Okay so so far we have focussed on single iteration examples.
And so what I mean we that is we've only had to do one distribution and carry over.
But remember at the moment distribution method involves locking all the joints and then sequentially
releasing and balancing each joint while all of the others remain locked.
And then the next step after that is to carry a proportion of the balancing moment over two adjacent
joints to carry over moment over two adjacent lot joints.
Now the carry over moment will on balance a previously balanced joint.
Okay so we balance our joints with the and the process of carrying over a moment to an adjacent fixed
joint.
Well then on balance a joint we previously balanced and so that will mean we going to have to go through
a second iteration a further balancing iteration.
So this really comes into play when we have larger structures with more more joints.
And so we're considered the next evolution of over of our study of them distribution method by considering
one of these larger structures and looking at what a multi iteration example would involve.
So let's start off by considering this example structure.
Here we have a structure that is similar very similar to what we've looked at previously it's just a
little bit larger.
So essentially there is an extra joint in there and it's directory joint that's gonna kick us in to
a multi iteration distribution.
So we'll start off as we always start off by looking at the differences first.
So let's consider those differences right.
B Right it's fixed and a it's continuous orbit's quarter B and so that would mean that that's for Eli
or else or rotational fixes that boat supports or resistances for rate.
These are again we're going to see any I guess constant.
So that's going to give us not point five.
E because we're dealing with a fixed continuous example.
Now kbc.
Well he is continuous over to support B and also continuous always caught at sea.
And so that's again going to be for you.
Over well.
So it's gonna be the same in this case.
And so it's going to be not point five.
We are because it is continuous continuous continuous.
Continuous Okay now kcet a seedy is going to be right it's continuous all with support at sea because
it flies on a connects to B when it's subpoena D does not rotational resistance at T.
And so it's going to be the lesser of two stiffness options three I over L. which is going to be three
over six years.
Which by coincidence ends up being an NOC point five.
Okay.
Because that one is continuous pinned.
Excellent stuff.
So all of our members have the exact same stiffness and so I can by inspection simply state my distribution
factors as being equal to an hour point five.
In each case sold by inspection the distribution factor for be a way apply any moment to joint B half
of that will going to be a half of it will go into B.C..
Because both members frame into joining the exact same stiffness and EXACT SAME that's true for c b
and c d distribution.
Factor for ACB point five the distribution factor for C do not point five.
Excellent.
OK so that's the first hurdle jumped.
Now next thing we have to do is lock up all of our joints and consider the fixed end moments.
So when I say lock or joints says look at the structure again.
A is already locked.
That's fine.
Joint B joint B is a roller support with the structure is continuous olver that support and so a moment
will develop there.
And so we're going to lock jointly the exact same is true.
Join C will lock join C would join.
Deep is a pin and and we've seen how to handle him down.
There can be no an applied are arada there can be no moment develop at D.
And so we're not going to lock that joint We're going to leave that as of him dead.
Okay so that informs that informs the models that we will use to cut clear fixed and moments of fixed
end.
Moments.
What do we have.
So let's look at.
Well there's only two sections that actually have loading on them so the fixed and moments when to consider
are going to be for nambour DC because it has a load of plus one hundred and mamber seedy.
It has a load of 50.
There's no loading on the structure between a and b so for B C.
Again from our tables of fixed end moments we have.
S e e n not P C F E M E C minus p e l o h.
The negative because it is an anticlockwise moment.
S E M C B plus P L grace.
Excellent.
That would be equal to minus one hundred and plus 100.
And we know how to handle a pin and so we're going to model a proper cantilever.
Fixed and moment for c d is minus 3 p l over 16.
Extend.
Moment for D C is 0 because it's a pin.
That would be minus 56 point to 5.
All right.
So we have a fixed end one was another hurdle jumped.
Now what we're gonna do is we're going to consider how this area of process develops are why there is
an initiative process.
And we going to see that.
I going to make it much more clear for you now.
So the first thing we're going to do is consider each joint.
So what do we have.
Let's look at joint B.
We have we look at the imbalance where the imbalance is.
The joint is locked.
And there is an outer balance moment.
Minus one hundred killiney meters.
That's me out of balance moat and joins C.
My balance moment.
Is a positive forty three point seventy five.
That's my out of balance moment.
Point C..
So let's consider it d distribution the moment distribution process.
Step by step.
So step one will be to unlock Joint be unbalanced.
So step 1.
Is unlock.
Joint.
B.
And balance.
So we'll sketch it out.
Joint C remains locked joint B I have on locked and so it has reverted back to being that role or support.
And I'm going to apply a balancing clockwise 100 Killen events are one hundred metres because distribution
factor for B and theses is equals not point five.
We know that half of that balancing moment clockwise 100 will go in to each of the members.
OK.
So my bending moment diagram will look like this with a 50 on the top and with a 50 on the bottom the
whole moment they're being applied as 100.
So that's gonna be 50.
We'll dispense with the positive and negative here.
And the positive are rodded a negative for hog moments because we just might focus on the numbers for
the time being.
So we apply 50.
Let me see.
This will also be 50.
OK so in total the balancing moment is 100 are supplied.
