We want, in what follows, to take

the example of an elevator, because it is a simple

example, which corresponds to the knowledge, on

structures and on internal forces that we have learned up to now.

So, on an elevator, we have a cable

which links the elevator to the upper part of the building.

And this cable,

is used both to make the elevator go up and go down, but

also to support it, and thus, to insure the safety of its occupants.

Well, let's think for a while, what does it

mthe service limit state mean for an elevator ?

Well, it clearly means to be able to easily enter in

and exit from the elevator.

Let's say that if when the elevator is full and that it

arrives at a given floor, it is two or three centimeters too low, there is

a significant risk that the people are going to trip over

the ground at the exit, and this, it is

something which is not good for the usage.

So we want that the elevator arrive quite accurately at the level of the floor.

Conversely, we absolutely want to avoid, for the ultimate limit state,

we absolutely want to avoid failure.

That is to say, the fracture of the cable, and obviously,

with as consequence, the risk of fall of the elevator.

So, it is clear enough.

Now, we are going to see, how we can,

with the tools that we know, solve this problem.

Let's start by the service limit state. I remind you that it is abbreviated ELS.

We have, here, put some data about this elevator.

First, we have two elements in the elevator, the self-weight

G which corresponds to the self-weight of the cabin.

Its value is given as 10 kiloNewtons.

That is to say that the maker told us,

well, the elevator that we are going to deliver,

will weigh 10 kiloNewtons. And then, there is the number of persons

that we can put in the elevator, who are 10 persons of 80 kilograms, it

corresponds to 800 kilograms. That is to say 8 kiloNewtons of force.

Let's now think about the case of the cabin.

It is very easy to adjust the cabin when it

is empty, for it to stop, exactly at the level of the floors,

isn't it ?

What we wish, is now that, if this, if in

this empty cabin, we add 10 persons, then, 8 kiloNewtons.

Well, this cubicle, goes down to not more, to not more than delta l equal to 10 millimeters.

We consider than 10 millimeters, it is acceptable.

People are going to raise a little bit their foot.

And if the cabin is too high, or too low of 10 millimeters,

then there will not be any problems for the use of the elevator.

Now, we need another data.

It is the length of the cable, and for this

exercise, we are going to take a length of 50 meters.

Then, when the cabin is in the

stationary position and it is empty, it is at position 0.

And then, it is going to go down of delta l,

when we are going to place the loads Q inside.

I put only one man.

Obviously, there would be ten persons.

And delta l, we know, has to be less than or equal to 10 millimeters.

How can we solve this ?

Well, using the formulae which we have seen previously.

We know

that epsilon is equal to delta l over l.

Then, delta l, we know it, not more than 10 millimeters.

l, 50 meters.

Thus, we can calculate epsilon.

What we know, on the other hand, it is that E the modulus of elasticity, our cabin,

I forgot to mention it, is made by steel.

Then the modulus of elasticity

is equal to 205 000 Newtons per square millimeter.

As we know, it is equal to sigma over epsilon.

Then epsilon, we know it, we just have seen it before.

But, we do not know sigma. Then, we can get sigma

equals to E times epsilon equals

to E times delta l over l.

Sigma being the stress in the cable.

How can we get it ? The stress in the cable is, is equal

to the internal force in the cable, divided by the section of the cable.

And it is thus equal

to E times delta l over l. What is the value of N ?

In our case, N is equal to the load of the occupants.

That is to say the 8 kiloNewtons that we

are going to add from the stationary position.

The weight of the cubicle does not play a role in this, in this case.

Since we know N, what, what are we looking for ?

We are looking for A, we get

that A is equal to N over E

times L over delta l. Here, I believe that we know everything.

Thus, we can substitute. N is equal to 8,000 Newtons.

We have to be careful, with the units. I am going to do

everything, in Newtons and in millimeters. The length is thus equal to 50,000 millimeters

divided by the modulus of elasticity of steel which is equal to

205,000 times the limit of 10 millimeters.

And all this, if I calculate it,

it is equal to 195 square millimeters

Then, how to get

the diameter of a cable whose we know the area ?

So, everyone knows the area of a cable, as pi r squared.

For us, it is much easier, for the

rest of this course, to use a very

similar formula which says that the area of a circle

is pi times the diameter squared, divided by 4.

If you try it mathematically, you will see that

it is equal to pi r squared, we do not use the radius, because we cannot

really measure the radius easily. So, what does it mean ?

It means that D is equal to the square root of 4 times A over pi.

And this number is roughly equal to 16 millimeters.

Thus, for the service limit state, it is necessary

that the diameter of the cable of the elevator should be 16 millimeters.

We can now deal with the ultimate limit state.

You remember of the idealized stress-strain

diagram, with, here, the value fs for steel.

But it is clear that when we buy steel,

we do not know if it is going to be exactly as

good as it should, as it should be.

On the other hand, it is very true within the framework of the case of the elevator.

Maybe it is going to wear out, with time.

Then, what we are going to do, it is that the

dimensioning is going to take place at a lower level.