Elephants on a Bridge
April 8, 2021 7:34 PM   Subscribe

Mini McGee says, "I learned you can ask physics really weird questions! Like, how thick would a cardboard bridge have to be if it was over a 40-foot gap and could support the weight of an elephant?"

I said, "I don't know, do I look like an engineer?" and he said, "But you could Ask MetaFilter, I bet they like weird physics questions!"

He said it's corrugated cardboard, and I decided on an adult male Asian elephant weighing 8,000 pounds, which seems to be right in the middle of possible elephant sizes.

I will gratefully take either actual answers to this question that I can show to him how you would figure that out, or pointers to bridge/materials equations that we can work out ourselves (as an 11-year-old and a woman who did pass calculus many years ago but is only reliable up through algebra anymore). This is not homework or anything he's learning in school, this is just him having recently discovered Mythbusters and us all having been trapped in a house together for a solid year. :)

(His follow-up example of "weird questions you can ask physics" was "How thick would walls made of cheese have to be to hold up the water in an above-ground swimming pool?" And that one I knew was, "First you need to specify your cheese and second it probably depends on the size of the pool.")
posted by Eyebrows McGee to Science & Nature (29 answers total) 12 users marked this as a favorite
I have nothing to provide to this but I'm eagerly looking forward to the answer to this question and any other questions Mini McGee will need AskMe to figure out for him!
posted by ashbury at 7:48 PM on April 8 [15 favorites]

Here's a bridge design exercise that might make a good start.

"With a single load acting at the mid-span of a beam, the equation is Force x Length ÷ 4 = Fy x Zx. As before, force equals the highest or most critical load combination pounds (lbs). Length is the total length of the beam that is usually known [in your case, 40ft]. Usually, units of length are given in feet (ft) and often converted to inches. Fy is the tensile strength or compressive strength of the material as described above [corrugated cardboard]. Zx is a coefficient that involves the dimensions of the cross-sectional area of the member. Therefore, Zx = (Force x Length) ÷ (Fy x 4), where Zx has units of cubed inches."

Sadly, the world's largest wheel of cheese is only 6ft in diameter and is also gouda, which seems too wobbly to support the contents of a kiddie pool without any wax or wrapping, although aged gouda is one of the most rubbery specimens tested in this study of commercial cheese torsion gelometry, so that idea might actually hold water.
posted by All hands bury the dead at 8:17 PM on April 8 [12 favorites]

I found the specifications the US National Archive uses for corrugated cardboard which seems to contain material properties. They may take some work to convert into units useful for bridges though.
posted by Zalzidrax at 8:21 PM on April 8 [4 favorites]

I would buy him blocks of different cheese and encourage him to try building a tiny mouse-sized swimming pool with something like paper towels lining a small container as the waterproof check - I bet gouda would be at a certain thickness be impermeable, while chedder would actually let water through at the same thickness. It's a good hypothetical test and he can use crackers or pretzels for struts and then you have the excuse of eating all the building materials afterwards.
posted by dorothyisunderwood at 9:35 PM on April 8 [3 favorites]

It's not just about the material properties of the cardboard; it depends on how you construct the bridge. Just putting on layers and layers of cardboard won't work as well as constructing a system of struts and trusses that minimize the total weight of the cardboard while maximizing the weight the structure can support. It gets complicated; hopefully a structural engineer will come by to provide more insight.
posted by mr_roboto at 9:42 PM on April 8 [10 favorites]

I don't have answers to either of his weird physics questions (sorry!), but do you know Randall Munroe's (author of xkcd) side project "What If?" where he tries to answer weird physics questions? Seems like it'd be up your alley!
posted by invokeuse at 10:02 PM on April 8 [8 favorites]

... what happens if it rains on the bridge?
posted by stormyteal at 10:35 PM on April 8

