>> These are iron structures.
And the consequences of industrialized iron is that it's lighter than structures of stone,
which now we have the portent for major failure.
At first, engineers didn't really understand this idea of how to design
with iron, how to make those connections.
And this concept of buckling and stability in members,
was something new that was being studied by those engineers.
So, unfortunately, sometimes we did see failures happen with these bridges.
And one of them was a bridge in Scotland called the Firth of Tay.
This bridge collapsed in 1879, killing all aboard the rail line.
There were 2 great barriers to Scotland's east coast travel.
They were the Firth of Forth and Firth of Tay,
both stormy estuaries on the east coast of Scotland.
After the Firth of Tay Bridge collapsed, the next time the Scots had to build a bridge
over a stormy estuary, they wanted to make sure it wouldn't fail.
And so, they built the Firth of Forth Bridge, designed by Benjamin Baker.
And we see how massive this structure is and it's still standing today, spanning 1,710 feet.
This was the longest spanning bridge in the world and it's also a railroad bridge.
So, it was a great achievement by the engineer, Benjamin Baker.
To give you a sense of scale, as to how large the members of this bridge are,
if you zoom in close to the supports, we see containers.
We can see the relative size of those containers to the members of the bridge.
It is a massive structure.
This close up image also gives you a sense
of the different perspectives one can get from a bridge.
So, close up, the Firth of Forth looks like a massive bridge full of clutter.
Whereas from far away, the bridge looks much lighter
and you don't get that sense of heaviness.
Now let's look at this bridge and dissect it from a scientific point of view.
The form for this bridge is called a horizontal cantilever.
And to simplify the analysis, I'm only going to look at one span of the Firth of Forth.
The cantilever arm spans from the support towards the center.
And the back span is called the anchor arm.
It anchors that center support towards the anchors at the end.
And in the center, we have what we call a suspended span.
We could think of this as essentially 2 seesaws.
So, we've all been kids playing on seesaws and the seesaw has, let's say a center support,
representing in this bridge, that center tower.
If we suspend a weight between these two seesaws,
we know that it's not going to be stable.
The seesaw will tend to rotate and it will no longer be horizontal.
To make those seesaws horizontal again, we know that the tips of them have to be pulled down.
And that is what those anchors do.
So, we can think of this Firth of Forth bridge as essentially 2 seesaws
with a suspended rate between them.
Let's define the reaction at the anchor, that downwards reaction, as Ra.
And let's define the suspended weight, a downwards reaction, W. So,
will the reaction at the seesaw support be up or will it be down?
We need equilibrium.
The sum of the forces in the vertical direction have to equal zero.
Therefore, the reaction at the seesaw supports must be up.
Let's define this seesaw reaction Rs.
Since the arms of the seesaw, meaning the size of the seesaw to the right and to the left
of the support are of equal length, Rs of S, must equal W. Meaning,
the seesaw support must equal that weight that's suspended.
In that case, what is the magnitude of the reaction at the anchor Ra, in terms of W?
Do you think that the reaction at the anchor Ra, is equal to W, W over 2, 2W or 2/3W?
We can solve it in 1 of 2 ways.
The algebraic solution tells us that the forces in the upwards direction,
equals the forces in a downwards direction.
So, 2W is going up.
And W plus 2Ra is going down.
And solving that, we get Ra, equals W over 2.
Another way to look at it is to divide that system into 2 seesaws.
So that weight W, half of it is going to 1 seesaw
and the other half is going to the other seesaw.
So that you know that if you're friend weighs W over 2, you must also weigh W
over 2, to keep that seesaw horizontal.
This double seesaw example is exactly how the Firth of Forth Bridge acts.
So, we have the suspended weight in the center W. And then we have the supports
at those center towers, so to speak, is going up W. And then it's anchored down W over 2.
Now that we understand the reactions, let's look at the internal forces in the arms
of the cantilever and anchor arm.
In this lecture, we're not going to try to solve for the magnitude
of the stresses or forces in those arms.
But we're going to try to define is it intention or is it in compression?
Benjamin Baker did a physical demonstration to illustrate
to the public how the Firth of Forth Bridge acts.
So, he had 2 men sit on a chair.
And they were holding another man in the center, who was the suspended weight.
And then they had some bricks anchoring down.
They acted like the anchors pulling down.
Just like in that seesaw example I just gave you.
So, do you think that those men's arms are in tension or in compression?
And those wood pieces that they're holding between their fingers and the seat,
are those wood pieces in tension or compression?
We did a similar example to this in my classroom,
where I asked my students the same question.
This is an easy experiment to do on your own and to build.
So, do you think that these students’ arms are in tension or in compression?
After the experiment, I asked them, were your arms being stretched or compressed?
And they knew for sure that their arms were being stretched.
And that means that their arms were in tension.
Meaning, that the upper cord of this cantilever is in tension.
And the bottom pieces of wood, the reason we used wood and not rope,
is that that wood is in compression.
If we had used rope instead of wood, the experiment wouldn't have worked.
So, the answer is the top cord of these horizontal cantilevers are in tension.
And the bottom cords of these horizontal cantilevers are in compression.
So, in this lecture, we looked at some big metal bridges for railroads.
We looked at the Britannia Bridge, made of iron.
The Saltash Bridge, also made of iron and the Firth of Forth Bridge,
which was actually made of steel.
What I didn't have time to talk about is the Eads Bridge in Saint Louis, which is also made
of steel and the Garabit Bridge designed by Eiffel.
Eiffel is famous for his tower, but Eiffel is also a famous bridge designer.
And the Garabit is probably one of his most famous.
Next time, we cross the Atlantic and come to America,
where we're going to see John Roebling is designing some magnificent bridges
for railroads as well.
I hope you'll join us.