introduction to the analysis and design of offshore structures

7/28/2019 Introduction to the Analysis and Design of Offshore Structures

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EJSE Special Issue: Loading on Structures (2007)

1 INTRODUCTIONThe analysis, design and construction of offshorestructures is arguably one of the most demandingsets of tasks faced by the engineering profession.Over and above the usual conditions and situationsmet by land-based structures, offshore structureshave the added complication of being placed in an

ocean environment where hydrodynamic interactioneffects and dynamic response become major consid-erations in their design. In addition, the range of

possible design solutions, such as: ship-like FloatingProduction Systems, (FPSs), and Tension Leg Plat-form (TLP) deep water designs; the more traditional

jacket and jack-up (space truss like) oil rigs; and thelarge member sized gravity-style offshore platformsthemselves (see Fig. 1), pose their own peculiar de-

mands in terms of hydrodynamic loading effects,foundation support conditions and character of thedynamic response of not only the structure itself butalso of the riser systems for oil extraction adopted

by them. Invariably, non-linearities in the descrip-tion of the hydrodynamic loading characteristics ofthe structure-fluid interaction and in the associatedstructural response can assume importance and need

be addressed. Access to specialist modelling soft-ware is often required to be able to do so.

This paper provides a broad overview of some ofthe key factors in the analysis and design of offshorestructures to be considered by an engineer uniniti-ated in the field of offshore engineering. Referenceis made to a number of publications in which furtherdetail and extension of treatment can be explored bythe interested reader, as needed.

Figure1: Sample offshore structure designs

Introduction to the Analysis and Design of Offshore Structures AnOverview

N. HaritosThe University of Melbourne, Australia

ABSTRACT: This paper provides a broad overview of some of the key factors in the analysis and design ofoffshore structures to be considered by an engineer uninitiated in the field of offshore engineering. Topicscovered range from water wave theories, structure-fluid interaction in waves to the prediction of extreme val-ues of response from spectral modeling approaches. The interested reader can then explore these topics in

greater detail through a number of key references listed in the text.

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2 OFFSHORE ENGINEERING BASICSA basic understanding of a number of key subjectareas is essential to an engineer likely to be involvedin the design of offshore structures, (Sarpkaya &Isaacson, 1981; Chakrabarti, 1987; Graff, 1981;DNS-OS-101, 2004).

These subject areas, though not mutually exclu-sive, would include:

Hydrodynamics Structural dynamics Advanced structural analysis techniques Statistics of extremes

amongst others.In the following sections, we provide an overview

of some of the key elements of these topic areas, byway of an introduction to the general field of off-shore engineering and the design of offshore struc-

tures.

2.1HydrodynamicsHydrodynamics is concerned with the study of waterin motion. In the context of an offshore environ-ment, the water of concern is the salty ocean. Its mo-tion, (the kinematics of the water particles) stemsfrom a number of sources including slowly varyingcurrents from the effect of the tides and from localthermal influences and oscillatory motion from waveactivity that is normally wind-generated.

The characteristics of currents and waves, them-selves would be very much site dependent, with ex-treme values of principal interest to the LFRD ap-

proach used for offshore structure design, associatedwith the statistics of the climatic condition of the siteof interest, (Nigam & Narayanan: Chap. 9, 1994).

The topology of the ocean bottom also has an in-fluence on the water particle kinematics as the waterdepth changes from deeper to shallower conditions,(Dean & Dalrymple, 1991). This influence is re-ferred to as the shoaling effect, which assumes

significant importance to the field of coastal engi-neering. For so-called deep water conditions (wherethe depth of water exceeds half the wavelength ofthe longest waves of interest), the influence of theocean bottom topology on the water particle kine-matics is considered negligible, removing an other-wise potential complication to the description of thehydrodynamics of offshore structures in such deepwater environments.

A number of regular wave theories have been de-veloped to describe the water particle kinematics as-sociated with ocean waves of varying degrees ofcomplexity and levels of acceptance by the offshoreengineering community, (Chakrabarti, 2005). Thesewould include linear or Airy wave theory, Stokessecond and other higher order theories, Stream-

Function and Cnoidal wave theories, amongst oth-ers, (Dean & Dalrymple, 1991).

The rather confused irregular sea state associatedwith storm conditions in an ocean environment is of-ten modelled as a superposition of a number of Airywavelets of varying amplitude, wavelength, phaseand direction, consistent with the conditions at the

site of interest, (Nigam & Narayanan, Chap. 9,1994). Consequently, it becomes instructive to de-velop an understanding of the key features of Airywave theory not only in its context as the simplest ofall regular wave theories but also in terms of its rolein modelling the character of irregular ocean seastates.

