EJSE Special Issue: Loading on Structures (2007)

Introduction to the Analysis and Design of Offshore Structures– An

Overview
N. Haritos
The University of Melbourne, Australia

ABSTRACT: This paper provides a broad overview of some of the key factors in the analysis and design of
offshore structures to be considered by an engineer uninitiated in the field of offshore engineering. Topics
covered 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.

1 INTRODUCTION mands in terms of hydrodynamic loading effects,

foundation support conditions and character of the
The analysis, design and construction of offshore dynamic response of not only the structure itself but
structures is arguably one of the most demanding also of the riser systems for oil extraction adopted
sets of tasks faced by the engineering profession. by them. Invariably, non-linearities in the descrip-
Over and above the usual conditions and situations tion of the hydrodynamic loading characteristics of
met by land-based structures, offshore structures the structure-fluid interaction and in the associated
have the added complication of being placed in an structural response can assume importance and need
ocean environment where hydrodynamic interaction be addressed. Access to specialist modelling soft-
effects and dynamic response become major consid- ware is often required to be able to do so.
erations in their design. In addition, the range of This paper provides a broad overview of some of
possible design solutions, such as: ship-like Floating the key factors in the analysis and design of offshore
Production Systems, (FPSs), and Tension Leg Plat- structures to be considered by an engineer uniniti-
form (TLP) deep water designs; the more traditional ated in the field of offshore engineering. Reference
jacket and jack-up (space truss like) oil rigs; and the is made to a number of publications in which further
large member sized gravity-style offshore platforms detail and extension of treatment can be explored by
themselves (see Fig. 1), pose their own peculiar de- the interested reader, as needed.

Figure1: Sample offshore structure designs

55
 EJSE Special Issue: Loading on Structures (2007)

2 OFFSHORE ENGINEERING BASICS Function and Cnoidal wave theories, amongst oth-
ers, (Dean & Dalrymple, 1991).
A basic understanding of a number of key subject The rather confused irregular sea state associated
areas is essential to an engineer likely to be involved with storm conditions in an ocean environment is of-
in the design of offshore structures, (Sarpkaya & ten modelled as a superposition of a number of Airy
Isaacson, 1981; Chakrabarti, 1987; Graff, 1981; wavelets of varying amplitude, wavelength, phase
DNS-OS-101, 2004). and direction, consistent with the conditions at the
These subject areas, though not mutually exclu- site of interest, (Nigam & Narayanan, Chap. 9,
sive, would include: 1994). Consequently, it becomes instructive to de-
• Hydrodynamics velop an understanding of the key features of Airy
• Structural dynamics wave theory not only in its context as the simplest of
• Advanced structural analysis techniques all regular wave theories but also in terms of its role
• Statistics of extremes in modelling the character of irregular ocean sea
amongst others. states.
In the following sections, we provide an overview
of some of the key elements of these topic areas, by 2.1.1 Airy Wave Theory
way of an introduction to the general field of off- The surface elevation of an Airy wave of amplitude
shore engineering and the design of offshore struc- a, at any instance of time t and horizontal position x
tures. in the direction of travel of the wave, is denoted by
η ( x, t ) and is given by:
2.1 Hydrodynamics
η ( x, t ) = a cos(κx − ωt ) (1)
Hydrodynamics is concerned with the study of water
in motion. In the context of an offshore environ- where wave number κ = 2π / L in which L repre-
ment, the water of concern is the salty ocean. Its mo- sents the wavelength (see Fig. 2) and circular fre-
tion, (the kinematics of the water particles) stems quency ω = 2π / T in which T represents the period
from a number of sources including slowly varying of the wave. The celerity, or speed, of the wave C is
currents from the effect of the tides and from local given by L/T or ω/κ, and the crest to trough wave-
thermal influences and oscillatory motion from wave height, H, is given by 2a.
activity 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, associated
with the statistics of the climatic condition of the site
of interest, (Nigam & Narayanan: Chap. 9, 1994).
The topology of the ocean bottom also has an in-
fluence on the water particle kinematics as the water Figure 2: Definition diagram for an Airy wave
depth changes from deeper to shallower conditions,
(Dean & Dalrymple, 1991). This influence is re- The alongwave u ( x, t ) and vertical v( x, t ) water
ferred to as the “shoaling effect”, which assumes particle velocities in an Airy wave at position z
significant importance to the field of coastal engi- measured from the Mean Water level (MWL) in
neering. For so-called deep water conditions (where depth of water h are given by:
the depth of water exceeds half the wavelength of
aω cosh (κ (z + h ))
the longest waves of interest), the influence of the u ( x, t ) = cos(κx − ωt ) (2)
ocean bottom topology on the water particle kine- sinh (κh )
matics is considered negligible, removing an other-
wise potential complication to the description of the aω sinh (κ ( z + h ))
hydrodynamics of offshore structures in such deep v ( x, t ) = sin (κx − ωt ) (3)
water environments. sinh (κh )
A number of regular wave theories have been de-
veloped to describe the water particle kinematics as- The dispersion relationship relates wave number
sociated with ocean waves of varying degrees of κ to circular frequency ω (as these are not inde-
complexity and levels of acceptance by the offshore pendent), via:
engineering community, (Chakrabarti, 2005). These
would include linear or Airy wave theory, Stokes ω 2 = gκ tanh (κh ) (4)
second and other higher order theories, Stream-

