh) against modern buildings is that they’re shoddy, lasting only a handful of years before they need to be torn down or replaced. Modern buildings are generally designed to have a 50 year lifespan, and many are torn down before they even reach that. Compare this to ancient buildings, some of which have lasted for hundreds or thousands of years, and continue to impress even today.

But it’s actually a good thing that our buildings don’t last that long. The reason is something called Net Present Value, which is how organizations decide the value of projects they’re considering funding. Net Present value is essentially the current value of all future cash flows, minus the initial cost. The higher the number, the more valuable the project is. The formula is below:

Where C0 is initial investment, Ct is annual cash-flow, r is the discount rate, and t is the number of years. Let’s plug in some numbers and compare a typical, 50 year life building with a theoretical building designed to last 1000 years.

First, the initial investment. We’ll say our typical building will cost 1,000,000 dollars. This might cover, say, a small (7-8000 square foot) commercial building like a store or a warehouse. It’s hard to say how much a hypothetical thousand year building would cost, but it would be FAR more expensive. NONE of our modern materials – steel, concrete, wood etc, would suffice, as they’re all subject to corrosion and decay. The only buildings that have lasted for a thousand years are ones constructed of stone. But let’s be conservative and say it costs three times as much – 3,000,000 dollars.

Next, the annual cash flow. For our typical building, we’ll assume a gross rent multiplier of 8 (which might make this a fairly lucrative building), which gives an annual rent of 125,000 dollars. We’ll give the same annual rent to the 1000 year building, because nothing about a 1000 year building would make it better for tenants (if anything, it’d be worse, as increasing durability tends to decrease usability).

But, our cashflow won’t be constant. Rents increase over time, even above inflation. This suggest an average increase of around 1.3% a year over inflation. This suggests a commercial rent increase of around 2-3% a year, though that isn’t inflation adjusted. We’ll assume an annual rent increase of 1.5% over inflation (though it’s unclear how realistic this is long-term – towards the end of a thousand years an initial 1000 dollar per month rent would turn into just over a billion, which seems…unlikely).

The discount rate is the theoretical rate of return you could get if you just held on to your money. For our discount rate, well, the 10-year treasury bond rate averages around 2% or so. The corporate bond rate has been steadily marching downward over the years, but is currently at 3.5%. The average return from the S&P 500 since it’s inception is 7% (inflation adjusted). This page suggest that 4-5% is a good number to use for a discount rate. We’ll go with 4%

Plugging these values in to a net present value calculator yields $2,571,612 for the typical building, and $2,075,000 for the 1000 year building.

In other words, the typical building is much more valuable, to the tune of ~25%, than the 1000 year building. And let’s be clear: I HEAVILY stacked the deck in favor of the 1000 year building by only making it 3 times more expensive, and assuming a perpetual rent increase of 1.5%.

It doesn’t matter that our 1000 year building lasts 50 times as long. Building it to last is so much more expensive that the “cheap” building comes out ahead. And keep in mind, assuming a building would actually last 1000 years, even if we designed it to, is a bold prediction. Vanishingly few buildings, even ones designed for longevity, have done so – the majority of them are rubble.

These numbers shouldn’t be that surprising. A building is just not that great of an investment. Buildings are nicer than they were 100 or 200 years ago, but they fulfill the same fundamental purpose they did then. They haven’t fundamentally changed the way we live our lives or the structure of civilization like the car or the computer or the plane or the cell phone have. In a dynamic, growing economy we should expect there to be much better places – more useful places – to put your money than a building. You don’t see the Vatican financing too many massive churches designed to last the ages these days, despite having a few billion dollars to invest. In some ways, ancient people’s predilections for long-lasting buildings reflects the fact that their economies were stagnant, with little innovation or growth.

How A Flying Buttress Works

August 28, 2017August 28, 2017by briancpotterLeave a Comment

The flying buttress is an iconic architectural element, used most famously on Notre Dame and other large, historic churches. But what exactly does a flying buttress do?

Ancient buildings, the sort that were designed to last anyway, were almost always built out of masonry: solid blocks of clay or stone fitted together. Masonry has high strength in compression, but almost no strength in tension. To get around this limitation, builders had basically one engineering trick: the arch.

So how does an arch work? Well, the easiest way of spanning between two points is with a simple beam. A beam works by having the top portion in compression in and the bottom portion in tension. An arch cheats this by switching out internal tensile forces for external compressive forces: instead of bending like a beam does, an arch pushes on its supports. This makes it perfect for using with materials with low tensile strength. The drawback is that using them requires a strong, stable surface that the arch can push against.