Now we're going to carry over half of that moment to end.
Because air is still fixed so we're going to have this.
K. And that's gonna be 25 and were going to carry half of them warmed over to join C because joint C
is still lock.
Remember we're only on locking and balancing one joint to her time so it's gonna be something like that
that's gonna be 25 right.
So that might carry over a moment and that's my coryell Moment.
So what we can say now is jeudy is balance so we can lock it up again.
So I joined B.
It's balanced.
Therefore look at Oak and move onto joints.
So steps you.
We'll be on lock and balance.
See.
All right so we have applied a balancing moment which is gonna be a negative forty three point seventy
five joint C. And so half an eye gets distributed into each member.
And so we have.
Twenty one half of it is twenty one point eighty eight and twenty one point eight eight.
And so that will look like.
So zero Harry over to join me and we have 50 percent carried over to the fixed joint a bit remember.
BS has been relocked again after we balanced.
So last gonna be ten point nine four.
So I that's my carry over here and the carry over.
0 0 to appear in.
Carry over is zero.
Because it's in we've balanced the initial imbalance.
Bosh when he was balanced the step one there was 25 kilometres carried olert to see making it own again
right.
And then similarly when C was balanced in step two.
There was ten point nine for kiln it was carried over to be making it on balance again.
And so the carry over the carry over moments.
25.
That means years.
Ten point nine four.
Kuna meters to be sold those to carry over moments have unbalanced joined B and C..
And it's because of that balance that we have to go in and do a second round of balancing in case of
the moral of the story here is that the carryover Mormons require a second this raison of balancing
to take place.
So the khairy.
Oh over moments.
Cause
a second round.
Bouncing.
To be required.
All right consult we can't move on to Step 3.
We can rest assured that the process of doing this practically is much quicker.
Where we're we're going through it a little bit slow right now just to make sure you can understand
this.
Where there's need for multiple iterations is coming from but it's not always this tedious that can
assure you.
So we're going to rebalance joint B.
Rebars straight be and for our own reference we will just note that the balance moment.
The total balance moment is a minus ten point nine for kill and needers.
All right then.
So let's balance out that minus ten point nine four.
So we will dispense with the structure again and we will just draw on our positive balancing moment.
And again it's split evenly between the two members.
And that's going to be half.
Ten point nine four five point four seven and five point four seven.
And again we noter carry over half of that is going to get carried over to the fixed support day and
half is gonna get carried over to the locked joint.
See.
So that's going to be two point seventy four and two point seven four.
So b is now balanced again.
And we're going to relock it OK and move on to joints.
And again we will not eat at all.
Balance moment at sea is plus 2 5 colonies and meters plus 25.
Okay let's balance that one out.
We're going to have twelve point five.
Again this is our Bouncing moment in the anticlockwise direction and that will induce tension on the
top of the right hand side of the joint which is moving down to zero and tension on the bottom of the
left hand side of the joint where we carry over half.
So if that was twelve point five and that was twelve point five This would be six point two for it right.
So now C is balanced and we can lock it back up.
Once again we can see that the carry over moments haven't balanced to neighbouring joints right.
And this is not technically this is a never ending process because you're always going to have a carry
over moment in balance or balance the joint that you just balanced.
OK.
But what you should notice is to carry over more ones getting smaller and smaller and smaller with each
iteration.
And that's the key.
That's the key to this.
So we will do one more balancing just for the purposes of demonstration we'll say for step five let's
rebalance jointly again.
It's going to be a clockwise positive balancing moments to counter the negative six point twenty five.
It's split evenly between boalt members.
That'll be three point worn three here and three point one three here.
Carrying over 50 percent to the fixed support.
And the locked joint at sea.
We've got one point five six one point five six and we can again say bees now balance and we can lock
it up and we can move on to Step Six which will the balance or rather rebalance.
Rebalance joints.
That's a positive at a balance moment which means it needs a negative balancing moment which will be
one point three seven and one point three 70 get are adding two points for curry or a moment to appear
in is zero Kerry or mom to a locked joint is harth which is not with six nine roughly.
Well right now Joyce it's all as usual Stargate Joyce's balance.
Lock it up.
Move on.
Right.
At this point right.
Let's consider the magnitude of the carry over moments.
OK.
Let's think about how big and why carry over moments.
So what I want to say is if because just doesn't the natural thing to do is jump back to joint and say
OK you're out of balance again.
I jump in and bounce you ok but if I carry over that moment jointly.
So if we carry over.
If we carry wife needed there.
If we carry over the imbalance.
Imbalance.
Joint B would be.
Minus nine point six nine.
Killing it in reverse.
Alright.
And similarly the imbalance joint.
C. ward the plus one point five six kill and eat meat.
Alright so common sense has to jump in here and prevail.
Right.
Let's think about how big that imbalance is compared to the initial imbalance.
Right.
We work out 69 as a percentage of the initial imbalance at joint B which was at 100.
It was 100.
Well then we can say that the imbalance is not is now not point seven percent of the initial imbalance.
OK.
And then we do the same thing in the in the imbalance that currently exists at point C is tree point
six percent of the initial imbalance.
