Frequency domain analysis of dynamic response of drag dominated offshore structures

E L S E V I E R

P I I : 8 0 1 4 1 - 1 1 8 7 ( 9 7 ) 0 0 0 3 8 - 2

Applied Ocean Research 19 (1997) 251-262 © 1998 Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0141-1187/97/$17.00 + 0.00

Frequency domain analysis of dynamic response

of drag dominated offshore structures

A. Naess^'* & A. A. Pisano*"

''Department of Structural Engmeering, Nonvegian University' of Science and Technology, N-7034 Trondheim, Norway ^Department of Structural Engineering, University of Pavia, 1-27100 Pavia, Italy

(Received 8 October 1997; accepted 21 November 1997)

The paper describes a method for stochastic representation of the hydrodynamic drag forces on offshore structures subjected to irregular waves. It is shown that, for the case of zero current, it is possible to construct a genuinely quadratic representation of the drag force which reproduces the statistical properties of the standard formulation of the drag force very closely, and which at the same time has sufficient flexibility to ensure a spectral density that accurately approximates the desired force spectrum. The distinct advantage of the new representation is that it brings dynamic analysis of extensive linear structures back into the frequency domain. © 1998 Published by Elsevier Science Limited. A l l rights reserved.

Keywords: frequency domain analysis, hydrodynamic drag, offshore structures.

1 I N T R O D U C T I O N

The search f o r efficient methods o f response analysis o f dynamic offshore structures subjected to M o r i s o n type wave loading i n random seas has represented a research challenge f o r many years. The main problem has been that the standard representation of the drag force component of the M o r i s o n f o r m u l a t i o n [ l ] o f the hydrodynamic wave load makes i t d i f f i c u l t to set up an efficient procedure f o r estimating the statistics o f the dynamic response o f even linear offshore structures i n cases where the drag component is important. The usually adopted approach to achieve this goal is to linearize the drag force. Since the (short-term) random wave f i e l d is generally represented as stationary and Gaussian, i t can be concluded, due to linearity, that the response w i l l also be stationary and Gaussian. This allows the response statistics to be f u l l y determined by a frequency domain analysis where efficient numerical methods are available. However, i t has been demonstrated that linearization of the drag force leads to erroneous wave load statistics[2,3], and therefore, i n general, to erroneous response statistics. Specifically, both extreme response estimates and derived stress range distributions may be sub-stantially i n error.

*To whom correspondence should be addressed.

Today, the only generally accepted method o f dynamic analysis o f an offshore structure that is f a i t h f u l to the non-linear character o f the drag force is the time domain inte-gration o f the equations o f motion. Then, non-linearities i n the external loading terms, as w e l l as i n the equations o f m o t i o n themselves, present no obstacle. T o derive response statistics, this has led to the development o f time domain Monte Carlo simulation programs tailored to the case o f offshore structures subjected to M o r i s o n type wave forces. B y this, a sample o f response time histories is obtained, and various statistical descriptors may be estimated depending on the specific purpose o f the analysis. However, the c o m -putational burden i n v o l v e d i n application o f this procedure f o r the estimation of, f o r example, long-term fatigue is almost prohibitive at present f o r a detailed analysis. Even though the computational capacity of computers is increas-ing at a rapid pace, i t is still considered to be o f interest to investigate the possibility o f developing a more e f f i c i e n t method o f analysis. Recently, Naess[4] proposed a new method f o r representing the drag force on an offshore struc-ture i n random waves. I n i t i a l efforts to investigate the accu-racy and efficiency o f this method f o r dynamic response analysis have been reported i n the literature[5,6]. I n this paper we shall extend these analyses and elaborate o n some particular issues o f relevance to the practical numer-ical implementation o f this method.