2.1.1Airy Wave TheoryThe surface elevation of an Airy wave of amplitudea, at any instance of time tand horizontal positionx

in the direction of travel of the wave, is denoted by),( tx and is given by:

( )txatx = cos),( (1)

where wave number L/2= in which L repre-sents the wavelength (see Fig. 2) and circular fre-quency T/2= in which T represents the periodof the wave. The celerity, or speed, of the wave Cisgiven by L/Tor/, and the crest to trough wave-height,H, is given by 2a.

Figure 2: Definition diagram for an Airy wave

The alongwave ),( txu and vertical ),( txv waterparticle velocities in an Airy wave at position z

measured from the Mean Water level (MWL) indepth of waterh are given by:

( )( )( )

( )txh

hzatxu

+= cos

sinh

cosh),( (2)

( )( )( )

( )txh

hzatxv

+= sin

sinh

sinh),( (3)

The dispersion relationship relates wave number to circular frequency (as these are not inde-

pendent), via:

( )hg tanh2 = (4)

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where g is the acceleration due to gravity (9.8 m/s2).

The alongwave acceleration ),( txu is given bythe time derivative of Equation (2) as:

( )( )( )

( )txh

hzatxu

+= sin

sinh

cosh),(

2

(5)

It should be noted here that wave amplitude, a, isconsidered small (in fact negligible) in comparisonto water depth h in the derivation of Airy wave the-ory.

For deep water conditions, h >, Equations (2)to (5) can be approximated to:

( )txeatxu z = cos),( (6)

( )txeatxv z = sin),( (7)

g=2 (8)

( )txeatxu z = sin),( 2 (9)

This would imply that the elliptical orbits of thewater particles associated with the general Airywave description in Equations (2) and (3), would re-duce to circular orbits in deep water conditions asimplied by Equations (6) and (7).

2.2Higher order and stretch wave theoriesA number of finite amplitude wave theories have

been proposed that seek to improve on the restrictionof the negligible wave amplitude compared withwater depth assumption in the definition of Airywaves. The most notable of these include second andhigher order (eg fifth order) Stokes waves, (Chakra-

barti, 2005), waves based upon Fentons stream

function theory (Rienecker & Fenton, 1981), andCnoidal wave theory (Dean & Dalrymple, 1991).

The introduction of the so-called stretch theoryby Wheeler (1970), as implied in its name, uses theresults of Airy wave theory under the negligible am-

plitude assumption as a basis, to map these resultsinto the finite region of their extent from the sea bot-tom to their current position of wave elevation. (Thisis essentially achieved by replacing z with )/1/( hz + in the Airy wave equations presentedabove).

Chakrabarti (2005) refers to alternative conceptsand some second order modifications for achievingstretching corrections to basic Airy wave theoryresults, though not commonly adopted, can nonethe-less be used for this purpose. with Le Mahautes

original description and that of a numbers of otherauthors in this field)

Le Mahaute (1969) provided a chart detailing ap-plicability of various wave theories using wavesteepness versus depth parameter in his description,reproduced here in Figure 3. (The symbol for depthof water is taken as dinstead ofh to be consistent.)

Figure 3: Applicability of Wave Theories

2.3Irregular Sea StatesOcean waves are predominantly generated by windand although they appear to be irregular in character,tend to exhibit frequency-dependent characteristicsthat conform to an identifiable spectral description.

Pierson and Moskowitz (1964), proposed a spec-

tral description for a fully-developed sea state fromdata captured in the North Atlantic ocean, viz:

4

5

2

)(

=

o

eg

S (10)

where = 2f,fis the wave frequency in Hertz, = 8.1 10

-3, = 0.74 ,o= g/U19.5 and U19.5 is

the wind speed at a height of 19.5 m above the seasurface, (corresponding to the height of the ane-mometers on the weather ships used by Pierson andMoskowitz).

Alternatively, equation (10) can be expressed as:4

4

5

5

0005.0)(

= ffp

ef

fS (11)

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in which fp = 1.37/ 5.19U , is the frequency in Hertzat peak wave energy in the spectrum and where

2

19.50.021U4 == sH . (Note that the variance ofa random process can be directly obtained fromthe area under its spectral density variation,hence the basis for the relationship for

2

19.50.005U , from the P-M spectral descrip-tion quoted above). Figure 4 depicts sample plotsof the Pierson-Moskowitz (P-M) spectrum for aselection of wind speed values, 5.19U .

