Today we are going to talk about the role
of calculus of variations in mechanics we
had discussed the beginning of this lecture
but now that we have discussed calculus of
variations in detail that is with constraints
and then two types of constraints functional,
constraints function type constraints or global
and local constraints and the general variation.
Now let us revisit that in the context of
beams first let us look at something that
we had discussed earlier about mechanics seen
in two in three different ways.
So if you look at mechanics in calculus of
variations we have three views of mechanics
so we have first one it is a three views of
mechanics we have we are discussing both statics
and dynamics, right. In the case of statics
if you just want to use for force balance
that is perfectly alright alternatively one
can use principle of virtual work that and
then if you want to find another alternative
that is a minimum potential energy principle
right, so this leads us to calculus of variations.
If we start from calculus of variations where
we want to minimize potential energy where
the variable is the deformation lot of possible
deformations only one will be chosen which
minimizes the potential energy that is the
one that is the static equilibrium solution.
If you start from here then you can move up
and we will get principle of virtual work
which is nothing but the weak form of differential
equation which you get from the weak form
we can get the strong form that becomes force
balance, okay.
So if you start from minimum potentially principle
you can get what we call weak form of the
differential equation of the differential
equation are force balance which we call strong
form of the differential equation so DE when
I write, I mean differential equation that
governs the static equilibrium condition.
If you start from force balance you can actually
come here which we also is known as our real
names D’Lambert method a mathematical method
gives you the weak form and from there you
can actually come to a minimization principle
under certain conditions which will be discussing
later.
So if you start over here you can come back
that is if you start here you can come back
to this and you can come back to this or if
you start here you can go back to this and
go back to this or if you start from the weak
former principle of virtual work you can get
the strong form or you can get this also,
so all three of these are equivalent see if
you know one you can get the other two so
you can believe in any one of these and we
will be fine this all for statics.
Similar thing exists for dynamics also so
if you start from Newton's second law everybody
knows this F=ma that is equivalent to D’Lambert
principle which is like a weak form of that
and from there you can come to something called
Hamilton's principle right, are you can start
here and go there and from here you go there
are from this you can go to this or that these
are three equivalent principles in mechanics
and calculus of variations in a way ties all
of them together.
Because when you take let us say statics minimal
energy principle from that you can get the
weak form or the principle of virtual work
from there you can get the force balance which
I intend to show using the example of a beam
that you would have learned in your mechanics
of materials course.
So where we have sorry, where is this can
you just come I open this but I do not see
the pen thing oh this is not that way okay,
let us take the beam problem let us take the
beam problem okay, in fact let us not worry
about constant cross section beam we can take
a variable cross section b I will only draw
the neutral axis okay and as far as the cross
section is concerned let us actually take
intentionally a variable one which are normal
mechanics of materials class you would not
take, right.
So here we have at every point there is a
cross-section whatever shape whatever size
parameter they may be there let us say that
is A(x) and our x starts here this is our
x, x=0 0 here X =L and A(x) is a function
right, if you take up the beam problem let
us start from minimizing the potential energy
so if I say minimize the potential energy
of this beam which has let us say some load
acting on it okay, so if i say a line diagram
now so I will just write the beam even it
has cross section I will just show this that
the force is acting on it in some fashion.
Let us call it q(x) is everywhere force acting
as per the function q(x) and the boundary
conditions do not matter I am just showing
a cantilever beam but they will come out of
calculus of variations calculation that we
are going to do. If you have this so we want
to minimize this potential energy with respect
to W(x,) W(x) is that this beam is going to
deform under the load and this displacement
is W(x) okay that is a transverse displacement.
So when a beam has a loading it can deform
in many other ways but it is going to take
a particular deformation what we can call
let us say W*(x) that is a solution of this
problem minimize potential energy problem
no constraints here just minimal pressure
the beam with respect to W(x). Now let us
say what this potential energy for a beam
is in general potential energy is strain energy
plus work potential as we had already stated
once strain energy is this SE and work potential
we can actually call it work potential energy
if you just want to say that it is energies
I also want to see energy explicitly stated
there its work potential energy that is potential
energy to the work done by the external force
that is exactly what it is but it is negative
of the work done by external forces here q(x)
is the external force on the beam a lot of
internal forces we do not consider that it
is only the external forces okay.
