[00:01] Flaming Gorge Dam rises from the Green River in&nbsp; northern Utah like a concrete wedge driven into&nbsp;&nbsp; the canyon, anchored against the sheer rock&nbsp; walls that flank it. It’s quintessential,&nbsp;&nbsp; in a way. It’s what we picture when we think&nbsp; about dams: a hulking, but also somehow graceful,&nbsp;&nbsp;
[00:18] wall of concrete stretching across a&nbsp; narrow rocky valley. But to dam engineers,&nbsp;&nbsp; there’s nothing quintessential about it.&nbsp; So-called arch dams are actually pretty rare.&nbsp;&nbsp; For reference, the US has about 92,000&nbsp; dams listed in the national inventory.&nbsp;&nbsp;
[00:34] I estimate that we have maybe around 50 arch&nbsp; dams - it’s less than a tenth of a percent. The only reason we think of arch dams as&nbsp; archetypal is because they’re so huge.&nbsp;&nbsp;
[00:48] I counted 11 in the US that have their own&nbsp; visitor center. There just aren’t that many&nbsp;&nbsp; works of infrastructure that double as tourist&nbsp; destinations, and the reason for it is, I think,&nbsp;&nbsp; kind of interesting. Because an arch dam isn’t&nbsp; just an engineering solution to holding back&nbsp;&nbsp;
[01:02] water, and it’s not just a solution to holding&nbsp; back a lot of water. It’s all about height,&nbsp;&nbsp; and I built a little demo to show you what I mean.&nbsp; I’m Grady, and this is Practical Engineering.
[01:23] Engineers love categories, and dams are no&nbsp; exception. You can group them in a lot of ways,&nbsp;&nbsp; but mostly, we care about how they handle&nbsp; the incredible force of water they hold back.&nbsp;&nbsp; Embankment dams do it with earth or rock, relying&nbsp; on friction between the individual particles that&nbsp;&nbsp;
[01:39] make up the structure. Gravity dams do it&nbsp; with weight. Let me show you an example. I have my tried and trusted acrylic flume with&nbsp; a small plastic dam. Once this is all set up,&nbsp;&nbsp; I can start filling up the reservoir. This&nbsp; little dam is a little narrower than the&nbsp;&nbsp;
[01:55] flume. It doesn’t touch the sides, so it&nbsp; leaks a bit. The reason for that will be&nbsp;&nbsp; clear in a moment. And hopefully you can&nbsp; see what’s about to happen. This gravity&nbsp;&nbsp; dam doesn’t have much gravity in it, so it&nbsp; doesn’t take much water at all before you&nbsp;&nbsp;
[02:08] get a failure. I’m counting failure as&nbsp; the first sign of movement, by the way.&nbsp;&nbsp; That’s when the stabilizing forces are overcome&nbsp; by the destabilizing ones. And the little dam&nbsp;&nbsp; by itself could hold until my reservoir&nbsp; was about a quarter of the way to the top.
[02:23] Gravity dams get their stability against&nbsp; sliding from… you guessed it… friction. Bet&nbsp;you thought I was going to say gravity.&nbsp; And actually, it kind of is gravity,&nbsp;&nbsp; since frictional resistance is a function of just&nbsp; two variables: the normal force (in other words,&nbsp;&nbsp;
[02:38] the weight of the structure) and a&nbsp; coefficient that depends on the two&nbsp;&nbsp; materials touching. Engineers analyze the&nbsp; stability of gravity dams in cross-section,&nbsp;&nbsp; essentially taking a small slice of the&nbsp; structure. You want every slice to be able&nbsp;&nbsp;
[02:53] to support itself. That’s why I didn’t want&nbsp; the demo touching the sides of the flume;&nbsp;&nbsp; it would add resistance that doesn’t actually&nbsp; exist in a cross-section. The destabilizing&nbsp;&nbsp; force is hydrostatic pressure from the reservoir,&nbsp; which increases with depth. And the stabilizing&nbsp;&nbsp;
[03:08] force is friction. There are some complexities&nbsp; to this that we’ll get into, but very generally,&nbsp;&nbsp; as long as you have more friction than pressure,&nbsp; you’re good; you have a stable structure. So let’s add some normal force to&nbsp; the demo and see what happens.
