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Hello. In this video, we will look at which

shape takes a cable according to the type of load to which it is subjected.

What we have seen until now, it the case of loads

which are distributed along the axis of

the structure, often horizontally.

But it is not a necessity, as it was the example of the case of the Golden Gate bridge.

There is also the possibility that

the load is uniformly distributed along the length of the

cable, and we will see that it leads to a

slightly different shape which has a particular name.

In this video, you can see a

cable subjected to 13 loads uniformly distributed,

and equal, with an equal spacing, which thus take the shape of a parabola.

Then, we superimpose on this white cable, a

red cable which is simply subjected to its self-weight.

In the case of the self-weight, the self-weight acts

regularly, not along the horizontal axis, but along the axis of the cable.

The cable weighs, for example, one kilogram per

meter, but per meter of length of the cable.

If we take a close look on these two cables, we can indeed notice that the

red cable is lower than the white cable, than

the cable which has loads uniformly distributed along the horizontal axis.

In the middle, they are approximately at the same level.

This, it is OK.

But here, on the left and on the right, it is always the red cable which is lower.

Why ?

Well, because if we look at a small segment of cable,

I am going to draw it in red, which as a certain small weight which

acts along its length, let's say one Newton per meter.

Well, when this segment of cable is inclined, to cross

the same horizontal distance, I would need a longer segment of cable.

Thus, this cable here will be heavier

for an equal horizontal distance, for

example, the distance between two red weights.

We can see this in this figure.

On the left, we have a cable which is subjected to a uniform load,

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along the axis

x. On the right, we have a load

distributed

along the length of the

cable. If we represent it

along x, we obtain, here, in the

middle, the same load than on the left, q, one kiloNewton per meter.

But as the inclination increases, we get a

load which is larger than q, one kiloNewton per meter.

And since this load is larger

at the ends, well, it is logical that this cable should go down

more vertically. On the left, we have

a second order parabola which

is well-known.

On the right, we have a curve which we call the catenary.

Not very

surprising, since it is really the shape that

a chain (catena in latin) takes under its self-weight.

And it is a curve that some of you may have

studied, which has a mathematical definition a little bit more complicated.

Well, I do not want to go into further detail for this course.

What is important is to be able to distinguish

that according to the nature of the loads, from where they

come, the shape of the structure,

the proper shape that the structure should take,

will be slightly different.

If we compare these two shapes, we can notice that,

for what I did on the blackboard,

we have maximized, here, this distance between the black cable,

so the second order parabola and the red cable, so the catenary.

In reality, our structures will usually be much flatter, and

the difference between a parabola and a catenary is quite small.

So, I would like that you

clearly keep in mind that these two shapes are not the same,

and that they exist and

you must be able to identify them.

However, at the same time, if you are looking for

the proper shape to give to a structure, it is not very important.

If you you draw second order parabolas, and that afterwards

we have to build catenaries, the geometric difference will be quite small.

To conclude, two examples of structures.

First, the exhibition hall of Portugal,

for the Universal Exhibition of 1998 in Portugal.

We have here, an extremely thin structure, with an

approximately 130 millimeters thick blade of concrete,

thus, something which is extremely slender, taut between two supports.

What we can immediately notice, is that the supports themselves are significant.

Well, it is not very surprising, since we know that if

we have an extremely taut cable, like this, what does it mean ?

It means that there will be large forces at the supports, particularly large

horizontal components of the forces at the supports, and these horizontal

components must be carried by structures.

That is why these vertical structures

have numerous walls, a lot of intermediate walls.

But it is a very beautiful structure which

covers a public space in a very enjoyable way.

The second example it is this airport hall

in the United States, with also the requires to have a

large surface with the least amount of supports possible, which

is made possible by a cable-type structure.

Indeed, we have a structure with a quite

low thickness.

We do not see it well here, because there is an edge beam which has been introduced,

but we have a structure with a quite low thickness which is

taut between the supports that we have on the left and on the right.

Note that the supports, the more or less

vertical columns, are themselves quite significant to be able to carry,

the horizontal component of the internal force which is

very significant, since it is a structure which is

also quite flat.

These two structures take the shape, essentially, of catenaries.

In this video, we have seen that the loads can be either

distributed along the axis of the structure, either distributed along the length of the cable.

It induces differences in the loads which act on

the cable, and in the shape which it takes.

If the load is uniformly distributed along the axis

of the main structure, then the cable takes the shape of

a of second order parabola, as we have seen in the previous video.

If on the contrary, the loads are, in a predominant way, distributed along the

length of the cable, then the cable takes the shape of a catenary.

These two shapes are however quite similar and it is not

very important, in any case at this stage of the design, to distinguish

which one is which one.

However, within the framework of this course, you will be

asked questions about the type of shape that a cable takes.

Thus, it is important to be able to make the distinction at the theoretical level.