How does it work ?
So, I am just going to talk about the case of the variable loads, the distributed loads.
You will see, it can be solved in the same way than usually.
So, if I have, here, a big variable
load which acts, the funicular polygon, with all
the loads, should go up here and go down here, but that is not the case.
Now, what is the shape that the arch would tend to take ?
The arch, when we add it a big force, it
tends to want to go down, here, and to go up in the right part.
But in the right part, we precisely have this reinforcement masonry which
says : "no, no, I am not ready to let you go up".
And thus, this masonry has as effect to add
a pressure on the arch, and this pressure is going
to stabilize the arch, making it come back to its place.
If now, my load moves to come on the other side,
then, that is the right part which now, tends to want to go down.
But the left part would like to go up, but on
the left, there is also this reinforcement masonry.
Thus, the structure is stable.
It also means that if you have such a bridge,
well, there is no question of unwisely taking off this
reinforcement masonry, because
it could easily lead to its collapse.
So, you have to proceed with caution during the restoration.
We have, here, the example of a big Gothic cathedral
which is surrounded by all a series of flying buttresses.
We are going to see that these elements are very important for
the stabilization of our cathedral. The vault, or the dome
which we have, here, tends to exert an inclined outwards force.
If there is nothing to resist to
this force, in all likelihood, the structure is going to collapse.
So, that is why the architects of the Middle-Age have invented this concept of
flying buttresses, and why they have put them.
The flying buttress is going to act by its weight; to simplify, I only make it act
in the middle of the length of the flying buttress,
but it obviously acts in a distributed way.
What is the effect of this weight ?
If we look at the funicular polygon, well, the force which comes from the dome
pushes in this way, until it meets the yellow
force, which is going to change its direction, and to make
this force go down in an inclined way, and we can see
that, the funicular polygon stays
inside the matter and our structure remains stable.
But you have already seen in the lower part,
the line of the polygon of forces tends to want to get out of the matter.
Then, how does it work ?
Well, it works in two parts.
There is first, here, the weight of this column which is not negligible.
We can see that this column is quite
thick, and moreover, here, above, we have an element which we often
consider as decorative, but which actually has a structural function. It is called the
pinnacle, and the function of the pinnacle is to procure an additional weight
to the column.
Both together have the effect of significantly
changing the direction of the new funicular polygon, in such
a way that it stays within the matter, and thus, a flying buttress
is a stabilizing element for a vault, or a dome of cathedral.
Approximately ten years ago,
we could read in the local newspaper
of Lausanne, "the cathedral stands by strings".
What was happening ?
If we look at the cathedral, we can see that in
this zone, and it is the same thing on the other side,
there is a certain number of flying buttresses which have the
function of, precisely, stabilizing the main nave of the cathedral.
However, these flying buttresses were made of a relatively friable rock
which was degrading, and it was necessary to replace them.
But, as I told you, it is not
a good idea to remove flying buttresses.
So, what did we do ?
Well, we have replaced them by cables
which I call internal
cables. Why internal ?
Because these cables link together two
parts of the arch, but are not linked
to the supports, unlike the first solution with
cables which I showed you.
Here we have a photo of the cathedral,
when the cables were present in the cathedral.
There were four cables according to the paper, but
we can see three, here, and in the infographics
of the newspaper, we can see, well, maybe it is not very accurate, we are not
totally sure that these cables were located at these places, but they had
the effect of exerting a inward force,
in this way, on the cathedral, as a replacement for
the effect of the flying buttresses which pressed
in this way on the vault to stabilize it.
That is clear that it is not exactly the same effect,
but was sufficient during the maintenance works for
the construction not to suffer, and for these
flying buttresses to be replaced, and the cables have been removed since.
We do not need to keep thinking in two dimensions.
We can also look at what happens in the third dimension,
and here, we have a very nice example, with the
dome of the cathedral St. Peter of Rome, built in the 16th century.
Soon after its construction, cracks were observed
in this dome which were approximately horizontal.
I am going to show you, later, how they were exactly.
These cracks,
if we look at a vertical section of the cathedral, we can see that
actually, we have two domes, one internal
dome, one external dome, that is
the one which we can see, and between both, we have an empty space.
This, it corresponds to the idea. It is necessary that the matter should be quite wide
for the funicular polygon to be able to stay inside the matter.
But it was not enough, and thus, it was observed, since
there are stairs within the dome, that there were cracks in the
internal dome. It lasted some
time, and in the 18th century, the mathematician
Poleni was given the task of studying this problem and of finding a solution.
What he did was to
look at how a sector of the cathedral works.
A sector, it is a slice which has not a constant thickness.
It is a little bit like a piece of cake, that is to say that in the middle
it is very, very thin, and on the edges, it is very large.
If we looked at
that in a plan view, it would have a triangular shape.
He represented the behavior of this arch, in a reverse
way using a chain, on which he placed weights.
You can see, here, circles of varying diameters which represent
the various points. Thus, he has obtained, for this dome,
this shape, which is the shape
of the funicular polygon corresponding to the weight of the
structure. Afterwards, he reversed this drawing,