Exercise 2007 SH – 2
Problem 1 Re-assessment of old platforms
You are employed by a consultancy company. An oil company has given the job of reassessing two of their oldest jacket structures to this consultancy company. Their motivation
for this reassessment is that the platforms have nearly reached their original design life time of
25 years, but the oil company would like to extend the life time with another 20 years. In
practise, the fatigue assessment will be a challenging job in connection with life extension.
However, here we shall concentrate on checking the extreme values.
Some information about the platforms and metocean conditions are:
Platform A:
Deck mass: 510 6 kg
Water level: 75m
Freeboard (distance from mean water level to underside of cellar deck): 23m
Platform B:
Deck mass: 1510 6 kg
Water level: 200m
Freeboard (distance from mean water level to underside of cellar deck): 25m
Design wave data:
10-2 – probability crest height: 16m
10-4 – probability crest height: 20m
10-2 – probability wave height: 28.5m
10-4 – probability wave height: 35.5m
Wave period for 10-2 – probability wave height: 13 – 16s
Wave period for 10-4 – probability wave height: 14 – 17s
Highest tidal amplitude: 1m
10-2 – probability storm surge: 1m
10-4 – probability storm surge: 1.2m
a) Here we will not calculate loads explicitly. We do therefore not consider wind and
current. However, in practise one would typically introduce wind load as a static force.
For that purpose one would possibly need the 3-second wind gust at say a height 40m
above sea level. Assume that the Example Metocean Report found on the web site is
representative for the wind climate and estimate the 10-2 – probability 3-second wind
gust.
b) A updated assessment of the environmental conditions have shown that the wave
climate has been slightly worsen since the original design, in particular, when it comes
to the recommended 10-4 – probability crest height. The oil company contact
recommends that one should use the original metocean design data in connection with
re-assessment. Your project manager seems to find that acceptable. Can you support
the approach they seem to agree on? If not – why? Where can you find support for
your view?
c) Exposing the structures to a horizontal load at deck level of 10MN, the corresponding
deck displacement for Platform A is 0.05m, while it is 0.25m for Platform B. Use this
information, the deck mass as given above and neglect the mass of the jacket
construct, and calculate the natural periods for the two platforms.
d) In view of the results from c) which method would you recommend regarding an
estimation of the 10-2 – probability deck displacement? Give the background for your
recommendations. One of your colleagues suggests after reviewing governing rules
that you also should carry out an estimation of an accidental environmental load, i.e.
10-4 – probability load. Will that be necessary here?
e) Describe briefly the steps you need to carry out for the method/methods you suggested
under d).
f) As the job is completed, the oil company is informing you that the latest water depth
measurements have shown that the reservoir subsidence is 3m at both locations, i.e.
the freeboard height is reduced by 3m. Will this have any impact on the work you
have done? If so – what will you suggest as further actions?
Problem 2 Prediction of 10-2 – annual probability response
You are sitting in the design group of a consultancy company. Another group of the company
has been establishing the statistical distributions for the important response parameters. Let us
here consider a response quantity, X. The group have provided you with three different
distribution functions for the target response quantity:

Storm maximum response, Xs
Your colleagues give the following distribution for the storm maximum response:

 x    s 
FX s ( x)  exp  exp 

 s 


where  ,  s ,  s are given parameters. In addition to the distribution function, they
also say that their data base shows that 80 storms have occurred during 40 years.

Annual extreme response, Xy
The annual extreme value distribution is given by:



 x  y 


FX y ( x)  exp  exp 

 y 




where  y ,  y are known parameters.

Global maxima of the response process, Xm (Global maximum is the largest maximum
between adjacent zero-up crossings):
The marginal distribution function for an arbitrary maximum (long term distribution of
maxima) is given by:
m

  x  

FX m ( x)  1  exp    


  m 

where  m ,  m are given parameters. The long term average zero-up crossing
frequency is 0.125 Hz.
Write down the expressions for the 10-2 – annual probability value using the three different
distributions functions.
Would you have any preference regarding choice of distribution to be used for obtaining the
characteristic response? If so – why?
How do you find the design response when you know the characteristic response (i.e. the 10-2
– probability value) determined above? How can you estimate the annual probability of
exceeding the design response.
Problem 3 Deck displacement due to wave-deck impact
A sister platform of platform B in problem 1 is located at a location 3m water depth. Tide and
storm surge is as given in Problem 1, but the wave environment is slightly worse. Updated
value for the 10-4 – probability crest height is 23m in this position, i.e. a wave-deck impact
can be expected in connection with the 10-4 –probability wave event when updated
environmental conditions are used. A rough estimate has come up with a maximum load
during the impact event of 30MN. The impulse load history increases rapidly to this level and
decays almost linearly to zero during 2.5s. Use the information you have for Platform B in
Problem 1 and come up with a rough estimate for the maximum deck displacement in
connection with the impact event when assuming that the mechanical system (platform) is
behaving elastically.
Problem 4 Estimating long term extremes using contour lines
You are going to estimate the 10-2 annual probability deck displacement of a jack-up. The
natural period is around 5.5s. The jack-legs are made slender members.
a) Your boss suggest that you do a linear (frequency domain based) long term response
analysis. You disagree with him – why? Why is a frequency domain not a good idea for this
case? If a fatigue analysis had been the issue could you then follow the recommendation of
your boss?
We will here decide that we shall use time domain simulations in combination with the
environmental contour line.
b) Describe the various steps you have to perform in order to come up with a reasonable
estimate for the 10-2 – probability deck displacement using the contour line method. Contour
lines for hs and tp corresponding to annual exceedance probabilities of 10-1, 10-2 and 10-4 can
be assumed known. (Just tell in brief the various steps you have to do, you do not have to
explain how you will do the various tasks.)
c) An important part is to select a proper short characteristic. Why will we for most cases
have to select a percentile around 0.9? What must be fulfilled if the mean or median 3-hour
extreme should be a good estimate?
Problem 5 Long term analysis using contour lines (See Appendix 7A for some guidance)
For the area we are considering, the long term distribution for the significant wave height of
an arbitrary 3-hour period is given by:
  h   
FH s (h)  1  exp     ,   2.16,  1.35
    
