analysis of jack-ups units using a constrained newwave methodology

7/29/2019 Analysis of Jack-Ups Units Using a Constrained NewWave Methodology

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Analysis of jack-up units using a Constrained NewWave methodology

M.J. Cassidya,*, R. Eatock Taylorb, G.T. Houlsbyb

aCentre for Offshore Foundation Systems, University of Western Australia, Crawley, WA 6009 Australia

b Department of Engineering Science, Oxford University, UK

Received 25 February 2000; revised 12 March 2001

Abstract

There is a steadily increasing demand for the use of jack-up units in deeper water and harsher environments. Con®dence in their use in

these environments requires jack-up analysis techniques to re¯ect accurately the physical processes occurring. This paper is concerned with

the models appropriate for the dynamic assessment of jack-ups, with a balanced approach taken in considering the non-linearities in the

structure, foundations and wave loading. A work hardening plasticity model for spud-cans on sand is used for the foundation model. The

spectral content of wave loading is considered using NewWave theory, and the importance of random wave histories shown by constraining

the deterministic NewWaves into a completely random background. A method for determining short-term extreme response statistics for a

sea-state using Constrained NewWaves is detailed. q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: NewWave; Constrained NewWave; Jack-up; Spudcan foundations; Work hardening plasticity; Random wave

1. Introduction

As jack-up units are used for most offshore drilling opera-

tions in water depths up to 120 m, they are of signi®cant

economic importance within the offshore industry. Since

their ®rst employment in relatively shallow sections of theGulf of Mexico, there has been a steadily increasing demand

for their use in deeper waters and in harsher environments [1].

There is also a desire for longer commitment of a jack-up at a

single location, especially in the role of a production unit [2].

Before a jack-up can operate at a given site, an assess-

ment of its capacity to withstand a design storm, usually for

a 50-year period, must be performed. In the past, with jack-

ups used in relatively shallow and calm waters, it has been

possible to use overly simplistic and conservative analysis

techniques for this assessment. However, as jack-ups have

moved into deeper and harsher environments, there has been

an increased need to understand jack-up behaviour anddevelop analysis techniques. The publication of the `Guide-

lines for the Site Speci®c Assessment of Mobile Jack-Up

Units' [3] is an example of the offshore industry's desire both

to standardise and to develop jack-up assessment procedures.

The most signi®cant developments in the modelling of

jack-ups have been in the areas of

² dynamic effects,

² geometric and other non-linearities in structural modelling,

² environmental wave loading,

² models for foundation response.

There has been considerable diversity in this development,

with studies using varying levels of complexity for different

aspects of the analysis. This is shown in Table 1, which details

a representative set of published studies from the last 15 years.

The four areas of development highlighted have been broken

into components of increasing degrees of sophistication (and

accuracy). Table 1 shows that single studies often employ

complex methods in one or two of theareas but use the simplest

of assumptions in the others. This is especially true for founda-

tion modelling, with many studies using detailed structural

models or advanced wave mechanics, whilst still using the

simplest of foundation assumptions (i.e. pinned footings).

In calculating wave loading on jack-ups deterministic regu-

lar wave theories,such as Airy andStokes V, together with the

Morison equationare still widely used. However, by assumingall wave energy is concentrated in one frequency component

rather than thebroad spectrum of the ocean environment regu-

lar wave theoriesgives an unrepresentativedynamic response.

For jack-ups, which are dynamically responding structures, it

is important to simulate the random, spectral and non-linear

properties of ocean loading. The extreme dynamic response

depends not only on the load currently being applied, but is

also affected by its load history. Therefore, the most accurate

methods of estimating extreme response are based on time

domain simulation of a pseudo-random ocean surface and

calculation of the corresponding kinematics.

Applied Ocean Research 23 (2001) 221±234

0141-1187/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0141-1187(01) 00005-0

www.elsevier.com/locate/apor

* Corresponding author. Tel.:161-8-9380-3732.

E-mail address: [email protected] (M.J. Cassidy).