[03:30] You can see my little reservoir gets&nbsp; a little higher before the dam fails,&nbsp;&nbsp; And we can try&nbsp; it again with more weight.
[03:42] But the result&nbsp;gets a little more interesting… Turns out gravity dams have two major failure&nbsp; modes: sliding and overturning. Resistance to&nbsp;&nbsp;
[03:55] sliding comes from friction, which really&nbsp; doesn’t depend on how the weight of the dam&nbsp;&nbsp; is distributed. That’s not true for overturning&nbsp; failures. Let’s look back at our cross-section.&nbsp;&nbsp; For a unit width of dam, the hydrostatic pressure&nbsp; from the reservoir looks like this. Pressure&nbsp;&nbsp;
[04:11] increases with depth. And the area under this line&nbsp; is the total force pushing the dam downstream. We&nbsp;&nbsp; can simplify that distribution and treat it&nbsp; like it’s a single force, and it turns out&nbsp;&nbsp; when you do that, the force acts a third of the&nbsp; way up the total depth of water. Most dams want&nbsp;&nbsp;
[04:28] to rotate about the downstream toe, so you have&nbsp; a destabilizing force offset from the point of&nbsp;&nbsp; rotation. In other words, you have a torque, also&nbsp; called a moment. The dam has to create an opposite&nbsp;&nbsp;
[04:40] moment around that point to remain stable. Moment&nbsp; or torque is calculated as the force multiplied by&nbsp;&nbsp; its perpendicular distance from the point of&nbsp; rotation. So, the further the center of mass&nbsp;&nbsp;
[04:52] is from the downstream toe, the more stable&nbsp; the structure is, and the demo shows it too. Here’s where we left the weights the last&nbsp; time, and let’s see it happen again. The&nbsp;reservoir makes it about two-thirds of the way&nbsp; up the walls before the dam overturns. Let’s&nbsp;&nbsp;
[05:07] make a simple shift. Just move the weights further&nbsp; upstream and try again. The reservoir reaches about three-quarters&nbsp; the way up before we see a sliding failure,&nbsp;&nbsp;
[05:23] but shifting the weights did increase the&nbsp; stability. And this is why a lot of gravity&nbsp;&nbsp; dams have a fairly consistent shape, with most&nbsp; of the weight concentrated on the upstream side,&nbsp;&nbsp; and usually a sloped or stepped downstream face.
[05:36] itself in a way. Watch what happens&nbsp; when I turn my little model around.&nbsp;&nbsp;
[05:48] Now the hydrostatic pressure applies both&nbsp; a destabilizing and stabilizing force,&nbsp;&nbsp; so you get more resistance for a given depth.&nbsp; A lot of deployable temporary storm barriers&nbsp;&nbsp; and cofferdam systems take advantage of this&nbsp; kind of configuration. You can imagine if I&nbsp;&nbsp;
[06:04] extended the base even further, I could&nbsp; create a structure that was self-stable&nbsp;&nbsp; just from its geometry alone. The weight of&nbsp; the water on the footing would overcome the&nbsp;&nbsp; lateral pressure. But there’s a catch to&nbsp; this. This is fully stable now, but watch&nbsp;&nbsp;
[06:18] what happens when I give the dam just a bit of&nbsp;a tilt. This might seem kind of intuitive, but I think&nbsp; it’s important to explain what’s actually going&nbsp;&nbsp;
[06:30] on. Hydrostatic pressure from the reservoir&nbsp; doesn’t only act on the face of a dam. With&nbsp;&nbsp; smooth plastic on smooth plastic, you get a pretty&nbsp; nice seal, but as soon as even a tiny gap opens,&nbsp;&nbsp; water gets underneath. Now there’s upward&nbsp; pressure on the bottom of the dam as well. If&nbsp;&nbsp;
[06:46] you’re depending on the downward force of a dam&nbsp; from its weight for stability, it’s easy to see&nbsp;&nbsp; why an upward force is a bad thing. And it’s so&nbsp; dramatic in the example with the upstream footing&nbsp;&nbsp; specifically. In that case, the downward pressure&nbsp; of the reservoir is acting as a stabilizing force,&nbsp;&nbsp;