I can't answer this mathematically. However, if you're going to build or imagine something, an idea of the forces and materials involved, and the general approach, has to come before the math.
In the 1960's Frank Gehry made a line of furniture he called Easy Edges, made from sheets of corrugated cardboard laminated together and then cut into various shapes. He made things you couldn't have made with another material, and from the convoluted shapes he made it's obvious that they were very strong. A search on the web will produce a lot of good pictures.
I worked with a fellow some years back who was dissatisfied with his desk, which sat in a space which was too big, and yet too small for a table. He collected cardboard and the tubes we got plotter paper on, basically large, very thick mailing tubes, and one day came in with a roll of packing tape and a knife.
That didn't work, so the next day I brought in my glue gun and we attached everything securely, put fillets of hot glue around the joints, and then added triangular gussets where the tube legs joined the top. Then he taped it securely to a pillar and a wall.
This irritated the office staff unreasonably, but there are some people in any setting who you don't mess with, for reasons nobody can express. When he quit, a year later, the furniture people descended on his office and grabbed the cardboard desk, happy to finally be able to dispose of it.
Nothing happened. Cardboard is much stronger than you'd think. They were reduced to laboriously cutting the tape and tearing chunks off it until nothing was left. I watched from my office, delighted.
So cardboard is strong. I have several sheets of three-ply laminated stuff, and it's like plywood, except that if you bend plywood it'll splinter and break, and cardboard will buckle. If you put a bolt into plywood it'll take a couple of hundred pounds. It'll pull out of cardboard.
I'd laminate a big arch of several hundred layers of cardboard, say twelve feet wide, and probably no less than three feet thick at the middle of the span. I'd make sure the base at either end rested on a flat surface so there were no pressure points to start it collapsing. Then I'd walk across it, and if it felt solid, send the elephant.
Most of the engineers before about 1850 did their work by eyeballing everything. Our evolutionary history of living in trees and working with materials has given us a good instinctive understanding of what we can get away with. Also we tend to over-build everything and use materials which are very strong, or which are, like wood, evolved to do pretty much what we want to do with them.
Cardboard is something we're not exactly familiar with, but we can assume that it's similar to wood, though less stiff. Exotics like carbon fiber require more calculations because we don't instinctively know how they'd behave.
You should ask your child how one would make a submarine out of glass. There are answers to this one, because it's been done.
posted by AugustusCrunch at 10:56 PM on April 8 [10 favorites]

Cylinders! I used to teach Intro Biology for sporty people and inherited a practical about bones:
1) look down a microscope at a cross section of a long bone [humerus femur] to note that it is composed of bundles of osteons - cylinders 0.25mm across.
2) look at a cross section of a sheep humerus to note that the mid section is another, bigger hollow cylinder.
3) I added an engineering guesstimate section asking the students to have a punt at how much weight [reams of xerox paper = 2.5kg] a toilet roll [7g] could support . . . and then test the hypothesis. It was easier to make small 'elephants' with 4 toilet rolls for legs.
Spoiler answer: a thin walled cylinder can support 1000x it's own weight IF the weight pushes along the height axis - much less if the weight crushes the diameter axis.
4 of those cardboard tubes from the AugustCrunch office could manage a baby elephant, so long as t'bugger didn't frisk about. Shear stress is a problem with bridges so cross bracing will be necessary.
posted by BobTheScientist at 2:36 AM on April 9 [2 favorites]

what happens if it rains on the bridge?

Not a problem if you've used waxed corrugated cardboard.

Seconding the laminated arch design. I'd use waxed double wall board, cross-grain laminated like plywood with the same wax that impregnates the board used to bond the layers together, built up to a final thickness equal to the desired width of the roadway over the bridge. So the arch would have the ply layers oriented vertically.

I wouldn't put less than six feet of material in the centre of the span between the flat roadway on top and the peak of the arch below, and I wouldn't make the foundations any smaller than six feet in any dimension.

I'd cut big circular holes of various sizes through the arch to reduce wind loading and save material, again making sure that there was at least six feet of material between any of the resulting surfaces.

As BobTheScientist notes, cylinders are hella resistant to axial compressive forces, and a sheet of corrugated double wall board is a pretty good approximation to a flat array of cylinders. Laminating those sheets with the corrugations in alternate laminations run at right angles gives you a material that's ridiculously resistant to edge-to-edge crush forces in any direction, and if you laminate it to great thicknesses it will never bow either.

An arch bridge design keeps pretty much all the forces inside the arch compressive, and waxed corrugated cardboard is a pretty dead material that's good at absorbing and dissipating vibration. So I'm pretty sure you could run a thundering herd of elephants over the result with no concern for their safety. Their feet might crush the roadway surface a little to begin with, but I would expect the wax to stick the crushed zone together and end up making it into a load-spreading skin well coupled to the bulk material below.