0

20

40

60

80

100

0 0.05 0.1 0.15 0.2 0.25

Frequency (Hertz)

SpectralDensity

U =20m/s

U =18m/s

U =16m/s

U =12m/sU =10m/s

Figure 4: Sample Pierson-Moskowitz Wave Spectra

An irregular sea state can be considered to becomposed of a Fourier Series of Airy wavelets con-forming to a nominated spectral description, such asthe P-M spectral variation.

Then wave height )(t can be expressed as

=

=2/

0

2sin

)(.2)(

N

n

nT

nt

T

fSt

(12)

where (t) is represented by a series of points (1,2, 3, , M) at a regular time step of dt for Mpoints where T = M.dt here represents the timelength of record, and n is a random phase angle be-tween 0 and 2. (Equation (12) offers a convenientapproach towards numerically simulating sea states

conforming to a desired spectral variation via theFast Fourier Transform. Such sea state descriptionscan then be adopted in numerical studies that takeinto account non-linear characteristics and featuresthat would otherwise not be considered for conven-ience).

3 ENVIRONMENTAL LOADS ON OFFSHORESTRUCTURES

3.1 Wind LoadsWind loads on offshore structures can be evaluatedusing modelling approaches adopted for land-basedstructures but for conditions pertaining to ocean en-vironments. The distinction here is that an open seapresents a lower category of roughness to the free-

stream wind, which leads to a more slowly varyingmean wind profile with height and to lower levels ofturbulence intensity than encountered on land. As aconsequence, wind speed values at the same heightabove still water level (for offshore conditions) asthose above ground level (for land-based structures)for nominal storm conditions, tend to be stronger

and lead to higher wind loads. (Figure 5 provides adiagrammatic representation of this mean windspeed variation).

250m

300m

400m500m

Figure 5: Variation of mean wind speed with height

For free-stream wind speed, UG, at gradientheight,zG (the height outside the influence of rough-ness on the free-stream velocity), the mean windspeed at level z above the surface, )(zU , is given bythe power law profile

G

ref

ref

G

G Uz

zU

z

zUzU

=

=

)( (13)

where is the power law exponent and ref refersto a reference point typically chosen to correspond

to 10m.Table I compares values for key descriptive pa-

rameters, and zG, for different terrain conditions,including those for rough seas.

Table I Wind speed profile parameters

Terrain Rough Sea Grassland Suburb City centre

0.12 0.16 0.28 0.40

zG (m) 250 300 400 500

The drag force, )(tFW , exerted on a bluff body

(eg such as the exposed frontal deck area of an off-shore oil rig), by turbulent wind pressure effects canbe evaluated from

)(2

1)( 2 tAVCtF DaW = (14)

where a is the density of air (1.2 kg/m3), A is the

exposed area of the bluff body, CD is the drag coef-ficient associated with the bluff bodyshape/geometry, and V(t) is wind speed at the loca-tion of the bluff body.

3.2 Wave LoadsThe wave loads experienced by offshore struc-

tural elements depend upon their geometry, (the size

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2.5 3 4 5 6 7 8 9 10 15 20 30 40 50 100 150 200

0.4

0.3

0.5

1.0

1.5

3.0

2.0 Re x 10-3

KC

CM

2.5 3 4 5 6 7 8 9 10 15 20 30 40 50 100 150 200

0.4

0.3

0.5

1.0

1.5

3.0

2.0 Re x 10-3

KC

CM

2.5 3 4 5 6 7 8 9 10 15 20 30 40 50 100 150 200

0.4

0.3

0.5

1.0

1.5

3.0

2.0 Re x 10-3

KC

CD

2.5 3 4 5 6 7 8 9 10 15 20 30 40 50 100 150 200

0.4

0.3

0.5

1.0

1.5

3.0

2.0 Re x 10-3

KC

CD

of these elements relative to the wavelength andtheir orientation to the wave propagation), the hy-drodynamic conditions and whether the structuralsystem is compliant or rigid. Structural elements thatare large enough to deflect the impinging wave (di-ameter to wavelength ratio,D/L > 0.2) undergo load-ing in the diffraction regime, whereas smaller, more

slender, structural elements are subject to loading inthe Morison regime.

3.2.1Morisons EquationThe alongwave or in-line force per unit length

acting on the submerged section of a rigid verticalsurface-piercing cylinder, ),( tzf , from the interac-tion of the wave kinematics at position z from theMWL, (see Fig. 6), is given by Morisons equation,viz:

),(),(),( tzftzftzf DI += (15)

where ( ) ),(4/),( 2 tzuDCtzf MI = and),(),(5.0),( tzutzuDCtzf DD = represent the iner-

tia and drag force contributions in which CM and CD

represent the inertia and drag force coefficients, re-

spectively,is the density of sea water andD is the

cylinder diameter.