Table 1

Level of complexity used in the analysis of jack-up units

Degree of complexity and accuracy

Structural Foundations Dynamic

Assessment

grade

g b b a g b b a g b b a

Author Year Linearelastic

Non-lineareffects as

correction

factors

Non-linearstructural

algorithm

Pinned Linearsprings

Non-linear

springs

Plasticitytheory

Quasistatic

DynamicAmpli®cation

Factor

Frequencydomain

Timdom

P±D Euler

Grenda [26] 86 u u u u

Bradshaw [10] 87 u u ub u u

Schotman [39] 89 u u u

Kjeùy et al. [40] 89 u u u

Brekke et al. [23] 90 u u u

Chen et al. [41] 90 u u u u

Mommass and

Grundlehner [42]

92 u u u u u

Karunakaran et al. [43] 92 u u u uSpidsùe and

Karunakaran [9]

93 u u u

Harland [17] 94 ug u u

Martin [20] 94 u u u

Taylor et al. [4] 95 ug u u

Manuel and Cornell [44] 96 u u u u u

Thompson [5] 96 u u u

Hoyle and Snell j [45] 97 u uk u

Morandi et al. j [46] 97 u u u

Daghigh et al. j [47] 97 u u u um u

Cassidy [7] 99 u u u

a Investigated Dynamic Ampli®cation Factors for different levels of damping for 1000 waves (representing one 3 h event).b Uncoupled springs.c

For random time domain simpli®ed the structure to a stick model.d Fitted extreme response of 3 h storm with 3 parameter Weibull distribution from one 40 min and two 20 min simulations.e Ten 2.3 h simulations.f Ten 3 h simulations.g Used single degree of freedom stick model.h Used Constrained NewWave.i Six 0.57 h simulations. j Studies on jack-up reliability.k Implemented the non-linear stiffness model recommended in the SNAME (1994) procedures.l Used a 200 s segment of one 3 h storm to scale and represent the extreme case.

m Used dynamic analysis of a simpli®ed stick model to evaluate Dynamic Ampli®cation Factors at speci®c H s values for use in a quasi static push over anal



Acquiring con®dence in random time domain simulation

results is, however, computationally time consuming, with

only a few of the waves in each time series capable of

producing the extreme result. For this reason the majority

of the studies in Table 1 used simplifying assumptions. In

contrast to this, by constraining a NewWave with a pre-

determined large crest in an arbitrary random time series,

a method for calculating extreme statistics without the same

degree of computational burden can be devised [4].

The purpose of this paper is twofold. Firstly, a more

balanced approach to the modelling of jack-up platforms

will be described. The major non-linearities in the analysis

of jack-up platforms are considered using a dynamic struc-

tural analysis program, named jakup. The principle char-

acteristics of the structural, foundation and wave loading

models used in jakup are given in this paper, however,

further details can be found in Thompson [5], Williams et

al. [6] and Cassidy [7]. Secondly, a methodology for using

Constrained NewWaves for the evaluation of extreme

response statistics of jack-up units is outlined. Examplenumerical calculations of a jack-up unit subjected to

short-term (three-hour) storms are shown.

2. Outline of the model used

2.1. Example structure

Fig. 1 shows a schematic diagram of the idealised plane

frame jack-up used in this paper. The mean water depth was

assumed to be 90 m with the rig size typical of a three-

legged jack-up used in harsh North Sea conditions. Fig. 1

represents an equivalent beam model, with the correspond-

ing stiffnesses and masses of the beams shown. The hull is

also represented as a beam element with a rigid leg/hull

connection. Though non-linearities in the leg/hull jack

houses are recognised as signi®cant [8,9], they vary from

rig to rig, and depend critically on the details of the leg

jacking and locking mechanisms. Accurate modelling of

these non-linearities can involve, for instance, use of special

elements to model gapping effects (see for example

Ref. [10]). No attempt was made to include these effects

in this study. Example structural node locations and hydro-

dynamic modelling coef®cients for the leg sections are

shown in Fig. 1.

