So, we will continue with the discuss analysis
of static cable. So, we are doing the
mooring analysis. Now there are the 2 basic
equations that we have derived last class.
.
That is your d T prime, this is equal to W
sin phi d s and the other one that we have
derived was T prime d phi is equals to W cos
phi d s. Now if you divide these two you
will get d T prime by T prime you write in
this form sin phi over cos phi. So, this we
have derived. Now, in this equation this you
integrate now sin phi d phi; you can write
as
this I think you can write this as d cos phi
let us put a negative sign. So, this expression
becomes. So, minus d cos phi over cos phi
so now you can integrate this. So, this will
be
your expression if you integrate 
this will be log of T prime this will be minus
log cos phi
I have to find out the constant C plus C.
So, this is your expression if you integrate
this
expression I think this was equation number
7; this was 8 probably you just check and
this was 9. Now, starting from our, this chain
cable.
..
The diagram that we had drawn now, there are
2 points which are your boundary
conditions. So, boundary conditions will be
at the sea bed under the water line. So, this
is
your chain cable it is assuming A, it is a
shape of a catenary and at this juncture it
becomes horizontal with this sea bed. So,
now, your, it is anchored at this point. So,
this
is your T, T prime there is a tension in the
cable and out here at the water surface this
becomes this is T W or T prime W just at the
water plain it will become that. Now, here
at the, this juncture it meets the sea bed.
So, if you draw a tangent and the tension
will be
T prime subscript 0 and this angle will be
phi equal to phi 0. So, these we have to these
boundary conditions in our problem in order
to get the expression.
Now here at this point this will be T prime
will be equal to tour prime 0. So, now, let
us
find out. So, here are the 2 objectives we
have, we have to find out in previous diagram
with d x value. This is what we were aiming
at first you find out the length of the chain
cable then distance along the x axis and along
the z axis. So, we have to find out the
length of this chain cable and this horizontal
distance is x. And we have to find out that
the length of the chain cable from here the
total length you subtract the portion that
is
lying in the seabed. So, this will be a 0.
So, the total length is x minus X 0 you have
to
calculate this find out x h. Of course, you
know and calculate the tension T. So, this
is the
flux of the problem what you have to solve.
.Now, the equation that you get out here so
you give boundary conditions and boundary
conditions are the one that I have restricted
and here you substitute. So, how much what
is your equation? So, the equation that we
have got is log T prime. So, you say this
is at
A and this is let us suppose this is A point
B. So, you at A, you substitute boundary
conditions. So, boundary conditions are phi
equal to phi m and another is T prime is
equals to T prime dot. So, then what happens
to this equation.
So, log of what you will get say log T prime
naught. So, we have to calculate the value
of
C. So, this is minus log cosec phi naught.
So, now, you can get your C naught from this
T
if you have T naught and phi m so but how
to calculate you know these values. So, this
is
see T T naught actually it will be slant at
this point that is a horizontal force or tan
curve
and the inclination of this point. So, those
I think you have to find out from instruments
otherwise you cannot measure.
Now, here so from here value of C what is
the value of c. So, this is log of so if want
to
analyze this chain cable see you have to find
T prime naught and phi naught. Now, what
is normally done in this you have to have
certain instrument? So, you have to do some
line instrumentation; you calculate the horizontal
force of tan curve at this point and T
prime you can put some this thing what is
call stargaze out weal and calculate the friction
in at others various points. So, normally
in various things you know various structure
or
this they do this kind of instrumentation
the similar thing you will find at the 2 end
find
out the stargaze in piddles. So, I think probably
in your affected also you will talk about
the severe piddling there also you have 2
instrument.
So, here you so now, we are getting C from
this. So, this you can simplify then what
is
expression? So, this will be log of T prime
naught whatever by cos phi naught A into B.
So, now, you are getting the expression. So,
your expression is coming like this log of
T
prime. So, this is equals to minus log cos
phi another value of C. So, C will be log
of T
prime naught cos phi naught. So, now, here
you can simplify. So, this will be log of
T
prime naught cos phi naught over cos phi.
So, now, we have simplified this. So, from
this
equation you can calculate your since both
sides are log.
..
So, this T prime that is your tension becomes
equals to this is T prime naught cos of phi
naught over cos phi this is your expression
for the tension at any point of the chain
cable
now this prime indicates what real of the
first equation that we had done what was the
significance of the prime actually we are
connected for hydrostatic pressure. So, if
you
look at this equation. So, what I have written.