Being the only play in the playbook, arches tended to get a lot of use – lots of ancient architectural elements are variations on it. A dome is essentially an arch wrapped around 360 degrees*. If a flat surface was needed, a flat arch could be used. And the “flying” part of a flying buttress is just an arch tilted on its side.

Churches tended to have ceilings made of large, ornate masonry vaults. A vault is, once again, just a slightly different flavor of arch, and these would create outward thrusts at the top of the wall. Handling these forces is a difficult task for a comparatively thin wall – something larger is needed to resist them.

One way is just a regular old buttress – a thick heavy section that is big enough to contain the outward thrust (i.e.: keep the member entirely in compression).

You could also move the buttress outside and connect them back to the support walls. This is perhaps a more elegant architectural choice, but it doesn’t improve your actual design much – you’ll need just as large of a buttress wall as you did before.

But if you slope your arch downward, the outside wall will be at a lower elevation. This lowers the overturning moment (force wanting to topple the wall over), and allows for a smaller buttress, and a lighter, more airy church design.

So, to sum up: a flying buttress was an arch that transferred lateral thrust from the roof vault. Transferring the force like this allowed the use of smaller, shorter buttresses on the outside and produced a lighter, more elegant design.

The flying buttress, like every other flavor of arch, isn’t used much anymore. With the development of steel it became economical to design structures that could carry large tensile forces, making the arch unnecessary. In some ways this is the main difference between ancient and modern (post 1850 or so) construction – the former are designed to only be in compression, while the latter can handle both tension and compression with ease. For this reason, it’s unlikely we’ll see a revival of the flying buttress any time soon.

US vs Japanese Building Codes

August 24, 2017August 24, 2017by briancpotterLeave a Comment

Japan has sort of famously strict building codes (to the extent that any building codes can be ‘famous’, anyway). During the 2011 earthquake, Japanese buildings suffered relatively little damage, a fact credited to diligent enforcement of very strict building codes.

But how strict are these codes, anyway? How would a typical Japanese building compare to a typical US one?

Let’s look at a sort of “typical” building that might be built, and see how it’ll differ under US and Japanese seismic design. For this example we’ll use a 4 story steel building, with composite deck flooring and concentric braced frames to resist lateral forces. This could be an apartment complex, or a hotel, or a classroom, or something else — it’s an extremely common method of construction. Since earthquake forces are a function of a buildings’ weight, we’ll assume it weighs 100,000 pounds (100 kips).

e738a-17nexjkujwt-h5avuqo2s5q — Our building might look something like this before the architects get a hold of it.er a caption

In both codes, the seismic design procedure is relatively similar. First, calculate the amount of force an earthquake is likely to inflict on a building. Scale this value using various factors based on construction type, building height, type of lateral system, etc. Design the lateral system to withstand this much force. Finally, perform checks to make sure the building doesn’t sway too much, and that, should it fail, it fails slowly and safely, allowing people time to get out.

For the US codes I’ll be using ASCE 7–10 and AISC 341–10. For the Japanese code I’ll be using the design method from the 1981 Building Standard Law Enforcement Order.

(If you don’t feel like following along with these calculations, go ahead and skip to the TLDR at the end).

We’ll start with the Japanese code. Here’s the Japanese formula for lateral seismic forces:

To get the seismic force Q, we multiply the building weight W by a shear coefficient Ci, which is a function of the following factors:

Z is the seismic zone factor, which comes from the map below. (I believe the numbers represent the ground acceleration, as a fraction of gravity, that would be experienced in a 500-year earthquake. Most countries use similar criteria, but the Japanese code is somewhat unusual and I couldn’t find good information on this). The lowest number on the map is 0.7, or 70% of g, which is an extremely high seismic force. We’ll use a value of 1.0.
R is the vibration characteristics factor. We’ll conservatively set this to 1.
A is a factor for evaluating the force at different levels. For this example we’ll just consider the force at the ground (called the ‘base shear’), so we’ll use 1. (Japanese and US codes use similar methods for evaluating force as a function of building height).
C is the standard shear coefficient. For a building less than 31 meters tall and without any major irregularities, we can use a shear coefficient of 0.2.

The Japanese code also has some provisions for steel buildings specifically, the most relevant being one that requires multiplying the calculated seismic force by up to 1.5. (The other provisions are about member size/strength and building shape, and seem to have similar analogues in the US code).