2.2. The wave loading model

NewWave Theory and Constrained NewWave have beenimplemented in jakup to evaluate the surface elevations and

wave kinematics, with the extended Morison equation

incorporating relative motions used to calculate the hydro-

dynamic loads on the jack-up legs. Horizontal particle kine-

matics are calculated at the unde¯ected beam position, with

the equivalent nodal loads found by integrating the dis-

tributed load with the corresponding shape function (using

Gaussian integration techniques).

NewWave theory, a deterministic method described by

M.J. Cassidy et al. / Applied Ocean Research 23 (2001) 221± 234 223

Fig. 1. General layout of idealised jack-up used in the analyses.



Tromans et al. [11], accounts for the spectral composition of

the sea, and can be used as an alternative to both regular

wave and full random time domain simulations of lengthy

time histories. By assuming that the surface elevation can be

modelled as a Gaussian random process, the expected eleva-

tion at an extreme event (for example a crest) can be derived

theoretically. The surface elevation around this extreme

event is modelled by the statistically most probable shape

associated with its occurrence. It is shown by Tromans et al.

[11] that the surface elevation is normally distributed about

this most probable shape, with the surface elevation

described by two terms, one deterministic and one random.

As a function of time, the surface elevation can be

written as

h t ar t 1

1 gt 2

; 1

where t t 2 t 1; the time relative to the initial position

of the crest (i.e. t 1 is the time when the extreme event

occurs). In Eq. (1), term (1) describes the most probablevalue, where a is the crest elevation de®ned as the

vertical distance between the wave maximum and the

mean water-level, and r t the autocorrelation function

for the ocean surface elevation. The autocorrelation

function is proportional to the inverse Fourier Transform

of the surface energy spectrum, allowing the surface eleva-

tion to be determined ef®ciently.

Term (2) of Eq. (1) is a non-stationary Gaussian process

with a mean of zero and a standard deviation that increases

from zero at the crest to s , the standard deviation of the

underlying sea at a distance away from the crest. Therefore,

as the crest elevation increases, term (1) becomes dominant

and can be used alone in the derivation of surface elevationand wave kinematics near the crest.

The continuous time autocorrelation function is de®ned

as:

r t 1

s 2

Z10

S hh v eivt dv 2

with the time history of the extreme wave group propor-

tional to r t at the region around t 0: For a time lag of

t 0; the autocorrelation function of Eq. (2) reduces to one,

allowing the surface elevation of the NewWave to be scaled

ef®ciently. This is shown below in Eq. (3).The NewWave shape as de®ned by the autocorrelation

function (Eq. (1)) can be discretised by a ®nite number ( N )

of component sinusoidal waves. As there exists a unique

relationship between wave number and frequency, spatial

dependency can also be included, leading to the discrete

form:

h X ; t a

s 2

X N

n1

S hh v n dv cosk n X 2 v nt ; 3

where k n is the wavenumber of the nth component. (An

approximate solution method for the dispersion relation

for plane waves in a constant water depth, as outlined by

Newman [12], has been implemented in jakup.) As de®ned

previously, a is the crest elevation, S hh v n dv the surface

elevation spectrum and s the standard deviation corre-

sponding to that wave spectrum. X x2 x1 is the distance

relative to the initial position with X 0 representing the

wave crest. This allows the positioning of the spatial

®eld such that the crest occurs at a user-de®ned position

relative to the structure, a useful tool for time domain

analysis.

This deterministic formulation of NewWave allows it to

be conveniently and ef®ciently implemented into structural

analysis programs, such as jakup. Furthermore, because it is

based on linear theory, the water wave particle kinematics

can be easily obtained once the water surface has been

established. However, care must be taken in describing

their values in a crest with a number of stretching

approaches commonly used. These include Delta stretching

[13] and Wheeler stretching [14]. Although there are numer-ous other stretching or extrapolation methods described in

the literature, there is no clear preferred option. Due to the

substantial separation of jack-up legs the ability to evalu-

ate accurately the time varying surface elevation and

kinematics at two spatial positions is important.

NewWave theory provides a realistic deterministic alter-

native to regular wave theories. For structures which

respond quasi-statically, NewWave theory has been used

successfully in the prediction of global response [15]. It

has also been validated against both measured global load-

ing and conventional random wave modelling on a real plat-

form by Elzinga and Tromans [16] and on standard columnexamples [11]. However, when used as a time-history for

analysing dynamically sensitive structures, NewWave's

single pro®le does not account for the randomness of the

sea in which a large crest is found and therefore the effect

this randomness has on the structure's dynamic response.

