In this lecture what we want to do is Takao the procedure for analyzing sway frames.
So it's a reasonably logical procedure but it takes a little bit of thinking to get it clear in our
mind.
So in summary to start off we said that in a way framed the bending moments are derived from two sources.
The first source was the external loads and the second source was the sway itself.
Okay so we said the moments.
Derived from two sources.
The first is the external loads
and the second is the sway itself.
And were carling's sway doubt degree clarac delta.
Okay so now to perform this analysis we're going to use a principle called superposition this is a very
common principle use and structural analysis and it's basically summarized like this let's imagine we
have a frame.
So that frame is subject to a lateral load of 10 killens promesa.
And that's going to induce a sway in the frame.
This is a classic example of a sway frame and in this case it's support afren.
OK.
So if John did a fleck that shape and we've shown the size on which tension is occurring on the members
of that frame now the bending moments that are generated in that sway frame can be determined as the
combination of the bending moments from two separate frames superimposed onto each other.
So the first frame is going to be the exact same frame that we started with.
But we're propping the frame so we're turning it from a sway frame into a propped frame by adding support
at this location here.
That's going to generate a propping force which will just label p for the time being.
So we very obviously can't stop there.
We can't simply say that that's way frame on the left at the equal sign is equal to the proper frame
on the right.
The eagle sign up plainly makes no sense to us.
So we have to add to that the bending moment that would be generated from this frame which I'm going
to call stage 1.
So the bending moment that they generate from my stage 1 frame have to be added to the bending moment
from a second frame.
So what essentially has to do is cancel the effect of that propping force P..
OK.
So now I have.
Well and Carling might stage to ring.
So this is this is essentially visually represented here.
The the principle of superposition.
I take my frame I start with let's call that stage zero just for the purposes of this discussion here.
So I take my stage zero frame and I got a I got to represent that by the combination of two separate
frames.
Or if I want the bending moment for the stage zero frame I have to take the bending moments from the
stage when the stage to frame.
So I'm starting off with a propped frame and that's the type of analysis which we've already completed
and we're very comfortable with analyzing prop frames and determining the magnitude of that propping
force and then we have to add to that this second frame or stage to frame where we're essentially cancelling
out the effect of that propping force.
So the combined effect of Stage 1 and Stage two is equal to stage zero.
And that is essentially superposition.
So what we can do is summarize that.
So one would say is superimpose.
That just means add together.
Superimpose the moments.
Free stage.
To.
On to stage one.
Stage one moments.
To get.
The final sway moments.
Sway moments.
And that would be the moment for the frame which I have labelled stage zero.
OK.
So that essentially is is laying out forest a two step process.
We're going to have to follow when we're analyzing and sway frame.
Stage one is to analyze a proper version of that frame and get the bending moments resulting from that
analysis and stage 2.
There's going to be analyzing the frame with a low P opposition to the propping force essentially cancelling
out the propping force.
So we have to analyze that second that second stage too.
And that's really what we're going to focus on here because stage one we've covered in this course already
that's just analyzing prop frames.
So what we now need to start thinking about is how do we analyze that sway frame with that propping
force p.
Notice that the stage one frame has the external loading applied to us.
That's where we're capturing the effect of external loading on the frame.
And Stage 2 does not have any external loading it just has the propping force or rather the reverse
of the propping force and using sway.
And so what Stage 2 that will get those swaying moments from we can start thinking now they have how
would we go about analyzing stage 2.
So we're going to develop a sort of a logical argument in our minds here.
So let's give it a heading.
Stage 2 we're going to have to apply the propping force.
That's the first that's the starting point.
So apply the prop in force piece or remember from our stage 1 analysis we will have determined the magnitude
Fritzy So we know the magnitude of the force we have to apply.
So apply propping force to frame.
Right now there's a problem we run straightaway into a problem because when we think about it when we're
applying a propping force to frame we're going to get we're going to generate moments due to the sway
of that frame.
And you remember previously we said that if I have sway the moments that I get as follows.
So let's imagine I had a member of simple mamber a be fixed up all ends and it undergoes a sway which
has a magnitude of doubt.
That's essentially what we have in our frame we're going to have to work at the bending moment that
generates Duse that's where.
Now we said previously that those could be calculated as f e m from A to B was going to be six times
e in the flesh.
The rigidity of the mamber times the swaim magnitude Delta over the length squared and we were going
to get the exact same on the bottom.
We can work it fixed and moments but we strayed away.
Notice that they depend on the magnitude of this way Delta.