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 EJSE Special Issue: Loading on Structures (2007)

where g is the acceleration due to gravity (9.8 m/s2). original description and that of a numbers of other
authors in this field)
The alongwave acceleration u ( x, t ) is given by Le Mahaute (1969) provided a chart detailing ap-
the time derivative of Equation (2) as: plicability of various wave theories using wave
aω 2 cosh (κ ( z + h )) steepness versus depth parameter in his description,
u ( x, t ) = sin (κx − ωt ) (5) reproduced here in Figure 3. (The symbol for depth
sinh (κh )
of water is taken as d instead of h to be consistent.)
It should be noted here that wave amplitude, a, is
considered small (in fact negligible) in comparison
to water depth h in the derivation of Airy wave the-
ory.
For deep water conditions, κh >π , Equations (2)
to (5) can be approximated to:

u ( x, t ) = aω eκz cos(κx − ωt ) (6)

v( x, t ) = aω eκz sin (κx − ωt ) (7)

ω 2 = gκ (8)

u ( x, t ) = aω 2eκz sin (κx − ωt ) (9)

This would imply that the elliptical orbits of the

water particles associated with the general Airy
wave description in Equations (2) and (3), would re-
duce to circular orbits in deep water conditions as
implied by Equations (6) and (7).

Figure 3: Applicability of Wave Theories

2.2 Higher order and stretch wave theories

A number of “finite amplitude” wave theories have 2.3 Irregular Sea States
been proposed that seek to improve on the restriction
of the ‘negligible wave amplitude compared with Ocean waves are predominantly generated by wind
water depth’ assumption in the definition of Airy and although they appear to be irregular in character,
waves. The most notable of these include second and tend to exhibit frequency-dependent characteristics
higher order (eg fifth order) Stokes waves, (Chakra- that conform to an identifiable spectral description.
barti, 2005), waves based upon Fenton’s stream Pierson and Moskowitz (1964), proposed a spec-
function theory (Rienecker & Fenton, 1981), and tral description for a fully-developed sea state from
Cnoidal wave theory (Dean & Dalrymple, 1991). data captured in the North Atlantic ocean, viz:
The introduction of the so-called “stretch” theory ⎛ ωo ⎞ 4
by Wheeler (1970), as implied in its name, uses the αg 2 − β ⎜ ⎟
results of Airy wave theory under the negligible am-
S (ω ) = 5 e ⎝ ω ⎠ (10)
ω
plitude assumption as a basis, to map these results where ω = 2πf, f is the wave frequency in Hertz, α
into the finite region of their extent from the sea bot- = 8.1 × 10-3, β = 0.74 , ω o = g/U19.5 and U19.5 is
tom to their current position of wave elevation. (This the wind speed at a height of 19.5 m above the sea
is essentially achieved by replacing “z” with surface, (corresponding to the height of the ane-
“ z /(1 + η / h) ” in the Airy wave equations presented mometers on the weather ships used by Pierson and
above). Moskowitz).
Chakrabarti (2005) refers to alternative concepts Alternatively, equation (10) can be expressed as:
and some second order modifications for achieving −5 ⎛ f p ⎞
4
⎜ ⎟
“stretching” corrections to basic Airy wave theory 0.0005 ⎜
4⎝ f ⎠ ⎟
results, though not commonly adopted, can nonethe- Sη ( f ) = e (11)
f5
less be used for this purpose. with Le Mahaute’s

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 EJSE Special Issue: Loading on Structures (2007)

in which fp = 1.37/ U 19.5 , is the frequency in Hertz stream wind, which leads to a more slowly varying
at peak wave energy in the spectrum and where mean wind profile with height and to lower levels of
H s = 4σ η = 0.021U 19.5
2
. (Note that the variance of turbulence intensity than encountered on land. As a
a random process can be directly obtained from consequence, wind speed values at the same height
the area under its spectral density variation, above still water level (for offshore conditions) as
hence the basis for the relationship for those above ground level (for land-based structures)
σ η ≈ 0.005U 19.5
2
, from the P-M spectral descrip- for nominal storm conditions, tend to be stronger
tion quoted above). Figure 4 depicts sample plots and lead to higher wind loads. (Figure 5 provides a
of the Pierson-Moskowitz (P-M) spectrum for a diagrammatic representation of this mean wind
selection of wind speed values, U 19.5 . speed variation).
100 500m
400m
U = 20m/s
80 300m
250m
Spectral Density