Figure 6: Wave Loading on a Surface-Piercing Bottom-Mounted Cylinder

Force coefficients CM and CD are found to be de-

pendent upon Reynolds number, Re, Keulegan-Carpenter number, KC, and theparameter, viz:

KC

Re

D

TuKC m == ; (16)

where um = the maximum alongwave water particlevelocity. It is found that forKC< 10, inertia forcesprogressively dominate; for 10 < KC< 20 both iner-

tia and drag force components are significant and forKC> 20, drag force progressively dominates.Sarpkayas (1976) original tests conducted on in-

strumented horizontal test cylinders in a U-tube witha controlled oscillating water column remain to bethe most comprehensive exploration of Morisonforce coefficients in the published literature.

Figures 7 and 8, derived from these results pro-vide an indication of the variation of these force co-efficients with respect to KC and Re. As a rule ofthumb, it can be stated that CM decreases as CD in-creases, and vice versa, and that both values gener-

ally lie in the range 0.8 to 2.0. The drag force coeffi-cient is also influenced by roughness on thecylinder. (Marine growth is particularly troublesomein this regard as it not only increases the effectivediameter of a cylinder, but the increase in roughnessgenerally leads to an increase in drag coefficient,CD).

Figure 7: Inertia force coefficient dependence onflow parameters

Figure 8: Drag force coefficient dependence onflow parameters

The Morison equation has formed the basis for de-sign of a large proportion of the worlds offshoreplatforms - a significant infrastructure asset base, soits importance to offshore engineering cannot be un-derstated. Appendix I provides a derivation of the

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Morison wave loads for a surface-piercing cylinderfor small amplitude Airy waves and illustrates keyfeatures of the properties of the inertia and dragforce components.

3.3 Transverse (Lift) wave loadsTransverse or lift wave forces can occur on offshorestructures as a result of alternating vortex formationin the flow field of the wave. This is usually associ-ated with drag significant to drag dominant condi-tions (KC> 15) and at a frequency associated withthe vortex street which is a multiple of the wave fre-quency for these conditions. The vortex sheddingfrequency, n, is determined by the Strouhal number,

NS, whose value is dependent upon the structuralmember shape and Re, (typically ~0.2 for a circular

cylinder in the range 2.5 x 10

2

< Re < 2.5 x 10

5

), andwhich is defined by

m

SU

nDN = (17)

where Um is the maximum alongwave water particlevelocity and D is the transverse dimension of themember under consideration (eg diameter of the cyl-inder).

The lift force per unit length,fL, can be defined via

mmLL UUDCf 2

1

= (18)where CL is the Lift force coefficient that is depend-ent upon the flow conditions. Again, Sarpkayas(1976) original tests conducted on instrumentedhorizontal test cylinders in a U-tube with a con-trolled oscillating water column, also provide acomprehensive exploration of the lift force coeffi-cient, from which the results depicted in Figure 9have been obtained. (It should be noted here, that inthe case of flexible structural members, when thevortex shedding frequency n coincides with the

member natural frequency of oscillation, the resul-tant vortex-induced vibrations give rise to the so-called lock-in mechanism which is identified as aform of resonance).

Figure 9: Lift force coefficient dependence on flow parameters

3.4Diffraction wave forcesDiffraction wave forces on a vertical surface-piercing cylinder (such as in Fig. 6) occur when thediameter to wavelength ratio of the incident wave,

D/L, exceeds 0.2 and can be evaluated by integrat-ing the pressure distribution derived from the time

derivative of the incident and diffracted wave poten-tials, (MacCamy & Fuchs, 1954). Integrating thefirst moment of the pressure distribution allowsevaluation of the overturning moment effect aboutthe base. Results obtained for the diffraction forceF(t) and overturning momentM(t) are given by:

( ) ( ) ( )

= thaA

HgtF costanh

2)(

2 (19)

( )

( ) ( )[ ] ( )

+

=

thhhh

aAHg

tM

cos1sectanh

.2

)(2 (20)

where

( ) ( ) ( )[ ] ( )( )

=+=

aY

aJaYaJaA

1

11tan

211 ;

2

1

2(21)

in which a is the radius of the cylinder (D/2), () de-notes differentiation with respect to radius r, J1 andY1 represent Bessel functions of the first and second

kinds of 1

st

order, respectively. It should be notedthat specialist software based upon panel methods, isnormally necessary to investigate diffraction forceson structures of arbitrary shape, (eg WAMIT,SESAM).

