Principle of Virtual Work Video Lecture - Engineering Mechanics - Civil

greetings welcome to this video in this
video we're going to be looking at the
principle of virtual work and
specifically our goal is to use the
principle of virtual work to compute
deflections in beams and trusses but
before we do that we're going to have to
go over the theory of it and get an
understanding of where these ideas came
from this idea of virtual work so we're
going to have to understand concepts of
work and strain energy so that's what
this video deals with we're going to be
going over these concepts and
specifically work is called Big W strain
energy big U and we're going to derive
the principle of virtual work equation
for trusses and then we'll just kind of
assume that the one from beams came from
a similar place so here I've shown this
truss member which is as you know a
single force member can only carry force
along its axis and the work done is by
this force P that I'm pushing at the top
of the roller the displacement as a
result of P must be in the same
direction for it to be considered work
as you may know work is defined as force
times displacement in this case the work
I'm doing on this truss member is from
the force P and the displacement the
corresponding displacement is lower case
Delta so if I want to figure out the
full amount of work done I'm going to
integrate from what I wasn't pushing on
the structure to the point now at which
I'm pushing with a load P and that will
give me this triangular shape if I
integrate it W is P times lower case
Delta over 2 the formula looks different
if you're applying a moment to create an
angle change so generally the work
expression is P times Delta but it could
be M times some angle theta the
important thing to remember is that a
point load tries to displace a structure
in X Y and a moment data location
tries to create an angle change now work
is a form of energy and the energy or
the work that's going into the structure
is stored in the structure in the form
of strain energy so deformations and we
can relate these two because they're
equal whatever we put in is what is
stored in there so that's this is called
the conservation of energy this is one
of the ways we can state that so W
equals u your external work done on a
structure is equal to the internal
strain energy stored in the structure
and just how we plotted the work being
done we can also plot the deformation
being stored so the internal force in
this truss member is F and the
deformation is big Delta L so we can say
similar to the W equation that U is f
times Delta L over two from the time
that we aren't doing any work till the
time that we're pushing with the load P
and we have created this displacement
small Delta and this deformation big
Delta L we know from an axial loaded bar
that Delta L can be equated to FL over
AE so we can write U is f over two times
F L over a e if we like if we equate
this to the work done we come up with
this equation P small Delta over two
equals F squared L over 2 AE eliminate
the twos and then we left with P small
Delta equals F squared L over AE and
this equation is useful it can be used
in the real work method and also it's
used in Castiglione's theorem so these
are both energy methods that we can use
to complete compute displacements but
neither of them are really as good as
the virtual work method and that's why
we're learning this it's very powerful
ok so I hope you're ready for a good
old-fashioned derivation now there's no
need to really be scared of this one
we're just going to be using a bunch
shapes to find an equation and it's
actually pretty cool in the end so let's
start by drawing this thing we call the
P system or the real system of loads and
on the other side we'll have the Q
system or the virtual system of loads
but let's first focus on this P system
so as you recall from the previous slide
we can take this truss member you can
subject it to a load P at this top
roller and it creates a displacement
Delta P and an elongation Delta L P now
for some reason we've chosen to use
deltas for both these deformations well
one is a displacement the other is a
deformation and it's kind of confusing
but oh well so we can label this work
plot and this energy plot just as we did
before but in this case we're going to
include the small subscript P signaling
that this work is done by the real force
the load P and so is the energy it's
stored due to the load P and the
deformation and the elongation are all
as a result of P and we can equate W p
equals u P moving over to this Q system
this is where things start to get a
little strange so we're going to apply a
load in the direction of the
displacement we want to compute what
we're really after is this small Delta P
on the left side but we're going to
apply this Q on the right side and we're
going to choose a value for it typically
we choose 1 kilonewton or 1 Kip but the
key thing about Q is it must be in the
direction of the desired displacement
Delta P as we said but it can be any
value you want the equations will still
work out and in the bottom here very
similar to the left side we have this
work done by Q and the you done and you
the energy stored as a result of Q and
we can say that W Q equals UQ so my
advice here would just be to maybe pause
the video and just look at what we have
here and make sure you're good with it
so make sure that all these labels on
these axes the Q the Delta Q the F Q the
Delta
to make sure it all makes sense flip
back to earlier in the video if you need
to and as usual comment when I make a
mistake I'll say because it's not an
offensive line okay so moving along I've
written here on the left side or shown
here on the left side everything from
the previous slide so there's no reason
to move back in the video you might have
to make the video big on your screen
because there's some little writing so
we have the P system the Q system and
now we're going to make a P plus Q
system so we're going to combine the two
so our force is now F total is FP + fq
I'm forcing I remember and we can see in
the work and energy strain energy plots
below now we've actually applied load Q
first and then load P so we have these
triangles kind of stacked up on one
another and we can label these different
areas in the triangles so we have our WQ
and our UQ as before and our WP and our
u P as before but now we have this
rectangular area which I'm going to call
W Q star and u Q star and these two
areas must be equal why because those
shaded areas didn't change sides and we
already said they were equals from
before so what is this WQ star for
example well WQ star is the work done by
Q over the displacement caused by the
real load P so this is kind of a strange
thing that Q can do work that it's
really taking credit for some of the
displacement that P caused but because
it's pushing there it's still considered
work by the definition so we can write W
Q star is Q times Delta P just looking
at the area of the shape there it's a
rectangle Q times Delta P and recall
that Delta P is of desire desired value
right that's the value that we really
wanted to compute here so our strain
energy UQ star we can say is F Q times
Delta L P
again just the rectangular shape putting
these together we can write that Q Delta
P is FQ Delta L P and recalling from
earlier that an axial loaded bar has a
Delta L P of FP l over AE the subscript
P just saying the P is causing the F and
the Delta L P we can sub this in and
then we'll just add this little
summation symbol being is there that
there could be more than one member and
there often is in the truss for example
so if you blink too fast you'll miss it
but we actually just wound up with a
meaningful equation so let's take a look
at that we have our Q virtual load which
is generally chosen B 1 kilonewton er
kip and our real displacement Delta P
the value were after and we multiply
those two together this equation is
telling us that's equal to our member
force due to our virtual load Q or FQ at
times Delta L P the real deformation and
this real deformation we know it's real
because it has a subscript P everything
with the PS is real everything with the
Q's is virtual this real deformation
could be due to a force as if that's the
case that we've been looking at but it
could also be due to a temperature
change or a fabrication error all these
results in members getting longer or
shorter so I've written the equation
that we derived in the bottom left
corner here and this was for a truss
subjected to real forces or real force P
but more generally we can write the
following the sum of all the Q's times
Delta P plus Delta sub or support equals
the sum of fq x FP l / AE plus alpha
delta T L plus Delta alpha so that was a
bit of a mouthful so let's link up all
these terms with what they're caused by
the FPL over a
from a force as we know the temperature
produces a change in length of Alpha
Delta TL and fabrication error is just
simple it's just how much longer with
the element then it should have been
Delta alpha on the left side we have
this new thing that we have to introduce
this Delta support or Delta sup or
whatever you want to call it and this
corresponds to a support settlement so
this equation is that robust it can even
consider support settlements on the left
side here we didn't derive it for that
part but it's it's kind of intuitive I
guess you could say but not until you've
done examples examples are what makes
everything make more sense so that's
what we're going to do next some
examples and thanks for sticking with me
throughout this video all right
example time