By constraining a NewWave with a pre-determined height

within a completely random background, NewWave theory

can be used to produce a time series of a random surface

elevation. This is performed in a mathematically rigorous

manner such that the constrained sequence is statistically

indistinguishable from the original random sequence. The

details of a procedure achieving this are outlined by Taylor

et al. [4].Fig. 2 shows the surface elevation of a NewWave with a

crest elevation of 15 m embedded in a random sea-state

characterised by H s 12 m and T z 10 s: The wave has

been constrained such that its peak occurs at about 60 s. It is

shown in Fig. 3 that, for this example, the in¯uence of the

NewWave on the surface elevation is contained to within

40 s of the constrained peak.

Constrained NewWave allows for the easy and ef®cient

evaluation of extreme response statistics, provided the

required extreme response correlates, on average, with the

occurrence of a large wave within a random sea-state.

M.J. Cassidy et al. / Applied Ocean Research 23 (2001) 221± 234224



Constrained NewWave has been used for the calculation of

extreme response by Harland [17] and Harland et al. [18].

The application of Constrained NewWave to the study of

jack-up response is discussed further here.

2.3. The structural model

Structural non-linearities (P±D and Euler effects) are

considered in jakup by using Oran's [19] path-depen-

dent formulation of beam column theory to specify anincremental stiffness matrix. Additional modi®cations to

produce the additional end rotations on the beam due to

the presence of shear have also been implemented [20].

Both the mass and damping matrix are time invariant,

with the former derived using cubic Hermitian poly-

nomial shape functions and the latter by use of

Rayleigh damping. Structural damping coef®cients are

de®ned for the lowest two modes and have been set

at 2% of critical. This represents a typical value used

for Rayleigh damping [21,22] and is based on estima-

tions of measured jack-ups. For instance, Brekke et al.

[23] estimated structural damping of between 1.8 and

2.8% for the Maersk Guarian jack-up in approximately

70 m of water.

Conventionally, jack-up assessments have used the

same quasi-static analysis methods employed for ®xed

structures [24]; however, the need to consider dynamic

effects has long been acknowledged [10,25,26,]. Withuse in deeper water, the contribution of dynamic effects

to the total response has become more important as the

natural period of the jack-up approaches the peak wave

periods in the sea-state. Due to their ability to model all

non-linearities, time domain techniques using numerical

step-by-step direct integration provide the most versatile

approach to the analysis of the extreme response of

jack-ups. Within jakup, the Newmark b 0:25; d

0:5 method is used, since it is an unconditionally stable

and highly accurate algorithm. Further details of the

structural formulations and dynamic solution techniques

can be found in Martin [20] and Thompson [5].

2.4. The foundation model

The use of a strain hardening plasticity theory is seen as

the best approach to model soil behaviour with a terminol-

ogy amenable to numerical analysis. This is because the

response of the foundation is expressed purely in terms of

force resultants. Accounting for a level of foundation ®xity

reduces critical member stresses (usually at the leg/hull

connection) and other critical response values [27,28].

Furthermore, the natural period of the jack-up is also


Fig. 3. Shape of yield surface in Model C.

Fig. 2. Surface elevation of a NewWave of 15 m elevation constrained within a random background.



reduced, usually improving the dynamic characteristics of

the rig.1

jakup has the capabilities of modelling pinned, ®xed or

linear springs as the foundations of a jack-up. Furthermore,

a strain hardening plasticity model for spudcan footings on

dense sand has been developed and its numerical formula-

tion implemented into jakup. It is called Model C and its

main features are discussed further here, however, for a

detailed description reference can be made to Cassidy [7].

Model C is basedon a series of experimentaltests performed at

the University of Oxford and described in Gottardi et al.

[29] and follows `Model B', a strain hardening plasticity

model described by Martin [20] for spudcans on clay.