Why is it negative, negative because in thermodynamic
thinking work done on the system is negative
that is how we put this as negative energy,
negative potential energy directional forces
restrain energy the energy stored in the structure,
so this strain energy we can write or the
cumulative for the oval overall elastic body
the beam is an elastic body here so there
we integrate over the volume okay, what do
we integrate we integrate strain energy density
strain energy density which is simply strain
energy per unit volume.
This strain energy density is defined as the
area under the stress strain curve, okay if
you write s versus e around the curve for
the linear analysis which is what we are discussing
now he will be se/2 okay, so we have strain
energy density here we want to integrate the
entire volume, so then we ask the question
what is the relation between stress and strain
so we have s for a beam there is only one
stress s and there is one stress a one strain
e normal stress and normal strain they are
related by E which is Young’s modulus which
is material property here, okay.
Now if we substitute over there for our strain
energy okay, our strain energy here now this
strain energy is integral V coming back from
here s if i substitute it will become e times
e another e is there square by 2 or the volume
okay, now looking at the beam at any given
positions if I take let us say this one there
is some cross section area here and the length
dx let us I take, I want to cover the entire
volume so I take a dx a slice of the beam
and that if i look from this side it is going
to be some cross section, right.
In this cross section I can take a little
one something like that little square call
it dA so I am splitting the dv that is the
overall volume that I have this little disk
okay, of the thickness dx this is dx that
is dx, dx and then dA I will have a little
thing that we are to integrate over the area
first and then the length also so if I take
that I can split this as two integrals one
over the area another one over the length
so that I can cover the entire beam okay.
If I want to cover the disk that we just do
where integrate over the area so that is this
disk we have to integrate over this entire
area here and then we are integrate from x
=0 to x=L then record entire volume of the
beam so I put 0 to L and A over the area of
the cross section and then we have Ee2/2 dA
and then dx so basically dv is split into
dA and dx right, next thing we recall is what
is e that is a strain normal strain in the
beam which if you recall from mechanics of
materials is given by –y/?.
y is something that is measured from the neutral
plane to the point under consideration that
is I have taken that d at certain height if
this is the point through which neutral axis
close let us I put this in yellow right, so
over here is neutral axis the dA is that in
certain height maybe the other color is better,
so this is y you know that if I take a point
over here from the neutral axis to this at
that little distance is y as the beam bends
the way we take our convention to for the
beam that is straight like this deforming
like this is considered positive so if I neutral
axis is the center like this if I take a point
above the neutral axis that would be here
these things will be under compression, compression
and hence time will be negative and that is
why we have the minus sign over here okay,
another thing we know is a note that strain
is linearly proportional to distance from
the interlocks that is why it is –y/? and
why that is linear interfere the strain other
thing we know is that 1/? is the w function
that we have this second prime of it that
is d2w/dx2 divided by 1+w'2 raise to 3/2 okay.
But in cases where w' is spa that is different
the slope of the beam that is deforming is
not significant we can neglect that so we
can neglect, neglect this compared to one
that we are adding to so this can be approximated
as simply w'' okay, if I do that this is going
to become minus y into w'' because 1/? is
approximated as w'' right, so now we come
back to this we can write this as 0 to l Ee
is now square y2w''/2 dA dx okay.
Now we will rearrange a few things to take
out the things that do not depend on that
do not change on the area which if you assume
E is number two is of course and then the
w''2 also because that is something that does
not change on the area of cross section because
a one-dimensional model the beam one-dimensional
model of the dimensional solid beam we just
have a neutral axis so if we take a slice
at a point particular X the W there is the
same because if you recall the way we do this
here.
We just show this we are not saying what exactly
happens to the beam as it is width is there
so across the width we do not say anything
all of that deforms like you are only worried
about a neutral axis deformation right.
