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friends welcome to the ninth lecture titled
environmental loads in the first part of this
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lecture we will discuss some complexities
that arise because of conventional environmental
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loads that acts on offshore structures once
we understand this we will slowly move on
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the special loads their complexities and the
response behavior of platforms under special
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loads then we will lead towards fire resistance
design etcetera so today we are going to talk
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about environmental loads as the first part
of the lecture
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environmental loads . . have actually two
components one is they vary with space . they
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vary with time also
they are generally classified based on two
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factors one physical phenomena causing them
the second factor is uncertainty of the loads
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. which is generally accounted in the design
using safety factors when we look at the variation
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of the environmental loads with respect to
time
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again there are two kinds of variations one
can be your macro scale variation which generally
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do not affect the structural response
the . second one is the micro scale variations
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which affect the structural response directly
under macro scale variations
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they can give certain examples lets say average
wind velocity over a specific period of time
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.
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usually this duration of loading is about
ten minutes the second variation in macro
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scale could be tidal current it could be significant
wave height and peak period of the wave spectrum
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and the fourth could be peak ground acceleration
of the earthquake motion so they all come
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under what we call . macro scale variation
they do not actually have direct influence
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on the response of the structural systems
on the contrary if look at micro scale variations
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these variations are so rapid that it makes
significant influence
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on the response of the structure lets say
in our case the platform .
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generally micro scale variations give rise
to dynamic effects on the platform lets say
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under conventional environmental loads let
us talk about wave loads the wind generated
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sea surface waves can be represented by a
. combination of regular waves so what you
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mean by a combination regular waves of different
amplitude
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different period and different wave directions
are combined to represent the wave load acting
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on offshore platforms the question comes how
actually they are represented wave forces
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are generated based on . the water particle
kinematics
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like horizontal water particle velocity horizontal
water particle acceleration vertical water
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particle velocity vertical water particle
acceleration
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the foremost theory which we all know is airys
theory which is also called as linear wave
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theory because it assumes linearity between
the kinematic quantities . and wave height
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it usually assumes a sinusoidal form with
wave height h
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which is very very small compared to wave
length lambda and very very small compared
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to water depth so these are some basic assumptions
which classifies airys theory as linear wave
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theory so according to . this theory if the
sea surface profile is explained as a sinusoidal
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wave with the water depth represented as a
small d the sea surface elevation eta where
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x and t varies in space and time is given
by where h is the wave amplitude k is called
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wave number . which generally given by two
pi by lambda where lambda is called wave length
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which is approximately one point five six
t square where t is called wave period omega
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is wave frequency which is two pi by t and
c p is called phase speed which is omega by
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k which is lambda by t
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once i know the sea surface profile when the
water particle is moving horizontally . in
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x with respect to the space and respect to
time t it generates as the water particle
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moves because of the wind action on sea surface
elevation they generate horizontal water particle
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velocity and vertical water particle velocity
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lets say in meter per second horizontal water
particle velocity is given by omega h by two
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cos hyperbolic k y by sin hyperbolic . k d
cos kx minus omega t
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vertical water particle velocity is given
by omega h by two sin hyperbolic ky by sin
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hyperbolic kd sin kx minus omega t differentiating
this with respect to time we get horizontal
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water particle acceleration which is omega
square h by two . of course cos hyperbolic
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ky and sin hyperbolic kd will remain unchanged
there is a minus sign and that becomes cos
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kx minus omega t there is another minus sign
this omega that becomes plus and omega square
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whereas v double dot x of t is minus omega
h by two sin hyperbolic ky by sin hyperbolic
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kd of cos kx minus omega t
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so there are some limitations where this theory
can be applied airys theory is valid . only
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upto mean sea level
but interestingly there is a complexity here
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when you have a cylinder or a leg of a platform
whose diameter is very large there may be
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a possibility part of the wave maybe in submerged
position of this particular cylinder so there
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is a possibility that the cylinder may occupy
a position which may be in the trough part
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so the differential submergence effect . of
the cylinder with respect to the mean sea
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level is not accounted so we call variable
submergence effect because airys theory computes
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a water particle kinematics only upto mean
sea level the variation because of the sea
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surface elevation with respect to the mean
sea level is not accounted in airys theory
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in that case the solution is people have used
something called stretching modifications
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.
