Okay so Marmer distribution is an analysis process that we can use to analyse statically terminal structures.
So up until now you've probably focussed on study determined structures and with this analysis procedure
were going to broaden massively the the type or the number if you like of different structures you're
going to be able to analyse.
So the process focuses on determining the internal bending moment at joints first so that's gonna be
our starting point as opposed to when we're dealing with statically determinate structures.
We got to determine reactions first.
Well this time we're gonna determine moments at the joints.
So it's a very procedural and intuitive process.
And it's an excellent way by hand to check let's say for example computer analysis.
So if I was to give you a quick overview of the process the very first thing we do is we stay out of
you can imagine in your mind's eye structure statically in terms of structure details out to you in
court and you're with the first thing we do is we go around a structure and we fix all the joints and
that what that means is we locked them against rotation so we stopped them from rotation.
So we essentially create an IF YOU LIKE THE that fixed or in or cantilevers type joint at every one
of our joints.
So we artificially alter it a structure.
Then what we do is we determine the bending moments that would develop at those joints and in response
to the external loading understructure.
Okay.
Then we analyzed the moment the joints and saw what were gonna have is each of these different joints
depending on the loading that's on the adjacent members.
You may have a moment in at the joint because remember now you've got this joint.
So the structure on the right will be subject to some loading and that loading will have no influence
on the structure to the left of the joint.
So essentially it's like having isolated structures that are just bottled up or pushed up against each
other.
So those individual joints you're going to have moment in balance is solved for each joint What we do
is we go around the structure and joint by joint we on like a joint We we we balance the imbalance so
we add a balancing moment to the joints or whatever the in balancing moment the joint is we go ahead
and add the opposite sense of rotation moment of equal magnitude and we're essentially balancing that
joint and then we locked the joint.
But before we locked the joint we have to carry over a proportion of the Hmong which that we just applied
to the adjacent air support.
OK so today support on the other end of the member and that's what's called carry over moment.
And that's essentially the process complete for one joint.
Now we move on to the next joint in the structure and we say right.
It's on lock.
We're balancing.
We are propagator our carry over moments and we locked the joint again and then we go onto the next
joint.
And so one around structure we go.
So it's an iterative process which means that we got you all of the joint once and by virtue of having
carried over moments two adjacent joints we will have imbalanced joints that we just balanced out previously.
So we're going to have to go you in a second iteration.
And we've got you around and around on subsequent iterations and with each iteration we're more closely
approximating the actual bending moment diagram for structure are or more corrosive approximating the
bending moment that the supports for and particular structure.
So what we have is a situation then where we have this very nice procedural process and it's logical
it's intuitive.
And the last thing to say though in terms of sorry is that this whole thing although it might sound
a little bit complicated and a little bit abstract at the minute we track the whole process with a table
or case we lay it all out on a table and what can look like very complicated structures.
When you put them into a table and you know the steps and you follow the procedure it all works out
relatively relatively easily.
So again I don't know whether it has helped or hindered you giving you that kind of a summary.
But what I would suggest is we're going to write you an example now which demonstrates this.
But what I'd have to say to you is when you get to the end of this lecture and we've gone through this
example more you'll understand much more what it was I was talking about at start.
So when we finished the lecture you should go back and listen to everything I've just said in the overview
because it will make an awful lot more sense after we've been sure a worked example right so let's crack
this working example and it's gonna be a very basic structure a very simple structure but what it does
is helps us demonstrate all of the key components of this analysis procedure.
So we're sketching out.
Okay so this is a structure to start with and straight away you would look at that and you'd say it
right.
Unless I know this this procedure this morning distribution procedure or some other procedure frustratedly
Internet structures you could analyse because there are too many or known reactions so it statically
indeterminant.
And we can we can confirm that if we look at the number of reactions in pace that the number of reactions
is three at record 83 at support c and worn at B.
So that's seven.
We have seven reactions so or minus three.
Three is the number of equations of equilibrium that we have at our disposal is equal to thaw.
Okay so it's statically indeterminate to four degrees case so we can start with determining the reactions
because there are too many nodes and this is where the moment distribution method steps in.