60
U = 18m/s

40
U = 16m/s
20 Figure 5: Variation of mean wind speed with height
U = 12m/s
U = 10m/s
0 For free-stream wind speed, UG, at gradient
0 0.05 0.1 0.15 0.2 0.25 height, zG (the height outside the influence of rough-
Frequency (Hertz) ness on the free-stream velocity), the mean wind
speed at level z above the surface, U (z ) , is given by
Figure 4: Sample Pierson-Moskowitz Wave Spectra the power law profile
α α
An irregular sea state can be considered to be ⎛ z ⎞ ⎛ z ⎞
U ( z ) = U G ⎜⎜ ⎟⎟ = U ref ⎜ ⎟ ≤ UG (13)
composed of a Fourier Series of Airy wavelets con- ⎜z ⎟
⎝ zG ⎠ ⎝ ref ⎠
forming to a nominated spectral description, such as where α is the power law exponent and “ref” refers
the P-M spectral variation. to a reference point typically chosen to correspond
Then wave height η (t ) can be expressed as to 10m.
Table I compares values for key descriptive pa-
N / 2 ⎛ 2.S ( f )
⎛ 2πnt ⎞⎞ rameters, α and zG, for different terrain conditions,
η (t ) = ∑ ⎜ η
sin⎜ − φn ⎟ ⎟ (12) including those for rough seas.
⎜ T ⎝ T ⎠ ⎟
⎝ n =0 ⎠
where η(t) is represented by a series of points (η1, Table I Wind speed profile parameters
η2, η3, …, ηM) at a regular time step of dt for M Terrain Rough Sea Grassland Suburb City centre
points where T = M.dt here represents the time α 0.12 0.16 0.28 0.40
length of record, and φn is a random phase angle be- zG (m) 250 300 400 500
tween 0 and 2π. (Equation (12) offers a convenient
approach towards numerically simulating sea states The drag force, FW (t ) , exerted on a bluff body
conforming to a desired spectral variation via the (eg such as the exposed frontal deck area of an off-
Fast Fourier Transform. Such sea state descriptions shore oil rig), by turbulent wind pressure effects can
can then be adopted in numerical studies that take be evaluated from
into account non-linear characteristics and features 1
that would otherwise not be considered for conven- FW (t ) = ρ a C D AV 2 (t ) (14)
2
ience). where ρ 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 body
3 ENVIRONMENTAL LOADS ON OFFSHORE shape/geometry, and V(t) is wind speed at the loca-
STRUCTURES tion of the bluff body.
3.1 Wind Loads
Wind loads on offshore structures can be evaluated 3.2 Wave Loads
using modelling approaches adopted for land-based
structures but for conditions pertaining to ocean en- The wave loads experienced by offshore struc-
vironments. The distinction here is that an open sea tural elements depend upon their geometry, (the size
presents a lower category of roughness to the free-
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 EJSE Special Issue: Loading on Structures (2007)

of these elements relative to the wavelength and u mT Re

their orientation to the wave propagation), the hy- KC = ; β= (16)
D KC
drodynamic conditions and whether the structural
system is compliant or rigid. Structural elements that where um = the maximum alongwave water particle
are large enough to deflect the impinging wave (di- velocity. It is found that for KC < 10, inertia forces
ameter to wavelength ratio, D/L > 0.2) undergo load- progressively dominate; for 10 < KC < 20 both iner-
ing in the diffraction regime, whereas smaller, more tia and drag force components are significant and for
slender, structural elements are subject to loading in KC > 20, drag force progressively dominates.
the Morison regime. Sarpkaya’s (1976) original tests conducted on in-
strumented horizontal test cylinders in a U-tube with
3.2.1 Morison’s Equation a controlled oscillating water column remain to be
The alongwave or in-line force per unit length the most comprehensive exploration of Morison
acting on the submerged section of a rigid vertical force coefficients in the published literature.
surface-piercing cylinder, f ( z , t ) , from the interac- Figures 7 and 8, derived from these results pro-
tion of the wave kinematics at position z from the vide an indication of the variation of these force co-
MWL, (see Fig. 6), is given by Morison’s equation, efficients with respect to KC and Re. As a rule of
viz: thumb, it can be stated that CM decreases as CD in-
creases, and vice versa, and that both values gener-
f ( z, t ) = f I ( z, t ) + f D ( z, t ) (15) ally lie in the range 0.8 to 2.0. The drag force coeffi-
cient is also influenced by roughness on the
where f I ( z , t ) = (π / 4)ρ C M D 2 u ( z, t ) and cylinder. (Marine growth is particularly troublesome
f D ( z , t ) = 0.5 ρ C D D u ( z , t ) u ( z , t ) represent the iner- in this regard as it not only increases the effective
tia and drag force contributions in which CM and CD diameter of a cylinder, but the increase in roughness
represent the inertia and drag force coefficients, re- generally leads to an increase in drag coefficient,
spectively, ρ is the density of sea water and D is the CD).
cylinder diameter. CM 3.0

2.0 Re x 10-3
1.5

1.0

0.5
0.4
0.3
2.5 3 4 5 6 7 8 9 10 15 20 30 40 50 100 150 200

Figure 7: Inertia force KC coefficient dependence on

flow parameters

CD 3.0

2.0 Re x 10-3
1.5

1.0

0.5
0.4
0.3
2.5 3 4 5 6 7 8 9 10 15 20 30 40 50 100 150 200

Figure 6: Wave Loading on a Surface-Piercing Bottom- Figure 8: Drag force KC coefficient dependence on
Mounted Cylinder flow parameters