I have no numerical modelling to back any of this. I just drew some pictures and they look about right.
posted by flabdablet at 2:59 AM on April 9 [3 favorites]

I would subscribe to Ask Physics! A column by Mini.
posted by chesty_a_arthur at 5:15 AM on April 9 [4 favorites]

First, the secret to bridges is triangles. Second, the secret to the egg drop contest is surface area to slow down the descent. Third, I think Mini McGee needs the David Macaulay books.
posted by Ms Vegetable at 5:22 AM on April 9 [3 favorites]

Trusses would certainly be stronger, but it would take a lot of repetitive math to calculate for each junction, which may be beyond your son's tolerance.

I would start with a simple beam and apply Classical Beam theory. There is a lot of theory in there, but it actually boils down to a single equation if enough assumptions can be satisfied.

The beam thickness would be calculated at:
thickness [inches]=2 * tensile failure strength of cardboard [psi] * second moment of area of beam cross section (I) [in^4] / Bending Moment Load (M) [lb*in]

As noted above, for a simply-supported (not clamped) beam, M=Elephant weight*Bridge Span/4. This assumes all the elephant feet at one point, which is a bit of an approximation.

Tensile failure strength of cardboard (parallel to the direction of the flutes) will take some digging or experimentation to find out.

Second moment of area is given by the equation I=1/12*b*h^3, where b is the width of the beam (not the span), and h is the height. This is complicated by corrugated cardboard not being solid, so you'd need to account for he holes. Since the holes are relatively evenly distributed, I would approximate this by scaling the width parameter b by the ratio of cardboard to empty space (if there is 25% cardboard and 75% air by area, b=.25 of the actual width.

This model assumes tensile failure on the bottom face of the beam, meaning the cardboard doesn't buckle in compression. This depends on material properties that I am not sure of. It also neglects the weight of the beam. With enough cardboard (more than 500 lbs say), you would want to model its weight as a distributed weight across the span. We can leave this for exercise #2.

Another thing to look at is whether the elephant's feet would crush the cardboard. The flat crush value in the cardboard specs quoted above would be the place to start, knowing the weight of the elephant and the area of its feet.
posted by cardboard at 5:25 AM on April 9 [8 favorites]

Response by poster: posted by cardboard

I ... I have questions
posted by Eyebrows McGee at 6:19 AM on April 9 [19 favorites]

Tensile failure strength of cardboard

isn't really an issue for a nicely proportioned arch design. And cardboard is so cheap compared to other building materials that trying to minimize the amount of it you use by making beams and trusses isn't really worth the trouble, especially given cardboard's limited ability to cope with the kinds of concentrated loads that would inevitably occur at the joints between truss sections. Whack up a hugely overbuilt and therefore reassuringly massive looking monolithic arch and call it Job Done.
posted by flabdablet at 6:31 AM on April 9

His follow-up example of "weird questions you can ask physics" was "How thick would walls made of cheese have to be to hold up the water in an above-ground swimming pool?" And that one I knew was, "First you need to specify your cheese and second it probably depends on the size of the pool."

I think the design of the pool also matters, yes? Like, is the containing receptacle squared-off, or is it like a bowl that's been carved into a block of cheese?
posted by EmpressCallipygos at 7:24 AM on April 9

I’d be particularly worried about the wall to floor joint with the cheese pool.

Funny thing is, I think I’ve seen that napkined, but I’ve just flipped through Monroe’s what if? and not found it.

Building bridges out of cardboard (flat, tubes) and Lego and spaghetti and then testing them is wildly entertaining and can be useful through life.
posted by clew at 8:39 AM on April 9 [1 favorite]

Interior corners make brittle material weak - here’s a nice short write up of an engineering student testing how much stronger the corners are with rounding of different radiuses.
posted by clew at 8:47 AM on April 9 [1 favorite]

I don't know anything abouot cardboard, bridges, elephants or building with cheese, but I do so appreciate a curious, creative kid. This one is a keeper. Congrats EM!
posted by AugustWest at 8:48 AM on April 9 [1 favorite]

I’d be particularly worried about the wall to floor joint with the cheese pool

Now I'm wondering what the best shield gas would be for TIG welding cheese. Or would I get a better result with a soldering iron? What if I fed cheese sticks through my hot glue gun? So many questions.
posted by flabdablet at 8:54 AM on April 9 [2 favorites]

Using ASME standards for a cylindrical pressure vessel, a 5 foot deep and 5 foot radius pool, an arbitrary "joint efficiency" of 0.8, and a value of around 0.05 MPa for yield stress of some particular samples of English Cheddar from a random paper, I get something like 25 inches for the pool walls. But, that's making a lot of assumptions that almost certainly aren't true. (Especially reinforcing the top rim of the tank, and of course the joints, and not worrying about what happens to cheese soaking in water. . .) Double check my numbers and add a significant safety margin when you build it.