[07:03] but if water can get underneath that footing,&nbsp; it basically cancels out. The pressure on the&nbsp;&nbsp; bottom is the same as the pressure on the top.&nbsp; But this isn’t only an issue in that case. The ground isn’t waterproof. In&nbsp; fact, I’ve done a video all about&nbsp;&nbsp;
[07:17] the topic. Soil and rock works more&nbsp; like a sponge than a solid material,&nbsp;&nbsp; and water can flow through them. That’s how&nbsp; we get aquifers and wells and springs and&nbsp;&nbsp; such. But it’s a problem for gravity dams,&nbsp; because water can seep below the structure&nbsp;&nbsp;
[07:31] and apply pressure to the bottom, essentially&nbsp; counteracting its weight. We call it uplift. Looking back at the cross-section,&nbsp; we can estimate this. Of course,&nbsp;&nbsp; you have the triangular pressure distribution&nbsp; along the upstream face. But at this point&nbsp;&nbsp;
[07:46] you have the full hydrostatic pressure also&nbsp; pushing upward. And at the downstream toe,&nbsp;&nbsp; you have no pressure (it’s exposed to the&nbsp; atmosphere). So, now you have a pressure&nbsp;&nbsp; distribution below the dam that looks like&nbsp; this. Of course, this part can get a lot more&nbsp;&nbsp;
[07:59] complicated since most dams don’t sit flush&nbsp; with the ground, and many are equipped with&nbsp;&nbsp; drains and cutoff walls, so definitely go check&nbsp; that other video out if you want to learn more. But let me show you the issue this causes with&nbsp; some recreational math on our cross-sectional&nbsp;&nbsp;
[08:14] slice of the dam. The taller the dam, the greater&nbsp; the uplift force. That happens linearly. In other&nbsp;&nbsp; words, the force is proportional to the depth&nbsp; of the reservoir. But look at the lateral force.&nbsp;&nbsp;
[08:26] Again, remember it’s the area under this&nbsp; triangle. Maybe you remember that formula:&nbsp;&nbsp; one-half times base times height. Well, the height&nbsp; is the depth of the water. And the base is also&nbsp;&nbsp; a function of the depth. More specifically, it’s&nbsp; the unit weight of water times depth. Multiply it&nbsp;&nbsp;
[08:42] together, and you see the challenge: the force&nbsp; increases as a function of the depth squared.&nbsp;&nbsp; So for every unit of additional height you want&nbsp; out of a gravity dam, you need significantly more&nbsp;&nbsp;
[08:54] weight to resist the forces, which means&nbsp; more material and thus a lot more cost. Hopefully all this exposition is starting to&nbsp; reveal a solution to this rapid divergence&nbsp;&nbsp; of stability and loads as a reservoir&nbsp; increases in height. Dams don’t actually&nbsp;&nbsp;
[09:09] float in space like my demonstration and&nbsp; graphics show. You know, by necessity,&nbsp;&nbsp; they extend across the entire valley and usually&nbsp; key into the abutments on either side. Naturally,&nbsp;&nbsp; that connection at the sides is going to&nbsp; offer some resistance to the forces dams&nbsp;&nbsp;
[09:25] need to withstand. And if you can count on that&nbsp; resistance, you can significantly lower the mass,&nbsp;&nbsp; and thus the cost, of the structure. But, again,&nbsp; this gets complicated. Let’s go back to the demo.
[09:38] Now I’m going to replace my gravity dam&nbsp; with something much simpler. Just a sheet&nbsp;&nbsp; of aluminum flashing, and, to simulate&nbsp; that resistance provided by socketing the&nbsp;&nbsp; structure into the earth, I’ve taped it to&nbsp; the bottom and sides… with some difficulty,&nbsp;actually.
