Hello. In this video, we want to talk about

the efficiency of the various systems of trusses with a constant depth

with V-shaped, N-shaped, X-shaped and K-shaped diagonals,

as we have seen before.

After having seen all these systems of trusses,

you maybe asked yourself the question,

"but, which configuration is the most efficient ?

There is maybe one which is terribly unfavorable,

and another one which is very favorable !"

We are also going to see principles of dimensioning of the chords

and also of the diagonals, for trusses,

and then we will look at the bending moments.

This picture shows a calculation which has been done

by Professor Muttoni to illustrate

the amount of matter which is necessary to build a truss,

with chords with a constant cross-section, or with a variable cross-section.

Let's quickly come back to the cover picture to figure out what it is.

Here we can see that the chords have constant dimensions

over all the length; such as the diagonals and the posts.

Of course, it can hide something, because if the cross-section is --

and it is probably the case here --

if the cross-section is a tube, then it is possible

that the tube has much thicker walls near the supports

than the tube, for example, used for the diagonals which are at mid-span;

such as here, we use a larger tube,

and maybe with very thick walls compared to here.

However, you can realize that it is going to have a certain cost

to change, for each diagonal, either the size of the profile used,

either the thickness of the tube.

That is thus something which is not always done,

and it is absolutely possible to have profiles which have chords,

for instance, with a constant size, the dimension of the cross-section would be choosen

for the largest internal force, generally at mid-span,

and then kept constant till the end.

This is what there is here, when we say that the chords

have a constant cross-section or a variable cross-section.

The variable cross-section will lead us

to be more economical, but however, to have to work more in the workshop.

Here, we have the volume,

the amount of matter used to create the structure,

and it is quite proportionnal to the price of this structure.

So, the larger the value is, the more expensive the structure is.

Here, we have the slenderness ratio, L/h, that we have already seen earlier,

which qualifies the ratio between the span of our truss, L,

and its depth.

So, the more slender the truss is, that is to say the larger the ratio L/h is,

the more the price increases.

So, how do the various types of trusses compare ?

Well, we can see that, for the part of chords with a constant cross-section,

we have a very narrow gap here, represented in blue,

on the top of which we can find, using a bit more matter,

the K-shaped and N-shaped trusses,

and then, on the bottom, the X-shaped and V-shaped trusses.

But you can see that the difference is not very significant.

What is decisive is the slenderness ratio:

with a larger slenderness ratio, we use more matter,

the type of diagonal does not have a lot of influence.

If we optimize the size of the chords, that is to say, if we often change

the size of the bars used,

we will use significantly less matter;

but it is not obviously cheaper, because we are going to use more labor:

in certain countries, labor is cheap,

we are going to save money; maybe in some others,

it will not be the case.

And we can see that here, again, most of our solutions

are located in a quite narrow strip.

On the left part, here, we have, depending on the slenderness ratio,

we have the deformability, or the deformation.

All this is related to the service limit state,

we can roughly talk about the comfort of the structure.

So we can see that the more slender a structure is,

the more it is going to deform.

We find again here our two families of configurations,

the family with constant chords; and since we have used more matter

in the chords, it is logical that the structure deforms less.

And then the sky blue family, which uses more matter, deforms less,

while the pink familly, which uses less matter,

here, deforms more.

But here again, you can see that the differences

between the various systems are quite unsignificant.

We have already seen that the internal force in the chords times the depth

is equal to the bending moment, which is approximately equal

to the shape of the arch-cable. It means that the internal force in the chords is equal

to the shape...

... of the arch-cable...

... divided by the depth of its cross-section,

that is to say the depth of our beam in each point.

So here, I have a graph which shows this depth; here, if I have a truss

which has a constant depth, well, I will have a constant depth...

I take the shape of the arch-cable, I divide it by its depth,

and then I obtain the internal force in the chords;

in the upper chord, I have a compressive internal force.

This internal force is going to have the same shape

than the arch-cable, then a parabolic shape, which is going to increase

to reach its maximum in the middle, and to be roughly equal to zero

at the level of the supports, and then for the lower chord,

it will be the same, except that it will be a tensile internal force.

When we see this, we think:

"there are maybe other solutions, when we have a constant chord,

if we keep the cross-section of the chord constant,

while we dimension it for the place where there is the largest internal force,

then, near the supports, or even here, the section is much larger.

Maybe it would be possible to also play with the depth of the cross-section.

This is what we are going to see in the next videos.

We have seen, about the configurations with V-shaped,

K-shaped, X-shaped, etc. diagonals, that the efficiency, for these systems,

is comparable.

Likewise, the internal forces in the chords are comparable,

which means that we can choose the type of diagonal,

for trusses with a constant depth, in a rather free manner,

according to, for example, other constraints,

whether it be geometrical constraints, to make pass

certain installations or people, if the trusses are very large,

or else simply linked to a personal choice.

1215 Efficacité et moments

L'Art des Structures 2 : treillis, poutres, dalles et cadres

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