Force coefficients CM and CD are found to be de- The Morison equation has formed the basis for de-
pendent upon Reynold’s number, Re, Keulegan- sign of a large proportion of the world’s offshore
Carpenter number, KC, and the β parameter, viz: platforms - a significant infrastructure asset base, so
its importance to offshore engineering cannot be un-
derstated. Appendix I provides a derivation of the

59
 EJSE Special Issue: Loading on Structures (2007)

Morison wave loads for a surface-piercing cylinder 3.4 Diffraction wave forces
for small amplitude Airy waves and illustrates key
features of the properties of the inertia and drag Diffraction wave forces on a vertical surface-
force components. piercing cylinder (such as in Fig. 6) occur when the
diameter to wavelength ratio of the incident wave,
D/L, exceeds 0.2 and can be evaluated by integrat-
3.3 Transverse (Lift) wave loads ing the pressure distribution derived from the time
derivative of the incident and diffracted wave poten-
Transverse or lift wave forces can occur on offshore tials, (MacCamy & Fuchs, 1954). Integrating the
structures as a result of alternating vortex formation first moment of the pressure distribution allows
in the flow field of the wave. This is usually associ- evaluation of the overturning moment effect about
ated with drag significant to drag dominant condi- the base. Results obtained for the diffraction force
tions (KC > 15) and at a frequency associated with F(t) and overturning moment M(t) are given by:
the vortex street which is a multiple of the wave fre-
quency for these conditions. The vortex shedding 2ρ g H
frequency, n, is determined by the Strouhal number, F (t ) = 2
A(κ a )tanh (κ h )cos(ω t −α ) (19)
NS, whose value is dependent upon the structural κ
member shape and Re, (typically ~0.2 for a circular 2ρ g H
cylinder in the range 2.5 x 102 < Re < 2.5 x 105), and M (t ) = 2
A(κ a ).
κ (20)
which is defined by
nD [κ h tanh(κ h ) + sec h(κ h )−1]cos(ωt −α )
NS = (17)
Um where
where Um is the maximum alongwave water particle ⎛ J ′ (κ a ) ⎞
[ (κ a )+Y1′ 2 (κ a )] 2 ; α =tan −1⎜⎜
1
velocity and D is the transverse dimension of the A(κ a )= J1′
2
−
1 ⎟ (21)
member under consideration (eg diameter of the cyl- Y1′ (κ a ) ⎟
inder). ⎝ ⎠
The lift force per unit length, fL, can be defined via in which a is the radius of the cylinder (D/2), (′) de-
notes differentiation with respect to radius r, J1 and
1 Y1 represent Bessel functions of the first and second
fL = ρ CL DU m U m (18) kinds of 1st order, respectively. It should be noted
2 that specialist software based upon panel methods, is
where CL is the Lift force coefficient that is depend- normally necessary to investigate diffraction forces
ent upon the flow conditions. Again, Sarpkaya’s on structures of arbitrary shape, (eg WAMIT,
(1976) original tests conducted on instrumented SESAM).
horizontal test cylinders in a U-tube with a con-
trolled oscillating water column, also provide a
comprehensive exploration of the lift force coeffi- 3.5 Effect of compliancy (relative motion)
cient, from which the results depicted in Figure 9
have been obtained. (It should be noted here, that in In the situation where a structure is compliant (ie
the case of flexible structural members, when the not rigid) and its displacement in the alongwave di-
vortex shedding frequency n coincides with the rection at position z from the free surface at time t is
member natural frequency of oscillation, the resul- given by x(z,t), then the form of Morison’s equation
tant vortex-induced vibrations give rise to the so- modified under the “relative velocity” formulation,
called “lock-in” mechanism which is identified as a becomes:
form of resonance). π π
f ( z, t ) = ρ .C M .D 2 u ( z, t ) − ρ .(C M − 1).D 2 x( z, t )
4 4 (22)
1
+ ρ .C D .D.(u ( z, t ) − x ( z, t ) ) u ( z , t ) − x ( z , t )
2
Consider the structure concerned to be of the
form of the surface-piercing cylinder depicted in
Figure 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-
Figure 9: Lift force coefficient dependence on flow parameters
tained from the equation of a single-degree-of-
freedom (SDOF) oscillator, given by:

