[Note that this article is a transcript of the video embedded above.]

Imagine the room you’re in right now was filled to the top with gravel. (I promise I’m headed somewhere with this.) I don’t know the size of the room you’re in, but if it’s anywhere near an average-sized bedroom, that’s roughly 70 tons of material. Fill every room in an average-sized apartment, and now we’re up to 400 tons. Fill up an average-sized house. That’s 900 tons. Fill up 30 of those houses, that’s roughly 25,000 tons of gravel. A city block of just pure gravel. Imagine it with me… gravel… chicken soup for the civil engineer’s soul. And now imagine you needed to move that material somewhere else several hundred miles away. How would you do it? Would you put it in 25,000 one-ton pickup trucks? Or 625 semi-trucks? Imagine the size of those engines added together and the enormous volume of fuel required to move all that material. You know what I’m getting at here. That 25,000 tons is around the upper limit of the heaviest freight trains that carry raw materials across the globe. There are heavier trains, but not many.

I’m not trying to patronize you about freight trains. It’s not that hard to imagine how much they can move. But it is harder to imagine the energy it takes. Compare those 625 semi trucks to a handful of diesel locomotives, and the difference starts to become clear just by looking at engines and the fuel required to move that mountain of material. We’re in the middle of a deep dive series on railway engineering, and it turns out that a lot of the engineering decisions that get made in railroading have to do with energy. When you’re talking about thousands of tons per trip, even the tiny details can add up to enormous differences in efficiency, so let’s talk about some of the tricks that railroads use to minimize energy use by trains. And I even tried to pull a railcar myself. I’m Grady, and this is Practical Engineering. In today’s episode we’re running our own hypothetical railway to move apartments full of gravel (and other stuff too, I guess).

By energy, I’m not just talking about fuel efficiency either. If it was that simple, do you think there would be a 160-page report from the 1970s called “Resistance of a Freight Train to Forward Motion”? I’ll link it below for some lightweight bedtime reading. Management of the energy required to pull a train affects nearly every part of a railroad. Resistances add up as forces within the train, meaning they affect how long a train can be and where the locomotives have to be placed. Resistances vary with speed, so they affect how fast a train can move. Of course they affect the size and number of locomotives required to move a train from one point to another and how much fuel they burn. And they even affect the routes on which railroads are built. Let me show you what I mean. Here’s a hypothetical railroad with a few routes from A to B. Put yourself in the engineer’s seat and see which one you think is best. Maybe you’ll pick the straightest path, but did you notice it goes straight over a mountain range?

If you've ever read about the little engine that could, you’re familiar with one of the most significant obstacles railways face: grade. A train moving up a hill has to overcome the force of gravity on its load, which can be enormous. Grade is measured in rise over run, so a 1% grade rises 1 unit across a horizontal distance of a hundred units. There’s a common rule of thumb that you need 20 pounds or 9 kilograms of tractive effort (that’s pull from a locomotive) for every ton of weight times every percent of grade. By the way, I know kilograms are a unit of mass, not weight, but the metric world uses them for weight so I’m going to too in this video. And metric tonnes are close enough to US tons that we can just assume they’re equal for the purposes of this video.

A wheelchair ramp is allowed to have a grade of up to 8.3 percent in the US. Pulling our theoretical gravel train up a slope that steep would require a force of more than 5 million pounds or 2 million kilograms, way beyond what any railcar drawbar could handle. That’s why heavy trains have locomotives in the middle, called distributed power, to divide up those in-train forces. But it’s also why railway grades have to be so gentle, often less than half a percent. Next time you’re driving parallel to a railway, watch the tracks as you travel. The road will often follow the natural ground closely, but the tracks will keep a much more consistent elevation with only gradual changes in slope.

You might think, “So what?” We’ll spend the energy on the way up the mountain, but get it back on the other side. Once the train crests the top, we can just shut off the engines and coast back down. And that’s true for gentle grades, but on steeper slopes, a train has to use its brakes on the way down to keep from getting over the speed limit. So all that energy that went into getting the train up the hill, instead of being converted to kinetic energy on the way down, gets wasted as heat in the brakes. That’s why direct routes over steep terrain are rarely the best choice for railroads. So let’s choose an alternative route.