(1)
We shall consider the 3-hour maximum of a response quantity, X, X3h. The distribution of X3h
is found to we well modelled by the Gumbel model:

 x   (h) 
FX 3 h | H s ( x | h)  exp  exp  
,  (h)  20  h,   3  h
 (h) 


(2)
a) Estimate the 10-2 – probability value of the significant wave height. Calculate
thereafter the conditional median (50-percentile) 3-hour maximum of the response
quantity, X, given that Hs is equal to the 10-2 – probability value.
b) The long term distribution of X3h is given by:
FX 3 h ( x)   FX 3 h | H s ( x | h) f H s (h) dh
(3)
h
Introduce the conditional density function for X3h given Hs, and show that the long
term probability of exceeding x can be written:

Q X 3 h ( x)  1  FX ( x)    f X 3 h H s ( s, h) ds dh
(4)
0 x
c) Let us assume that failure occur as X3h > xc. Draw a simple figure where the
sample space for X3h and Hs is indicated. Shade also the failure region of the
sample space and clearly mark the failure line.
d) Instead of solving Eq. (4) numerically, we shall find the long term probability of
exceeding xc, by transforming the problem over to a plane defined by two
independent and standard Gaussian variables, U1 = N(0,1) and U2=N(0,1).
Establish the relations between the physical variables, X3h and Hs, which are
neither Gaussian nor independent, and the standard Gaussian variables U1 and U2.
This is conveniently done using the Rosenblatt transformations:
FH s (h)  (u1 )
FX 3 h | H s ( x | h)  (u 2 )
(5)
Show how the failure surface looks in the transformed space (u-space).
e) In the transformed the contours of the joint distribution of U1 and U2 will be
circles. Explain briefly why? Find from the figure the distance from origin to the
point on the failure surface closest to the origin. Draw a tangent to the contour at
this point. What is the probability of being outside the tangent? The tangent is an
approximation to the failure line used in reliability calculations and referred to as
First Order Reliability Method.
f) The probability calculated above is an approximate value for the long term
probability of exceeding xc since we have utilized the FORM approximation. The
adequacy of the FORM approximation could have been found by solving the
integral numerically in the physically parameter space where the failure line is
simple. If the estimated probability corresponds to an annual probability of 10-2,
that means that xc would have been the 10-2 - probability value of X3h.
Since we are working with an arbitrary 3-hour period, what is the target probability
corresponding 10-2 per year? What is the radius for the contour corresponding to
this exceedance probability in u-space?
The contour with this radius defines all combinations of u1 and u2 corresponding to
an annual exceedance probability of 10-2. Use the transformations found under d)
and determine the corresponding contour in the h-x space.
g) The contour when drawn in the physical parameter space (h-x space) includes all
combinations of h-x that corresponds to an annual exceedance probability of 10-2
per year. What is the probability per 3-hour of exceeding the largest value of x
along the contour? What is the 10-2 – annual probability value of X3h, x0.01? How is
x0.01 compared to the median you calculated under a)? Have you any comments to
the result?
h) If you are told to use the 10-2 – annual probability value of Hs as the sea state,
what could you do in order to obtain a proper estimate of the underlying 10-2 –
annual probability value of X3h.
Discuss this results in view of what you did in Problem 4.
Problem 6: Distribution function of the largest response maximum in a given storm
A problem in connection with applying the POT (or Random Storm) approach is to establish
the distribution function of a given storm event. Let us assume that we consider a given storm
event, where storm is defined as events with significant wave height above some threshold,
h0. Let us approximate the storm by a step function where the duration of each stationary step
is 30 minutes. The particular storm considered herein consists of M steps.
A colleague of you has been working with this problem. He has concluded that the response
process is very well modelled as a Gaussian process. However, he has furthermore found that
important weather characteristics regarding determining the variance of the process are: wind
speed, wind speed direction relative ship heading, significant wave height for wind sea,
spectral peak period for the wind sea, and wave direction relative to the ship heading, swell
significant wave height, swell spectral peak period and swell wave direction relative ship
heading. All the environmental characteristics are given in hindcast data bases. Your
colleague has done a great job. He has found the ship heading for each step of each storm and
calculating the response variance,  2 , and the expected zero up-crossing response period, tz ,
for all 30-minute steps of all hindcast storms exceeding 8m.
(The approach discussed above could in principle be used for obtaining 10-2 rolling angle of a
FPSO which typically will have a heading defined by aiming to have wind in 15 degrees of
the bow. The assumption of linearity, however, will have to be replaced by piecewise
linearity.)
At this time your colleague leaves for another position and you are given the job to show how
to predict 10-2 – annual probability values for the response.
a) Show how you can determine the distribution function for the storm maximum
response with M steps of 30-minutes from the information given above.
b) Assume that you have determined the storm maximum distribution, how can you
establish the distribution function of the storm maximum response of an arbitrary
storm exceeding 8m significant wave height. (Long term distribution of storm
maximum response.)
c) Assuming the expected number of storms exceeding 8m per year is  and set up the
expression of the 10-2 – annual probability response.
d) Do you have any comments to what you have done above comparing it what you do in
a classical (all sea state) long term analysis? If you identify a gap of importance
regarding the final results, can you suggest how to bridge the gap?