3.5Effect of compliancy (relative motion)In the situation where a structure is compliant (ie

not rigid) and its displacement in the alongwave di-rection at positionz from the free surface at time tis

given byx(z,t), then the form of Morisons equationmodified under the relative velocity formulation,becomes:

( )

( ) ),(),(),(),(...2

1

),(.1.4

),(..4

),( 22

tzxtzutzxtzuDC

tzxDCtzuDCtzf

D

MM

+

=

(22)

Consider the structure concerned to be of theform of the surface-piercing cylinder depicted inFigure 6. Consider the displacement at the MWL tobe xo(t) and the primary mode shape of response of

the cylinder to be (z), with (0) = 1, then the cyl-inder motion can be considered to satisfy that ob-tained from the equation of a single-degree-of-freedom (SDOF) oscillator, given by:

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++0

),()()()()(h

ooo dztzfztkxtxctxm (23)

where the integration has been taken to the MWL inlieu of(t), and at x = 0, as an approximation. Coef-ficients m, c, and k represent the equivalent mass,viscous damping and restraint stiffness of the cylin-

der at the MWL. (Note that allowing for forcing tobe considered at x(z,t) via u(x,z,t) produces non-linearities that normally have only a minor effect onthe character of the response (Haritos, 1986)).

When equation (22) for f(z,t) is substituted

into equation (23) above, the so-called added

mass term is identified for the cylinder viz:

dzzDCmh

A =0

22 )(4

'

(24)

in which CA (= CM 1) is the added mass coeffi-cient.

This is an important result as it suggests that forall intensive purposes a body of fluid surroundingthe cylinder appears to be attached to it in its iner-tial response, and hence the coining of the labeladded mass effect.

Equation (23) can be re-cast in the form

'

)()(2

2

mm

tFtFxxx DIoooooo +

+=++ (25)

where

dzztzutFh

I )(),()(0

= (26)

and

( )

dzzztxtzu

ztxtzutF

o

oh

D

).(.)()(),(

.)()(),(.)(0

=

(27)

in which2

4DCM

= and DCD21= , o

is the natural circular frequency of the first modeand o is the critical damping ratio of the structure inotherwise still water conditions.

In the case of )()( txz o small compared to u(z,t)an approximation that can be made for this interac-tive term is of the form:

( )

)()(),(2),(),(

)()(),()()(),(

ztxtzutzutzu

ztxtzuztxtzu

o

oo

(28)

Under these circumstances, equation (25) can befurther simplified to:

( )'

)(')(2 2

mm

tFtFxxx DIoooHooo +

+=+++ (29)

where

dzztzutzutFh

D )(),(),()('0

= (30)which is interpreted as the level of equivalent drag

force at the MWL in the case of rigid support condi-tions (negligible dynamic response).

The term H in equation (29) is the contribution

to damping due to hydrodynamic drag interaction

viz

( ) oh

Hmm

dzztzu

'

)(),( 20

+

(31)

(In the case of large diameter compliant cylinder in

the diffraction forcing regime, analogous expres-sions can be derived for added mass effects and ra-diation damping due to structure-fluid interaction ef-fects).

4 RESPONSE TO IRREGULAR SEA STATES4.1Inertia Force

Since the inertia force term FI(t) in equation (29)is linear it generally poses little difficulty in model-ling under a variety of hydrodynamic conditions.

Consider an irregular sea state composed of aFourier Series of Airy wavelets conforming to a P-Mspectral description. Then ),( tzu can be obtainedfrom the expression

( )

+=

=

n

N

n n

nn

T

nt

T

fS

h

hztzu

2cos

)(2

.)sinh(

)(cosh),(

2/

0

2

(32)

in which n satisfies the dispersion relationship ofequation (4).

In the case of(z) being a power law profile, asin Figure 10, then

N

h

zz

+= 1)( (33)

and FI(t) can be shown to be obtainable via the ex-pression given by (Haritos, 1989),

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N

N=4

N=2

N=1

N=0

IN(h)

h

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

% 25

10

100

5

4

3

2

1

01 2 4 8 16

xxs

fofp

(

= =

n

N

n

nNMI

T

nt

T

fS

hIDgCtF

2cos

)(.2

).(4

)(2/

0

2

(34)

where:

1,)cosh(

11

1)()(

0,)tanh()(

2,)(1

1)()(

01

0

20

=

=

==

=

Nhh

hIhI

NhhI

NhIh

N

h

NhIhI

nn

nn

nn

nN

nn

nnN

(35)

Figure 10: Compliant vertical surface-piercing cylinder

Figure 11 presents the variations in )( hI nN for arange of power exponents N in mode shape (z).The result for N = 0 is consistent with the derivationfor the inertia force component acting on a rigid cyl-

inder due to an Airy wave made in Appendix I.

