Influence Lines with Muller Breslau Principle Video Lecture | Structural Analysis - Civil Engineering (CE)

what up everybody welcome back to
structure free but today what we're
going to do is going to continue our
little foray into influence lines and
structural analysis and specifically
talk about the Muller Breslau principle
and which is and and we'll focus our
discussion today on statically
determinate structures and what will
hope to do in this video is just help
explain how to construct influence lines
using Muller Breslau principle and this
is this might be one of the concepts
that even faculty have a hard time
understanding or just because you don't
do it all the time and and whatever you
know it just it's a challenge it's just
something that you have to do it if
you're using a lot of like bridge design
or if you're doing that kind of stuff
for a lot of moving loads and this might
be something that you deal with on a on
a daily basis but really the Muller
Breslau principle it's just you know
it's like the Mercedes Benz of influence
line approaches if you will right it's a
it's a graphical technique and the
influence line for any action
essentially here this is the statement I
got this from golly and Nevel some
old-school structural analysis book
right when I was a grad student but here
the influence line for any action
reaction the action can be a reaction
internal shear or internal moment
whatever right in a structure is equal
to the deflection curve when remove the
action and replace it with a
corresponding unit displacement or
rotation it's like what what does that
mean right but here what it means is
what they discovered and I'll maybe I'll
show you a little proof in a second is
that if I want to find the influence
line let's say for this simply supported
beam and specifically I want to find the
influence line for the reaction at a the
vertical reaction a Y here so I want to
find the influence line influence line
for a Y okay so the first thing you got
to do is be able to identify what you
need right so where is this thing
identify that the quantity or the action
that you need and the location and
really the the hard part about this is
is trying to figure out what you need to
remove and then to if you can identify
the action then you have to redraw
the structure redraw the structure
redraw structure without the capacity
without the capacity for that action and
what does that mean that me just means
you know we have to redraw this removing
a Y for that action okay so a lot of
words here but here
so if I want to so essentially I found I
want to find a y I got to remove it so
in it and I want to redraw the structure
with without any resistance in this at
location a in the vertical direction so
let me see here what that would turn
this thing into is I need to be able to
resist motion or I have a force this ax
has to remain but I have to be able to
move up and down now so this thing would
look like like a roller here there'd be
a giant roller and the beam would be
attached to this roller like this okay
like this right here and here's another
roller right here and this thing
essentially this is like a bunch of
walls right here and and this thing at
Point a is able to move up and down and
here I still retain a reaction in the X
and B Y right here okay I still have
that just now I've lost a y I've lost
any capacity in the vertical direction
at it then oops
then what I want to do to find the
influence line is apply a unit
displacement or rotation but unit
displacement at the location where I
have removed the the capacity or the
action okay or the force or whatever it
is okay let's see apply unit
displacement okay at location in place
of the action in place of action in
place of action that's probably better
okay but really you know just to
basically it's identify the action
redraw the structure and then apply that
unit displacement and then here so I
would apply a unit displacement which
means one displacement of one whatever
your units are and I draw that deformed
shape that deflected shape here
which looks like this right here this
deflected shape is essentially an
influence line for a unit load moving
you know on the structure somewhere this
is this right here this blue line this
deflected shape is the influence line
for a unit look for a Y okay so for a Y
here this is the influence line and
that's amazing right that's all it takes
and this is that influence line here and
in the way that it's proven or the way
that these guys Muller Breslau
discovered it is that they use the
method the proof is in the pudding well
not really pudding but the method of
virtual work method of virtual work and
some of you're going oh my gosh right
and it's cool you know I would go like
that too but right here I wish it was as
good as pudding but it's not but here
method of virtual work here so what they
said is if you can imagine here this
deflected shape here so the idea is that
you know all the work done by an
imaginary X T displacement an imaginary
unit displacement is equal to zero okay
that's the idea right of the the
principle of virtual work or the method
of virtual work you know that's a whole
nother video set you know for for us
later on maybe ok but here the idea
would be that you know if I had this
displacement and with this reaction a Y
here right here and I have to look at
all the work that my external forces
acting on this so let's say I have an
external force P here as well moving
through in this displacement Delta Y ok
so this Delta Y is the displacement
under the load
wherever my unit force is or whatever my
this point load is right here and if I
look at the sum of all the work done by
the external forces let's see here I
would have a Y times 1 which is the unit
displacement plus let's see here P and
this there would be a negative here does
everyone agree with that it would be a
negative because the force so in this
case a y
it was a force going in the upward
direction in the displacement was in the
upward direction so that's a positive
work whereas here the force is pointing
downwards and there's a and the
displacement is upward so that would
have been a negative and put a negative
P times Delta Y here and then plus B Y
and B Y experiences no displacements so
times 0 so that's ox is 0 and then plus
ax times 0 which is 0 2 equals 0 okay so
all my external work which is this left
side is equal to by virtual you know a
virtual work or virtual whatever
displacement all my work due to a
external work due to a virtual
displacement is equal to 0 and here this
just 0 0 and and here this whole
business a Y is equal to P times Delta Y
and what that would even mean that what
that means further is if P is 1 right
here that means if this is a unit force
that's applied here the value or the the
the value of a Y is equal to the
displacement or it's proportional right
and so here when when I have the unit
for the value of a Y is equal to that
virtual displacement or that
displacement that's going on right there
so that's that that's essentially the
proof all right so anyway this is a
different proof this is pretty simple
proof you know that just says that that
that this dis you know that the reaction
at a of a Y is related to that that
displacement or the deflected shape
under the point load okay they're
selected value and so but that's the
simple proof whether or not you buy it
you know what you got to do is remember
that that you gotta identify the action
be able to redraw that structure and
apply their unit displacement and draw
that deflected shape that's that's
probably the hard part on that deflected
shape so we'll come back at that next in
part 2 of this video or these videos and
and do a bunch of cases and talk about
what it means to draw that deflected
shape and and come up with influence
lines using Muller Breslau