But, also, it could be a fun experiment to actually do on the scale of one small block of cheese on a table. e.g., how thick would your slice of cheese need to be to hold an upturned glass full of water without breaking? A wider glass with the same height of water? A taller class with the same radius? (Scaling that to a pool takes some work, but could be fun.)

If you haven't come across them already, cardboard bridge contests might be of interest.
posted by eotvos at 9:59 AM on April 9 [1 favorite]

(On reflection, regarding my above comment about a table-top version, you might get cleaner results using a cylinder with a hole in the top of rather than a glass with no way for air to get in. Aluminum food cans or some PVC pipe scrap, for example.) This is a fun question. Sounds like a fun kid.
posted by eotvos at 10:07 AM on April 9

cardboard bridge contests might be of interest

A bridge building challenge given to the civil engineering class of a boyfriend of a housemate many years ago:

You have two tables of equal height, set some distance apart. Construct a bridge between the tables that's capable of supporting a full 375ml (12oz US) ringpull beer can half way across the gap between the tables, at such a height that no part of the can is below tabletop level.

Allowable materials and tools: one whole sheet from a broadsheet newspaper (New York Times would do), one foot of half-inch-wide sticky tape, one small wire paper clip, scissors.

Widest span wins.

The best design achieved by the engineering students had apparently spanned half a metre (20 inches). I had a few tries at home and was pleased to come up with a design that spanned one metre (39 inches). Can you do better?
posted by flabdablet at 10:50 AM on April 9 [1 favorite]

19 inches! Using this massively simplified method that ignores lots of critical things:
Stress = Mc/I for a beam in bending
I= width * (thickness^3)/12 (second moment of area for a rectangular cross-section)
M = Force * length / 8 (moment for a beam fixed at both ends with a concentrated load in the center)
Put it all together: Stress = (Force * length/8) * (t/2) / (width * t^3 / 12)
Simplify and rearrange: t= sqrt [(3/4 * Force * length) / (stress * width)]
Force = 8000 lbs, length = 40 ft (480 in), width = 48 in (how wide is an elephant?)
The max stress is challenging, but to estimate why not use the burst strength from the link Zalzidrax shared, 170 psi
This method treats cardboard like a homogenous material like steel that would fail by tearing in tension, ignores how it can crush locally and that the bridge would be composed of individual layers glued together, treats the elephant like a point load equally distributed across the width of the bridge, assumes you're just piling up cardboard rather than making clever trusses, incorrectly uses burst strength for tensile strength, and I'm sure lots of other equally terrible assumptions...
posted by Gravel at 12:36 PM on April 9 [1 favorite]

Not a direct answer, but I bet your kid would get a kick out of Wolfram Alpha

posted by momus_window at 12:43 PM on April 9 [2 favorites]

I was going to suggest Wolfram Alpha as well; it can do some pretty impressive stuff, and it supports some really fantastic units (including the weights of elephants, etc.).

The subreddit /r/theydidthemath may also be of interest. It has questions like "How many miles of highway could 1 Tomahawk Missile adopt?" and "How long it would take him to chug a bathtub of milk?".
posted by Kadin2048 at 1:23 PM on April 9 [1 favorite]

Qualifications: I'm not a structural engineer, but I'm trained as an architect and currently working in the structural engineering department of a large architecture/engineering firm. I am not your structural engineer/this is not structural engineering advice.

It seems like Mini Mcgee has enough of a grasp of the basic intuitive concepts that he knows the absurdity of supporting an elephant on bridge made of cardboard or making a pool out of cheese - he's right that they're both certainly possible, but due to the qualities of the materials proposed, neither is an efficient or desirable solution to the problem of elephant support or water containment. Both questions lead into some pretty fundamental issues for structural engineering and why the decisions are made to use certain materials given their inherent strengths and weaknesses.

The cheese one is pretty easy, so let's start there. As others have noted, the strength of "cheese" is largely dependent on what kind of cheese is used. Could you have a slab of brie 5' high that covers an acre and somehow carve out a large enough space in the middle of it to form a pool even after the brie oozes into whatever final form it wants to take? Sure! Any other kind of cheese will basically be the same thing, just with a steeper final angle of repose . I think your model of thinking about it is an earthen dam vs a concrete one (like Hoover Dam). With an earthen dam, you're really not expecting the earth to do any work besides sit there and retain water by its mass, and the slope of the dam is such that the earth isn't going to go anywhere. A concrete dam is more designed, and is essentially a barrel vault spanning between two canyon walls, like a beam carrying the water in a way. With a cheese pool, you're kind of doing a lake completely surrounded by an earthen dam; how thick the walls need to be depends somewhat on how deep the water is and how large the pool is, but the mass of the cheese is doing most of the work rather than any structural property of the cheese, like how it performs as a beam (not well!). "Stronger" cheeses that hold their own shape better than brie may be able to have thinner walls.