So let's say let's just let's just try title down here moment distribution.
The actual method itself.
So we start off and we initially initially assume of joints fixed against rotation.
OK.
So all joints are going to be fixed against rotation and it encourages you to follow along with me and
write your own notes as I'm going.
There is definitely some value in this.
Right.
So if I was to drop the structurer gambit and fixing all joints Well I noticed a joint day is already
fixed.
That's fine joint C is already fixed so what I'm going to be doing is fixing joint B.
So this was already fixed.
This one was ready fixed.
OK.
So we've imposed this is the key bit here right.
We have imposed.
Rotational fixity like a rotation on us the whole thing here we're dealing with moments.
And so it's rotational fixity we're interested in so we've imposed rotational fixity at be OK.
So that's the first thing.
OK.
Now look at this structure right.
The loading just label it up a b c d right the loading on a B is now completely isolated from the loading
and B C and vice versa.
Okay.
So what we essentially have is essentially it's like having two separate beams that are just happened
to be sort of slid open and pushed against each other at point B.
So what we have to do now is determine what I call the fixed end moments the question we're trying to
answer is what moments for this modified structure what moments would develop at a b and c and the source
of this process is known as determining the fixed and moments.
So remember we've got two separate structures so we have something that will look like this.
OK.
Now if we think of these as two independent structures Well we know that a reaction moment would develop
can't support be like so this will be a clockwise moment and anticlockwise reaction mode would develop
that support.
Similarly we would have an anticlockwise on the other side of support B for the right hand structure
and a clockwise to the right hand support of the right hand structure.
Now what we have now is two structures.
Individually these are statically interments And at this stage what we're gonna do is and what you will
typically do is resource to tables of what are called fixed and moments.
OK.
And if you're an engineer it's tuned for unliked time.
You'll be well familiar with these tables and there will be one provided with this course.
In any case what you will pull out from a table for a structure like this with a load P and a span of
L That would be L O to an O the L over to you could identify from a table pretty easy at the moment.
Here would be minus P L or 8 and minus here indicating the anticlockwise rotation.
Similarly we'd have a plus P L overate.
On the opposite side rippy is the magnitude the point load and span and you would pull from the same
table.
The moment for the other side or the other a structure W L squared or 12 and plus to be well squared
over 12.
Not k.
So we can actually put the we can put the numbers in here and we would end up with this Ruby.
Let's call it the fixed and moment for a B is equal to.
Let me see that will end up being minus 100 q. in meters and the fixed and moment B A OK.
The first letter in each case as indicating what side of the structure that we're on.
So the 6 and mine would be a is obviously going to be plus one hundred Kill and meters dissenting.
The first side we end up with.
Exit.
Okay so we're at work.
We're rapidly advancing through this.
So what do we don't.
We took our structure and he joined that was narrowly fixed.
We fixed it that identified foris to isolated structures to isolated substructures we'd identified those
we pulled those two over and we looked up what reaction Hmong would develop at the score.
In each case.
So these at the fixed end moments.
So that guy that guy that guy and that guy at the fixed end moments.
Great.
Okay so now we have to do is look at the imbalance.
Okay so there is an obvious imbalance here at Point B.
So this is this is really the heart of the Momi distribution met that right.
So now what we do is we now we release release joint B and allow it to rotate and allow it to rotate.
Okay what have we got.
Let's just focus in on joint be forsaken and how we think about what's going on.
Joint B stars are locked in there with a member kimono it.
Now the moment on the left hand side a joint B was this guy.
So that was a plus 100.
Right.
So he's got a plus.
One hundred at the joint the moment on the opposite side all members speak.
Is this guy which is gonna be a minus 2 0 8 point 3 wise 2 0 8 point three.
Shown Shannon wine cereal.
It's not necessary if I've also shown the eat the arrow.
But just for a demonstration we leave it in.
Right now if I was to consider what would the bending moment like rallied a joint look like it would
look something like this.
Right.
We would have a moment coming up that we left.
Answer I put down the right answer I did have a moment quinel to a higher peak.