So that can be taken out what is left here
we will have y2 da over so here also di is
there so I should put over the area carrying
there so this is over the area of cross section
okay and then we have dx now if you see this
quantity that we have is what we denote as
I our second moment of area second moment
of area of the cross section sometimes it
is called moment of inertia that is actually
incorrect because there is no mass associated
with this is purely a geometric property so
calling it second moment of area is appropriate.
So if you had learnt it as inertia change
it to call it second moment of area or not
inertia okay so this gives us 0 to l will
call this I now W´´ 2 / 2 dx okay all this
we discussing again to potential energy we
have written versa energies.
Work potential both of that we do.
So we get so I want I would now right minimize
potential energy which is strain 80-pluswork
potential strain energy as we have just derived
0 to L EI W ´´/ 2 DX thurston energy plus
work potential so let us write work potential
in blue color that is negative or the work
done by external force external force is here
q and then it is being acted upon through
a distance w that is the transverse displacement
so this will be the work potential this also
goes from 0 to L dx because q(x) is acting
w of x is the consequent deformation product
of these two so together we can write it as
one functional so we have problem minimization
with respect to W(x) of potential energy which
we can write as 0 to L ELW ´´2 / 2- q w
dx.
Now we got a calculus of variation problem
or other minimization of potential energy
that is a principle so here what do we do
first thing we do is to take the variation
of potential energy that is our functional
here with respect to W equate it to 0 if we
do that what we get is that 0 to L I want
to take variation so that will become EI W´´
this square and ½ will go away so that to
here and the two that you get w?2 this to
these two will get cancelled into ? W ´´ - q
and then we are taking variation so ? W will
be there this whole thing dx should be equal
to 0 okay.
Now this particular thing if you see if I
if I just write it in a slightly different
form I will say0 to l IW ´´ ?? W ´´ equal
to because the minus is there I take this
to the other side q into ?? w dx okay because
that is equal to zero I have taken this term
to the other side right so now we see what
this ?? W ?? W is for us from the calculus
of variation it is the variation of W of x
?? W of x is a variation of Wx meaning if
the beam has deformed here we are perturbing
around that that perturbation little perturbation
that we have here that is our ?? W (x) right
but now we can also think of this as virtual
displacement virtual displacement W (X) is
the real displacement ?? W(x) the virtual
displacement.
So what is this quantity here this quantity
here if ?? W is virtual displacement this
is the because q (x) is external force this
is external virtual work okay so what we are
saying is that somebody says that this particular
thing for a straight beam if this is the equilibrium
deformation under some loading q(x) okay we
want to verify is it really we go back to
Chris Berman operation energy and imagine
a virtual displacement as if it is perturbed
when it is perturbed to first order it should
be equal to 0 that is exactly what our variation
predations respect w says sectional work and
this portion is actually internal virtual
work internal virtual work okay.
So these two being equal to is nothing but
the principle of virtual work principle of
virtual work that is if you have a structure
if you believe that a particular deformation
is in equilibrium deformation then if you
imagine a virtual displacement it is a virtual
not real then the imaginary virtual work due
to external forces is equal to the internal
virtual work because this is nothing but strain
right that is what we had just written for
abeam.
that e is proportional to W ´´ here so the
e is w´´ y is there of course for a beam
across the cross section strain varies linearly
that is y is there otherwise it is proportional
to ?? W ´.
So that is like real strain this real strain
and ?? W ´´is the virtual strain okay so
if we imagine ?? W virtual displacement they
will be strained so real strain real stress
and then we have of course this here because
stress and strain if you multiply you get
the internal virtual work basically force
and stress are equivalent displacement and
strain or equivalent if force times displacement
has units of energy stressed I am strain also
use of energy so this is the work energy both
have units of joules right.
So internal virtual work external watch work
what you will work that is what we got so
this one tells us.
That from principle of minimal energy we can
derive the principle of virtual work if you
go one step further which was our integration
by parts so if I start from principle of virtual
work 0 to L ELW´´ ?? W´´ - I'm bringing
it to the left side again 0 to L q ?? w well
I have to write dx right so we should not
forget that let me whenever have integration
with respect to what right sinter is respect
to x here so if we have this here so dx equal
to zero now we need to because ?? W is the
one that is arbitrary.