So, what was the correction this
expression. So, this expression was your T
prime so; that means, that is just the
correction for you hydrostatic pressure I
think I would. So, this is your collection
for T
prime is the tension in the chain cable corrected
for hydrostatic pressure. So, this is
expression, and remember we are taking the
weight of the chain cable that is taking into
account to your bouncy the bouncy term. So,
that is the net weight which is acting
downwards.
So, now the other terms that are coming is
we have the first equation that we started
out
the 2 most important ones are this. And these
two expressions you should remember the
other equation will follow from this now from
8 you write now, we have to calculate the
length of the chain cable. So, you may use
this equation. So, from 8 write the serial
numbers you just check, because if you put
down the correct serial number. Now, what
you can find out from this is the length of
the chain cable d s. So, this is equation
8. So,
from here we can calculate the first you find
out the expression for d s. So, d s is d prime
d phi divided by W cos phi now, you integrate
this expression; you put the boundary
conditions and integrate now T prime what
is a expression for T prime.
.So, T prime you substitute this expression.
So, d s will become. So, max here I will the
max is not that difficult. So, you can easily
follow the only thing we have certain spells
od integration that is all. So, in differential
equation are not comb. So, this a very simple
analysis so this will be your d phi over W
cos phi. So, now, your d s is coming as cos
square phi. So, if you want to find out S
you just integrate this, the right hand side
W cos
phi. So, 1 W comb will be there now it is.
So, now, you integrate this expression. So,
we
are interested in calculating the length of
the chain cable, but for which portion that
is of
the catenary portion.
So, you put the limits of integration as S
and s naught. So, this will be you just integrate
this. So, what are your variables? So, you
bring this the T prime naught that is your
initial condition at phi naught. So, this
way you are supposed to know this from
instrumentation or from physical examination
of the chain cable by diverse going down.
So, this you put this divided by W and this
especially you integrate between phi naught
and phi. So, that is d phi over cos square
phi now if you integrate you check your
integration terms for this will be…
.
What is the expression for integration of
cos square phi? So, this will be S minus S
naught and at right hand side you will get
tan phi. So, do the few you just check on
any
book on integration and differentiation you
will get this. So, this will be from phi phi
naught. So, ultimately your equation is coming
like this S minus S naught this is equals
.to this will be cos phi naught over W. And
this expression will be tan phi minus tan
of
phi naught for you are getting this expression.
So, this is the catenary part you write in
brackets catenary length now, please do not
talk this is your catenary length. Now, you
find out the horizontal component of x say;
this is your expression for your x the
horizontal total and again your X naught you
subtract from the ankle length. So, this is
your X naught. So, you find out this distance.
So, this distance is X minus X naught. So,
what is the expression say take a element
of length about this line.
So, you call it d x above and this length
is say d s now can you get the expression
the
angle is you write phi it is a phi angle.
So, from this you can calculate d x. So, that
is d x
is d x cos phi good. So, the expression is
d x over d x. So, that is equal to something
cos
phi. So, from this d x is equals to d x cos
phi. So, now, you substitute the expression
for d
x. So, what was the expression d s that we
had got? So, d s was this expression.
So, you write cos phi you bring it forward.
So, multiply it by d s. So, d s is T prime
naught cos phi naught W cos square phi naught.
So, this again we multiplied by d phi
now you simplify this how much we get and
then you integrate. So, this becomes T
prime naught over W is this the constant term
the other term that you will get is the
variable term is d phi over cos phi . So,
now, you integrate if you integrate the expression
for x. So, we can write in this form. So,
that is X naught X d x. So, this will be T
prime
over W cos phi naught now you integrate this
expression d phi over cos phi. So, this is
your phi naught to phi. So, you integrate
his you will get X minus X naught. Now, what
is the integral of this 1 by cos phi?
..
So, X minus X naught is equal to. So, integration
of 1 by cos phi how much? You check
your again your, you will get this expression.
So, this is little bit large. So, this minus
X
naught is now similarly, you find out you
calculate Z minus Z naught you calculate Z
minus z naught. If you want to calculate Z
minus naught then what is the expression that
you should hey [fl] stop talking you find
out Z minus Z naught I have shown you how
to
calculate X minus X naught 
and after that you calculate the tension t.
So, what is the
value of d z over d s? Now similarly, we have
started in this expression that is d x by
d s.
So, it is just simply cos. So, this will be
cos or sin. So, this will be sin phi.
So, now, from this expression you can calculate
Z minus Z naught that is z particle
projection of a cable line. So, you have calculated
the horizontal projection that is x
minus X naught. So, this is called the horizontal
projection of the catenary this is the
horizontal projection. So, this expression
is simple you take d z over d s is equal to
sine
phi. So, from this expression if you integrate
what you do substitute the value of this d
s.
