Hello students today we are starting
with a new chapter in in sum the name of
the chapter is Strain energy so let us
start with this chapter we are starting
with this new topic now let us
understand what is Strain energy Strain
energy is the energy which is stored in
any material when it is loaded it means
when we are loading we are applying a
load on a material that material will
get deform and because of the
deformation there is some energy stored
in that material and the moment we
remove the load that material would be
releasing its internal energy it means
here I will write the definition of
strain energy strain energy is the
internal energy which is stored in any
material which is loaded within its
elastic limit
so here is the basic definition the most
simple definition of strain energy that
it is defined as the internal energy
which is stored in any material which is
loaded within its elastic limit it means
the condition for strain energy is that
the material should be loaded within
elastic limit it should not go into
permanent deformation that the material
will be deformed only temporarily it
will store the energy and the moment we
are removing the load the material will
it will release the energy so that is
the concept of strain energy now strain
energy is also called as the work done
by a loaded material so here strain
energy we can also say that it is the
work done by a loaded material next it
is also called as
resilience the other name for strain
energy is resilience and since we are
talking about energy E the unit of
strain energy is either it will be in
terms of newton meter or newton
centimeter Newton mm or kilo Newton
meter
kilo Newton centimeter or kilo Newton mm
since it is energy so here we have load
into displacement and next for strain
energy to be stored material must deform
within elastic limit that is for strain
energy to take place material should
change its shape and the change of shape
should be within elastic limit that is
it should not get permanently deformed
next point strain energy depends upon
applied load the amount of strain energy
stored in any material depends upon the
load which is applied this applied load
it can be gradually applied load it can
be suddenly applied load or it can be
impact load that is a load which is
falling from a height now as far as
application is concerned
strain energy principle is used in
Springs as we all know Springs are
elastic members when we are applying
loads spring will get d-formed
and when the load is removed the spring
will regain its original shape so you
can treat a spring as a material which
has strain energy or you can say stored
energy because the other Memphis strain
energies it is a terminal energy or you
can say the stored energy so as we know
spring when we are applying load it will
change its shape this change in shape is
called a strain and because of that
whatever work is stored inside the
spring that would be called a strain
energy next when the load is removed
material will regain its original shape
next the load applied
on a material can be we have three
different cases in which the load can be
applied the first one is gradually
applied load next suddenly applied load
and at last we have impact loads impact
loads are also called as load falling
from a height so we see that there are
three methods or three ways in which
load will be applied on a machine member
or you can say any material and the load
is gradually applied gradually applied
means the load will be in the form of
steps suddenly applied load the total
value of load would be acting together
on a member in impact load the load will
fall from a height so all these in all
these three cases the strain energy
value will be different because as we
know that strain energy the stored
energy which is there inside a material
it depends on the type of loading next
if we say that it is resilience means
strain energy is the resilience then you
can say that resilience
is the energy which is stored within
elastic limit it is given by little
capital you means we will be denoting
resilience Australian energy by letter U
next I will give you the formula of
strain energy strain energy it is given
by the formula u is equal to Sigma
square 0.2 e multiplied by volume so
here I have written the formula for
strain energy so here we have written
the formula for strain energy where I
can say that Sigma it indicates stress
in the material
unit will be Newton per mm square
capital e is the Youngs modulus of
material and V is the volume of the
material and strain energy unit it will
be Newton mm if I put these values no
after this I'll write how this formula
has come I'll say that strain energy in
and actually loaded member so how the
strain energy formula is developed here
if I consider a bar or a rod which is
loaded actually and I will consider the
load as tensile so here I can see that
the stress
which is stored in the bar that is Sigma
is equal to P upon a and this stress
which is stored I will call this as from
this stress because of this there will
be strain energy which I will denote it
by later capital u now when this member
is loaded we will be getting the
behavior of this rod and I will plot it
on a graph on y-axis we would be having
load that is P load on x-axis we would
be having the deflection deflection is