60
 EJSE Special Issue: Loading on Structures (2007)

0
FI (t ) + F ' D (t )
mxo (t ) + cx o (t ) + kxo (t ) ≈ ∫ Ψ ( z ) f ( z , t ) dz
−h
(23) xo + 2ω o (ζ o + ζ H )x o + ω o2 xo = (29)
m + m'
where the integration has been taken to the MWL in where
lieu of η(t), and at x = 0, as an approximation. Coef- 0
F ' D (t ) = ∫ β u ( z, t ) u ( z , t ) Ψ ( z ) dz (30)
ficients m, c, and k represent the equivalent mass, −h
viscous damping and restraint stiffness of the cylin- which is interpreted as the level of equivalent drag
der at the MWL. (Note that allowing for forcing to force at the MWL in the case of rigid support condi-
be considered at x(z,t) via u(x,z,t) produces non- tions (negligible dynamic response).
linearities that normally have only a minor effect on The term ζH in equation (29) is the contribution
the character of the response (Haritos, 1986)).
to damping due to hydrodynamic drag interaction
When equation (22) for f(z,t) is substituted
viz
into equation (23) above, the so-called “added 0
mass” term is identified for the cylinder viz:
ζH ≈
∫ −h
β u ( z , t ) Ψ 2 ( z ) dz
(31)
0 π
(m + m')ωo
m' = ∫ ρ C A D 2 Ψ 2 ( z ) dz (24)
−h 4
(In the case of large diameter compliant cylinder in
the diffraction forcing regime, analogous expres-
in which CA (= CM – 1) is the “added mass” coeffi-
sions can be derived for added mass effects and ra-
cient.
diation damping due to structure-fluid interaction ef-
This is an important result as it suggests that for
fects).
all intensive purposes a body of fluid surrounding
the cylinder appears to be “attached” to it in its iner-
tial response, and hence the coining of the label
4 RESPONSE TO IRREGULAR SEA STATES
“added mass” effect.
Equation (23) can be re-cast in the form
4.1 Inertia Force
F (t ) + FD (t ) Since the inertia force term FI(t) in equation (29)
xo + 2ω oζ o x o + ω o2 xo = I (25)
m + m' is linear it generally poses little difficulty in model-
ling under a variety of hydrodynamic conditions.
where Consider an irregular sea state composed of a
Fourier Series of Airy wavelets conforming to a P-M
0 spectral description. Then u ( z , t ) can be obtained
FI (t ) = ∫ α u ( z , t ) Ψ ( z ) dz (26) from the expression
−h

and N /2 ⎛ − ω n2 cosh (κ n ( z + h) )
u ( z , t ) = ∑ ⎜
⎜ .
β .(u ( z, t ) − x o (t )Ψ ( z ) ).
0
FD (t ) = ∫ n =0 ⎝ sinh(κ n h)
−h
(27) (32)
u ( z, t ) − x o (t )Ψ ( z ) .Ψ ( z ).dz
2 Sη ( f ) ⎛ 2πnt ⎞⎞
cos⎜ − φn ⎟ ⎟
T ⎝ T ⎠ ⎟⎠
π 1
in which α = ρ C M D and β = ρ C D D , ωo
2
in which κn satisfies the dispersion relationship of
4
is the natural circular 2 first mode
frequency of the equation (4).
and ζo is the critical damping ratio of the structure in In the case of Ψ(z) being a power law profile, as
otherwise still water conditions. in Figure 10, then
In the case of Ψ ( z ) x o (t ) small compared to u(z,t)
an approximation that can be made for this interac- ⎛ z⎞
N

tive term is of the form: Ψ ( z ) = ⎜1 + ⎟ (33)

⎝ h⎠
and FI(t) can be shown to be obtainable via the ex-
(u ( z, t ) − x o (t )Ψ ( z ) ) u ( z, t ) − x o (t )Ψ ( z ) pression given by (Haritos, 1989),
(28)
≈ u ( z , t ) u ( z , t ) − 2 u ( z , t ) x o (t ) Ψ ( z )

Under these circumstances, equation (25) can be

further simplified to:

61
 EJSE Special Issue: Loading on Structures (2007)

π N /2
FI (t ) = ρgC M D 2 ∑ (I N (κ n h). 5
2
4 n =0 ζ%
(34) 4 5
2.Sη ( f ) ⎛ 2πnt ⎞⎞ σx
cos⎜ − φn ⎟ ⎟
T ⎝ T ⎠ ⎟⎠ σxs 3 10
where: 2
N ⎛ N −1 ⎞
I N (κ n h) = I 0 (κ n h) − ⎜⎜1 − I N −2 (κ n h) ⎟⎟, N ≥ 2
κnh ⎝ κ nh ⎠
1
100 fo
I 0 (κ n h) = tanh(κ n h) , N = 0 (35) 0
fp
1 2 4 8 16
1 ⎛ 1 ⎞
I1 (κ n h) = I 0 (κ n h) − ⎜1 − ⎟ , N =1 Figure 12: Influence of Dynamic Properties on Response
κ n h ⎜⎝ cosh(κ n h) ⎟⎠ (Inertia dominant forcing in deep water)

N
The levels are quoted as the ratio of the standard de-
viation in the response of a cylinder exhibiting a
natural frequency of fo to that of a near weightless
cylinder with the same stiffness for which fo ap-
proaches infinity. It is clear from direct observation
of Figure 12, that response levels are controlled by
both damping and the amount of relative energy
available near 'resonance' for a dynamically respond-
ing cylinder in an irregular sea state.
Figure 10: Compliant vertical surface-piercing cylinder