So, your previous expression what we have
substituted d s was this is sin phi multiply
by
what was the, you find out what is the expression
for d s? Now remember in doing
expression for d s we had substituted T prime
is not it?
So, ultimately with T prime naught that is
at the sea bed we are getting this now, you
do
this do the integration is it expression for
d s is it right. So, now, you what it is called?
So, you take out the, your constant terms
outside. So, this is coming as dived by this
is
.cos phi naught then the other term that you
get is sin phi. So, we are getting sin phi
out
here. So, this will be cos square phi, cos
square phi cos phi will not be cos phi squaring
again d phi. So, if you integrate this you
will get Z minus z naught. So, that is your
vertical projection of the catenary, so that
here if you integrate Z minus Z naught
integration will be d Z.
.
Z Z naught and what is your after integration;
what you will get for this? So, this will
be
phi naught limits will be phi naught phi now
you integrate this expression. So,
integration is 
if you remember here you put simply put sin
phi d phi equal to this
expression put r equal to cos phi. Then you
will get what is your d of cos phi it is minus
sin phi d phi. So, this expression you can
write, but you change your limits by phi and
phi naught. So, this expression will be integral
of how much say minus d of cos phi and
this will be cos square phi.
So, now it is simple. So, this will be how
much? So, now, can you do it? So, if you do
this, this will be 1 by r. So, our expression
is coming as Z minus Z minus z naught. So,
this will be how much. So, this is the expression
for Z minus z naught. So, has to be you
have found out all the items, we have found
out the length of the chain cable is 1 is
S
naught. Then we have found out x minus X naught
your x minus X naught is not delta
your x minus X naught you have found out.
So, the most complicated expression is the
.horizontal length on the sea bed is not it?
This one because you are having this log term
the other is the Z minus Z naught now, you
calculate your tension.
.
Now, phi naught phi naught is the angle 
made at sea bed 
you write during contact. Now
if you want to simplify this equation; you
can write phi naught equal to 0. Now, what
was your expression for the tension corrected
for your hydrostatic pressure cable line
tension? What was the equation we had got
remember that first we started out with? So,
I
have given you the expression for d T prime
is not it? Then from that we have derived
the expression for T prime. So, what was your
expression for T prime? So, T prime you
have got as this expression this is T prime
naught cos phi naught over cos phi. So, this
expression was coming from that 2 equations
that d T and T prime equations which we
have divided and we have found out.
So, now can you find out from this expression
here you substitute this phi naught equal
to 0 since we are signifying this you put.
So, cos phi naught is going to be one. So,
your
expression for T prime coming as T prime naught
over 1 by cos phi. So, this and this
expression you remember now, next you calculate
horizontal component 
horizontal
component of tension at water line. So, what
was the horizontal component at seabed?
So, horizontal component with tension you
write at water line that is your winch; winch
tension was coming, how much winch tension
you have written this as T prime W now
.how much is the angle? So, angle was given
as phi W. So, here you find this is T H. So,
we will come across a simple tension you know
it will come from free body mechanics.
So, how much is the horizontal tension cable?
It is quite simple you just take the
component. So, at water line the winch tension
is T W. So, how much is T H T H;
obviously, will be T W multiplied by cos phi
W. But you have to know both T W and phi
W now this T W and phi W. You have to know
T W is actually you can straight away find
out from anchor winch, but phi W how are you
going to find out. So, phi W also you
have to find out from ship dick. So, this
is fine now, what was your expression for
T
prime?
.
So, we will come back to this equation that
is our T H later on, but we you tell me the
expression for T prime. So, T prime was the
corrected tension minus what the hydrostatic
pressure. So, rho g Z A, A is your cross section
area of chain cable now here what is the
expression for T prime naught 
you tell me this T prime naught at water plain
what are the
expression for T prime naught. So, T prime
naught is T prime multiplied by cos phi now
water plain what is going to happen this expression
of water plain if you write.
So, this T prime naught will be equals to
T prime you just simple put subscript W. So,
this will be cos phi W. So, this is your at
water plain or at water surface you simply
put T
prime W at here and cos phi W that is all
now you compare. Now I have to water surface
this expression that is T prime naught now,
you just see that you have put a prime out
.here; that means, you have corrected for
hydrostatic pressure. So, this will be T W
minus
how much 
you write the expression and you then you
put Z equal to 0. So, these
expressions this is cos phi W.
So, I have simply written the expression for
T prime naught now, at water surface 
at
water surface what is the value of Z, Z at
water surface is 0. So, then your expression
for
T prime naught becomes what? So, this is simply
T W cos phi W now, you compare the
expression the expression that we have obtained
earlier. So, what we are getting now?
