Arch Forces

How do forces work to produce these curves of arches and cables? We'll see, using the time-honored method of starting simply.

Suppose you're lying still on your bed. What are the forces on your body? There's no net force, since you stay still, but there are forces, which balance each other out:

Your weight, the force of gravity, pushing you down.

The restoring force of the mattress springs, compressed by your weight and trying to spring back, pushing you up.

The two forces are equal and opposite, canceling each other out and you just lie there, blissfully unaware of all this.

Now this is a general situation, even when the springiness isn't obvious. If you have a soft drink can on the table next to you, the same forces are present. The seemingly rigid table is actually, like the mattress springs, slightly compressed by the weight of the can, and the springy restoring force balances the force of gravity on the can. Even steel or stone surfaces work this way when they support an object, though the compression is way too small to see.

It's very similar in the inverted situation, where the support is above the object; that is, something is hanging. The weight of the object stretches the support, which tries to spring back, and this upwards force of tension balances the weight of the object. You can see this stretch in one of those handheld spring scales weighing what you hang on the bottom, but the stretch is always there when hanging, even if too small to see.

Compression and tension can thus work to the same end, of balancing gravity. By the way, this part of physics where the forces all balance is called statics, as opposed to dynamics where unbalanced force rules.

So, what about those catenary curves we started with?

Don't be so impatient, we're getting there.

Next, we'll consider forces where there's a stack of objects. If you have a stack of blocks, say alphabet blocks, the forces are slightly more complex. Say that the A block is on top of the stack, the B block is right under A, the C block is next, and so on. (This child passed the exam)

The forces on the top A block are as before: the weight of A directed down, the restoring force of its support (B) directed up and canceling gravity. But when we move down the stack and consider the forces on the next block B, we see there are three forces.

the weight of B,

the restoring force of its support C, and

the weight of what's resting on B, namely the top block A.

That is, two down forces and one up force, which cancel out to zero since the stack is stationary. We can write an equation for this:

         WeightA  +  WeightB  = RestoringC

If we look at forces on the next block C, we see three similar forces: the weight of C, the restoring force of its support D, and the combined weight of everything above:

         WeightAbove  =  WeightA  +  WeightB 

         WeightAbove  +  WeightC  = RestoringD

The inverted situation works similarly: if you have a chain hanging free from one end, the forces on each link can be written down. We can call the bottom link Z, the next link up Y, and so on. Then the forces on link Y are

         WeightBelow  =  WeightZ  

         WeightBelow  +  WeightY  = RestoringX

So, uh, what about those catenary curves we started with?

The point of this analysis is that the same thing is happening in an arch or a cable! There are two differences:

• The continuous curve is just the limit as the individual elements get smaller and smaller.
• There's a sideways force as well as a vertical force, since the curve is sloping.

People first made arches out of stone blocks, and that's still a good place to start analyzing the forces. A block partway up the left side of the arch is being pushed down and out by the blocks above it. There's a restoring force from the block underneath, pushing up and in. The sum of the these two forces and the block weight must be zero, since the arch isn't going anywhere, but the "sum" is now more complicated, since the forces aren't all vertical.

You may know that physicists use mathematical things called vectors for this situation, but here we only need to think in terms of the vertical and horizontal components of the force. The "down and out" force from above is exactly equivalent to a pure "down" force plus a pure "out" force of the correct size. The diagram shows these horizontal and vertical force components, as they are called. The correct sizes are such as to make a triangle, as shown.

The static condition is then then that the sum of the vertical components are zero, just like the stack of blocks before, and that the sum of the horizontal components is zero. (This was also true in the stack of blocks. since there weren't any horizontal forces.)

To get a feel for how this works, here's another natural arch. If you have Java, we've superimposed an interactive arch. Sure, you passed the course in playing with blocks, but this is even better (except for being unable to throw them). If you click and drag on the arch, you can change the shape by moving the nearest joint inwards or outwards. To return to the starting configuration, press the "reload" button on your browser, or the back and then forward buttons. To make this a physics arch, the red arrows show the downwards forces at the joints between the blocks. The forces are, of course, due to the weight of the blocks above, so the force arrows get longer as you go from the top to the bottom of the stack of blocks. The poor block at the bottom is under all that weight!

Why are the longer arrows more vertical? This is a little more subtle, but it's key. Remember we said that the horizontal force components on each block must add to zero? Since each block's weight force is purely vertical, the horizontal forces are the same for every block! Only the vertical forces increase as you go down the arch. Think about the component triangles of the red arrows: the horizontal components are the same, and the vertical components get bigger and bigger closer to the bottom.

You might have noticed while fooling around with the Interactive Arch that the blocks would sometimes change color. This is actually not a software bug, as you may have thought, but a cunning way to show you why a catenary arch is special: its structural stability. We shall see what this means.

If you arrange the blocks so that the force arrows are aligned along the blocks, you get a catenary! (Go back any try it. You should notice that the blocks are now all gray.) Now, if you slowly move one block so that its load vector (the force arrow at its top, from the blocks above it) points further and further away from the block, eventually the vector points outside the base of the block, and the block turns pink.

Why? Well, such a structure is in danger of collapse! Worst case, the block could just fall out. The nature of any joints in the arch is important: we've shown blocks with straight line joints at right angles to the load arrow, so they don't slip. Even if there are no joints or the joints are such that the block can't fall, the block is now under a sideways, or shear, force as well as compression, and materials are much more prone to fracture under shear. Any outside stresses like earthquake or forces from other parts of the construction in which the arch is embedded are more likely to cause cracking and collapse.

