In this video, we explore arches and chains.
Arches are a popular and beautiful structural component.
If you know where to look,
you can find them in bridges,
buildings,
tunnels,
wells,
dams,
and many other places.
Chains are also used widely,
and they can be found in lifting equipment,
playgrounds,
and many other settings.
As we will show, close parallels exist
between arches and hanging chains.
And since chains are easier to understand, let's talk about them first.
Consider two chains side by side.
One is light in weight while the other is heavy.
As you can see, the weight of a free hanging chain
does not affect its shape,
but the spacing of its ends does.
Each shape in this family is called a "catenary",
and each catenary is similar to a parabola.
If we add external loads to a light-weight chain,
its shape changes considerably.
What do you think will happen
if we add the same weights to the heavy chain?
It turns out that those weights have little effect on the shape of the heavy chain.
Chains are often referred to as tension systems
because each of their links carries tension.
Notice that no bending forces or moments are transferred from one link to the next.
If we make a mirror image of a hanging chain, we get an arch.
In an arch, the component pieces push against each other,
unlike chain links which tend to pull apart.
To ancient builders, this represented a huge advantage.
If they placed a bunch of stones in just the right positions,
they would push against each other and stay in place.
This made for structures that were both simple to construct and highly durable.
Here is another difference between chains and arches.
Because chains are tension systems,
they automatically reshape to better carry whatever load is applied.
Because arches are compression systems,
they do the opposite.
So, if they are not shaped appropriately for the loads they are asked to carry, they can collapse.
As a result, designers must tailor each arch to the loads that it will need to carry.
And how did we figure out the shape for the arch?
That was easy, we just hung it upside down,
so that it worked like a chain system,
applied the loads,
and maintained those angles when we turned it right side up.
In the late 1800s,
a famous architect by the name of Antoni Gaudi
used exactly this technique to design the arches
of the famous Basilica of the Sagrada Familia in Barcelona, Spain.
To figure out how to shape the arches,
he built a precision, upside-down model of the basilica.
His model was like this chain model.
It changed shape as he added miniature weights
corresponding to the loads of the roof
and the other features that the arch had to support.
The resulting final profiles showed him
the exact shape to use for each arch.
In many real-world structures,
the weight of the arch is much greater
than that of any applied moving loads.
In cases like this,
the arch shape does not need to take into account
the exact positions of those loads.
This idea is consistent with the heavy chain
that did not change shape when small weights were added.
Now, let's take a look at some common arch designs,
starting with the popular semi-circular shape.
What do you think,
can a thick semi-circular arch support its own weight?
As you may have guessed, it can.
Here is another semi-circular arch.
It is the same size and shape as the previous one,
but its members are thinner.
Can this thin, semi-circular arch
support its own weight?
You might have guessed that it would collapse,
but can you explain why?
To make you think even harder,
suppose we modify the ends of this thin arch
so that they lean further outwards.
Can the new shape support its own weight?
It can, but are you able to explain this curious result?
The mystery of which arches will stand under their own weight
and which will not
can be resolved with the aid of hanging chains.
Remember how arches and chains are mirror images of each other?
And how chains tell you the right shape to use?
Well, suppose you want to evaluate a particular arch design.
If you can hang a chain entirely inside the profile of that arch,
then it is a shape that can carry its own weight.
We call this the "chain test."
Does this thin semi-circular arch pass the chain test?
It does not, because it is impossible to position a hanging chain
so that it lies entirely inside the profile of the arch.
And that is why it collapses.
One way to make the arch pass the chain test is to make it thicker.
Another solution would be to change the shape of the arch so that it follows the chain shape.
For this arch, we could do that by tilting the bottom pieces of the arch outwards.
As you can see,
when the arch shape follows the chain,
it stands up just fine.
Suppose, instead, that we added forces to the chain
to make it follow the shape of the semi-circular arch.
As you can see, outwards forces
applied at these points on the chain would do the job.
They change the catenary shape
into something more like a semi-circle.
Recall that a chain is a tension system,
while an arch is a compression system.
Thus, we have to reverse those external chain forces when applying them to the arch.
When the appropriate forces are applied to the arch,
it stands.
And if we take them away, it falls.
The forces acting on the ends of the arch
are also important to its stability,
and it is easy to demonstrate this fact.
If you prop your head on your hands,
and place your elbows apart like this,
you might be able to feel the forces
that are keeping your elbows in place.
Those forces have a vertical component,
which you can feel
as the table pushing up on your elbows.
They also have an inwards horizontal component
that is carried by friction
between your elbows and the table.
You'll notice that these forces change
depending on the position of your elbows.
When your elbows are closer together,
the forces become more vertical,
and their horizontal components become smaller.
However, if you move your elbows further apart,
you will discover that the horizontal forces increase.
In doing these tests, it is important that you not use the muscles in your arms
to stop your elbows from moving outwards.
If you spread your elbows wide enough,
the horizontal forces needed to support your elbows
can become greater than the available friction forces,
and your model arch collapses.
Adding a rubber mat
can increase the friction forces available at your elbows
and thereby prevent collapse.
Arches can take many shapes,
from tall and narrow,
to shallow and wide.
And shallow arches like this one
can often carry a surprising amount of load
if their supports provide sufficient horizontal forces.
In the real world, arch supports are often angled
so as to better transfer these horizontal forces.
Here are a couple of other interesting facts about arches.
The horizontal forces that semi-circular arches need
can be provided by material that fills the space around them.
We used blocks for our fill,
but in the real world, stones or soil are typically used.
In addition, if multiple arches are placed end-to-end,
the horizontal forces at their ends
can be made to balance,
and they can be supported
on surprisingly thin columns.
Arches can be laid on their sides
and used to support large horizontal forces
like the water forces that act on the Hoover dam.
As you can see in this top view,
even the Hoover Dam has a shape
rather like a hanging chain.
Oh, and one last point.
Not all structures that follow an arch shape
actually function as one.
In this bridge, for example,
the arch shape is strictly decorative.
Much more could be said about arches and chains,
but here we only have time to introduce the basics.
As our team worked on this video,
we actually learned quite a bit about arches and chains.
We hope that you learned something, too.
Thanks for watching.
We hope you will enjoy some of our other
"Engineering Models" videos.