Now this is not a valid bending moment I.
I mean I'm going to see it's a step in the bending moment ihram like that eyelid joint here where only
two members are coming in to meet at the joint.
So there's this essentially is our moment in balance.
That's 2 0 8 point 3 and this valley down here is 100.
So this is another way of visually identifying or visually I guess they're displaying the moment and
Balance natsir moments in balance.
Excellent.
So now what we do is dust the dust situation right now.
Then we go ahead and we release let's release this joint release the joint.
Now what do we have.
Well it's gone back to being a roller coaster.
But we have this moment in balance which is minus 1 0 8 point 3 1 or eight point three kiln metres Okay.
So our balance moment our balance for future reference is going to call that 0 0.
We are at a balance moment.
It is minus 1 0 8 points 3 so if it's not obvious are ready.
I'm I'm assuming clockwise moments are positive and anticlockwise moments and negative.
So that's minus 1 or 8 23 is going to be important that you get yourself a clear sign convention and
stick to it.
In this method co-signs are gonna be very important.
I'll k now what we do is we apply a balancing moment k so we are structure again.
Our role or support.
We have our air of balance moment of minus one or eight point three.
What I do is I come along and say I want a balanced stop because I can't have a moment in balance.
I joined.
There must be moment equal European after joint so I apply a positive one a quaint three plus one not
quite three.
And it is a balancing moment.
So now what we have here is the song of the moments.
B is equal to zero tick.
Were happy with that we know we must have moaned liberal at the joint.
But now here is the next little nugget which is key to this method.
We have to balance the moment or rather we have to distribute the moment between the members that meet
at the joint in proportion to their stiffness arcade daddy's daddy's kit and when I'm going right down
and I would strongly suggest you also write it out.
So it's it's important right.
Or we got the balancing moment bouncing moment is distributed hence the name normal distribution method
which it's distributed
between.
The members.
Members Meeting.
At that point the joint.
This is it.
In proportion
to their stiffness is.
Stiffness is OK.
Now there is a procedure which we will cover which goes into identifying the stiffness of the members
meeting at the joint and it's very straightforward it's going to be afraid of we're going to get there.
But to go in and explain that now would kindly kick us off course a little bit so what I do is for the
purposes of just illustrating the whole process.
I'm going to say for now for now.
For just the rest of this we'll nesher let's assume let's assume member a B is twice as stiff is twice
as stiff.
And what that just means is it's harder to bend it's twice as hard to bend harder to bend.
In simple terms as member of b.c.
OK.
Right.
What would this mean.
So this basically means the balance moms are divided between a B and B c in a ratio of two to one.
OK.
So the balance between a B and B c is a ratio of two to one.
OK.
Now again this is just temporary for now for this lecture.
The next lecturer will talk about how do I actually churman stiffness.
So let's jump back to the example.
We had our joint B and we had the roller.
Right.
What we have now at the fixed end moments were let me see four dart this way fixed and moments were.
One hundred here and minus two oh eight point three here.
So that is a fixed end moment.
And at a fixed end moment their starting point.
Then we have a balancing moment of loss one way point three now.
Two thirds of that going to mamber a B which would mean that that is gonna be plus.
Seventy two point two.
And that's the bouncing moment.
And one third of it is gonna come into the right hand mamber which is going to be thirty six point one.
Again it's a plus.
That's right at the top.
Plus thirty six point one.
And that's a balancing moment.
Right.
So that and that together add up to my When I weigh eight point three my plus one or eight point three.
So now what I actually have a thigh just zone the moments up the joint irada of are some of the moments
on each side of the joint.
And so joint B is balanced.
Okay.
Right now it's almost finished almost on the final final step is to now propagate through what we call
carry over moments.
Okay.
So if your way if you if you've got everything up until now Grace you're at you're at your 90 percent
of the way to understanding the moment distribution method to get your last hampering sense left hip.
Let's talk about the what I call carry over moments carry over moments.
Right.
We're going to take a sidestep from our example of two seconds just to talk about what is a carry over
moment against our staff with a simple structure.
Now if I am I'm down here.
Join me here.