So, previously we have obtained T H is equals
to T W cos phi. So, this you put some
number out here. So, this is 9 or 10 whatever
we are forgetting. So, compare with 9 this
is 10. So, this you write this as 10.
So, compare with 10 how much we are getting?
So, you write this as 11 comparing with
10 we get T H equal to T prime naught. So;
that means, what is our inference that is
we
are balancing the horizontal force at d follow
so; that means, in your free body diagram.
So, your if your T H is in this direction
you are simple getting T naught in the opposite
direction. The horizontal forces has to be
balanced what about the vertical forces that
T
W sin phi is used to balance the weight now,
what is the condition at seabed?
.
Now, you choose the conditions at sea bed
your mathematics is not that difficult, but
you
have to work out. So, X 0 you put this; you
started from X 0 equal to 0 now, what is the
value of Z 0 sea bed minute? Sea bed is minus
h or plus h minus h and you set S naught
.this you start at 0. In order to simplify
the problem you are making all these assumptions
phi naught will be equal to for chain lying
horizontal on the sea bed. So, your phi naught
equal to 0 now, you write down the expression
for S minus naught. See what are the 3
equations that we have just now derived? The
3 equations X minus X naught. And your
this S minus S naught Z minus Z naught now,
your x minus X naught that we have
derived is it is a quite a long expression.
So, X minus X naught this is equal to T prime
dot over W now these are fundamental equations
which we have to use now.
So, this multiplied by log of derive you will
get long expression. So, this is sec phi plus
tan phi you put some number else otherwise
you will become late to write this down. So,
this is sec phi naught plus tan phi naught,
now you tell me the expression the expression
is Z minus z naught. So, this is one relationship
which we should not forget. So, whereas,
write this as 12. The other expression that
we have derived is Z minus Z naught. So, what
was you expression for Z minus Z naught that
is the vertical this thing. So, multiplied
by
1 minus cos phi naught now, you tell me S
minus S naught. So, that is the expression
we
require now.
So, this is say equation number thirteen.
So, S minus S naught how much we have
derived this expression? We have just derived
now I see this papers lose what was your
expression? So, T prime over multiplied by
cos phi naught over W now you in all these
expression; you substitute this value equal
to this sea bed conditions and see how much
you get. So, they are the 3 important equation
12 13 and 14. So, these 3 equations
actually give you the diameters of the chain
cable the length of the cables. And there
projections at the horizontal and vertical
axis now, you simplify with this sea bed
conditions. So, sea bed conditions if you
in this expression for S minus S naught cos
phi
naught is how much? So, this will be cos phi
naught I remember you are starting with S
naught equal to 0. So, you simplify 14. So,
put boundary conditions in 14.
..
Or rather you write put sea bed conditions.
So, this will be simply S, your S naught is
0
and there on the right hand side we get 0
cos phi naught you will get 1. So, this will
become T prime naught over W and tan phi naught
will be how much your phi naught is
we have assumed phi naught to be 0. So, this
expression simplifies to obey this
expression tan phi. Now, we have found out
T prime naught is equals to what was your
expression for T prime naught that you have
derived that is equal to T H somewhere I
have written this. So, you substitute that.
So, this equation let us write this as see
how
much and these sequence are not following
actually see from 15. So, S becomes equals
to T H over W multiplied by tan phi.
So, now, we have obtained a very simple equation
for S able to see that these all simple
sea bed conditions and mathematical algebra
that is all. Now, you derive this for X minus
X naught, what were the expression for X minus
X naught? And you substitute this sea
bed that is all. So, x so this we have you
just tell me what is the equation what should
I
put equation number 15? I have already put
inside let us put this as sixteen 15 then
you
do not put any number out there. So, your
X expression for X minus X naught was T
prime naught over W. Now, you signify that
you put boundary conditions you will get the
answer. So, this will be log of. So, remember
this is only study kind of this is we have
not started dynamic analysis the other one
was cos static is not it.
.So, here you substitute how much this becomes
T prime naught is T H is not it. So, you
just write T H, T H over W your cos phi naught
becomes 1 and here this will be log of
the other term will be how much. So, this
will be simply log of 1 by cos phi plus tan
phi.
So, this is equation number how much is this
16 or what? Now, you tell me what is the
expression for? So, this is your X X naught
we have started as 0. So, your x is equals
to
this now you find out z what you substitute
in this expression when you substitute you
do
not put Z naught equal to 0. Because your
sea bed condition what were sea bed
conditions sea bed condition; you put Z said
Z naught equal to minus h do not put Z
naught equal to 0 in the other equation. So,
this you complete it in your home. So, will
complete this and then we will find out the
expression for Z plus h there is still a long
way to go.
.