Model C has four major components:

(1) An empirical expression for the yield surface in three

dimensional vertical, moment and horizontal loading space

V ; M = 2 R; H (note: moment is normalised by the diameter

of the spudcan, 2 R). This envelope represents a yield locus

de®ning permissible load states and its `rugby ball' shaped

surface is shown in Fig. 3. The shape is best described as an

eccentric ellipse in section on the planes of constant V , and

approximately parabolic on any section including the V -

axis. Once the yield surface is established, any changes of

load within this surface will result only in elastic deforma-

tion. Plastic deformation can result, however, when the loadstate touches the surface.

(2) An empirical strain-hardening expression to de®ne the

variation of the size of the yield surface with the plastic

component of vertical displacement. Though the shape of

the yield surface is assumed constant, it expands with verti-

cal plastic penetration (as the footing is pushed further into

the soil). A strain hardening law for spudcans has been

derived by combining the experimental observations for

¯at circular footings and the theoretical predictions of coni-

cal footings. Details of this `combined' strain hardening law

will not be given here, but can be found in Cassidy [7] or

Cassidy and Houlsby [30].

(3) A model for elastic load-displacement behaviour

within the yield surface. Finite element work has shown

that cross coupling exists between the horizontal and

rotational footing displacements [33], with a linear elas-

tic incremental force±displacement relationship of the

form

dV

dM = 2 R

dH

0BB@

1CCA 2GR

k v 0 0

0 k m k c

0 k c k h

2664

3775

dwe

2 Rd u e

due

0BB@

1CCA 4

where w, u and u are the vertical, rotational and hori-

zontal displacements respectively. G is a representative

soil shear modulus, estimated as:

G

pa

g

2 Rg 0

pa

s ; 5

where pa is atmospheric pressure, g 0 the submerged unit

weight of sand and g a non-dimensional shear modulus

factor. Eq. (4) represents elastic behaviour in Model C

with typical values of the stiffnesses k v; k m; k h; k c and g

given in Table 2.

(4) A suitable ¯ow rule to allow prediction of the footing

displacements during yield. The experimental evidence did

not support the application of associated ¯ow, and a plastic


Table 2

Example values used in the foundation model

Constant Dimension Explanation Typical value Notes

R L Footing radius (various)

g 0 F / L 3 Unit weight of soil 10 kN/m3 Saturated sand

g ± Shear modulus factor 4000 For calculation of the ShearModulus (G) using Eq. 5.

k v ± Elastic stiffness factor (vertical) 2.65 For calculation of elastic

behaviour within the yield

surface using Eq. 4. These

dimensionless elastic

coef®cients were evaluated by

Bell [33] for ¯at circular

footings using ®nite element

methods. These typical values

represent results at the surface

with a Poisson's ratio of 0.2.

k m ± Elastic stiffness factor (moment) 0.46

k h ± Elastic stiffness factor (horizontal) 2.3

k c ± Elastic stiffness factor

(horizontal/moment coupling)

20.14

1 It is still widely accepted practice to assume pinned footings (in®nite

horizontal and vertical and no moment restraint) in the analysis of jack-ups

[31,32]. This results in over-conservative analyses. Another approach often

used in jack-up analysis, and an improvement on use of pinned footings, is

the use of linear springs. Unfortunately, linear springs, while easy to imple-

ment into structural programs, do not account for the complexities and non-

linearities of spudcan behaviour, and this simplistic method can render

unrealistic results, which may be unconservative.



potential function g was de®ned to relate the ratio

between the plastic displacements. Details are given in

Cassidy [7].

3. Example response of a jack-up subjected to a

constrained NewWave

An example jack-up analysis for the structure shown in

Fig. 1 subjected to a Constrained NewWave will be

outlined. As in the rest of this paper, wave loading is the

only environmental force, with no current or wind applied.

Fig. 4 displays the wave as in Fig. 2 constrained such that its

peak is located at the upwave leg of the jack-up. The surface

elevation time history at the downwave leg, as evaluated in

jakup, is also displayed. The corresponding deck displace-

ments with time are shown in Fig. 5 for the cases of linear

springs, Model C and pinned foundations. Pinned footings

represent in®nite horizontal and vertical, but no rotational

stiffness. Model C is the strain hardening plasticity model

described above, whilst linear springs represents the elastic

region of the Model C case (Eq. (4)) with plastic strains

excluded.