If I apply a moment am jointly I'm gonna get some sort of deflect that shape that's going to look like
this.
No pushing here at the moment are they.
It is support.
Is a fixed support and the positive fix support.
I know that a this squinched attention to attention bottom if I apply a moment am that joint be so small
that moment has got to be propagated all over the joint.
So there isn't a threat.
There is an influence this moment of am I joined B has an influence on the support at joint day.
And if we are bending benchwarmer diagram we can see much more clearly what that is.
So we've got here bending moment of an as applied a joint am and I've always complete the bending moment
diagram for that structure.
It would look like so we would have a moment at the support of an older two kids not a huge amount to
this whole carry over Mom business it's pretty pretty straightforward.
So if I'm down here at the joint B and I apply some moment to it.
If the opposite of the opposite support it's fixed.
And in a moment distribution in most cases it will be fixed.
Because remember we're locking on of the joints with some minor exceptions.
So if the opposite end is locked.
Well then if I apply an I get a carry over of AM or two.
So that's one situation.
The only other situation is if I am again simple structure what this time I have my roller on the right.
Be that's fine.
But instead of a fixed support let's say I have a patient support at the opposite end at a well if I
apply a moment that bit again we have no carry over amoled because we know at this stage.
If we study bending die Ramsey we know that there can be no moment develop that appeared support and
solve the pain if I get an applied here I have no carry over moment.
OK.
So no carry over to a pen.
OK.
And that's an important goal.
OK.
So let's just what are we saying with this guy over here.
If if we were to just encapsulate what I've shown in this diagram in words if the opposite ends.
OK.
Which in this case was end day is fixed.
So if the obsidian is fixed we apply a moment and we when we apply a moment a B when we apply a moment
at B.
We must we must carry 50 percent over to eight.
OK.
All right that's that's it.
That's all there is carry over moments.
So let's think about that now in the context of our example.
So for our structure so for our structure what are we up let's just draw what we had.
Right.
So we applied arm the left hand side of this support.
We applied a clockwise balancing moment of seventy two point two.
Now the opposite ends a fixed end.
Right.
So that let me let's just say the opposite and it's fixed.
So that would mean that we have to apply half of this moment to seventy two point two different so if
that was seventy two point two well then down here would be thirty six point one.
Senator Kerry a moment.
So that is the carry over moment.
And then he exact same thing is applied as happens on the other side.
We applied a moment.
Let me see.
Thirty six point one.
Which would give me a bending moment diagram like this.
When asked thirty six point one and then half of that was carried over to the fixed end.
Which is eighteen point one.
And again that's my carry over moment.
So remember this guy here and this guy here our simply.
Are Simply this one and this will Kate read that was jointly.
So that is that it that older is to the carry over moments at the last last thing we got to do is so.
What are our final support moments.
Our fixed end moment we start laugh with.
Let me see that's a we won't walk we won't worry about drawing the loads down.
We started out with a minus 100 over here.
We had a plus one hundred here.
We had a minus two always on the far side.
Remember these are fixed end moments that we started with.
We locked our joint and plus 2 0 8 point three over here.
Then we added our balancing moments to joint B which brass or on this side we had a seventy two point
two added and on the other side we had a plus thirty six point one down we carried over a half of that.
To hear this fixed the quarter which was thirty six point one and again heart attack carried over here
which is plus eighteen point one.
And the final thing to do is just so all of the moments at the joints.
And we'd end up with.
So we have an anticlockwise moment.
Which is what you would expect a clockwise moment at c which is what we would expect.
And the moments that be at balance each other which is exactly what we would expect for this structure.
Self that is all there is to us.
Now what I would say to you is what I have shown you is that the barebones of the procedure but essentially
that old does not a lot more to it are a few kinks and a few sort of exceptions that we have to go through
in the following lectures.
What if what you should do now is go back to the start of this lecture.
Listen to my summary again probably watched as lecturer at least one more time just to make sure you're
really it's all sinking in and the various different steps are sinking in.
What that is more or less all there is to it what we're going to be doing now is just polishing it.
Over the following lecturers.