In this video, we explore arches and chains.
In this video, we explore arches and chains.

Arches are a popular and beautiful structural component.
Arches are a popular and beautiful structural component.

If you know where to look,
If you know where to look,

you can find them in bridges,
you can find them in bridges,

buildings,
buildings,

tunnels,
tunnels,

wells,
wells,

dams,
dams,

and many other places.
and many other places.

Chains are also used widely,
Chains are also used widely,

and they can be found in lifting equipment,
and they can be found in lifting equipment,

playgrounds,
playgrounds,

and many other settings.
and many other settings.

As we will show, close parallels exist
As we will show, close parallels exist

between arches and hanging chains.
between arches and hanging chains.

And since chains are easier to understand, let's talk about them first.
And since chains are easier to understand, let's talk about them first.

Consider two chains side by side.
Consider two chains side by side.

One is light in weight while the other is heavy.
One is light in weight while the other is heavy.

As you can see, the weight of a free hanging chain
As you can see, the weight of a free hanging chain

does not affect its shape,
does not affect its shape,

but the spacing of its ends does.
but the spacing of its ends does.

Each shape in this family is called a "catenary",
Each shape in this family is called a "catenary",

and each catenary is similar to a parabola.
and each catenary is similar to a parabola.

If we add external loads to a light-weight chain,
If we add external loads to a light-weight chain,

its shape changes considerably.
its shape changes considerably.

What do you think will happen
What do you think will happen

if we add the same weights to the heavy chain?
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.
It turns out that those weights have little effect on the shape of the heavy chain.

Chains are often referred to as tension systems
Chains are often referred to as tension systems

because each of their links carries tension.
because each of their links carries tension.

Notice that no bending forces or moments are transferred from one link to the next.
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.
If we make a mirror image of a hanging chain, we get an arch.

In an arch, the component pieces push against each other,
In an arch, the component pieces push against each other,

unlike chain links which tend to pull apart.
unlike chain links which tend to pull apart.

To ancient builders, this represented a huge advantage.
To ancient builders, this represented a huge advantage.

If they placed a bunch of stones in just the right positions,
If they placed a bunch of stones in just the right positions,

they would push against each other and stay in place.
they would push against each other and stay in place.

This made for structures that were both simple to construct and highly durable.
This made for structures that were both simple to construct and highly durable.

Here is another difference between chains and arches.
Here is another difference between chains and arches.

Because chains are tension systems,
Because chains are tension systems,

they automatically reshape to better carry whatever load is applied.
they automatically reshape to better carry whatever load is applied.

Because arches are compression systems,
Because arches are compression systems,

they do the opposite.
they do the opposite.

So, if they are not shaped appropriately for the loads they are asked to carry, they can collapse.
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.
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?
And how did we figure out the shape for the arch?

That was easy, we just hung it upside down,
That was easy, we just hung it upside down,

so that it worked like a chain system,
so that it worked like a chain system,

applied the loads,
applied the loads,

and maintained those angles when we turned it right side up.
and maintained those angles when we turned it right side up.

In the late 1800s,
In the late 1800s,

a famous architect by the name of Antoni Gaudi
a famous architect by the name of Antoni Gaudi

used exactly this technique to design the arches
used exactly this technique to design the arches

of the famous Basilica of the Sagrada Familia in Barcelona, Spain.
of the famous Basilica of the Sagrada Familia in Barcelona, Spain.

To figure out how to shape the arches,
To figure out how to shape the arches,

he built a precision, upside-down model of the basilica.
he built a precision, upside-down model of the basilica.

His model was like this chain model.
His model was like this chain model.

It changed shape as he added miniature weights
It changed shape as he added miniature weights

corresponding to the loads of the roof
corresponding to the loads of the roof

and the other features that the arch had to support.
and the other features that the arch had to support.

The resulting final profiles showed him
The resulting final profiles showed him

the exact shape to use for each arch.
the exact shape to use for each arch.