Figure 11 presents the variations in I N (κ n h) for a 4.2 Drag force

range of power exponents N in mode shape Ψ(z).
The result for N = 0 is consistent with the derivation Whilst it is possible to deal with the u|u| term for
for the inertia force component acting on a rigid cyl- drag force numerically, the linearised approximation
inder due to an Airy wave made in Appendix I. to u ( z , t ) attributed to Borgman (1967), can be used
in the case u(z,t) Gaussian in a random sea state, so
I N(κh) that
1.0
8
0

u ( z, t ) ≈ .σ ( z ) (36)
=
N

0.8
1
π u
N=
2 where σu(z) is the standard deviation in water
0.6 N= particle velocity and where current is taken as
4
N=
zero-valued for all z.
0.4 This approximation can be used to simplify the
expressions for both ζH in equation (31) and FD(t)
0.2 in equation (30) and to thereby obtain closed-
κh form solutions in the case of nominated Ψ(z)
0 2 4 6 8 10 variations.
Figure 11: Variation of IN(κh) for varying N This approximation would seem reasonable for de-
termination of ζH for “stiff” structures, but in situa-
It is observed that all variations for I N (κ n h) are as- tions when the drag force FD(t) is considered domi-
ymptotic to 1 and that I0(π) is close to this value to nant, this linearization can lead to significant errors
the order of accuracy associated with the “deep wa- in the modelling of both the non-linear drag force
ter” limit of Airy waves (ie κh = π) but for N>0, and the prediction of the resultant response, accord-
IN(κh) ≈ 1 for κh >> π. In general, the effect of ing to Lipsett (1985).
higher order mode shapes (N > 0) is to reduce the
level of inertia forcing of each Airy wavelet in an ir-
regular sea state. 5 EXTREME VALUES
Figure 12 depicts the results obtained for the re-
sponse of a vertical cylinder in deep water condi-
tions (IN(κh) = 1) for inertia only forcing in uni- In the case of random vibrations associated with lin-
directional P-M waves. ear systems, use can be made of upcrossing theory
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 EJSE Special Issue: Loading on Structures (2007)

in combination with spectral modelling of the proc- level y, Ny, divided by the time length of trace, T (ie
esses involved to develop a basis for prediction of νy = Ny/T). νo would correspond to the rate of up-
peak response values, (Nigam & Narayanan, 1994). crossings of the zero mean.
A linear filter has the characteristics described by It can be shown (Newland, 1975) that upcrossings
(Hy,η(f), φlag(f)) which apply to the Fourier compo- for such a trace would satisfy
nents of a random time varying quantity (such as a 1⎛ y ⎞
2

waveheight trace, η(t), conforming to say a P-M ν y Ny − ⎜

2 ⎜⎝ σ y
⎟
⎟
spectrum) at frequency f, to produce a modified re- = =e ⎠
(39)
sultant time varying quantity, y(t), that is linearly re- ν o No
lated to η(t), as follows
N /2
The concept of a “peak value” in a time period of
y (t ) = ∑ H y ,η ( f n ) .
T would correspond to a y value with an upcrossing
n =1 (37)
⎛ ⎛ 2πnt ⎞ ⎛ 2πnt ⎞⎞ count of 1 so that ymax can be estimated from
⎜⎜ a n cos⎜ − φ lag ( f n ) ⎟ + bn sin ⎜ − φ lag ( f n ) ⎟ ⎟⎟
⎝ ⎝ T ⎠ ⎝ T ⎠⎠ 2
1⎛ y ⎞
− ⎜ max ⎟
A 'zero-lag' filter would be one for which φlag(fn) 1 2 ⎜⎝ σ y ⎟
=e ⎠
(40)
= 0 for all frequencies fn. ν oT
The spectrum for y, given by Sy(f), can be ob-
tained from the spectrum of waveheight, Sη(f), via so that
S y ( f ) = H y2,η ( f ).Sη ( f ) (38)
y max = 2 ln(ν oT ) .σ y (41)

5.1 Extreme Wave Forces Because the value of ymax itself shows a statistical
variation, Davenport (1964) has suggested a small
Use can be made of the dispersion relationship of correction to equation (41) for the value of E(ymax)
equation (4) in conjunction with the separate de- so that
scriptions above for Inertia and Drag force, to obtain
the associated relationships for H FI ,η ( f ) and ⎛ 0.577 ⎞⎟
H FD ,η ( f ) respectively, and hence the total force y max = ⎜ 2 ln(ν oT ) + .σ y (42)
⎜ 2 ln(ν oT ) ⎟⎠
spectrum for the surface-piercing cylinder of Figure ⎝
6. A diagrammatic illustration of the concept is pro- Now the rate of “zero” upcrossings is given by:
vided in Figure 13.
∞
⎛ σ ( y ) ⎞
⎜⎜ ⎟ ∫f
2
S y ( f ) df
H2FI,η(f) ⎝ σ ( y ) ⎟⎠
SFI(f)
νo = = 0
(43)
X = 2π ∞