Factor for increasing seismic forces in steel buildings.

This gives us a base shear of 100 kips x 1.0 x 1.0 x 0.2 x 1.5 = 30 kips. Our lateral-resisting system has to be designed to withstand 30,000 pounds of force.

Now let’s consider the same building built in the US. Here’s the equation for lateral seismic forces (from ASCE 7–10):

Sds is roughly the equivalent of the Z factor in the Japanese code, the ground acceleration felt under a 500 year earthquake. We’ll use the same value as 1 (100% of g), the sort of acceleration which could be experienced along the west coast, or in places like Charleston or Missouri.
This isn’t in the above formula, but an acceleration this high will put our building into seismic design category ‘D’. This is a category for buildings which are likely to experience extremely high seismic forces. It will, among other things, force us to multiply our seismic forces by an overstrength factor of 2.0. (Courtesy of AISC 341–10).
R here is the response modification factor, which lets us reduce the design forces for systems which are highly ductile. Because we’re in seismic design category D and taller than 35 feet, our frames will have to be “Special Concentric Braced Frames”, which have an R value of 6. (The Japanese code also uses a ductility factor, but it’s values are generally lower, and it’s only used in a design procedure we’re able to skip).
I is importance factor, which is higher for buildings which meet certain importance criteria. We’ll assume our building is unimportant and use a value of 1.

This gives us a base shear for the US codes of 100 kips x 2.0 x 1.0 / (6 / 1) = 33.33 kips.

This is in fact slightly more than the force required under the Japanese code. We have being in seismic design category D and being taller than 35 feet to thank for that — those facts combined mean some extremely burdensome design requirements.

Besides lateral force, the other major component of seismic design codes are drift limits, which dictate how much the building can shake back and forth. Limiting drift prevents damage to drywall, windows, face brick, and other non-structural portions of the building that might be damaged during an earthquake even if the structure itself remains standing.

For this sort of building, the Japanese code limits story drift to 1/200th of the story height. For a story height of 12 feet, this is about 3/4 of an inch.

The US code limits deflection for this category of building to 2% of the story height. However, it also requires multiplying the calculated deflection by a “deflection factor”, which in this case is 5. This results in an allowable drift of 5/8 of an inch. Once again, the US code is slightly more strict than the Japanese code.

Formula for calculating deflection in ASCE 7–10.

So, to sum up: The US building code requires designing for a slightly higher lateral force, and slightly lower story drift. The Japanese code is less strict than the US one, though the requirements for each are very close.

A few caveats to this surprising conclusion:

We’d get slightly different results with different types of construction, and different building heights. Buildings taller than 31 meters in Japan require an additional, stricter design procedure. And buildings taller than 60 meters require special approval from the government, so it’s hard to tell how restrictive the design for them has to be (though US buildings of this height also have very restrictive provisions). But in both countries the majority of construction is going to be short buildings similar to our example.
I am not an expert on either the Japanese building code, or designing US buildings in high seismic areas. It’s very common for building codes to have a single, obscure line that completely changes the design requirements. It would have been very easy for me to miss something in the code that substantially alters the above calculations.
Building codes in the US vary from state to state. The calculations above were done with the latest versions of the code, and states that still use older versions would have less strict requirements.

However, for a building of moderate height in a seismically active area, the most recent US code provisions seem to be just as strict, if not stricter, than the Japanese ones.

The San Francisco Earthquake and Reinforced Concrete

August 3, 2017August 3, 2017by briancpotterLeave a Comment

Concrete has been used as a material for thousands of years. But reinforced concrete – concrete with steel embedded in it – is a much more recent invention. It didn’t start to be used until the mid 1800’s (it’s first use is usually traced to some reinforced garden tubs built in France), and it was years after that before people started using it effectively.

One of the first “modern” systems of reinforced concrete was the Ransome system, invented by Ernest Ransome in 1884. This system was distinguished by using twisted steel bars to improve their bond with the concrete.

Engineers are sort of skeptical of new technology by nature (and by incentive), and reinforced concrete (including the Ransome system) wasn’t any different. Up through the end of the 19th century it remained unpopular to use as a building material, being used for foundations but not much else. Most building codes didn’t even recognize it.

One of the few buildings made of reinforced concrete during this period was a museum on the Stanford campus, built in 1891 using the Ransome system. Reinforced concrete was chosen for it’s speed – it could be put up much quicker than a traditional masonry building. It was later enlarged with wings on either side, but these were built of conventional masonry construction and built to match the appearance of the original building.