this will account for water particle kinematics
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upto the actual level of submergence before
that lets see some classical definition which
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are again complexities and defining these
wave theories there are some classifications
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set in the literature as deep water
shallow water and intermediate water depth
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.
in this case a classical definition says if
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the water depth is greater than lambda by
two the lambda is the phase length then we
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call this is deep water the characteristic
of deep water is phase speed which is cp that
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is omega by k is independent of depth under
this condition
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shallow water classically says if the water
depth is less than lambda by twenty then its
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classified as shallow water the characteristic
of this is that phase speed . depends only
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on water depths but it is not a function of
lambda in case of intermediate water depth
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the definition says it is between these two
values in this case phase b will be influenced
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by both water depth and period or wave length
. having said this let us know talk about
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the stretching modifications
friends we are looking into the completeness
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of estimating wave loads to some extent so
that we really understand the complexities
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present in the conventional environmental
loads acting on offshore platforms before
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we discuss in detail about the special loads
acting on them and the response behavior of
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offshore platforms under these special loads
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so under the stretching modification the first
classification what we will see is wheelers
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modification this is given by wheeler . in
nineteen seventy irregular waves journal of
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petroleum technology page numbers three fifty
nine three sixty seven it actually modifies
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or modify the horizontal water particle velocity
and accelerations to account for the variable
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submergence effect . so the modified horizontal
water particle velocity is given by
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so you can very well see this is actually
the modification which has happened and the
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acceleration part omega square h by two cos
hyperbolic ky d by d plus eta . sin hyperbolic
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kd sin kx minus omega t but if this is my
sea surface elevation y is measured from here
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if this is my sea bed this is my water depth
t and this is my eta which is sea surface
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which is used here
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the second modification was given by chakrabarthi
called as chakrabarthis modification this
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was given by s k chakarbarthi in nineteen
seventy one .
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discussion of dynamics of single point moorings
in deep water . journal of water waves harbor
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and coastal engineering division asc ninety
seven w n three five fifty eight five ninety
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according to this modification suggested by
chakrabarthi ninety . seventy one the horizontal
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water particle velocity is modified as so
the modification is in the denominator as
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you see from this equation differentiating
to get the acceleration
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we get two equations now suggested by chakrabarthis
modification
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now interestingly let us extend this discussion
. to understand the further complexities in
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wave loads . in general when we calculate
wave forces wave loads are assumed to be based
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on a classic theory saying the sea state is
a short time period short time in sense which
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is typically of three hours duration its an
idealization . it is also assumed to be a
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zero mean process it is also assumed to be
following a gaussian distribution . and it
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is said to be an ergodic process short time
period is defined for a specific duration
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to some degree as engineers we accept this
zero mean process is essentially the statistical
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distribution of the mean distri[bution] of
the wave height where the amplitude is equal
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with respect to a mean line so lets say this
is my mean line gaussian distribution is useful
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because the statistical properties associated
with this are important in estimating the
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characteristics . of the wave loads let us
slightly talk more about these two to understand
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the complexities
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now to understand what is ergodic process
and stationarity in a given wave load lets
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us take an example lets say i have a wave
screen which is varying in amplitude time
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and direction . so lets say this is my t and
this eta in the vertical axis eta one of t
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let us take another history this is again
t two in the vertical axis of this this plots
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eta two of t let us take one more time history
which is t three on the vertical scale of
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this it plots theta three .