∫
Sη(f) SFtot(f)
S y ( f ) df
0
σ η2
+ =
σ 2
Ft
H2FD,η(f) SFD(f)
X = which can be determined from the spectral descrip-
tion. If y(t) is a narrow-banded process (ie energy is
concentrated at a peak frequency, fp), then νo ≈ fp.
Figure 13: Diagrammatic description of spectral modelling of
Morison wave loading
5.2 Extreme Response Values
The area under the total force spectrum equals the
variance, σ F2T , knowledge of which may be used to The concepts above can be applied to the dynami-
estimate peak Morison loading of the vertical sur- cally responding surface-piercing cylinder to esti-
face-piercing cylinder, under consideration. This mate the peak response at MWL, (xo)max.
peak load value may reasonably be expected to be of An additional stage is required for this purpose,
the order 3σ FT , but a more precise estimation is of- namely the linear transformation from Morison forc-
fered through the use of upcrossing theory. ing to dynamic excitation via the description of
For a Normally distributed trace y(t) with zero equation (29), which in terms of a spectral modelling
mean and variance σ y2 , the rate of upcrossings at approach, is diagrammatically depicted in Figure14.
level y, νy, equates to the count of upcrossings at

63
 EJSE Special Issue: Loading on Structures (2007)

Davenport, A.G. 1964. Note on the Distribution of the Largest

Value of a Random Function with Application to Gust
Loading, Proc. ICE, Vol. 28, No. 187.
DNV-OS-C101, 2004. Design of Offshore Steel Structures,
SFtot(f) H2x,F(f) Sx(f) General (LFRD Method). Det Norske Veritas, Norway.
X = DNV-OS-C105, 2005. Structural Design of TLPS, (LFRD
Method). Det Norske Veritas, Norway.
σ 2
Ft fo σ x2 fo DNV-OS-C106, 2001. Structural Design of Offshore Deep
Draught Floating Units, (LFRD Method). Det Norske Veri-
Figure 14: Diagrammatic description of spectral modelling of tas, Norway.
dynamic response Chakrabarti, S. K. (ed) 2005. Handbook of Offshore Engineer-
ing, San Francisco: Elsevier.
Chakrabarti, S. K. 2002. The Theory and Practice of Hydrody-
The Transfer Function for response from Morison namics and Vibration, New Jersey: World Scientific.
loading in irregular sea states for the dynamically re- Chakrabarti, S. K. (ed) 1987. Fluid Structure Interaction in
sponding surface-piercing cylinder of Figure 10 is Offshore Engineering, Southampton: Computational Me-
given by Hx,F(f), via chanics Publications.
Chakrabarti, S. K. 1994. Hydrodynamics of Offshore Struc-
tures, Southampton: Computational Mechanics Publica-
χ m2 ( f ) tions.
H x2, F ( f ) = (44)
(m + m')2 Dean R. G. & Dalrymple, R. A. 1991. Water Wave Mechanics
for Engineers and Scientists, New Jersey: World Scientific.
where Rienecker, M.M. & Fenton, J.D. 1981. A Fourier approxima-
1 tion method for steady water waves, J. Fluid Mechs, Vol
χ m2 ( f ) = (45) 104, pp 119-137.
⎛⎛ 2 2⎞ Graff, W. J. 1981. Introduction to Offshore Structures – De-
⎜ ⎜ ⎛ f ⎞ ⎞⎟ ⎛⎜
2
⎛ f ⎞⎞ ⎟ sign, Fabrication, Installation, Houston: Gulf Publishing
⎜ ⎜1 − ⎜⎜ f ⎟⎟ ⎟ + ⎜ 2ζ tot ⎜⎜ f ⎟⎟ ⎟⎟ ⎟ Company.
⎜⎝ ⎝ o ⎠ ⎠ ⎝ ⎝ o ⎠⎠ ⎟ Haritos, N. 1986. Nonlinear Hydrodynamic Forcing of Buoys
⎝ ⎠ in Ocean Waves, Proc. 10th Aust. Conf. on Mechs. of
Structs. & Materials, Adelaide, pp 253-258.
in which fo is the natural frequency of the cylin- Haritos, N. 1989. The Influence of Modal Characteristics on
der and ζtot is the total critical damping ratio (ζtot =ζo the Dynamic Response of Compliant Cylinders in Waves,
+ζH). Computational Techniques & Applications: CTAC-89, edit
W.L. Hogarth & B.J. Noye, pp 683-690, (Hemisphere).
The area under the response spectrum yields σ x
2
Holand, I., Gudmestad, O. T. & Jersin, E. (eds). 2000. Design
and after application of equation (43) to obtain νo, of Offshore Concrete Structures, London: Spon Press.
extreme value (xo)max can be obtained from equation ICE, 1983, Design in Offshore Structures, ICE, London: Tho-
(42), for the nominated duration of the irregular sea mas Telford Ltd.
state under consideration, (eg T = 3600 secs for a 1- Le Mehaute, B. 1969. An introduction to hydrodynamics and
water waves, Water Wave Theories, Vol. II, TR ERL 118-
hour storm). POL-3-2, U.S. Department of Commerce, ESSA, Washing-
ton, DC.
Lipsett, A.W. 1985. Nonlinear Response of Structures in Regu-
6 CONCLUDING REMARKS lar and Random Waves, Ph.D. Thesis, Univ. of British Co-
lumbia, Canada.
This paper has provided an overview of some of the MacCamy, R. C. & Fuchs, R.A.1954. Wave Forces on Piles: a
Diffraction Theory, U.S. Army Coastal Engineering Centre,
key factors that need be considered in the analysis Tech Memo No. 69.
and design of offshore structures. Emphasis has been Newland, D.E. 1975. An Introduction to Random Vibrations
placed on modelling of the hydrodynamic response and Spectral Analysis. Longman.
of a compliant vertical surface-piercing cylinder in Nigam, N. C. & Narayanan, S. 1994. Applications of Random
the Morison loading regime under uni-directional Vibrations, New York: Springer-Verlag.
Sarpkaya, T. 1976, In-line and transverse forces on cylinders in
waves. Reference has also been made to a number of oscillating flow at high Reynold’s number, Proc. 8th Off-
publications in which further detail and extension of shore Technology Conference, Houston, Texas, OTC 2533,
treatment can be explored by the interested reader. pp 95-108.
Sarpkaya, T. & Isaacson, M. 1981. Mechanics of Wave Forces
on Offshore Structures, New York: Van Nostrand
REFERENCES SESAM,http://www.dnv.com/software/systems/sesam/progr
amModules.asp
Stewart, R. H. 2006. Introduction to Physical Oceanography,
API, 1984. Recommended Practice for Planning, Designing http://oceanworld.tamu.edu/resources/ocng_textbook/PDF_
and Constructing Fixed Offshore Platforms, American Pe- files/book.pdf.
troleum Institute, API RP2A, 15th Edition. WAMIT, http://www.wamit.com/
Borgman, L.E. 1967. Spectral Analysis of Ocean Wave Forces Wheeler, J.D. 1970. Method for calculating forces produced by
on Piling, Jl WW& Harbors, Vol. 93, No. WW2, pp 129- irregular waves, Journal of Petroleum Technology, March,
156. pp 359-367.