What we don't know Delta.
Okay so let's let's say we don't know.
We don't know how much sway.
How much sway.
I.e. that's just Delta to our propping force will cause.
So we know the magnitude of the propping force but we don't know how much swayed at all and that will
cause and so we don't know the magnitude of the bending moment that that sway will generate propping
force causes.
Case of what now.
So now this is where this is the mental hurdles come in that we have to sort of wrap our minds around.
Let's assume right let's assume as the clever part.
So let's assume we apply an arbitrary sway.
An Arab lottery value doesn't matter.
Okay we're just gonna call it Delta pride.
So let's assume let's assume we apply
Arab literary literary sway.
And we've got we're just going to call Delta Prine.
Okay so this allows us to calculate an Arab country set or fixed and moments right away.
I'll just say basically this gives us gives us our arbitrary our arbitrary fixed and moments and others
not those by FEMA were a prime on the and on k.
So just arbitrary values.
So let's all rise where we at this point.
We have an arbitrary sway.
So we don't have an owner far.
We're just gonna say Delta prime.
That's gonna allow us to determine in Arab Atri set of fixed and moments FEMA prime K I can do a Mormon
distribution based on those are arbitrary moments which will give me a set of finer set of Arab literary
Arab shary moments that post distribution after doing a distribution and work on a label those An with
a subscript to a Roman numeral 2 and then a prime.
Okay just to indicate the arbitrary nature of those moments right.
And then at the very end of the process we're going to get an Arabic Shehri propping force in our treasury
propping force
which we will call p prime.
OK so that is that is the the process basically from stage to analysis.
Now when we have our arbitrary propping force then what do we do.
Well the key to this is as follows.
The key is as follows.
This is the bit that you have to get clear in your mind and once you realize this it's pretty straightforward.
The absence of propping force p prime has a linear relationship with at the moments the set of moments
to prime in case Day a day.
That's the set of moments we get from our moment distribution.
So there is a linear relationship between that propping force and those set of Mormons.
So let's just say we're just put in here linear for a linear relationship.
Well that simply means then that there's also a linear relationship between the actual propping force
that we know from our stage one analysis and the actual set of sway moments that we want to.
Okay so these are the moments due to the actual propping force.
Moments due to actual propping force which is what you're after in this stage to analysis and so you'll
be able to recognize.
Hopefully at this point that I really have to do is scale.
These are two stage two moments by an appropriate factor.
In order to arrive at these m-2 moments so we can simply we can simply scale
our M to two private moments.
By a scale factor or sway factor rather.
By this way factor.
We get so used to seeing a scale factor but it's actually a sway factor sway factor which I'm going
to call lambda where lambda is equal to p o p prime.
K And so therefore we can say I e.
The stage 2 moments which I am after which is the whole point of doing this analysis stage two moments
ago and equal lambda times my Arab street stage two moments.
OK so that's the process.
It almost certainly won't make a 100 percent sense to you if you're just seeing it for the first time.
So I'd suggest watching this video again.
What I'd also say is it's gonna make an awful lot more sense in the next video when you see me walk
through an example and explain as I'm going.
All of these different stages.
So let's just summarize the process.
Summary of Stage 2.
Moments.
Process.
Uh uh.
Okay.
First thing we do we honour apply an arbitrary sway force p prime which causes an arbitrary sway.
Apply our literary sway force.
P prime which causes an arbitrary sway Delta prime.
Then we do a moment distribution based on our crabtree's Fixed and immelmann sound we calculate we calculate
Arabic Treasury final moments.
Or stage two final moments based on those we go in quite late.
What is not our obituary propping force so we back calculate.
So we back calculate
a numerical value numerical value for P prime and K using our free body diagrams and then we can calculate
our final stage 2 moments which are the moments that are actually caused where the actual value of our
sway force p.
Not p prime but P that we got from our stage 1 analysis.
So calculate stage 2 moments.
So and to which we will subsequently add two and one to get the final moments first way frame is equal
to p over p prime which is lambda times an to prime rate.
So these are the moments.
Caused by the actual sway of our frame.
And that's what stage 2 analysis is all about.
It's trying to determine what moments are generated due to the actual sway of the actual frames.
I'm trying to analyze because Stage 1 analysis has given us the bending moments that arise due to external
loading directly.
So when the next lecture you're going to really you're going to flesh has this idea an awful lot more
and you going to see it much more clearly because you're gonna apply.
Everything that I've said here to an actual frame and hopefully that will clear up any confusion that
you might have.