We were lazy. and rather than animate the collapse of the arch, we've just colored the block to show the danger. Try moving the top slider in the figure: it controls the thickness of the arch blocks. If you repeat the block dragging with thick and thin blocks, you'll see how much less choice you have with thin arches. If you haven't arranged the blocks to be close to a catenary, as you thin the arch, some blocks will start turning pink. Load arrows that were within the base of fat blocks will miss the base of thin blocks. A shape that was "good enough" with all gray fat blocks may not be good enough with thin blocks. (Again, you can reset the arch to the initial position by hitting the "reload" button on your browser, or the back and then forward buttons. That triangular arch will be all gray with fat blocks and turn more and more pink as you thin the arch with the slider.)

For more intuition, back in the real world try pressing straight down on the top of a soft drink can using one finger. Nothing happens, of course. Press again at a slight angle from vertical, then at increasingly greater angles. At some angle, the object will start to tip and will eventually abruptly tip over (so I hope it can't spill or break something!). What's the story?

Well, if you try this again, replacing your finger with a pencil with the eraser end down (so it won't slide easily), watch where the pencil is pointing when the tipping starts. You should see the pencil pointing just beyond the edge of the base. What's happening, intuitively, is that the force causes the object to pivot around the edge of the base. When the force points inside the base, the table resists the force and nothing moves (as long as there's enough friction to prevent the object from sliding).

So the point of this for architects is that if the direction of the load on a block points outside its base, the block wants to tip over, causing the whole arch to collapse! In this case, as with the can, a block may first have to move up a little to pivot before falling, and this is resisted by the weight above it. But an arch designed that way is more fragile: a crack in the material, earthquake, or whatever can take it down more easily. This design constraint is more severe, of course, the smaller the base: a narrow block has to really follow the load angle, and this controls the whole shape of the arch. So thin arches (and dome cross sections) are the closest to catenaries. As the illustration shows, even thin arches can last a long time. Try catenary in stone. It's a nice real-world project to use blocks to physically build small catenary arches and other arches.

By the way, the old Roman arches were usually semicircular, not catenary, because they carried a lot of weight from above the arch stones themselves, which makes a different design issue. To bear a heavy load, the stones of a semicircular arch are slightly tapered so the weight jams all the stones towards the center, taking advantage of the compressive strength of stone. Here's a nice description of this, and here's cool-looking falsecolor computer analysis of a stone arch. Non-catenary arches/domes usually have dead weight above the sides to control the horizontal forces. There's an architectural jargon term for such weight, a surcharge. Architecture has zillions of old words for parts of things, like sailing does: here're definitions of arch parts (follow the links there ... he has a nice case study on what's happened to a certain 200-year-old stone arch bridge.) An extensive essay on arch bridges with lots of pictures. Do you know what a squinch is? Another design for an arch is the pointy kind called Gothic used in medieval churches, for both aesthetic and structural reasons ... follow that page's link for "fallen arches"! Can you guess why the St. Louis Arch is really a "modified catenary"? (hint: chains are uniform)

You probably never thought of a dome as an arch rotated about its center, but engineers have, because the forces are like that. You can build with a lot of materials: an architecture department built a 32 foot domed Ice Pantheon; follow the link at the bottom of that page for an insider's story of ice arches. On a small scale, the exact shape isn't critical for an igloo.

Getting back to our "automatic" physical curves, why is the chain hanging from both ends the shape it is, a catenary? The way the shape is defined physically is that the forces between the links are always directed along the links. Think what happens where two links join under tension: if the forces were not along the links, the links would slip in the direction of the net force: for example, if you pulled on a link. The defining property of the shape of a hanging cable is that the non-gravity forces be along the cable. The slope of each link is proportional to the weight it carries (since the horizontal force component is constant, remember.) For a chain, that load weight is the number of links below it, for a catenary the arc length below a point on the curve. But for a suspension bridge, there's also the weight of the roadway hanging from the cable. (Which is why the the bridge shape isn't a catenary.) Of course, the simpler rope bridge, like you see in the jungle adventure movies, where the weight is directly and uniformly on the cable is a catenary. (Except perhaps when there's enough load on the bridge to distort it ... here's a cool old photograph of an authentic rope bridge.) The criterion is uniform weight per unit length of curve.

Are you, dear reader, looking for something a little meatier, a little more advanced, the real thing? No, you aren't. Trust me. But we do have available something for people comfortable with this stuff, which actually demonstrates a claim we made on the opening page about suspension bridges. Try it if you dare, but don't say I didn't warn you. Elsewhere on the Web, there's an overview of suspension bridge engineering and here's how to build your own.

Let's summarize: the catenary is the natural shape of the cable or arch by itself, but the shape changes if additional external loads are also present. The arch often supports something, and the bridge has a roadway hanging from the cable.

Why is our pure thin arch and chain the same shape, a catenary? If you turn the chain upside down, the key insight is that the force pattern is the same: the forces are along the arch, not at an angle (with a vector component that would stress horizontally) Although, of course, the forces are now reverse direction: compression rather than tension, so a chain will just collapse. Remember, the elements of the structure have to be put together in a way that supports compressive forces, like a stack of blocks. Physically, you could take a wet fabric rope and let it freeze while hanging from its ends. If you turn it over, you have a catenary arch in pure compression. Over 300 years ago, Robert Hooke (of Hooke's Law of springs) pointed out this inversion.

Did you know this force analysis also applies to biological architecture? Just
follow the force arrow!

Return to CPO Home Page