710 Arcs-et-câbles

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En provenance du cours de École Polytechnique Fédérale de Lausanne

L'art des structures 1 : Câbles et arcs

23 notes

École Polytechnique Fédérale de Lausanne

L'art des structures 1 : Câbles et arcs

23 notes

L'art des structures propose une découverte du fonctionnement des structures porteuses, telles que les bâtiments, les toitures ou les ponts. Ce cours présente les principes du dimensionnement et les structures en câbles et en arcs. Un deuxième cours présente les structures en treillis, en poutres et en cadres. Après avoir suivi ce cours, vous serez capable d'identifier les structures en câble, en arc et en arc et câble. Vous pourrez déterminer les efforts dans les éléments de structure principaux et procéder au choix de leurs dimensions en fonction du matériau de construction choisi. Votre maîtrise des constructions de statique graphique ainsi que de l'applet de calcul i-Cremona vous permettra de comparer diverses solutions et de choisir la forme la plus appropriée pour un type de structure donné. De manière générale, vous pourrez, en observant les structures autour de vous, reconnaître leur mode de fonctionnement, expliquer le choix de leurs dimensions et de leurs proportions.

À partir de la leçon

Arcs-et-câbles

Les arcs-et-câbles sont des structures qui combinent un arc et un câble. Cela leur confère des propriétés intéressantes et particulières. En tant que type de structures, elles ne sont pas très communes, mais elles serviront d'élément déterminant pour la compréhension et le calcul des structures en treillis dans le cours l'Art des Structures II. Ne cherchez pas ce mot ailleurs : c'est nous qui l'avons inventé, mais c'est un bon nom, très utile.

710 Arcs-et-câbles17:29

715 Variations sur les arcs-et-câbles13:45

725 Papeterie Burgo10:17

720 Structures haubanées9:39

730 Forces aux appuis - arc-et-câble auxilaire10:09

Rencontrer les enseignants

Olivier L. Burdet
Dr
Laboratoire de Construction en Béton
Aurelio Muttoni

Hello.

In this lecture, I am going to introduce to you

a new type of structures, called arch-cables, which

will be the last type of structures which we will see within the framework of this course.

We will see that arch-cables are a

type of structures in which the arches'

thrust, which we know well, is taken in

a different manner, which is why we will need to introduce

a mobile support.

We will see how to solve this kind of

structures and also how to model them using the applet i-Cremona.

What is a mobile support?

Well, in my model, it is a support

which looks a little bit like a roller skate.

So, here, we have casters, which are going to be able to roll on the horizontal surface

on which I am going to place my cart.

I call that a cart. What can

we notice? A displacement is possible

in the horizontal

direction. This means that there cannot be

a force at the support in this direction.

Thus, the only force at the support, here, will be a vertical support force.

We can notice that, as before, this kind of support also allows a rotation.

Let's also note that, even if we will not necessarily see often within the

framework of this course, we can very well have a mobile support which rolls on

a surface which is not horizontal, for example

a vertical surface, but also, possibly, an inclined surface.

We are going to see, obviously, in the rest of this

course, what the chain is used for.

Here, I have introduced the mobile support on the right in my arch.

When I have introduced two loads on my arch, obviously, the arch

wants to push, and as it cannot push since the mobile support moves,

the whole structure collapses.

The solution, as we have already seen it in the picture, is

to link the mobile support to the fixed support by means of a chain.

In this case, we can see that we can normally load the arch.

Let's look at the internal forces which act in our arch.

So, we have loaded 10 Newtons on our arch. Here, I already solve the case of

the arch and we will see afterwards the one of

the arch-cable to be able to compare.

So, for the arch, I have introduced

2 forces of 10

Newtons, And the equilibrium, I do it very

quickly, because you have already done it several times, and we have done it together.

So the internal force in the right leg and in the left leg,

as well as in the horizontal part of the arch.

Thus, this arch is only under compression.

Then, if we take an interest in the forces at the support,

using the force in the right leg and in the

left leg, in the other direction, we obtain the usual forces at the support,

that is to say an horizontal thrust and a vertical force, which is equal to 10 Newtons

of course, here. And then, we have a force,

V and a force H at each support. How is it with the arch-cable?

We are also going to load it with twice 10 Newtons.

