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from Ekeeda Hello friends welcome back to the subject of machine design 1 we are
a numerical which is based on C frame design so this is a C frame which we are
familiar with given with this particular cross section the cross section here is
T the load is given which C frame is going to support which is 10 kilonewton
the distance between the point of curvature or the center point of
curvature and the endpoint of the load or the application of flowed given is
thousand millimeter the thickness term is given for the inner radius so inner
radius is basically twice the thickness this thickness term is basically
associated with the cross section the cross section has been given like this
where this lip thing is 3t this is the thickness T this is the base of 0.75
times the T the height given is 3t and the distance between the neutral axis
and this given is 2 T which is this one so basically this particular curve which
is the first fiber is nothing but this one and rest of the part of T is nothing
but this one so anywhere across this is considered this section X X looks like
this so the problem statement says that determine the dimensions of the C frame
shown this side in short all the dimensions are parametric there in terms
of e so we need to find out the thickness T it is sufficient to find out
thickness T so as to completely define the dimension of this given C frame they
have provided us with this material property which is the ultimate stress
200 Newton per millimeter square and factor of safety three students one
thing that we need to understand is so low value of the ultimate
indicates that this particular material must be a brittle material though this particular aspect doesn’t
impact this numerical directly but that gives us a better understanding about
different material properties so the low value of this particular ultimate
tensile strength indicates that it is a brittle material so let’s move ahead to
find out the thickness T so that it can define the parametric terms completely
so it can be seen that the cross-section is subjected to two types of stresses
the first one is direct stress and the second one is the bending stress it is quite obvious that bending stress
will be maximum at the inner fiber and the directors will act over the
complete surface or cross-section in the adverse case both these dresses
will act together and that’s why the total stress will be considered using
both of them together so it comes out to be VI which is the bending stress at the
inner surface the formula simply becomes bending moment to H I area of cross
section with eccentricity and the inner radius but the direct stress will be
given by the load P divided by area of cross section we have been given the
ultimate stress value and we have been given factor of safety so let us write
the expression here itself it’s going to be two hundred divided by three so this
is what the main expression that we are going to look at and evaluate the answer
for the unknown parameter T now let us proceed one by one let’s start with the
value R and for the given cross section now given cross section is a section and
from PhD data book we can easily figure out this particular
expression the substitute the values of the available parameters of certain
terms we need to understand precisely because the cross-section values are
given but we need to convert them in terms of the radii in different radii of
curvature so let us start one by one so before we input the expression let’s do
it beside it is understood that the radius value or the thickness value has
to be T in a case the next expression is the width the inner width will of course
will be three times the T the H size which is basically three times just T we
haven’t given the cross-section the inner radius is directly given which is
nothing but 2 times the T so it is quite obvious that outer radius which is
nothing but the inner radius plus the H value comes out to be 5 times 3t and the
last thing is the actual thickness expected in our problem is 0.75 times
the T value there’s mention it T – for the sake of understanding so these are
nothing but the parameters that we can figure out based on the given images or
figures let’s substitute these values here comes out to be T you in the log to the base e if we see this again each and every term
of this expression is in terms of T so T is the main common parameter that we
need to take out so the final our n value comes out to be three four times DT the value will also
be in millimeter and that’s a first finding let me quickly revise that this
particular expression which is quite lengthy need not mugga we can always
find him from PhD data book for the reference purpose moving ahead let us
find out the component R where RI is 2t the H value here is 3d pío values T Bahai values 3t minus 0.75 times DT
there’s nothing – component of T that we have here also it becomes you we seen that this term is also in terms
of tea and at the end we can find it out like odd is equal to three times the
expression number two so moving ahead with other parameters we
can easily figure out the value of e comes out to be the value of H however is comes out to be and area of cross-section for this
particular T is the wave area plus the flange area you so these are the important parameters
that we have found out now let’s move ahead to find out the bending stresses
direct stresses and equate them to the main equation so the bending stress will
be basically force into radius R the force that we have been given is 10 kilo
Newton which makes it 10 into 10 raised to 3 the value R here is 3 times DT so
it makes it newton millimeter and hence the bending stress at the
inner radius or inner fiber comes out to be you area of cross section and the inner radius which is two times
DT after evaluating this expression we get Sigma inner value is equal to so the
bending moment in this case will become this particular expression where R is
nothing but the radius from thee or the radius of curvature or the distance of
the curvature of radius from the center axis and hence it actually becomes 2
times T plus thousand and hence this particular expression comes out to be 2t
plus thousand newton per newton millimeter in that
case this expression does not remain the valid i heard of this when we are going
to find out the bending moment equation and find out the bending stress let’s
substitute the values the value here comes out to be 2t plus thousand after
using this expression we’ll get a very straightforward but lengthy expression
in terms of T let me write it down for you you the expression since is of the stress
we’ll have this unit Newton per millimeter square whereas the
directories in our case will become / area of cross section and then we have arrived with the
expression that we have already derived their substitute D values after
substituting the value of this and this we can understand that there is only one
unknown which is T and therefore solving this expression we will get the value of
T is equal to let me write down this another side which is two hundred
divided by three let me quickly revise after substituting
these two expressions here which is quite lengthy I am avoiding that step
and directly switching back to the next step
where we can find out thickness T and it comes out to be in terms of a quadratic
equation let’s write down the quadratic equation and that is you now this quadratic equation can be
expanded like this and after evaluating this expression
we’ll get T is equal to or T is equal to 100 millimeter – the
value our model is same we have to refer to thee since the thickness T is
directly proportional or inversely proportional with the value of stress
and hence we are to go for the lower value and therefore T is equal to 99
point two millimeter is the optimum solution for the given cross section so
there we end with this particular numerical it metrically revise what we
have done is you consider the C frame and its design analysis says that there
are two types of stresses that may induce which is direct stress and the
bending stress based on which we have written similar expression then we went
on finding Sigma bending and Sigma T in order to find out Sigma bending there
supposed to find out the bending moment and in order to find our we have gone
through different parameters like RN r b RI centricity e h i etcetera etcetera
after finding out those expression we obtain this particular quadratic
equation after solving which Yordy solution so that was from my side thank
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