The elephant question raises a number of issues. We actually had the cardboard bridge contest in my structures class in college - we had set sizes and quantities of materials, had to span a given distance, and were rated on how much load the bridge design would carry. None of the materials allowed were long enough to span the entire distance, so part of the problem was maintaining strength through connections between materials. The elephant question is somewhat similar, since you're really not going to be able to get a 60' long piece of cardboard (40' for the span, plus 10' for bearing on either end), so the cardboard bridge will require multiple pieces of cardboard stitched together somehow. I'd assume that the elephant is going to be in the middle of the bridge rather than at either end, so failure from bending will be the primary concern rather than shear, so you're worried about an 8000 pound load acting with a 20' long moment arm that the bridge will have to support, rather than just the bridge material itself being crushed by weight of the elephant. For a decent idea of how much easier shear is to handle than bending, office boxes at Costco come in a 10-pack, and each box supposedly can withstand 850 pounds of vertical load. If you can distribute the weight of the elephant effectively over a grid of those ten boxes on the ground, they'll support the elephant, so if the elephant was standing at one end of the bridge, that's about all you'd need and the rest of the bridge would just have to be enough to support the weight of the cardboard structure itself.

With bending, you effectively have the weight of the elephant on the end of a 20' lever trying to rip the cardboard, which is a bit tougher to withstand. So, you could theoretically have an infinitely tall piece of cardboard that would be able to take the vertical load of the elephant, but would fail in other ways - one sheet of cardboard is going to want to bend to one side or the other rather than take the full load of the elephant (like how if you push on the edge of a sheet of paper with your finger, it'll bend), so the cardboard needs some thickness to it as well to resist that, or it can be braced from either side, similar to how flying buttresses work in gothic cathedrals.

I'd guess that the best way to join different pieces of cardboard together for this purpose is just to laminate them with glue, overlapping seams in one layer with full sheets in the next and alternating the direction of the corrugation in each layer, just like Gehry did with his furniture and how layers are built up in plywood. You could mechanically tie them all together, like with a long bolt or tie rod, but then you'd also have to worry about the cardboard tearing at the penetrations through it. Although delamination is a concern, that should reduce your concerns to crushing at the ends of the bridge, failure through bending, and failure through buckling (the out-of-plane paper thing). If it were a real structure and we were concerned about the elephant's comfort level walking across the bridge, we might also worry about deflection - how much the bridge would sag without failing and maybe how "bouncy" it is - but that can be resolved through just making the bridge resist bending better.
posted by LionIndex at 8:21 PM on April 9 [4 favorites]

Part of the design problem here is how much you can customize the material. Are you just laminating flat, corrugated sheets? I think the strongest/lightest form of cardboard you could use for this purpose would be to make a honeycomb of tubes, which has the advantage of high stiffness and resistance to buckling under linear compression without as high a dead load of its own. The cross section of a structural honeycomb would be much larger than a simple laminate of the same compressive strength, but it would weigh a couple orders of magnitude less. If you can't use manufactured tubes you could fold a single corrugated sheet in three sections (parallel to the direction of the corrugation) to make an ersatz triangular tube, and then laminate your triangular tubes together into a honeycomb for similar structural strength.

I'd consider building an arched bridge similar to one of the wood examples here with a structural arch of honeycomb and a surface of laminate where the grain crosses. Tie the laminate surface to the supporting arch with honeycombed vertical members spaced a half elephant apart, and any span of the surface would never have to support the full elephant's weight. I'd have thumbnailed such a design at about 14" thickness at the surface, but then I'd round it up to 18" to be sure. To support just one elephant the honeycomb for the main structural arch would probably have to be about 2' in cross section. (It's also been about 30 years since I took a materials class and had to do the beam moment math above, so this is all a guess).
posted by fedward at 12:28 PM on April 13 [1 favorite]

Also, just because "hey let's build big stuff out of cardboard" really isn't that crazy of an idea: Shigeru Ban
posted by LionIndex at 10:01 AM on April 15

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