Figure 11: Variation of IN(h) for varying N

It is observed that all variations for )( hI nN are as-ymptotic to 1 and that I0() is close to this value tothe order of accuracy associated with the deep wa-ter limit of Airy waves (ie h = ) but for N>0,

IN(h) 1 forh >> . In general, the effect ofhigher order mode shapes (N > 0) is to reduce thelevel of inertia forcing of each Airy wavelet in an ir-

regular sea state.Figure 12 depicts the results obtained for the re-

sponse of a vertical cylinder in deep water condi-tions (IN(h) = 1) for inertia only forcing in uni-directional P-M waves.

Figure 12: Influence of Dynamic Properties on Response(Inertia dominant forcing in deep water)

The levels are quoted as the ratio of the standard de-viation in the response of a cylinder exhibiting anatural frequency offo to that of a near weightlesscylinder with the same stiffness for which fo ap-proaches infinity. It is clear from direct observationof Figure 12, that response levels are controlled byboth damping and the amount of relative energyavailable near 'resonance' for a dynamically respond-ing cylinder in an irregular sea state.

4.2Drag forceWhilst it is possible to deal with the u|u| term fordrag force numerically, the linearised approximation

to ),( tzu attributed to Borgman (1967), can be usedin the case u(z,t) Gaussian in a random sea state, sothat

)(.8

),( ztzuu

(36)

where u(z) is the standard deviation in waterparticle velocity and where current is taken aszero-valued for all z.

This approximation can be used to simplify theexpressions for both H in equation (31) and FD(t)in equation (30) and to thereby obtain closed-

form solutions in the case of nominated (z)variations.

This approximation would seem reasonable for de-termination ofH for stiff structures, but in situa-tions when the drag force FD(t) is considered domi-nant, this linearization can lead to significant errorsin the modelling of both the non-linear drag forceand the prediction of the resultant response, accord-ing to Lipsett (1985).

5 EXTREME VALUESIn the case of random vibrations associated with lin-ear systems, use can be made of upcrossing theory

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S(f) SFtot

(f)

SFI(f)

SFD(f)H2FD,

(f)

H2FI,(f)

X

X

=

=

=+2

2

tF

S(f) SFtot

(f)

SFI(f)

SFD(f)H2FD,

(f)

H2FI,(f)

X

X

=

=

=+2

2

tF

in combination with spectral modelling of the proc-esses involved to develop a basis for prediction ofpeak response values, (Nigam & Narayanan, 1994).

A linear filter has the characteristics described by(Hy,(f), lag(f)) which apply to the Fourier compo-nents of a random time varying quantity (such as awaveheight trace, (t), conforming to say a P-M

spectrum) at frequency f, to produce a modified re-sultant time varying quantity,y(t), that is linearly re-lated to (t), as follows

+

= =

)(2

sin)(2

cos

.)()(2/

1

,

nlagnnlagn

N

n

ny

fT

ntbf

T

nta

fHty

(37)

A 'zero-lag' filter would be one for which lag(fn)= 0 for all frequenciesfn.

The spectrum fory, given by Sy(f), can be ob-tained from the spectrum of waveheight, S(f), via

)().()( 2, fSfHfS yy = (38)

5.1Extreme Wave ForcesUse can be made of the dispersion relationship of

equation (4) in conjunction with the separate de-scriptions above for Inertia and Drag force, to obtain

the associated relationships for )(, fH IF and)(, fH DF respectively, and hence the total forcespectrum for the surface-piercing cylinder of Figure6. A diagrammatic illustration of the concept is pro-vided in Figure 13.

Figure 13: Diagrammatic description of spectral modelling ofMorison wave loading

The area under the total force spectrum equals thevariance, 2

TF , knowledge of which may be used to

estimate peak Morison loading of the vertical sur-face-piercing cylinder, under consideration. This

peak load value may reasonably be expected to be ofthe order

TF

3 , but a more precise estimation is of-fered through the use of upcrossing theory.

For a Normally distributed trace y(t) with zeromean and variance 2

y , the rate of upcrossings at

level y, y, equates to the count of upcrossings at

levely, Ny, divided by the time length of trace, T (iey = Ny/T). o would correspond to the rate of up-crossings of the zero mean.

