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EJSE Special Issue: Loading on Structures (2007)

1 INTRODUCTION

The 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 BASICS

A 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.

2.1 Hydrodynamics

Hydrodynamics 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.

2.1.1 Airy Wave TheoryThe surface elevation of an Airy wave of amplitudea, at any instance of time t and horizontal position x

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

( )t xat x κ η −= cos),( (1)

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

Figure 2: Definition diagram for an Airy wave

The alongwave ),( t xu and vertical ),( t xv water particle velocities in an Airy wave at position z

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

( )( )( )

( )t xh

h zat xu ω κ

κ

κ −

+= cos

sinh

cosh),( (2)

( )( )( )

( )t xh

h zat xv ω κ

κ

κ −

+= sin

sinh

sinh),( (3)

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

pendent), via:

( )hg κ κ ω tanh2 = (4)

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

2

19.50.021U4 == η σ s H . (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≈η σ , 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)

S p e c t r a l D e n s i t y

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 η can be expressed as

∑=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ⎟ ⎠

⎞⎜⎝

⎛ −=2/

0

2sin

)(.2)(

N

n

nT

nt

T

f S t φ

π η

η (12)

where η(t) is represented by a series of points (η1,η2, η3, …, ηM) at a regular time step of dt for M points where T = M.dt here represents the timelength of record, and φn is a random phase angle be-tween 0 and 2π. (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 Loads

Wind 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 sea presents 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, U G, 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 U z

zU

z

zU zU ≤⎟

⎟ ⎠

⎞⎜⎜⎝

⎛ =⎟⎟

⎠

⎞⎜⎜⎝

⎛ =

α α

)( (13)

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

Table I Wind speed profile parameters

Terrain Rough Sea Grassland Suburb City centre

α 0.12 0.16 0.28 0.40

zG (m) 250 300 400 500

The drag force, )(t F W , exerted on a bluff body

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

)(2

1)( 2 t AV C t F DaW ρ = (14)

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

exposed area of the bluff body, C D 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 Loads

The 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.

3.2.1 Morison’s EquationThe alongwave or in-line force per unit length

acting on the submerged section of a rigid verticalsurface-piercing cylinder, ),( t z f , from the interac-tion of the wave kinematics at position z from theMWL, (see Fig. 6), is given by Morison’s equation,viz:

),(),(),( t z f t z f t z f D I += (15)

where ( ) ),(4/),( 2 t zu DC t z f M I ρ π = and),(),(5.0),( t zut zu DC t z f D D ρ = represent the iner-

tia and drag force contributions in which C M and C D

represent the inertia and drag force coefficients, re-

spectively, ρ is the density of sea water and D is the

cylinder diameter.

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

Force coefficients C M and C D are found to be de-

pendent upon Reynold’s number, Re, Keulegan-Carpenter number, KC , and the β parameter, viz:

KC

Re

D

T uKC m == β ; (16)

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

tia and drag force components are significant and forKC > 20, drag force progressively dominates.Sarpkaya’s (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 C M decreases as C D 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,C D).

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 world’s offshore platforms - 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 loads

Transverse 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,

N S , 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

S U

nD N = (17)

where U m 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, f L, can be defined via

mm L L U U DC f ρ 2

1

= (18)where C L is the Lift force coefficient that is depend-ent upon the flow conditions. Again, Sarpkaya’s(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

3.4 Diffraction wave forces

Diffraction 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 moment M(t) are given by:

( ) ( ) ( )α ω κ κ κ

ρ −= t ha A

H gt F costanh

2)(

2 (19)

( )

( ) ( )[ ] ( )α ω κ κ κ

κ κ

ρ

−−+

=

t hhhh

a A H g

t M

cos1sectanh

.2

)(2 (20)

where

( ) ( ) ( )[ ] ( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

′

′−=′+′=

−

aY

a J aY a J a A

κ

κ α κ κ κ

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 , J 1 andY 1 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).

3.5 Effect of compliancy (relative motion)

In the situation where a structure is compliant (ienot rigid) and its displacement in the alongwave di-rection at position z from the free surface at time t is

given by x(z,t), then the form of Morison’s equationmodified under the “relative velocity” formulation, becomes:

( )

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

1

),(.1.4

),(..4

),( 22

t z xt zut z xt zu DC

t z x DC t zu DC t z f

D

M M

−−+

−−=

ρ

ρ π

ρ π

(22)

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

the cylinder to be ψ (z), with ψ (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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∫− Ψ≈++0

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

ooo dzt z f zt kxt xct xm (23)

where the integration has been taken to the MWL inlieu of η (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 to be 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:

dz z DC mh

A∫− Ψ=0

22 )(4

' ρ π

(24)

in which C A (= C M – 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 label“added mass” effect.

