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.
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L'art des structures 1 : Câbles et arcs
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Equilibre dans le plan
Après avoir vu comment les structures unidimensionnelles (sur une seule ligne) sont en équilibre et peuvent être dimensionnées, nous passons aux structures bidimensionnelles, qui se situent dans un plan. Les conditions d'équilibre sont présentées.
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Olivier L. BurdetDr
Laboratoire de Construction en Béton
Aurelio Muttoni
Hello.
In this video, we will see how to explain the equilibrium of two persons
who are practicing tug-of-war.
We will see how the weight of the persons intervenes, how
we can determine the internal force in the cable, and what
are the effects of the ground on the persons who
are practicing tug-of-war.
A very simple exercise, it is an exercise of tug-of-war in which both persons
are exactly identical. For this case, we have a
person on the left, with a weight of 800
Newtons, a person on the right, with a weight of
800 Newtons. Then we are going to have an internal force in
the rope, and an internal force under the feet of the person,
which we will be able to reveal by means of a
free-body. Actually, we want to take an interest in three
free-bodies, the free-body of the man on the left, cutting
just under his feet, and cutting the rope.
The free-body of the man on the right, which is exactly the same.
And finally, a little free-body with
a segment of cable on the middle, whose you will see that it is useful for us anyway.
So, first, we have the weight, the 800 Newtons
of each of these two persons. We
have the effect
of the rope on the man on the left, on
the intermediary segment and on the man on the right.
So, a priori, we are going to say that there is a tension T one, and a tension T two.
But if we have a look on the equilibrium of the free-body in
the middle, we will see that there are, acting on this free-body,
only two forces, the force T one which pulls towards
the left, the force T two which pulls towards the right.
Thus, necessarily, T one must be equal to T two, and then, is equal to
T, which we will use in the following. So, we needed this
free-body anyway. If we extend the line of action of the
force T on the left and on the right, it
will cut the line of action
of the force of the weight.
And, to have the equilibrium, it will be necessary
that the force under the feet should be converging with these two other forces.
Then, that these three forces are converging.
And we thus get, that the force, under the feet
of the person on the left, S one, passes by this point.
And likewise,
the force under the feet of the person on the right,
S two.
Let's draw the acting forces, for the
free-body on the left in the Cremona
diagram. So, here, we have
the weight of 800 Newtons.
We are going to have the force T
which acts on the horizontal axis. We are not totally sure
until where. Then, we know, on the other hand,
the direction of the force S. I make the transition,
a little bit quickly. Excuse-me.
So, we get the force S one, and the force
T which express the equilibrium of the free-body on the right.
Thus, this person is in equilibrium, and we can read
graphically the value of the internal force in the cable, which is equal to T.
Let's now take a look at the equilibrium of the person on the right.
To do that, we are going to reuse the segment T that we have already drawn.
In this way, we are not going to introduce any mistakes.
So, we are going to introduce the new force of 800 Newtons.
We are going to reuse the segment T, in the other direction,
and we are going to close the polygon of forces expressing the equilibrium of the
person on the right, by the force under the feet, S two.
This construction, in which we combine two polygons of forces,
together, with one or several shared forces, is called a Cremona diagram.
This Cremona
diagram will be a construction that we will use very often, thereafter.
So, it is very important to understand well how we construct it,
and to see well the meaning of the various components of this diagram.
Let's now look at a tug-of-war
a little bit more realistic, between two persons.
The person of 800 Newtons, that we know
well, and another person evidently lighter.
It is quite clear that the lighter person was not able to stay
in equilibrium, when the heavier person,
on the left, began to lean back.
But let's try to see why, in the Cremona diagram.
We have, here, on the left, a person of
800 Newtons, and on the right, a person of indeterminate
weight, but clearly
lighter. We draw the line of action of these two
forces, since we will need them after. And we
draw the line of action of the internal force in the cable.
This construction enabling us to obtain the
direction of the internal force under the feet.
We can see that this person, in this case, did not really
lean on the heels, on tip-toe.
So, here, we have S one.
Then, we can notice that the second person is more leaned back
with an internal force S two, under her feet. Let's now have a look at the Cremona diagram,
with the force of 800 Newtons, the
direction of the internal force T, the tension in the cable.
And, we copy the inclination of the person.
It gives us the complete solving of the equilibrium of the person on the left.
Then, with the internal force T in the cable, and the internal force S one under the feet.
Let's look at what is about the person on the right.
Using the construction of the Cremona diagram,
we already know the internal force T, and we directly
obtain the internal force S two,
under the feet of the person, and, we deduce from that what must
be the weight of this person. We do not need to
know it, but we could determine it graphically.
What we can notice
that the inclination of the person
on the left, alpha one, is lower than the inclination alpha two.
We can see it even more clearly here.
We have alpha one, here, alpha two.
Just a this moment, we still have the equilibrium,
but you have seen in the video that just
after, the person on the right loses her equilibrium, while
the person on the left leans down a little bit more.
Why is it
possible ?
Well, precisely because, if we
supposed that both persons had the same
friction capacity, then, the person on
the left could increase more his inclination without sliding.
Actually, if we take a closer look on the feet, we
notice that the person on the left had very favorable shoes.
So, in all likelihood,
the alpha max of one, is greater than the alpha
max of two. Thus, the person
on the left would have been able to lean back more than the person on the right and would have won.
But on the other hand, if we come back to
the previous diagram, what we can see is that
the person on the right, because she is
lighter, must, anyway, in all cases,
lean back more than the person on the left to offset the tensile internal force T.
I should have put the second arrow on T
for the Cremona diagram to be complete.
So, in these conditions, the person on the right has simply no chance to win.
Even if, actually, both friction
coefficients would have been equal, she would not have any chances.
The only
configuration, in which the person on the right could have a chance, that is if
the heaviest person wore, for example, roller skates.
In this way, the person on the right had a chance to win this game.
We can ask ourselves the question, when we see that, to know if tug-of-war
is really a sport, since actually,
what counts is a friction coefficient.
We can imagine that all the participants will have good shoes and
a good friction coefficient, and
on the other hand, it is the weight of the persons.
So, obviously,
the heavier team is going to win.
In conclusion, we have seen in this video on tug-of-war,
how to isolate the free-bodies which
enable us to determine the individual equilibrium of each person.
We have been able to construct the polygon
of forces which expresses the equilibrium of each person,
and combining these two polygons
together, we have created a new construction
which is called the Cremona diagram, which will be
very useful for the rest of this course.
We have also seen the dominating influence of the weight, and of the friction
coefficient in tugs-of-war.
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