How about the winding path that avoids the steep terrain by curving around it? Of course, the path is longer, and that’s an important consideration we’ll discuss in a moment, but those curves also matter. Straight sections of track are often called tangent track. That’s because they connect tangentially between curved sections of rail that are usually shaped like circular arcs. Outside the US, curves are measured by their radius, the distance from the center of curvature and the center of the track. Of course, in the US, our systems of measurement are a little more old-fashioned. We measure the degrees of curvature between a 100-foot chord. A 1-degree curve is super gentle, appropriate for the highest speeds. Once you get above 5 degrees, the speed limit starts coming down, with a practical limit at slow-speed facilities of around 12 degrees. In an ideal world, you only have to accelerate a train up to speed once, but on a windy path with speed restrictions, slowing and accelerating back up to speed takes extra energy.

But those curves don’t just affect the speed of a train, they also affect the tractive effort required to pull a train around them. Put simply, curves add drag. As you might have seen in the previous video of this series, the wheels of most trains are conical in shape. This allows the inside and outside wheels to travel different distances on the same rigid axle. But it’s not a perfect system. Train wheels do slip and slide on curves somewhat, and there’s flange contact too. Listen closely to a train rounding a sharp curve and you’ll hear the flanges of each wheel squealing as they slide on the rail. A 1-degree curve might add an extra pound (or half a kilogram) of resistance for every ton of train weight (not much at all). A 5-degree curve quadruples that resistance and a 10-degree curve doubles it again. When you’re talking about a train that might weigh several thousand tons, that extra resistance means several thousand more pounds pulling back on the locomotives. It adds up fast. So, depending on the number of curves along the route, and more importantly, their degree of curvature, the winding path might be just as expensive as the one straight up the mountain and back down.

Sometimes terrain is just too extreme to conquer using just grades and curves. There comes a point in the design of a railroad where the cost of going around an obstacle like a mountain or a gorge is so great that it makes good sense and actually saves money to just build a bridge or a tunnel! Many of the techniques pioneered for railroad bridges influenced the engineering of the massive road bridges that stir the hearts of civil engineers around the world. And then there’s tunnels. You know how much I like tunnels. There are even spiral tunnels that allow trains to climb or descend on a gentle grade in a small area of land. I could spend hours talking about bridges and tunnels, but they’re not really the point of this video, so I’ll try to stay on track here. Hopefully you can see how major infrastructure projects might change the math when developing efficient railroad routes.

Of course, I’ve talked about grades, curves, and acceleration, but even pulling a train on a perfectly straight and level track without changing speed at all requires energy. In a perfect world, a wheel is a frictionless device and an object in motion would tend to stay in motion. But our world is far from perfect. I doubt you need that reminder. And there are several sources of regular old rolling resistance. Let me give you something to compare to.

I put a crane scale on a sling and hooked it to my grocery hauler in the driveway to demonstrate. This car just keeps showing up in demos on the channel. Doing my best to pull the car at a constant speed, I could measure the rolling resistance. With no friction, my car would just keep rolling once I got it up to speed, but those squishy tires and friction in the bearings mean I have to constantly pull to keep the car moving. It was pretty hard to keep this consistent, so the scale jumps around quite a bit, but it averages around 30 pounds or 14 kilograms. Very roughly, it’s about a percent of the car’s weight. I put half the car on the gravel road to compare the resistance, and it took about twice the force to keep it rolling. 60 pounds (around 2% of the car’s weight) is a little much for a civil engineer, so I had to get some help pulling. We tried it with a lighter car, but the scale must not have been working right.

At slow speeds like in the demo, drag mostly comes from the pneumatic rubber tires we use on cars and trucks. They’re great at gripping the road and handling uneven surfaces or defects, but they also squish and deform as they roll. Deforming rubber takes energy, and that’s energy that DOESN’T go into moving the load down the road. It’s wasted as heat. At faster speeds, a different drag force starts to become important: fluid drag from the air. I didn’t demo that in my driveway, but it’s just as important for trains as it is for cars. Let’s take a look back at that 1970s report to see what I mean.