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

'

)()(2

2

mm

t F t F x x x D I oooooo +

+=++ ω ζ ω (25)

where

dz zt zut F h

I )(),()(0

Ψ= ∫− α (26)

and

( )

dz z zt xt zu

zt xt zut F

o

oh

D

).(.)()(),(

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

ΨΨ−

Ψ−= ∫−

β (27)

in which2

4 DC M ρ

π α = and DC D ρ β 21= , ω o

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

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

( )

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

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

zt xt zut zut zu

zt xt zu zt xt zu

o

oo

Ψ−≈

Ψ−Ψ−

(28)

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

( )'

)(')(2 2

mm

t F t F x x x D I ooo H ooo +

+=+++ ω ζ ζ ω (29)

where

dz zt zut zut F h

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

Ψ= ∫− β (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 ζ H in equation (29) is the contribution

to damping due to hydrodynamic drag interaction

viz

( ) oh

H mm

dz zt zu

ω

β ζ

'

)(),( 20

+

Ψ≈ ∫−

(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 STATES

4.1 Inertia 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 ),( t zu can be obtainedfrom the expression

( )

⎟⎟

⎠

⎞⎟ ⎠

⎞⎜⎝

⎛ −

⎜⎜⎝

⎛ +−= ∑

=

n

N

n n

nn

T

nt

T

f S

h

h zt zu

φ π

κ

κ ω

η 2cos

)(2

.)sinh(

)(cosh),(

2/

0

2

(32)

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

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

N

h

z z ⎟

⎠

⎞⎜⎝

⎛ +=Ψ 1)( (33)

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

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N

N = 4

N = 2

N = 1

N = 0

IN(κh)

κh

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

ζ

2

5

1

1

5

4

3

2

1

01 2 4 8 16

σ

σ

x

xs

f of p

(

⎟⎟

⎠

⎞⎟ ⎠

⎞⎜⎝

⎛ −

= ∑=

n

N

n

n N M I

T

nt

T

f S

h I DgC t F

φ π

κ ρ π

η 2cos

)(.2

).(4

)(2/

0

2

(34)

where:

1,)cosh(

11

1)()(

0,)tanh()(

2,)(1

1)()(

01

0

20

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −−=

==

≥⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −−−= −

N hh

h I h I

N hh I

N h I h

N

h

N h I h I

nn

nn

nn

n N

nn

nn N

κ κ κ κ

κ κ

κ κ κ

κ κ

(35)

Figure 10: Compliant vertical surface-piercing cylinder

Figure 11 presents the variations in )( h I n N κ for arange of power exponents N in mode shape Ψ(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 I N(κh) for varying N

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

I N (κ h) ≈ 1 for κh >> π. 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 (I N(κh) = 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 of f o to that of a near weightlesscylinder with the same stiffness for which f o approaches infinity. It is clear from direct observationof Figure 12, that response levels are controlled by both damping and the amount of relative energyavailable near 'resonance' for a dynamically respond-ing cylinder in an irregular sea state.

4.2 Drag force

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

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

)(.8

),( zt zuu

σ π

≈ (36)

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

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

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

This approximation would seem reasonable for de-termination of ζ H for “stiff” structures, but in situa-tions when the drag force F D(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 VALUES

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

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Sη(f) SFtot

(f)

SFI(f)

SFD(f)H2FD,η

(f)

H2FI,η(f)

X

X

=

=

=+2η σ

2

t F σ

Sη(f) SFtot

(f)

SFI(f)

SFD(f)H2FD,η

(f)

H2FI,η(f)

X

X

=

=

=+2η σ

2

t F σ

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

A linear filter has the characteristics described by( H y,η (f), φ lag(f)) which apply to the Fourier compo-nents of a random time varying quantity (such as awaveheight trace, η (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 η (t), as follows

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟ ⎠

⎞⎜⎝

⎛ −+⎟ ⎠

⎞⎜⎝

⎛ −

= ∑=

)(2

sin)(2

cos

.)()(2/

1

,

nlagnnlagn

N

n

n y

f T

nt b f

T

nt a

f H t y

φ π

φ π

η

(37)

A 'zero-lag' filter would be one for which φ lag(f n) = 0 for all frequencies f n.