In 1906, a 7.8 magnitude earthquake struck the San Francisco bay area. Thousands of buildings were damaged or destroyed by the shaking and the subsequent fires. The wings of the Stanford museum, built out of masonry, were reduced to rubble. But the original reinforced concrete structure suffered no damage at all.

Unreinforced masonry (masonry with no steel embedded in it) is perhaps the worst possible material to use if you want your building to survive an earthquake. As the building shakes back and forth, parts of it are put in tension which normally only see compression. Masonry is exceptionally weak in tension, but reinforced concrete, with it’s steel skeleton, is far more resilient. Engineers inspecting the Stanford Museum, built using the Ransome system, were impressed with how little damage it suffered.

Buildings are designed to survive worst-case loading that, in all likelihood, they’ll never see. Because of this, engineering practice tends to proceed one disaster at a time. The success of the Stanford Museum (and other reinforced concrete buildings in the bay area) helped popularize reinforced concrete as a building material. And modern steel reinforcing is specifically designed and shaped to grip the concrete, like Ransome’s bars were.

Why Roman Concrete Outlasts Ours

June 25, 2013June 25, 2013by briancpotter1 Comment

Recently, there’s been a flurry of news surrounding a new paper which examined the mineral structure of concrete samples taken from a 2000 year-old Roman breakwater. The articles range from measuredly pointing out it’s carbon efficiency, to extolling it’s near-mystical properties. The fact that these structures are still intact after millennia, while ours often decay to the point of uselessness after less than 50 years, obviously raises some questions. Namely, was Roman concrete better than ours? Why does ours fail so quickly?

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New Southern Pine Design Values

June 18, 2013June 18, 2013by briancpotterLeave a Comment

On June 1st, new design values for southern pine lumber came into effect. These results are based on full-scale testing of various lumber sizes, and supersede the interim results that went into effect last year, which only affected 2″-4″ sized lumber. The kick in the teeth is that the new values show a sizable decrease in capacity for compression, bending, and tension, with reductions ranging from 10-30%. More information can be found at the SPIB site.

Allowable design values vs. actual strength at failure.

The changes are the result of the large-scale destructive testing of thousands of pieces of southern pine lumber. Wood is a highly variable material, and so requires a large number of samples to reliably establish safe design values. This sort of testing first began in the late 1800’s, and is conducted every so often by lumber testing organizations. Testing standards have changed over the years, but currently must follow ASTM D 1990. Testing organizations must be certified by the American Lumber Standards Committee. There are currently seven organizations, which are responsible for various regions and wood varieties. Southern pine lumber is covered by the Southern Pine Inspection Bureau.

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Using The Tools You Have

June 12, 2013June 16, 2013by briancpotterLeave a Comment

Engineering is a game of optimization under constraints. Problems are never just “design a beam that can span a hundred feet“, but “design a beam that can span a hundred feet, is made of concrete, weighs less than 40 tons, and is less than five feet tall.” Or, more likely, “design a beam that can span a hundred feet as cheaply as possible”. Problems with only one requirement are easy to solve – it’s the ones with multiple, sometimes conflicting requirements that require clever solutions.

One of the most important of these requirements is “…and design it using only these tools“. This isn’t something that shows up in the design contract, but it’s a necessary reality. The tools humans have invented so far, be they wrenches or word-processors, are a limited subset of what’s theoretically possible to accomplish. And the tools any given engineer will have available are a limited subset of that. Much like MacGyver, we can’t solve engineering problems any way we’d like. We have to use whatever junk happens to be lying around.

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An Introduction to Graphic Statics

June 6, 2013June 16, 2013by briancpotterLeave a Comment

As I’m so fond of mentioning, engineering design required the use of a number of creative methods before the invention of calculators and computers. Some of the most important and widespread of these were graphic methods of analysis. Graphic methods essentially translate problems of algebra into geometric representations, allowing solutions to be reached using geometric construction (ie: drawing pictures) instead of tedious and error-prone arithmetic.

Unfortunately, these methods are slowly being forgotten. It’s extremely rare to ever see them used, outside of a select few occasionally taught in structural analysis courses. But understanding how, and more importantly why, they work unquestionably makes for a better engineer.

To remedy this, this post will lay out some of the basics of graphic statics. If there’s interest, more posts on more advanced methods will follow.

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