The peak displacements shown in this example are

not necessarily what would be evaluated for an analysis

considering just the equivalent NewWave. This is

because of the random background and the structural

memory caused, indicating that for dynamically sensi-

tive structures, such as jack-ups, the response is not

only dependent on the present applied load, but also

on the load history.

The assumption of pinned footings is clearly shown in


Fig. 4. Surface elevations at the upwave and downwave legs for a Constrained NewWave.

Fig. 5. Horizontal deck displacements due to the Constrained NewWave.



Fig. 5 to be overly conservative. The linear springs can be

seen to yield lower displacements than the Model C footing,

due to the greater stiffness exhibited. In addition, Model C

indicates a permanent horizontal displacement occurring in

the jack-up. There is a little less than a factor of four differ-

ences between the pinned and the linear springs cases. This

is expected as non-linearities in both the structure and load-

ing are included, and the linear springs case, though very

stiff, is not fully ®xed. Theoretically, if non-linear and

dynamic ampli®cation effects were not considered, there

would be a factor of four difference between a pinned and

®xed case for a jack-up with a very stiff hull and a rigid leg/

hull connection.

4. Application of Constrained NewWave in evaluating

response statistics of jack-ups

Though only one Constrained NewWave example has so

far been shown here, one of the main bene®ts of the

constraining technique is that the probability distribution

of the extreme response can be estimated without the need

to simulate many hours of real time, most of which is of no

interest. For a storm associated with one sea-state, shorter

time periods can be used with a logical combination of crest

elevations to simulate responses for the expected wave

sizes within that sea-state. Convolution with the prob-

ability of occurrences of crest elevations allows for the

compilation of response statistics. A method used for

evaluating short-term extreme response statistics will

be described here.

Within this paper, simulations of 75 s duration, with

the crests constrained to occur after 60 s, are used to

estimate the time history of response associated with

one crest height. A lead-time of 60 s was considered

to be long enough to give the same statistical response

properties (due to the crest) as longer lead-time periods.

For ®ve discrete crest elevations representative of the

full range of wave heights in a three-hour storm, 200

simulations per crest elevation were performed using


Fig. 6. Explanation of methodology adopted to evaluate the extreme response statistics for one short-term sea-state.



jakup. The largest response caused by the constrained

peak was recorded for each simulation. The extreme

response distributions were evaluated by convolving

responses from the ®ve constrained crest elevations

with the Rayleigh distribution of crest heights. This

was accomplished numerically using Monte Carlo tech-niques. The steps followed are outlined below and are

also shown in Fig. 6:

Step 1: For a short time period (for example 3 h), crest

heights may be randomly derived assuming a Rayleigh

distribution.2 One crest elevation is predicted as

apred

22s 2 ln12 rn

q ; 6

where rn is a random number generated from a uniform

distribution between 0 and 1. Therefore, a set of wave crests

ai; i 1; 2; ¼; N crest; can be estimated from a Monte Carlosimulation ( N crest is the number of wave crests, assumed as

the time period divided by T z).

Step 2: Using the response information from the 5 sets

of 200 jakup runs, lines of constant probability of

exceedence are constructed by ®rstly sorting the popu-

lation of responses at each of the ®ve NewWave crest

elevations into order from the lowest (or 1st) response

to the highest (or 200th) response. Following this, poly-

nomial curve ®tting of the ®ve responses representing

the ®ve NewWave elevations at each response level 1 !

200 gives 200 lines of constant probability.3 These linesallow response values to be estimated for crest elevations

between the ®ve NewWave amplitudes used in jakup.

Construction of the lines of constant probability will be

shown later in the example calculation.