The spectrum for y, given by S y(f), can be ob-tained from the spectrum of waveheight, S η (f), via

)().()( 2, f S f H f S y y η η = (38)

5.1 Extreme Wave Forces

Use can be made of the dispersion relationship ofequation (4) in conjunction with the separate de-scriptions above for Inertia and Drag force, to obtain

the associated relationships for )(, f H I F η and)(, f H DF η 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

T F σ , 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

T F

σ 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

yσ , the rate of upcrossings at

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

level y, N y, divided by the time length of trace, T (ieν y = N y /T ). ν 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

e N

N σ

ν

ν (39)

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

2

max

2

1

1 ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

= y y

o

eT

σ

ν (40)

so that

yoT y σ ν .)ln(2max = (41)

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

y

o

oT

T y σ ν

ν .)ln(2

577.0)ln(2max ⎟⎟ ⎠

⎞

⎜⎜⎝

⎛ += (42)

Now the rate of “zero” upcrossings is given by:

π

σ

σ

ν 2

)(

)(⎟⎟ ⎠

⎞⎜⎜⎝

⎛

= y

y

o

∫

∫∞

∞

=

0

0

2

)(

)(

df f S

df f S f

y

y

(43)

which can be determined from the spectral descrip-tion. If y(t) is a narrow-banded process (ie energy isconcentrated at a peak frequency, f p), then ν o ≈ f p.

5.2 Extreme Response Values

The 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


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

=X2

t F σ 2

xσ f o f o

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

=X2

t F σ 2

xσ f o f o

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 by H x,F (f), via

( )22

2

,'

)()(

mm

f f H mF x

+= χ

(44)

where

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

=222

2

21

1)(

o

tot

o

m

f

f

f

f

f

ζ

χ (45)

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

The area under the response spectrum yields2

xσ

and after application of equation (43) to obtain ν 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 REMARKS

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

and design of offshore structures. Emphasis has been placed 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, (LFRD Method). 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 Mechanics for Engineers and Scientist s, 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 Reynold’s number, Proc. 8

th Off-

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 1tT

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 1tT

FtotFI

Drag

Inertia

Total

(b)

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

),(),(),( t z f t z f t z f D I +=

ƒ I (z,t) = α .u ; (α =

π4

ρ CM D2)

ƒ D (z,t) = β u|u| ; (β =12 ρ CD D)

Inertia:

FI (t) = ⌡⌠ -h

o

α .u dz

⎝ ⎜⎜⎛

⎠⎟⎟ ⎞

= ⌡⌠ -h

o

ƒI (z,t) dz

= α ω2

sinh(κh) a.sin(κx-ωt) ⎪

⎪⎪⎪sinh( )κ(z + h)

κ -h

o

= α ω2

κ a . sin(κx - ωt) ⎪⎪ ⎪⎪sinh(κh)sinh(κh) - 0

= α g tanh(κh) . a . sin(κx - ωt)

= α g a sin(κx - ωt) (Deep Water)

Drag:

FD (t) = ⌡⌠ -h

o

β u|u|dz

⎝ ⎜⎜⎛

⎠⎟⎟ ⎞

= ⌡⌠ -h

o

ƒD (z,t) dz

= β a2 ω2

sinh2 κh cos (κx - ωt) | |cos (κx - ωt)

. ⌡⌠ -h

o

cosh2( )κ(z + h) dz

FD (t) = β a2 ω2

sinh2(κh) cos (κx - ωt) | |cos (κx - ωt)

. ⌡⌠

-h

o

1

2 (1 + cosh 2( )κ(z + h) ) dz

FD (t) = β a2 ω2

sinh2(κh) cos (κx - ωt)| |cos (κx - ωt) .

1

2 ⎝ ⎛

⎠ ⎞h + sinh(2κh)

2κ

FD (t) =

⎣

⎢⎡

⎦

⎥⎤

β a2 ω2 h

2 sinh2(κh)

+ β a2 g tanh(κh)

4

⎝

⎜⎛

⎠

⎟ ⎞sinh(2κh)

sinh2(κh)

. cos (κx - ωt)| |cos (κx - ωt)

=⎣⎢⎡

⎦⎥⎤

β a2g κhsinh2κh + β

a2g

2 cos(κx-ωt)| |cos (κx - ωt)

=β a2g

2 ⎣⎡

⎦⎤2 κ h

sinh2κh + 1 cos (κx - ωt)| |cos (κx - ωt)

Hence, as an alternative approximation

FD (t) ≈ β a2g

2 cos(κx-ωt)

Inertia:

α a g tanh(κh) sin (κx - ωt) → FI sin (κx - ωt)

Drag:

β a2g2

⎣⎡

⎦⎤2 κ h

sinh2κh + 1 cos (κx - ωt) . | cos (κx - ωt)|

→ FD cos (κx - ωt) | cos (κx - ωt)|

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

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