Now at this stage we do the usual thing that we do in engineering we say right.
That is so small that that imbalance is so small.
It's negligibly small.
I don't carry a sweat knock on a carry over it not really care that it's so small as to be insignificant.
So what we're going to say here is it is the imbalances negligibly negligibly small and because of that.
Therefore we don't carry it over.
So I'm saying right I'm going to make a judgment call here.
I'm not gonna in balance join be joined again by carrying over this small imbalance.
I'm just going to say right we're done here we're finished because the carryover moments are so small.
So negligibly small.
So don't carry over.
In other words leave joint's B and C balance leave.
Join us.
B and C balanced.
Okay.
So typically the rule at home here is typically when you're imbalance has been reduced to one to two
percent of its initial value.
You can stop it's raging.
And so the final moments.
So what's two percent so we're still a little bit high.
Three point six percent and joints.
But we're below and unjoin be.
So you know practically Would you go and do another round of in balancing.
You could put it's not going to make much difference.
So I'm making the judgment call here that at three point six percent we've reduced that imbalance in
North and I can draw a line on and say Right and off and off I'm gone.
So all of my moments what for you if you want a rule at home to live by use you want to target an imbalance
of less than 1 to 2 percent of the initial imbalance.
And at that stage you know.
Right.
You're down.
All right.
So at that stage we can sum up essentially we're setting something up all of these bending moment diagrams.
And again you can see this procedure much more clearly in the table.
We are seeing it codify everything we've done here and put it all into our table you would see very
clearly how the something or process works.
So we're going to consider it at that table now.
So consider.
This procedure.
Carried it on the table.
And in fact this is how you would do it.
You know usually you wouldn't gosh you know continue drawing all these bending moment I or take you
forever.
So you process the distribution in the table and it's much more is much more straightforward.
So it's quicker.
So process to consider the procedure carried out in the table.
Got to start right.
So our first balance let me see our first balancing it was at joint B.
And we added 50 to each size so we can put in our 50 and 50 and we can balance up joints C as well.
And we added a negative twenty one point eighty eight if you remember to both sides.
And we now look after our carry overs.
So we carry over to a 25.
And then we carry that we carry over from C to be half of that which was minus ten point nine four.
And we carry over 25.
And now you can see that if I draw a line under this distribution.
We can see that joint B is out of bounds by my ten point nine for my joints is out of balance by 25.
Again quickly just no there's no carryover to appear to support a deal.
So how is that a balanced this OK.
So this balanced joint B we worked out the what do we have half ten point nine for was going to be my
positive five point four seven on either side.
And then balancing the 25 were minus twelve point five on each side.
And so the joints were not balanced but we then had to carry over.
So we're carrying over.
We see two point seven four.
Let me see let's carry over to point seven for ear no carryover from C to B minus six point twenty five.
And again no carry over to support a deal.
And we can see okay were again off balance a B and C but that's okay because the read the D amount by
which read of balance has reduced less deal and other distribution.
We said we were going to balance that with three point one three three point one three minus one point
threes then and minus one point three seven which is half of the two point seventy four and the negative
to counteract the positive dewpoints core.
And then we in our bought this table we worked out that well if we carry over and we in balance jointly
bnc game the the magnitude of that imbalance was so small as to be negligible so we work on a bother
on balancing those joints again and we were going to draw a line on it here Id say right.
That is the end of the distribution.
And at this point we can say what in each column and we would get twenty seven point seven for at a
we have fifty eight point six for a b and minus fifty eight point six ribisi.
Lovely.
A week ninety one point nine nine for C B that's just something my one hundred minus twenty one point
eighty eight plus 25.
Mine's 25 plus duplex and or minus one seven gives me ninety one point ninety nine.
And similarly on the opposite side I think it's the irritant it gives us minus 92.
That's just a simple rounding error.
And we have zero at the pin which is exactly as we would expect.
So that is the complete solution as you can see when you have a situation like this.
Hopefully well hopefully now you can see that the table comes into its own tracking everything to a
table is so much quicker and it makes you less prone to errors.
So yeah so hopefully we can see the value of the table so as always at this stage the question continues
as it would normally and I will give you the solution I enjoy the sheer force bending moment diagram
and reactions and suggest that you work your way to the same solution or confirm my solution.
Okay then so youve got your reactions.
The sheer force diagram and your bending moment diagram.
Obviously the sheer force and bending moment diagram are woefully not to scale.
That's not something that you should worry about.
Provided you have all of the numbers shown this.
The scale is not really that important.
So.
Right.
That is how we handle a multi a multi iteration example.
You don't go through dry all the bending moment.
I Gram's like I did for demonstration purposes.
Once you've got your fixed end moments you jump straight into the table and get your support moments
out of that table and then you go ahead.
Considering free body diagrams of these subsections understructure to work out your reactions.
Sheer force is unbending almost like we've seen in a previous lecture.
So that is that is all we have to really introduce at Florida.
For the moment distribution method at this stage it's about practice.
And if you listen to many of my lectures are courses previously you'll know it's all about practice.
So over the next few An actual action where we're basically just going to try and implement and put
into practice everything we've learned so far with song worked examples.