Do not forget that there is a cable because, otherwise,

with the mobile support... You can see the symbol for the mobile support on the right, we will

come back to it in a little bit, it is a triangle which is on

1 or 2 casters, here, 2 casters, it symbolizes the fact that it can

move transversely, it is quite similar

to my little cart which you have seen before.

The Cremona diagram, with twice 10 Newtons,

is very similar to the one of the arch because, if we

make the equilibrium of the points of application of the

loads, we will obtain the same

thing. So, we are going to first take an interest

in the free-body which is just next to the support

on the right. What do we have in this free-body ?

Let's solve this free-body.

We obviously have an inclined leg of the

arch, we have the horizontal cable, and what

we know is that, we can only have a

vertical force at the support, which comes from the ground.

Well, let's come back to the Cremona diagram, it is this force,

here, in the leg along which we are going to travel in the other direction.

Then, we are going to add the tensile internal

force in the cable, and a vertical force at the support.

So, here, this is correct, we only have a vertical force

on this support.

Let's now look at the free-body close to

the left support, which I draw again.

What we have, acting on this free-body, is then an

inclined leg, which comes from the arch, the cable in

tension and, certain ground reactions which we will see a little bit later.

What we

know is that here, we have compression which acts and here,

we have tension, the force T which we have just determined before.

So, we have, acting on this free-body, the force T, well, I must draw it again

in the Cremona diagram, this time,

it is not possible to find a combination.

To this force, we add afterwards the diagonal internal

force in the leg and finally

we obtain that the vertical force at the support is the

only component which we have here, thus we only have a vertical component.

It is interesting to see that, just by introducing a single mobile support,

actually, we get the effect that on both supports, there are only vertical reactions.

Now, be careful, we got this effect because there

only are vertical loads acting on the system.

If we had an inclined load which acted

on the system, of course the support on the right

could only support a vertical component, however,

the support on the left could support an inclined component.

Let's see what happens if I now add an asymmetrical load

on my arch.

Well, as on a normal arch, we can see that

the arch tends to go down under the additional load and that actually

it is necessary to give it the opposite shape, that is to say to make the arch go up

here, where we apply the load for it to be stable.

What this means, is that arch-cables

behave, for the arch part, exactly as arches.

So, the stability problems are still here, and we

still need to stiffen them but we have seen before

that we know how to do it.

So, let's look at how to support the thrust

and let's see how it can be advantageous.

We have, here, a cathedral and if I indicate, here, the internal space

of the cathedral, we can see that it is actually quite limited.

There is a main nave and 2 secondary naves.

Everything we have around is essentially used for supporting the thrust.

On the right,

I have drawn the same cathedral, with still

the same volume which is useful for people

who are inside, and we have taken off everything which is used for supporting the thrust.

And, we can notice that it correspond to a lot of matter.

Of course, it makes the beauty and the

majesty of the cathedral, my purpose is not to take off

the flying buttresses, where they are, but if we

want to make modern and economical structures, it is interesting

to know that it is thrust.

If I simply introduce a cable, here, the

forces at the supports, under vertical loads, will only be

vertical and can thus directly go down in the

columns which we have on the left and on the right.

And all the flying buttresses and the other systems of buttress are not necessary anymore.

We do not need either to introduce a mobile support,

there are no casters because in this

case here, these columns, which are very long, very high, they

are flexible and thus they enable a slight transversal movement.

By the way, I think that you have a feeling, if we take off all

the matter of the flying buttresses, we feel a little bit that the arch will want to open.

Yes, it will want to open but when it will start

to open, the cable will be put in tension and

it will say : "I do not go further, there

will only have vertical internal forces from now on".

Let's now look at how to model an arch-cable with the applet i-Cremona.

I start by modelling an arch.

Well, you can see that it is really very similar.

So, I introduce a support on the left, a support on the right.

I activate the funicular polygon

and I give it the adequate shape.

There we go, it is probably the shape of the

funicular polygon of this vault.

Now, if I want to introduce a mobile support, I erase the second support.

If we introduce a mobile support, it will always be the second support.

I take it off, and I press the button "control"

on the keyboard or I click on the button "control" of the applet.

