Obviously, we could continue towards the
left doing exactly the same construction.
I spare you this construction but I
do it myself to have the complete drawing.
So, here, it will be C3 with
both loads, the two last
loads of 10 Newtons and then the
tensions T3 and
T4.
Obviously, the compressions are used in both
directions for each of these construction. We can make a few
interesting observations on the basis of this solving.
What we can see is that the compression is not constant in the
deck, it is lower at
the end and becomes greater in the middle.
Likewise, the internal force in the cables T1 is greater than the internal force in the cables T2.
So, the cables which are more inclined support
more loads than the cables which are less inclined.
On the other hand, we can obviously see that the cables T2 are
distinctly shorter than the cables T1.
In this picture of the transporter bridge in Nantes, designed by
the french engineer Arnodin, we can see that the stay cables have the
same configuration than in our
model, that is to say a configuration with
a fan shape where all the stay cables are hung on the top of the mast.
This configuration has the advantage, we have seen it, that the
internal forces are lower in the stay cables which are less inclined,
however, the stay cables are quit long and
it is especially quite difficult to all hang them on the top of the mast.
Thus, it is a configuration which has a few problems.
Independently of this configuration, because the
central span is much longer than the
adjacent spans, it has been necessary to use
vertical cables to anchor the structure in the ground.
On the bridge of this illustration,
we have another configuration of the stay cables which we call the
harp shape, that is to say that the stay cables are parallel to each other.
Which is striking in this configuration, it is
that the length of the stay cables varies very, very quickly.
The external stay cables are very long, while the internal stay cables are very sort.
Let's look at what is about the internal forces.
I just draw the right part of a cable-stayed
bridge, such as we have calculated it before,
except that this time, the stay cables have a harp
configuration and to get something, I also load each stay cable
with 10 Newtons. Then I can draw here this load of
10 Newtons for the free-body on the right.
I trace a parallel to the stay cable.
This is the tension in the external stay cable and this, it is the
compression, so we are going to call them, T1, C1,
T2 and C2. Here I have T1 and C1.
If I look at the second stay cable, there is
also 10 Newtons and its orientation
is still the same. Well,
what is interesting is that T2 is equal to T1,
So the internal force in the stay cables is
the same. However, the compression C2 is
quite a bit greater, in this case
double, but it is really greater than C1.
So, we can imagine that when we reach
the end, when we have the shortest stay cables,
we have a huge compression. However, we have
the advantage to have internal forces in the stay cables which are constant.
Note that before, the internal forces in the stay cables
rather tended to decrease, which was favorable.
An advantage of the harp configuration is
that we have more space to place the
anchors of the stay cables which are not easy to
hang up on the top of the mast in the harp
configuration.
As you have probably guessed it, it is possible
to make transitional configurations, as we can see it here.
Here, a half-harp configuration.
The stay cables are not absolutely
parallel but they are anyway well distributed
on the height of the mast, which enables
to have a little bit the advantages of both solutions.
It is also possible to make cable-stayed bridges with multiple spans.
However, if we remember of the necessity to anchor in the ground, as we had seen it
for the bridge of Arnodin, it is actually
necessary that the pillars should be stiff enough
to help support the vertical
loads which are very significant in the exceptional cases.
But the bridge of Millau, which is in service since 2004, have shown that it was really
possible to make this kind of structure in an efficient and elegant way.
In this video about the cable-stayed systems,
we have seen how to solve them by means
of graphical statics.
We have seen that there are several possible configurations
for the stay cables, particularly the fan shape or
the harp shape and we have seen that there is a
large compression in the deck of this type of structure.