One of the most commonly used methods for estimating train resistance is the Davis Formula, originally published in 1926 and modified in the 70s after roller bearings became standard on railcars. It says there are three main types of resistance in a train for a given weight. The first is mechanical resistance that only depends on the weight of the train. This comes from friction in the bearings and deflections of the wheels and track. Steel is a stiff material, but not infinitely so. As a steel wheel rolls over a steel track, they squish against each other creating a contact patch, usually around the size of a small coin. The pressure between the wheel and track in this contact patch can be upwards of 100,000 psi or 7,000 bar, higher than the pressure at the deepest places in the ocean. There is an entire branch of engineering about contact mechanics, so we’ll save that for a future video, but it’s enough to say that, just like the deformation of a rubber tire down a road, this deformation of steel wheels on steel rails creates some resistance.

The second component of resistance in the Davis formula is velocity dependent. The faster the train goes, the more resistance it experiences. This is mainly a result of the ride quality of the trucks. As the train goes faster, the cars sway and jostle more, creating extra drag. The final term of the Davis formula is air resistance. Drag affects the front, the back, and the sides of the train as it travels through the air. This is velocity dependent too, but it varies with velocity squared. Double the speed, quadruple the drag. Add all three factors together and you get the total resistance of the train, the force required to keep it moving at a constant speed.

But why use an equation when you can just measure the real thing. I took a little trip out to the Texas Transportation Museum in San Antonio to show you how this works in practice. Take a look at these classic Pullman passenger cars. You can see the square doors on the bearings where lubrication would have been added to the journal boxes by crews. This facility has a running diesel locomotive, a flat car outfitted with seats for passengers, and a caboose. This little train’s main job these days is to give rides to museum patrons, but today it’s going to help us do a little demonstration.

First [choo choo] we had to decouple the car from the caboose. Then we used the locomotive to move the flat car down the track. This car was built in 1937 and used on the Missouri Pacific railroad until it was acquired by the museum in the early 1980s. The painted labels have faded, but it weighs in the neighborhood of 20 tons empty (about 15 times the weight of my car). So I set up a small winch with the force gauge and attached it to the car. The locomotive provides an ideal anchor point for the setup. But on the first try, the scale maxed out before the car started to move. It turns out the rolling resistance of a rail car is pretty high if you don’t fully disengage the brakes first. Who would’ve thought?

Now that the wheels are allowed to turn, it’s immediately clear that the tracks aren’t perfectly level. Even without the car rolling at all, it’s pulling on the scale with around 100 pounds or 45 kilograms. Once I start the winch to pull the car, the force starts jumping around just like the car, but it averages around 150 pounds or 68 kilograms. If I subtract the force from the grade, the rolling resistance of the car, the force just required to keep it moving at a constant speed, is just about 50 pounds or 32 kilograms. That’s about the same force required to move my car on the gravel road even though this car is 15 times its weight. And it’s not far off from what the Davis Formula would predict either.

We tried this a few times, and the results were pretty much the same each time. This is an old rail car on an old railway, so there’s quite a bit of variation to try and average out of the results. Little imperfections in the wheels and rail make a huge difference when the rolling resistance is so low. A joint in the track can double or triple the force required to keep the car moving, if only for a brief moment. Kind of like getting a pebble under the wheel of a shopping cart: It seems insignificant, but if it’s happened to you, you know it’s not.

Watching the forces involved, I couldn’t help but wonder if I could move the car myself. But there was no safe way for me to start pulling the car once it was already moving. I would have to try and overcome the static friction first… aaaaand that turned out to be a little beyond my capabilities. If you look close, you can see the car budging, but I couldn’t quite get it started. On a different part of the track with the wheels at a different position, maybe I could have moved it, but considering most of the working out I do is on a calculator, this result might not be that surprising. Those joints between rails don’t only add drag, but maintenance costs too, but that’s the topic of the next episode in this series, so stay tuned if you want to learn more. It’s still remarkable that the rolling resistance between a 20 ton freight railcar and my little hatchback is in the same ballpark. And that’s a big part of why railways exist in the first place. Those steel wheels on steel rails get the friction and drag low enough that just a handful of locomotives can move the same load as hundreds or trucks with a lot less energy and thus a lot less cost.