For each crest elevation in step 1, i.e. ai; i

1; 2; ¼; N crest; a response corresponding to its elevation is

`randomly' chosen. By generating a random number

between 1 ! 200 (the size of the population of jakup

runs per NewWave crest elevation), interpolation along

that number's line of constant probability gives the

`random' response for that one crest. Repeating for all N crest

elevations completes a set of responses for that timeperiod. This is shown in step 2 of Fig. 6. Therefore,

for one random three-hour event, the distribution of

responses within that sea-state has been calculated and the

extreme event can be extracted. Importantly, the largest

crest in the sea-state does not necessarily create this extreme

response.

Step 3: By repeating steps 1 and 2, responses for different

three-hour events are evaluated and, by using the maximum


Fig. 7. Deck displacements calculated by jakup for ®ve Constrained NewWave elevations.

2 This is based on the assumption of a Gaussian sea and a narrow-

banded process (see Longuet-Higgins [34]). Cartwright and Longuet-

Higgins [35] have outlined the probability density functions for

maxima as a function of the bandwidth parameter. In the water depths

where the dynamic response of a jack-up if of concern, it is reason-

able to use the Gaussian assumption.

3 The 200 lines of constant probability need to be constructed only once

and contain all of the response information from the jakup runs.



of each, the distribution of extreme response can be

compiled. This is shown in step 3 of Fig. 6. This can be

repeated until the number of extreme responses required is

ful®lled.

In the example numerical experiments in this paper,

distributions are based on 2500 of these samples. Therefore,

this technique has allowed the results of 75,000 s of simula-

tion (21 h) to be used to generate statistics for 7500 h of

storms. Note that even though statistics for 2500 stormsmay seem excessive, it is the initial simulation of

Constrained NewWave that is computationally expensive

and not the Monte Carlo technique described in steps 1±3.

Therefore, to generate 1000, 2500 or even more extreme

response values make very little difference in computational

effort.

This method provides more information than 21 h of

conventional simulation. Although the latter gives an

indication of the mean value of the extreme response,

it would contain too few extremes to establish the CoV

of the extreme response with any accuracy. By focussing

attention on extreme events, the method here provides

more information about them, and in particular provides

information at the tails of the distributions, which can be of

most importance.

5. Example numerical results for one sea-state

An example of the methodology described to compile

short-term extreme response statistics is outlined here. A

sea-state described by the JONSWAP wave energy spectrum


Fig. 8. Lines of constant probability.

Fig. 9. Extreme deck displacement distributions for a three hour period of a sea-state represented by H s 12 m and T z 10:805 s:



with parameters H s 12 m and T z 10:805 s has been

chosen, conditions which could be representative of the

100 year sea-state in a central North Sea location.

Fig. 7 shows the deck displacements calculated by jakup

for the ®ve crest elevations: 3.5, 7, 10, 12 and 15 m. Though

Fig. 7 show a larger variation of deck displacements as the

NewWave crest elevation increases, this is not the case.

When the coef®cient of variation (CoV) is calculated,

de®ned in the usual way as the standard deviation divided

by the mean of the random set, there is actually a reduction

in CoV with increasing crest elevation. This implies that as

the crest elevations become higher, they also become more

dominant in the calculation of global response, thereby

reducing the response's variation.


Fig. 10. Repetition of the calculation of extreme response statistics.

Fig. 11. Comparison of extreme statistics evaluated using the Constrained NewWave method with 100 three-hour random simulations.



Constructed by simple polynomial curve ®tting, Fig. 8

shows four example lines of constant probability between

the levels of evaluated response. The actual number of lines

of constant probability used is not four, as implied in Fig. 8,

but is the number of simulations at each crest elevation

performed by jakup (in this case 200).

The extreme response distribution for deck displacement,

evaluated using the convolution procedure described above,

is shown in Fig. 9. The mean and 50% exceedence values

are 0.251 and 0.241 m, respectively, and the distribution has

a CoV of 22.26%.

5.1. Veri®cation of short-term extreme response results

The accuracy of this method has been examined by

repeating the calculations. Four new sets of 200 extreme

responses per crest elevation were evaluated by jakup,

and the convolution procedures repeated. With little differ-

ence between the resulting statistics, uniformity in succes-

sive tests is shown in Fig. 10. The question of whether 200responses per crest elevation is a large enough sample has

also been addressed. With convolution performed on all the

data (1000 jakup responses per crest elevation), the extreme

response statistics evaluated are consistent with results eval-

uated from the 200 jakup response data. This is shown in

Fig. 10 and validates the use of only 200 extreme responses

per crest elevation.