It can be shown (Newland, 1975) that upcrossingsfor such a trace would satisfy

2

2

1

==y

y

o

y

o

y

eN

N

(39)

The concept of a peak value in a time period ofT would correspond to a y value with an upcrossingcount of 1 so thatymax can be estimated from

2

max

2

1

1

= yy

o

eT

(40)

so that

yoTy .)ln(2max = (41)

Because the value ofymax itself shows a statisticalvariation, Davenport (1964) has suggested a smallcorrection to equation (41) for the value ofE(ymax)so that

y

o

oT

Ty

.)ln(2

577.0)ln(2max

+= (42)

Now the rate of zero upcrossings is given by:

2

)(

)(

=y

y

o

=

0

0

2

)(

)(

dffS

dffSf

y

y

(43)

which can be determined from the spectral descrip-tion. Ify(t) is a narrow-banded process (ie energy isconcentrated at a peak frequency,fp), then ofp.

5.2Extreme Response ValuesThe concepts above can be applied to the dynami-cally responding surface-piercing cylinder to esti-mate the peak response at MWL, (xo)max.

An additional stage is required for this purpose,

namely the linear transformation from Morison forc-ing to dynamic excitation via the description ofequation (29), which in terms of a spectral modellingapproach, is diagrammatically depicted in Figure14.

63


10/11


SFtot(f) H2x,F(f) Sx(f)

=X2

tF 2

xfo fo

SFtot(f) H2x,F(f) Sx(f)

=X2

tF 2

xfo fo

Figure 14: Diagrammatic description of spectral modelling ofdynamic response

The Transfer Function for response from Morisonloading in irregular sea states for the dynamically re-sponding surface-piercing cylinder of Figure 10 isgiven byHx,F(f), via

( )22

2

,'

)()(

mm

ffH mFx

+=

(44)

where

+

=222

2

21

1)(

o

tot

o

m

f

f

f

f

f

(45)

in whichfo is the natural frequency of the cylin-der and tot is the total critical damping ratio (tot=o+H).

The area under the response spectrum yields2

x

and after application of equation (43) to obtain o,extreme value (xo)max can be obtained from equation(42), for the nominated duration of the irregular seastate under consideration, (eg T = 3600 secs for a 1-hour storm).

6 CONCLUDING REMARKSThis paper has provided an overview of some of thekey factors that need be considered in the analysis

and design of offshore structures. Emphasis has beenplaced on modelling of the hydrodynamic responseof a compliant vertical surface-piercing cylinder inthe Morison loading regime under uni-directionalwaves. Reference has also been made to a number of

publications in which further detail and extension oftreatment can be explored by the interested reader.

REFERENCES

API, 1984. Recommended Practice for Planning, Designing

and Constructing Fixed Offshore Platforms, American Pe-troleum Institute, API RP2A, 15th Edition.

Borgman, L.E. 1967. Spectral Analysis of Ocean Wave Forceson Piling, Jl WW& Harbors, Vol. 93, No. WW2, pp 129-156.

Davenport, A.G. 1964. Note on the Distribution of the LargestValue of a Random Function with Application to GustLoading, Proc. ICE, Vol. 28, No. 187.

DNV-OS-C101, 2004. Design of Offshore Steel Structures,General (LFRD Method). Det Norske Veritas, Norway.

DNV-OS-C105, 2005. Structural Design of TLPS, (LFRDMethod). Det Norske Veritas, Norway.

DNV-OS-C106, 2001. Structural Design of Offshore Deep

Draught Floating Units, (LFRD Method). Det Norske Veri-tas, Norway.Chakrabarti, S. K. (ed) 2005.Handbook of Offshore Engineer-

ing, San Francisco: Elsevier.Chakrabarti, S. K. 2002. The Theory and Practice of Hydrody-

namics and Vibration, New Jersey: World Scientific.Chakrabarti, S. K. (ed) 1987. Fluid Structure Interaction in

Offshore Engineering, Southampton: Computational Me-chanics Publications.

Chakrabarti, S. K. 1994. Hydrodynamics of Offshore Struc-tures, Southampton: Computational Mechanics Publica-tions.

Dean R. G. & Dalrymple, R. A. 1991. Water Wave Mechanicsfor Engineers and Scientists, New Jersey: World Scientific.

Rienecker, M.M. & Fenton, J.D. 1981. A Fourier approxima-tion method for steady water waves, J. Fluid Mechs, Vol104, pp 119-137.

Graff, W. J. 1981. Introduction to Offshore Structures De-sign, Fabrication, Installation, Houston: Gulf PublishingCompany.