[09:55] When I fill up the reservoir with&nbsp; water, it holds just fine. There’s a little&nbsp;&nbsp; leaking past my subpar tape job, but this&nbsp; is a fully stable structure. And I think&nbsp;&nbsp; the comparison here is pretty stark. When you&nbsp; can develop resistance from the sides you can&nbsp;&nbsp;
[10:10] get away with a lot less dam. But it’s&nbsp; harder than you might think to do that. For one, the natural soil or rock at a&nbsp; dam site might not be all that strong.&nbsp;&nbsp; The banks of rivers aren’t generally known for&nbsp; their stability, so the prospect of transferring&nbsp;&nbsp;
[10:25] enormous amounts of force into them rarely makes a&nbsp; lot of engineering sense. But the other challenge&nbsp;&nbsp; is in the dam itself. Take a look back at this&nbsp; demo. See how my dam is bending behind the force&nbsp;&nbsp;
[10:37] of the water. It’s holding there, but, you know,&nbsp; we don’t actually build dams out of aluminum&nbsp;&nbsp; flashing. Resisting loads in this way basically&nbsp; treats the dam like a beam, like a sideways bridge&nbsp;&nbsp; girder. Except, unlike girder bridges that&nbsp; usually only span up to a few hundred feet,&nbsp;&nbsp;
[10:53] dams are often much longer. Even the stiffest&nbsp; modern materials, like prestressed concrete boxes,&nbsp;&nbsp; would just deflect too much under load to&nbsp; transfer all the hydrostatic pressure across&nbsp;&nbsp;
[11:05] a valley into the abutments. Plus we usually&nbsp; don’t like to rely on steel too much in dams&nbsp;&nbsp; because of issues with corrosion and longevity.&nbsp; So where a typical beam experiences both tensile&nbsp;&nbsp; and compressive stress on opposite sides,&nbsp; we really need to transfer all that load,&nbsp;&nbsp;
[11:21] creating only compressive stress in the material.&nbsp; I’m sure you see where I’m going with this. How have we been building bridges for ages&nbsp; from materials like masonry where tensile&nbsp;&nbsp; stress isn’t an option? It’s arches! The arch&nbsp; is a special shape in engineering because you&nbsp;&nbsp;
[11:37] can transfer loads by putting the material&nbsp; in compression only, allowing for simpler,&nbsp;&nbsp; cheaper, and longer-lasting materials like&nbsp; masonry and concrete. You basically co-opt&nbsp;&nbsp; the geology for support, reducing the need for&nbsp; a massive structure. For completeness’s sake,&nbsp;&nbsp;
[11:53] let me show you how it works in the demo.&nbsp; I’ve formed a little arch from my thin sheet&nbsp;&nbsp; of aluminum. Now when I fill up the reservoir,&nbsp; there’s no deflection like the previous example.&nbsp;&nbsp; And again, side by side, it’s easy to see the&nbsp; benefits here.
[12:08] than you do with an&nbsp;earthen embankment dam or a gravity structure. Of course, there are some drawbacks here. For&nbsp; one, arches create horizontal forces at the&nbsp;&nbsp;
[12:20] supports called thrusts that have to be resisted.&nbsp; Sites that use this design really require strong,&nbsp;&nbsp; competent rock in the abutments to withstand&nbsp; the enormous loads. And just like with bridges,&nbsp;&nbsp; the span matters. The wider the valley,&nbsp; the bigger the arch needs to be,&nbsp;&nbsp;
[12:35] so these dams generally only make&nbsp; sense in deep gorges and steep,&nbsp;&nbsp; narrow canyons. The engineering is a lot more&nbsp; complicated, too. You can’t use a simple 2D&nbsp;&nbsp; cross-section to demonstrate stability.&nbsp; The structural behavior is inherently&nbsp;&nbsp;
[12:50] three-dimensional, which is tougher&nbsp; to characterize, especially when you&nbsp;&nbsp; consider unusual conditions like earthquakes and&nbsp; temperature effects. And since they’re lighter,&nbsp;&nbsp; arch dams don’t resist uplift forces very&nbsp; well, making foundation drainage systems&nbsp;&nbsp;