Delta L now the graph would be linear
since I am considering the load to be
gradual so here I can say that I am
getting this area shaded area because
when you go on increasing the load
deflection will go on increasing this is
gradual load now from this I can say
that area under the curve will give me
the energy stored so I will say that
area is equal to half into P into Delta
L that is the area of this right angle
triangle now since we know that stress
is equal to node upon area so therefore
load is equal to stress into area so
this formula will be half into stress
into area into Delta L we know that
Delta L deflection is equal to PL upon
ei
so P upon a is equal to stress so it is
equal to stress into L upon e stress
into L upon e so therefore here I have
this as equal to Sigma square upon 2 e a
into L will give me the volume so here I
am getting two conditions from the first
condition I am considering that a member
is subjected to external load because of
this there will be internal stress and
because of that internal stress there
will be strain energy stored i give that
value as you next the same behavior if I
plot it on a graph I am getting the area
as Sigma square 1 2 e into V this area
is nothing but strain energy which is
stored so if I give you as equation 1
and Sigma square 2 e upon Sigma square 2
e into V as equation 2 so from 1 & 2
I will be getting the same formula that
is Sigma square 2 e into V so this is
how the formula of strain energy is
developed now after this I write down
what is proof li resilience proof
resilience it is
the maximum strain energy stored in a
material so in other words proof
resilience becomes maximum strain energy
and we can say that maximum strain
energy will be given by it is U max
which will be equal to Sigma max square
upon 2 e into volume so this is the
formula for maximum strain energy after
this I'll write another definition that
is called as modulus of resilience
modulus of resilience it is defined as
the strain energy per unit volume
mathematically it will be denoted by
modulus of resilience is equal to it is
equal to strain energy per unit volume
we know that strain energy is Sigma
square upon 2 e into V and here we are
dividing by volume so the formula which
I have of modulus of resilience
therefore I will say mod of resilience
that will be equal to just Sigma square
upon 2 e so here I have written the
definition of profe resilience and
modulus of resilience now since we know
the strain energy formula let me write
three different formula of stresses for
three different cases it means the
heading is stresses
in members for different cases here I
will first write down the first case is
gradually applied load as we know that
strain energy depends on the method by
which we are applying the load so first
we are seeing gradually applied load so
for gradually applied load stress is
given by Sigma is equal to P upon e in
Newton per mm square next for suddenly
applied load stress is given by Sigma is
equal to 2 P upon e and the last case we
have that is for impact loads that is
load which is falling from a height so
for that stress is given by Sigma is
equal to P Upon A plus square root of P
upon a whole square plus 2 e pH upon a L
so here I have the formula for impact
load for gradually applied load we have
stress as load upon area we can
illustrate this with a diagram so here
is the example of gradually applied load
where the moment load goes on increasing
you can say deflection will be
increasing here I have shown area
actually this is deflection so the graph
is between load versus deflection and
the moment you go on increasing the load
deflection will go on increasing that is
for gradually applied load next for
suddenly applied load here as the load
increases from 0 to a value suddenly
there is increase in deflection so here
we have the diagram for suddenly applied
load next for impact load I'll draw the
diagram on the next page I will say that
for impact loads or impact loading
for studying impact load here we have a
diagram in which there is a rod and one
end of the rod is fixed to the other end
we are attaching a collar now over this
rod there is a weight which is placed I
will denote it by P this P is at a
distance of H from the collar and the
moment we are releasing this load this
weight will try to strike the collar and
it will try to pull it so because of
this there is change in shape of the rod
and strain energy will be stored in the
rod so here I will write the formula of
strain energy as I have already written
previously also so strain energy for
impact load u it will be equal to Sigma
square upon 2 e multiplied by volume and
here stress for impact load is given by
P Upon A plus square root of P upon a
whole square plus 2 e pH upon a L so
here is the formula to calculate stress
for impact loads once we know the value
of load which is falling from a height
we know the area of the rod Young's
modulus for odd material H is the height
between the weight and the collar area
of the rod and length of this rod then
we can easily calculate the value of
stress and once this value of stress is
known we can put it in the formula of
strain energy and get the answer of
energy which is stored in this rod so
like these concepts would be there in
strain energy
and once we finish off the concepts we
can easily start with the problems