Haritos, N. 1986. Nonlinear Hydrodynamic Forcing of Buoysin Ocean Waves, Proc. 10th Aust. Conf. on Mechs. ofStructs. & Materials, Adelaide, pp 253-258.

Haritos, N. 1989. The Influence of Modal Characteristics onthe Dynamic Response of Compliant Cylinders in Waves,Computational Techniques & Applications: CTAC-89, editW.L. Hogarth & B.J. Noye, pp 683-690, (Hemisphere).

Holand, I., Gudmestad, O. T. & Jersin, E. (eds). 2000. Designof Offshore Concrete Structures, London: Spon Press.ICE, 1983, Design in Offshore Structures, ICE, London: Tho-

mas Telford Ltd.Le Mehaute, B. 1969. An introduction to hydrodynamics and

water waves, Water Wave Theories, Vol. II, TR ERL 118-POL-3-2, U.S. Department of Commerce, ESSA, Washing-ton, DC.

Lipsett, A.W. 1985. Nonlinear Response of Structures in Regu-lar and Random Waves, Ph.D. Thesis, Univ. of British Co-lumbia, Canada.

MacCamy, R. C. & Fuchs, R.A.1954. Wave Forces on Piles: aDiffraction Theory, U.S. Army Coastal Engineering Centre,Tech Memo No. 69.

Newland, D.E. 1975. An Introduction to Random Vibrationsand Spectral Analysis. Longman.

Nigam, N. C. & Narayanan, S. 1994. Applications of RandomVibrations, New York: Springer-Verlag.

Sarpkaya, T. 1976, In-line and transverse forces on cylinders inoscillating flow at high Reynolds number, Proc. 8

thOff-

shore Technology Conference, Houston, Texas, OTC 2533,pp 95-108.

Sarpkaya, T. & Isaacson, M. 1981.Mechanics of Wave Forceson Offshore Structures, New York: Van Nostrand

SESAM,http://www.dnv.com/software/systems/sesam/programModules.asp

Stewart, R. H. 2006. Introduction to Physical Oceanography,

http://oceanworld.tamu.edu/resources/ocng_textbook/PDF_files/book.pdf.WAMIT, http://www.wamit.com/Wheeler, J.D. 1970. Method for calculating forces produced by

irregular waves, Journal of Petroleum Technology, March,pp 359-367.

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-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1t

T

FtotFI

Drag

Inertia

Total

(a)-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1t

T

FtotFI

Drag

Inertia

Total

(b)

Appendix I. Base shear on a surface-piercingcylinder from Morison loading

),(),(),( tzftzftzf DI +=

I(z,t) = .u ; ( =

4

CM D2)

D (z,t) = u|u| ; ( =12 CD D)

Inertia:

FI (t) = -h

o

.u dz

= -h

o

I (z,t) dz

= 2

sinh(h)a.sin(x-t)

sinh( )(z + h)

-h

o

= 2

a . sin(x - t) sinh(h)sinh(h) - 0

= g tanh(h) . a . sin(x - t)

= g a sin(x - t) (Deep Water)

Drag:

FD (t) = -h

o

u|u|dz

= -h

o

D (z,t) dz

= a22

sinh2h cos (x - t) | |cos (x - t)

. -h

o

cosh2( )(z + h) dz

FD (t) = a22

sinh2(h)cos (x - t) | |cos (x - t)

.

-h

o

1

2(1 + cosh 2( )(z + h) ) dz

FD (t) = a22

sinh2(h)cos (x - t)| |cos (x - t) .

1

2

h + sinh(2h)

2

FD (t) =

a22 h

2 sinh2(h)

+ a2 g tanh(h)

4

sinh(2h)

sinh2(h)

. cos (x - t)| |cos (x - t)

=

a2g hsinh2h +

a2g

2cos(x-t)| |cos (x - t)

= a2g

2

2 h

sinh2h + 1 cos (x - t)| |cos (x - t)

Hence, as an alternative approximation

FD (t) a2g

2cos(x-t)

Inertia:

a g tanh(h) sin (x - t) FI sin (x - t)

Drag:

a2g2

2 h

sinh2h + 1 cos (x - t) . | cos (x - t)|

FD cos (x - t) | cos (x - t)|

Figures I.a and I.b depict representative varia-tions over one cycle of Airy wave of the Base Shearforce acting on a cylinder normalised with respect toFD/FI = 2, respectively, by way of illustration.

Figure I: Morison Base Shear Force components for (a): FD/FI = 0.8 and (b): FD/FI = 2

65

introduction to the analysis and design of offshore structures

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