Consistency is also shown in Fig. 10 for a very extreme

sea-state: H s 16:45 m and T z 12:66 s; corresponding to

a one in a million years. This is an important outcome, as the

method used is validated for cases that involve large

amounts of plasticity and non-linearities in the Model C

calculation. The results have also been veri®ed by con-

ventional random wave simulation for both sea-states. As

shown in Fig. 11, one hundred full three-hour simula-

tions correspond well to the Constrained NewWave

results.

6. Comparison of different footing conditions

Fig. 12 shows the extreme hull displacement statistics for

a three-hour time period for the same two sea-states withthree footing assumptions: linear springs, Model C and

pinned. The mean and CoV values of the extreme response

are also presented. Again, the pinned case is giving overly


Fig. 12. Extreme deck displacement distributions for three footing conditions.



conservative results when compared to the Model C predic-

tions. The yielding effects in Model C are giving only a

slightly larger mean extreme response than the assumption

of linear springs for the H s 12 m and T z 10:806 s case.

However, for the larger sea-state the effects of the plasticity

model are clearly observed. There is an interesting differ-

ence in CoV values for the three cases, with extra variation

in the Model C case due to the use of a plasticity model.

Jack-ups dynamic behaviour is period sensitive and often

more responsive to a sea-state other than the one with the

highest H s. This implies that consideration should be given

to the natural period of the jack-up (signi®cantly affected by

the rotational stiffness of the foundations) and the sea-states

analysed, especially when extrapolating to extreme events.

7. Conclusions

A method for determining short-term extreme response

statistics using the Constrained NewWave was demon-strated. The peak responses due to ®ve Constrained

NewWave elevations (for 200 random backgrounds each)

were used to evaluate numerically the short-term extreme

statistics of that sea-state using Monte Carlo methods. This

was found to be computationally ef®cient and the results

comparable with 100 full 3 h simulations of random seas,

even with the inclusion of signi®cant amounts of foundation

non-linearities.

In this paper, linear NewWaves were constrained within

random backgrounds. Second order corrections to single

non-constrained NewWaves have been suggested by several

authors [36±38]. By accounting for the effects of shortwaves riding on longer waves, both the free surface eleva-

tion and the horizontal ¯uid velocities can be modelled to

second order. The sensitivity of jack-up response to second-

order effects (as well as linear stretching procedures) has

been described for one case at this water depth in Cassidy

[7]. It is possible to constrain second-order waves in a simi-

lar way; however, considering the level of rigour for other

components of the model (as shown for published studies in

Table 1), this might be seen to be overly sophisticated.

Furthermore, even a complete second order analysis

would not adequately account for the observed non-disper-

sive behaviour of very large waves, which retain their form

over distances comparable to the leg separation in a typical jack-up. Further investigation of this effect is recommended.

With knowledge of long-term conditions, convolution

of short-term statistics into long-term probabilities can be

performed. The methodology of evaluating short-term

statistics outlined here allows this procedure to be

performed ef®ciently. Long-term extreme exceedence prob-

abilities for response design properties of interest in the

reliability of jack-ups (for example lower leg-guide

moments or deck displacement) can therefore be evaluated.

Model C, as implemented into the plane-frame structural

analysis program jakup, represents a signi®cant advance to

the response analysis of jack-up units. When compared with

techniques widely used in the jack-up industry, a signi®-

cantly different response is found, as was shown in this

paper. Model C together with the structural model and the

Constrained NewWave technique outlined here re¯ects a

balanced approach to the design uncertainties and achieves

what is believed to be a realistic modelling of a jack-up.

Acknowledgements

Support from the Rhodes Trust for the ®rst author is

gratefully acknowledged. The helpful advice and comments

from Dr Paul Taylor of Oxford University were appreciated

during the course of this research.

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analysis of jack-ups units using a constrained newwave methodology

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