And I introduce, well, I am going to introduce it on the right, in this way you can

see well its symbol, a mobile support, which does not have the small

casters but we clearly see that the support is separated from the ground, it

can slide a little bit as on an air cushion, it is another symbol.

We can now place this support just at the place where it was before.

And, we can notice that, indeed, the thrust is

supported as for the other elements.

If I place my cursor in the middle of the cable, I obtain the internal force which acts

in this cable. Another example of arch-cable

structure, by necessity, it is a an arch, it has been used

for the shuttering of a big bridge, a quite imposing structure.

And because this bridge crosses a river, the builders of

that time have decided to create this arch and to transport it

making it float on 2 barges which we have on the left and on the right.

So, here, we have a barge, and, here, we have another barge.

Obviously, in this case, the only load which acts on this

arch is its self-weight but it is enough to create a thrust.

But, in this case, it is very, very

clear, the only things that the barges can offer,

they are ships, are vertical thrust, it is

the Archimedes's force, which is necessarily upwards.

So, it is absolutely impossible to support the thrust.

The solution which

has been adopted in this case, and we can see it in the water, with small marks,

small waves, is to introduce a series of cables, which link these

2 barges, in such a way that the thrust is compensated and that there is

only a vertical thrust, since obviously, it was impossible

to take an horizontal thrust with water.

Here is a picture

of the finished bridge, you can see the very ingenious aspect

of the solution which has been chosen, the

barge not only enables to move the formwork but

it also enables, once we have finished the arches,

lowering the formwork, to move it, to bring it

at the place where we want to build the second one, and so

forth, and thus to have a system which has

certainly been very economical for the construction of this bridge.

Until now, we do not have,

actually, seen yet how to really make mobile supports.

You have, on the left, the symbol for the mobile support.

So, we can enable, here, an horizontal movement, and we will only

have a vertical force, perpendicularly to the surface of the support.

In practice, we sometimes use supports shaped like a roller

but it is important to say that if the displacement

which we must have, must be significant, it is not

very easy and on the other hand, the surface

on which the rollers roll can easily becomes a

little bit dirty, if there are small stones which fall, if there is a little

bit of dirt, and thus these supports get fulled up and

do not necessarily work very well in the long run.

So, we have tried to find new solutions.

On the left, I show you a relatively old

solution, that is what we call a pendulum support.

By all appearance, we can see something

that we know well, something which is circular.

But actually, you can see that there is a small

tooth, on the top and on the bottom, that is a shape which is even

more particular, but there is a small tooth

which enables to control the way this support moves.

So, how does it work in reality ?

We have fixed to the ground

a plate with a slit, likewise, there is the same plate

with a slit which is fixed to the bridge.

And, our system, is placed between both, so,

that is a circle with a tooth which enters into this slit.

When a displacement occurrs, both

slabs remained where they were before,

but the system has moved diagonally

the slits are still aligned, in such a way

that the system does not get skewed. Thus, we have a displacement.

This displacement is caused by the fact that the

cable must lengthen to enable to support the internal forces, on the one hand,

and on the other hand, by the thermal effects, if the

bridge lengthens or shortens, further to temperature differences.

Then mobile supports are common solutions

for bridges and various types of structures.

A more modern system,

that is what we call a sliding support, a

system which is quite common nowadays.

So, for this support, we have the same system, we have a piece which is fixed

in a permanent way to the foundation, another one

which is fixed in a permanent way to the bridge.

It is generally steel. Inside,

we have, each time, a block of

rubbery products, it is

generally a synthetic product, like neoprene,

a rubber which is very hard and which is very

durable. Then, between both, we have two

layers of a material which we have already

seen, which enables well the sliding, which is Teflon.

Well, both, neoprene and Teflon, are protected brands, there are non-protected

versions of these materials, but they really enable

the support to slide, without having any

friction or almost without having any friction.

In this lecture, we have seen a new type of structure, the arch-cable, which is

the combination of an arch and a cable, in which the cable supports the thrust.

The use of an arch-cable requires generally the use

of a mobile support, which is a support

which enables a movement alongside the line between the supports of the arch.

The consequence of the use of an arch-cable is that the

forces at the supports are vertical, in the presence of vertical loads.

We have also seen how to use

the applet i-Cremona to model an arch-cable.

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