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 EJSE Special Issue: Loading on Structures (2007)

a2 ω2
Appendix I. – Base shear on a surface-piercing FD (t) = β cos (κx - ωt)|cos (κx - ωt)| .
sinh2(κh)
cylinder from Morison loading
1 ⎛ sinh(2κh)⎞
2 ⎝h + 2κ ⎠
f ( z, t ) = f I ( z, t ) + f D ( z, t )
. π ⎡ a2 ω2 h a2 g tanh(κh) ⎛sinh(2κh)⎞ ⎤
ƒI (z,t) = α u ; (α = 4 ρ CM D2) FD (t) = ⎢β + β ⎜sinh2(κh)⎟ ⎥
⎣ 2 sinh2(κh) 4 ⎝ ⎠⎦
1
ƒD (z,t) = β u|u| ; (β = 2 ρ CD D) . cos (κx - ωt)|cos (κx - ωt)|

Inertia: ⎡ a2g κh a2g⎤

=⎢β + β 2 ⎥ cos(κx-ωt)|cos (κx - ωt)|
o ⎣ sinh2κh ⎦
⌠ . ⎛ o ⎞
FI (t) = ⎜= ⌠ ƒI (z,t) dz⎟
⌡ α u dz
⎜ ⌡ ⎟ β a2g ⎡ 2 κ h + 1⎤ cos (κx - ωt)|cos (κx - ωt)|
-h ⎝ ⎠ = 2
-h ⎣sinh2κh ⎦
o
ω2 ⎪sinh(κ(z + h))⎪
= α a.sin(κx-ωt) Hence, as an alternative approximation
sinh(κh) ⎪ κ ⎪ -h
β a2g
FD (t) ≈ cos(κx-ωt)
ω2 2
a . sin(κx - ωt) ⎪ - 0⎪
sinh(κh)
= α
κ ⎪sinh(κh) ⎪

= α g tanh(κh) . a . sin(κx - ωt) Inertia:

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

Drag: Drag:
o ⎛ o ⎞
FD (t) = ⌠
⌡ β u|u|dz ⎜= ⌠ ƒD (z,t) dz⎟ β a2g ⎡ 2 κ h
⎜ ⌡ ⎟ ⎤ cos (κx - ωt) . | cos (κx - ωt)|
-h ⎝ -h ⎠ 2 ⎣sinh2κh + 1⎦
a2 ω2
= β cos (κx - ωt) |cos (κx - ωt)| → FD cos (κx - ωt) | cos (κx - ωt)|
sinh2 κh
o
Figures I.a and I.b depict representative varia-
.⌡⌠ cosh2(κ(z + h)) dz tions over one cycle of Airy wave of the Base Shear
-h force acting on a cylinder normalised with respect to
a2 ω2 FD/FI = 2, respectively, by way of illustration.
FD (t) = β cos (κx - ωt) |cos (κx - ωt)|
sinh2(κh)
o
1
. ⌠ 2 (1 + cosh 2(κ(z + h))) dz
⌡
-h

1.5 2.5
Total Total
2
1 1.5
Drag Drag
1
0.5
Inertia Inertia
0.5
Ftot Ftot 0
0
FI 0 0.2 0.4 0.6 0.8
t 1 FI -0.5 0 0.2 0.4 0.6 0.8
t 1
-0.5 T T
-1
-1.5
-1
(a) -2 (b)
-1.5 -2.5

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

65