In many real-world structures,
In many real-world structures,

the weight of the arch is much greater
the weight of the arch is much greater

than that of any applied moving loads.
than that of any applied moving loads.

In cases like this,
In cases like this,

the arch shape does not need to take into account
the arch shape does not need to take into account

the exact positions of those loads.
the exact positions of those loads.

This idea is consistent with the heavy chain
This idea is consistent with the heavy chain

that did not change shape when small weights were added.
that did not change shape when small weights were added.

Now, let's take a look at some common arch designs,
Now, let's take a look at some common arch designs,

starting with the popular semi-circular shape.
starting with the popular semi-circular shape.

What do you think,
What do you think,

can a thick semi-circular arch support its own weight?
can a thick semi-circular arch support its own weight?

As you may have guessed, it can.
As you may have guessed, it can.

Here is another semi-circular arch.
Here is another semi-circular arch.

It is the same size and shape as the previous one,
It is the same size and shape as the previous one,

but its members are thinner.
but its members are thinner.

Can this thin, semi-circular arch
Can this thin, semi-circular arch

support its own weight?
support its own weight?

You might have guessed that it would collapse,
You might have guessed that it would collapse,

but can you explain why?
but can you explain why?

To make you think even harder,
To make you think even harder,

suppose we modify the ends of this thin arch
suppose we modify the ends of this thin arch

so that they lean further outwards.
so that they lean further outwards.

Can the new shape support its own weight?
Can the new shape support its own weight?

It can, but are you able to explain this curious result?
It can, but are you able to explain this curious result?

The mystery of which arches will stand under their own weight
The mystery of which arches will stand under their own weight

and which will not
and which will not

can be resolved with the aid of hanging chains.
can be resolved with the aid of hanging chains.

Remember how arches and chains are mirror images of each other?
Remember how arches and chains are mirror images of each other?

And how chains tell you the right shape to use?
And how chains tell you the right shape to use?

Well, suppose you want to evaluate a particular arch design.
Well, suppose you want to evaluate a particular arch design.

If you can hang a chain entirely inside the profile of that arch,
If you can hang a chain entirely inside the profile of that arch,

then it is a shape that can carry its own weight.
then it is a shape that can carry its own weight.

We call this the "chain test."
We call this the "chain test."

Does this thin semi-circular arch pass 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
It does not, because it is impossible to position a hanging chain

so that it lies entirely inside the profile of the arch.
so that it lies entirely inside the profile of the arch.

And that is why it collapses.
And that is why it collapses.

One way to make the arch pass the chain test is to make it thicker.
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.
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.
For this arch, we could do that by tilting the bottom pieces of the arch outwards.

As you can see,
As you can see,

when the arch shape follows the chain,
when the arch shape follows the chain,

it stands up just fine.
it stands up just fine.

Suppose, instead, that we added forces to the chain
Suppose, instead, that we added forces to the chain

to make it follow the shape of the semi-circular arch.
to make it follow the shape of the semi-circular arch.

As you can see, outwards forces
As you can see, outwards forces

applied at these points on the chain would do the job.
applied at these points on the chain would do the job.

They change the catenary shape
They change the catenary shape

into something more like a semi-circle.
into something more like a semi-circle.

Recall that a chain is a tension system,
Recall that a chain is a tension system,

while an arch is a compression system.
while an arch is a compression system.

Thus, we have to reverse those external chain forces when applying them to the arch.
Thus, we have to reverse those external chain forces when applying them to the arch.

When the appropriate forces are applied to the arch,
When the appropriate forces are applied to the arch,

it stands.
it stands.

And if we take them away, it falls.
And if we take them away, it falls.

The forces acting on the ends of the arch
The forces acting on the ends of the arch

are also important to its stability,
are also important to its stability,

and it is easy to demonstrate this fact.
and it is easy to demonstrate this fact.