So we have to get rid of this meaning we have
to do integration by parts we have to do twice
and that gives us the boundary conditions
as we had seen earlier right so if I do integration
by parts once then I will get the first function
that is this thing integral a second function
the heater w ´ that is that minus derivative
of first function d/ dx of EI´ the cyber
return d / dx our unit right prime here okay
d /dx of that integral of the second function
dx and this stays as it is that is Q ?? W
dx equal to zero.
Now we need to get rid of this by doing another
integration by parts so when I do that so
I have the same thing here EI W´´ times
?? W ´at 0 to L minus here again the first
function if I take that is d / dx of EIW´´
okay times integral a second function w 0
to L minus of minus plus minus of the minus
thing of the integral of the derivative of
the first function that will become second
derivative r / dx 2 EI W´ integral second
function that will be ?? W x minus 0 to L
q ?? w dx = 0 now we get the boundary conditions
and the differential equation so what for
differential equation we take this and this
I will write it in black 0 to L d2 / dx 2
of EI W ´´ okay minus q this whole thing
multiplied by ?? w dx plus we have the boundary
condition which I shall read in blue this
one and that one.
So that will be EI W´´ ?? W ´ that is a
virtual straying are variations strain minus
d / dx of EL W times ?? W 0 to L all of them
are 0 so individually we say this is equal
to 0 this is equal to 0 and this is equal
o 0 right unless somehow ?? W ´ and LW are
dependent on each other you can make up problems
of that kind then you have to put them together
and say it 0 otherwise you say individually
things are 0 okay so if you make them individually
0.
And the fundamental lemma of calculus of variation
say that this is arbitrary for if you make
that individually 0 which we have to hear
for boundary conditions and differential equation
part that is arbitrary fundamental lemma tells
us that what multiplies that is this thing
should be equal to 0 if I write EI´´ and
this d 2 / dx 2 I will just pull it as two
primes okay minus q = 0 this is what we get
so if I were to be uniform that is it does
not vary with x then that will become EI 40w
is equal to q r - q equal to 0 that is what
ever learnt in mechanics of materials when
you discussed learnt beam theory but this
is more general this is for variable cross
section.
And then we have the these two are the boundary
conditions okay these are the boundary conditions
because we have forth our differential equation
so we need enough boundary condition to solve
for the four constants that are there w let
us say e and I are not dependent on x then
you get the fourth derivative of W when integrated
you need four constants we have two sets of
boundary conditions at 0 and L and 0 and L.
So you get four of those to solve for these
boundary conditions so you can see the weak
form gives rise to strong form this is strong
form y is a strong form not strong worm it
is strong form strong form why is this wrong
form because where needs to be differentiable
four times whereas if you go back to our principle
of virtual work here w needs to be differentiated
only two times that is why it is called weak
form or the principle of virtual work.
Whereas the other one is strong form okay
so what we see now with the example of a be
That we can use principle of minimum potential
energy go back to week four more the principle
of virtual work from there we can go to force
balance our differential equation along with
the boundary conditions so what our boundary
conditions are there whether w specified or
slope is specified we can go back to these
conditions and write out our boundary conditions
deliciously essential are normal or natural
so when W is not specified this thing is not
zero.
So EIW ´´´ should be zero or d/ dx of EIW´´
B 0 where I can be version of x as we have
taken and this equation is very important
for us because now when we want to design
in a beam we actually make W ( x) as a state
variable and look at I the cross-section as
a function of x is what we have taken that
is what we design that becomes the design
variable and this equation that we have becomes
the governing equation for the state variable
which is W (x) governing equation for W W(X)
cannot be anything it has to obey this equation.
Because that is what controls are started
similarly for dynamics if you do the Hamiltons
principle will give you the dynamic equation
which we will discuss in a later lecture when
we look at the calculus variations role in
dynamics thank you.