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we are in the process of explaining ergodic
process let us take a specific scale here
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and draw a line and draw another line at intervals
of tau
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along this we try to work out the values
we try to work out the values this value and
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this value once i have these values statistically
with us for a good sample this is my sample
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which i can call as . ensemble in statistics
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so what we did is we do lines across the random
records at intervals tau then let us try to
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find the mean square value of this of these
records if this mean square value is unique
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across the ensemble then the process . is
called stationery
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so what is ergodic ergodic is a special stationery
process which has the same mean square value
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as that of the unique mean square value so
ergodic is a single sample so out of the sample
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eta one of t eta two of t eta three of t etcetera
you can pick up any sample this particular
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sample chosen for your analysis is called
as ergodic when the mean square value . of
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the sample is exactly same as that of the
unique mean square value which we found out
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at intimus of tau along the entire ensemble
so this is one of the basic assumption we
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have in terms of estimating wave loads on
offshore platforms
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the issue now comes here is wave loads have
wave amplitude as a variation wave period
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as a variation and of course wave direction
as a variation one is interested to find out
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the maximum force on a given member so the
interest is determine . the maximum force
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maybe on a member lets say on a offshore member
structural member ok how to do this
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consider a surface piercing cylinder
example could be pile of a structure leg of
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a jacket platform we know that on this member
combined drag and inertia . forces are acting
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both of them varies with time and it is interesting
that at a specific occasion this maybe maximum
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so one is interested to know that maximum
load therefore in order to find the maximum
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force
phase angle at which
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this will occur should be first determined
.
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so we all know that the total force on the
pile member is arising from the velocity and
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acceleration components of water particle
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let us obtain this force vector and integrate
it between the boundaries so let us say what
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are the boundaries within which we should
integrate this especially the boundaries . are
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from the sea surface to sea bed that is zero
to minus h so the total force is given by
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half rho c d dia pi square h square by t square
cos theta cos theta sin hyperbole square k
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h sin hyperbolic two k h by four k plus h
by two minus cm . rho pi d square by four
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two pi square by t square h sin theta by k
this equation is obtained by substituting
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the horizontal water particle velocity and
acceleration as obtained by airys theory until
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mean sea level without considering the stretching
modifications
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to find the maximum force i should find
the differential of this and equate it to
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zero because we know the variation of the
force total is with respect to the . angle
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theta so once we do this to substitute i get
theta max as sin inverse of minus pi d by
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h cm by cd twice of sin square k h by sin
hyperbolic two k h plus two k h once i know
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the maximum phase angle then i can find by
substituting theta max in equation one one
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can find f total which is going to the maximum.
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so this will have a specific occasion where
both inertia and drag forces are maximum for
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a given offshore cylinder the second complexity
which arise from computing wave forces and
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members is effect of phase lag on the wave
forces
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interestingly offshore structures have large
water plane area which means that spacing
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. between the members encountering forces
the lateral forces in this case it is actually
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the wave force could also be very high
lets take for example a tension leg platform
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we say the size of the platform is about ninety
meter by ninety meter resting on four columns
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so if this is the plan of a tlp if this is
my x axis this is plan so the centre to centre
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distance between the legs or the column members
. is about ninety meter in the wave direction
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so therefore the phase angle theta could be
two pi by lambda delta x where delta x is
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the distance between the locations where the
wave encounters a structure and lambda of
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course is the wave length so if delta x is
ninety meter as in the case of tlp example
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and for a wave period of ten seconds whose
lambda is approximately . one point five sixty
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square which is one fifty six meters then
theta actually is two pi by one fifty six
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into ninety which is about one point two pi
which can have a global amplification it may
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result in cancellation of forces because it
is one point two pi what does it mean
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it is important now to note it is important
to note that the geometric spacing of the
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members is also governed by .
by the wave forces acting on them please understand
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thats why i said offshore platforms are form
dominated design it is not that you assume
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a structural form and then find the stresses
or forces on the member it is you decide the
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structural form in sense of spacing of the
member such that the wave forces on the members
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gets cancelled so its form dominated design
so thats very interesting that one can use
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intelligently the choice of spacing of the
members so that the forces acting on these
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members can compromise on each other
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so friends in this lecture we are discussing
about the complexities on conventional environmental
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. loads we picked up a discussion the wave
loads and we saw how wave loads and stretching
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modifications can be found can be seen from
the ready equations and we also slightly understood
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the complexities arise because of the phase
angle and other factors that influence the
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mem[bers] forces on the offshore members
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thank you very much .