If you prop your head on your hands,
If you prop your head on your hands,

and place your elbows apart like this,
and place your elbows apart like this,

you might be able to feel the forces
you might be able to feel the forces

that are keeping your elbows in place.
that are keeping your elbows in place.

Those forces have a vertical component,
Those forces have a vertical component,

which you can feel
which you can feel

as the table pushing up on your elbows.
as the table pushing up on your elbows.

They also have an inwards horizontal component
They also have an inwards horizontal component

that is carried by friction
that is carried by friction

between your elbows and the table.
between your elbows and the table.

You'll notice that these forces change
You'll notice that these forces change

depending on the position of your elbows.
depending on the position of your elbows.

When your elbows are closer together,
When your elbows are closer together,

the forces become more vertical,
the forces become more vertical,

and their horizontal components become smaller.
and their horizontal components become smaller.

However, if you move your elbows further apart,
However, if you move your elbows further apart,

you will discover that the horizontal forces increase.
you will discover that the horizontal forces increase.

In doing these tests, it is important that you not use the muscles in your arms
In doing these tests, it is important that you not use the muscles in your arms

to stop your elbows from moving outwards.
to stop your elbows from moving outwards.

If you spread your elbows wide enough,
If you spread your elbows wide enough,

the horizontal forces needed to support your elbows
the horizontal forces needed to support your elbows

can become greater than the available friction forces,
can become greater than the available friction forces,

and your model arch collapses.
and your model arch collapses.

Adding a rubber mat
Adding a rubber mat

can increase the friction forces available at your elbows
can increase the friction forces available at your elbows

and thereby prevent collapse.
and thereby prevent collapse.

Arches can take many shapes,
Arches can take many shapes,

from tall and narrow,
from tall and narrow,

to shallow and wide.
to shallow and wide.

And shallow arches like this one
And shallow arches like this one

can often carry a surprising amount of load
can often carry a surprising amount of load

if their supports provide sufficient horizontal forces.
if their supports provide sufficient horizontal forces.

In the real world, arch supports are often angled
In the real world, arch supports are often angled

so as to better transfer these horizontal forces.
so as to better transfer these horizontal forces.

Here are a couple of other interesting facts about arches.
Here are a couple of other interesting facts about arches.

The horizontal forces that semi-circular arches need
The horizontal forces that semi-circular arches need

can be provided by material that fills the space around them.
can be provided by material that fills the space around them.

We used blocks for our fill,
We used blocks for our fill,

but in the real world, stones or soil are typically used.
but in the real world, stones or soil are typically used.

In addition, if multiple arches are placed end-to-end,
In addition, if multiple arches are placed end-to-end,

the horizontal forces at their ends
the horizontal forces at their ends

can be made to balance,
can be made to balance,

and they can be supported
and they can be supported

on surprisingly thin columns.
on surprisingly thin columns.

Arches can be laid on their sides
Arches can be laid on their sides

and used to support large horizontal forces
and used to support large horizontal forces

like the water forces that act on the Hoover dam.
like the water forces that act on the Hoover dam.

As you can see in this top view,
As you can see in this top view,

even the Hoover Dam has a shape
even the Hoover Dam has a shape

rather like a hanging chain.
rather like a hanging chain.

Oh, and one last point.
Oh, and one last point.

Not all structures that follow an arch shape
Not all structures that follow an arch shape

actually function as one.
actually function as one.

In this bridge, for example,
In this bridge, for example,

the arch shape is strictly decorative.
the arch shape is strictly decorative.

Much more could be said about arches and chains,
Much more could be said about arches and chains,

but here we only have time to introduce the basics.
but here we only have time to introduce the basics.

As our team worked on this video,
As our team worked on this video,

we actually learned quite a bit about arches and chains.
we actually learned quite a bit about arches and chains.

We hope that you learned something, too.
We hope that you learned something, too.

Thanks for watching.
Thanks for watching.

We hope you will enjoy some of our other
We hope you will enjoy some of our other

"Engineering Models" videos.
"Engineering Models" videos.