[13:04] more critical. All this means that it’s&nbsp; really only a solution that makes economic&nbsp;&nbsp; sense in a narrow range of circumstances,&nbsp; one of the most important being height. For smaller dams, the additional complexity and&nbsp; expense of designing and building an arch aren’t&nbsp;&nbsp;
[13:21] justified by the structural efficiency. Gravity&nbsp; and embankment dams are much more adaptable to&nbsp;&nbsp; a wider range of site conditions. And&nbsp; there are other types of dams, too,&nbsp;&nbsp; that blend these ideas. Multiple-arch dams use a&nbsp; series of smaller arches supported by buttresses,&nbsp;&nbsp;
[13:36] dividing the span into more manageable components.&nbsp; Even what is perhaps the most famous arch dam&nbsp;&nbsp; in the world - Hoover Dam - isn’t a pure arch&nbsp; structure. Technically, it’s a gravity-arch dam,&nbsp;&nbsp;
[13:48] meaning it resists part of the water load through&nbsp; mass while also distributing the forces into the&nbsp;&nbsp; canyon through arch action. The proportions&nbsp; are carefully balanced to take advantage&nbsp;&nbsp; of the unique site conditions and relatively&nbsp; wider canyon than most arch dams are built in.
[14:04] And so, when you look at the tallest dams on&nbsp; Earth, one structural form dominates. By my&nbsp;&nbsp; estimation, around 40 percent of the tallest 200&nbsp; dams in the world incorporate an arch into their&nbsp;&nbsp;
[14:17] design. There aren’t that many places where it&nbsp; makes sense, but when you compare what it takes to&nbsp;&nbsp; hold a reservoir back in a narrow canyon valley,&nbsp; I think the case for arches is pretty clear.
[14:29] I think these models help a lot to explain&nbsp; engineering principles, so I have built&nbsp;&nbsp; quite a few of these acrylic demonstrations&nbsp; in the garage to cover interesting topics.&nbsp;&nbsp; One of my favorite creators, Neo, does the same&nbsp; thing, but instead of demos, he uses beautiful 3D&nbsp;&nbsp;
[14:44] graphics. His video about the construction of the&nbsp; World Trade Center towers in New York City is a&nbsp;&nbsp; fascinating look into how much effort, care, and&nbsp; engineering went into these buildings before the&nbsp;&nbsp;
[14:56] 2001 attack brought them down. My favorite part&nbsp; was the design of the slurry wall foundation,&nbsp;&nbsp; and of course, the 3D animations. And&nbsp; it was produced as a Nebula original. You’ve heard me talk about Nebula before. It’s&nbsp; a streaming service built by and for independent&nbsp;&nbsp;
[15:12] creators, including a lot of my favorites like&nbsp; Neo, Wendover Productions, the Coding Train,&nbsp;&nbsp; and Branch Education. I don’t know about&nbsp; you, but independently-produced content&nbsp;&nbsp; is most of what I watch these days. I just&nbsp; like the authenticity and thoughtfulness of&nbsp;&nbsp;
[15:27] videos that haven’t been through ten levels of&nbsp; studio executives watering the information down&nbsp;&nbsp; to capture the widest audience possible.&nbsp; I just think passionate individuals and&nbsp;&nbsp; small teams make the most compelling work,&nbsp; and Nebula is the perfect place for it.
[15:42] Nebula’s totally ad-free, with tons of excellent&nbsp; channels and lots of original series and specials&nbsp;&nbsp; like Neo’s video on the Twin Towers. It’s also a&nbsp; great gift, especially because a yearly membership&nbsp;&nbsp;
[15:54] is 40% of the link in the description. At&nbsp; thirty-six bucks for a year, that’s pretty&nbsp;&nbsp; tough to beat. My videos go live on Nebula before&nbsp; they come out on YouTube. If you’re with me that&nbsp;&nbsp; independent creators are the future of great&nbsp; video, I hope you’ll consider subscribing.&nbsp;&nbsp;
[16:10] That’s go.nebula.tv/Practical-Engineering. Thank&nbsp; you for watching, and let me know what you think!