So now in stage two we want to take that propping force of people equal to forty point one one killiney
fans.
And we want to apply it directly to our frame.
Now we know we can supply forty point 1 1 killens because we don't know the amount of sway that that
would cause.
And we need to know the amount of sway in order to work get these fixed end moments to process truth.
Question.
So what we're going to do is we're going to apply in our arbitrary way.
Kate.
So what we're trying to do here is apply that people to forty point one one and that will induce a sway
which is Delta.
But as I've said we don't know what the magnitude of daudi is.
And so we can't work get fixed and moments.
So this is where we say let's apply an arbitrary sway.
So let's just say.
Dowds equals a question mark.
We have no idea.
So to process things forward let's apply.
In Arabic srey and Arabic Cheree sway which we're just going to call Delta prime.
Okay.
And then we're going to calculate so and calculate the corresponding p prime corresponding p prime.
OK.
So now let's just work on the basis that we've applied some arbitrary sway and work at some flixton
moments or some arbitrary fixed and moments or so.
Let's consider member a B first.
So member a B.
So we've locked joint B.
So we have a central line where the structure used to be and then with luck joint B with the plain arbitrary
way and so we have something like this now.
That sway will induce moments and those moments are going to look like so.
So that was a that was B.
And that would mean that that is f e n B to A.
Equal to right fixed on both ends with a sway of doubt.
So we can say that the fix and moment is six years.
I doubt a prime over L squared.
Now that's tree meters high so that's gonna be tree trees as 9 and solid on it and up with minus two.
He Delta prime over 3 and the exact same is going to be true for the bottom.
Now it's an anticlockwise moment and that's where the minus is coming from.
And it's going to be the same for fpn.
Maybe now the exact same if we look at mamber at c d.
We're gonna find the exact same magnitude Mormans OK.
So I've just read drawing and I will say we have the same four member saeedi says no need to draw all
over again.
So now this is where we want to take these fixed end moments and we want to turn them in two more easy
to manage numbers because all we're really interested in here is the relative magnitude of these fixed
and momence our obituary arbitrary.
Ms.
So let's summarize we had warned for member a B.
We had one for the other end of that mamber which was b.a. and we did the same for DC and c d and we
worked out that the values we can cancel out the E R Delta prime for all of them because remember the
relative magnitudes were interesting.
So that was minus 2 over 3 and in fact it was the same for each.
So mine's two or three minus two or three k now that we can simplify that and say look there are the
exact same magnitude.
Let's just call them all minus one minus one.
WAYNE SWAN and minds want because if I the same values.
So our summerize here and say that we're only interested we're only interested.
In relatives in the relative values of the arbitrary
past and moments because these are going to facilitate a moment distribution at the end of that moment
distribution we're going to take our arbitrary moments and we're going to scale them to bring them back
into reality if you like.
So let's now proceed through a standard moment distribution for that structure with these fixed end
moments.
So now that we finish the moment distribution we straight away can label these as two prime moments
and the task now is determine is to determine what is p prime.
So we want to determine determine p prime which caused or which causes.
Delta prime.
Right.
So we applied Delta prime and arbitrary sway and we're going to now back calculate to work out what
is p prime the value that would cause that's way the value of the force that would cause us sway.
So again we only do the bare minimum here.
We just want to work out what is the value of the sheer force at the top of member A B and at the top
of member C D.
So taking some of the moments at that point a.
By inspection we know that the sheer force of the top of member Cedi is the exact same because we're
dealing with a nambour that is the same length and it has the same end moment applied to us.
So we'll just say by inspection
the c l o the sheer force below joint C is going to be again Point 4 6 9.
All right or straight away we can say that p prime the force that is going to cause that sway is going
to be some of that shear and that shear p prime is going to be two times not point four six nine that
is the force that is acting.
Crew member of sea and therefore that's the force that's causing that's where we are right now.
We can work out our sway factor lambda our sway factor lambda is going to be p over p prime.
OK.
So what do we do in here.
We're basically we're saying that there is a linear relationship between the force that causes the sway
and the monmouth's that is generated within the within the frame.
So the linear relationship or the proportionality between the force causing the sway and the moment
has been given to us by identifying p prime and knowing the moments that cars are about the moment that
resulted from that.
So we can we can use the knowledge of that linear relationship to scale our tree stage to moments.
So let me see that's going to be forty point one one.
The original propping force over point 9 3 which is Ones which was from here now point 9 3 8.
And that's gonna let That's going to give me landham equal to forty two point seven six.
So remember our our our tree moments are our stage two moments were based on our arbitrary fixed and
moment so the actual values of those fixed end moments the absolute lies were not significant or important
it was what was important was the relationship between them the ratios between them.
So they were the the absolute values need to be brought back to reality so to speak.
And this is the factor that brings them back to reality.
The best way to summarize this stage of the analysis are to wrap up the analysis is again by a little
table.
So that's summariser more in moments we are going to summarize are two prime moments.
These were the our literary stage two moments that were caused by our our obituary sway.
We then turned those arbitrary stage two moments into our actual stage two moments which is the model
which at the moment that is generated by the actual sway force of forty point one one and we got those
by taking out our sway factor alarmed and multiplying it by an to prime so to prime get brought back
to reality.
Back to real values.
By multiplying by the sway factor.
Because of that linear relationship that gives us our stage two moments and then our final moments are
just stage one plus stage two.
So the final thing I think to be I one plus two.
So what did we have.
We had member a B.
It was just to summarize again the superposition part was taking these moments and adding them.
To these moments.
And that was superposition
and that gave us our final moments which was which was these guys
final sway frame moments.
At that point you revert back to the standard procedure of considering free body diagrams for each of
your sub elements within that structure and working out the sheer force diagram the bending moment diagram
under deflected shape so just to kinda close the loop on this one and wrap this one up.
I'll take the information in this table and just sketch out the bending moment diagram for demonstration
purposes.
Needless to say you should definitely try to tackle the bending moment diagram sheer force around and
complete deflect it shape yourself.
Okay so that's the complete bending moment diagram.
That results from the influence of external loading Plus the sway effect of the frame because it's not
a proper frame in the original question that we were asked.
So does a lot in there was a lot of different steps and stages but the logic is actually fairly straightforward
and it all boils down to the fact that there is a linear relationship between the sway force that we
apply to a frame and the moments of the bending moment that are generated as a result of that sway force.
And so the whole this whole idea of applying an arbitrary sway force is just simply to allow us to get
that linear relationship to identify that linear relationship for the particular frame.
The relationship between propping force or lateral sway force and the moment that are generated and
once we know that linear relationship we can then apply this to our notion propping force to generate
our actually will sway moments.
Once you've got that concept idea clear in your mind you're more or less have this team nailed it.
It's pretty much straightforward analysis techniques that we've used previously at all at once you've
understood that little nugget of information.
So do one more of these sort of.
Step by step examples in the next lecture.
Before you can then go on and tackle some questions yourself.