For the sake o f completeness, it should also be mentioned that recently a set o f alternative statistical representation techniques have been proposed. Donley and Spanos[7,8] proposed the method o f statistical quadratization, w h i c h includes terms up to second order i n a Volterra series expan-sion o f the drag force. I n its standard f o r m , i t reduces to the method o f statistical linearization when the current speed is zero. The method w i l l also miss third and higher harmonics o f wave frequencies present i n the drag force spectnam. B y the introduction of the statistical c u b i c i z a t i ö n method[9] this aspect was partly rectified. I n principle, higher order statis-tical representation methods can also be established, but their practical value deteriorates quickly. Even i f it is cleaiiy important to be able to include spectral components at higher harmonics of the wave frequencies i n cases where the eigenfrequencies o f a dynamic structure f a l l w i t h i n the range o f these higher harmonics, the serious drawback of such higher order representations o f the drag force is that dynamic analyses beyond the spectral level becomes rather complicated. I n practice this implies that only statistical moments up to f o u r t h order are calculated. T o provide stadstics of the dynamic response f o r fatigue calculations, it is then necessary to assume that such statistics can be derived f r o m the first f o u r statistical moments o f the response.

2 S T O C H A S T I C R E P R E S E N T A T I O N O F T H E D R A G F O R C E

The standard engineering model o f the hydrodynamic forces on offshore structures w i t h a non-negligible drag force com-ponent, is the Morison f o r m u l a [ l ] . F o l l o w i n g this model, the hydrodynamic force F(t) = F(t;x) at a given location X = {x, y, z) can be expressed as

m = K m + h{U{t) -Xit))\U{t) -Xit)I (1) where k^ = k^{x) and ki = ki{x) are appropriate constants,

and U{t) = U{t;x) denotes the water particle velocity i n the given direction at the specified location x. X{t)=X{t;x) denotes the corresponding displacement response o f the assumed structure subjected to wave forces. The over-dot signifies derivation w i t h respect to time. W e have chosen to express the drag force component i n terms o f the relative velocity U{t)-X{t). The reason f o r this is that the purpose o f this paper is to study dynamic structures, and i n such cases the relative velocity formulation accounts f o r addi-tional damping due to the interaction of the m o t i o n of the structure and the wave m o t i o n . I t is assumed that f o r most wave loading conditions, the water particle velocity w i l l substantially exceed the displacement velocity o f the struc-ture. M o r e specifically, it is assumed that {XlUf < 1 This leads to the f o l l o w i n g approximation

{U{t)-X{t)}\U{t)-X{t)\ « U{t)\U{t)\-2\U{t)\X{t) (2) This approximation reveals a term 2\U{t)\X{t), w h i c h can

be viewed as a time-variant damping contribution. I n some cases this term can contribute substantially to the overall damping. Hence, i t cannot i n general be neglected. H o w -ever, a time-variant damping model can only be handled by time domain methods o f analysis. T o obtain a model that w o u l d make calculations more manageable, and at the same time retain the general damping level, Penzien[10] pro-posed replacing \U{t)\ b y its mean value E{\U{t)\]. This clearly represents only one possible choice among many. For example, i n the case o f prediction o f extreme responses, one w o u l d expect that a different choice is appropriate. Instead o f going into details on this point, i t is assumed that a suitable choice o f a time-invariant damping coefficient has been made. I n the f o l l o w i n g we shall therefore adopt a hydrodynamic loading o f the f o r m

m=Km+hu{t)\u{t)\

I n agreement w i t h c o m m o n practice, the water particle velocity U{t) is assumed to be linearly related to the wave elevation process r;(?). B y assuming this to be a zero-mean, stationary Gaussian process, U{t) is then also stationary and Gaussian. I t w i l l also be assumed that E[U{t)\ — 0, that is, no current is present. For the subse-quent analysis, eqn (3) w i l l be rewritten i n the f o r m

F{t) = W{t) + tJ.V(t)\V{t)\

The f o l l o w i n g notation has been introduced

(4)

km<^u cTf/

(5) where al = V a r [ [ / ] and al = V a r [ ( / ] .

Naess[4] proposed representing the drag force Fdit) — iJiV{t)\V(t)\ i n a random sea way by a stochastic process $ ^ ( 0 defined as f o l l o w s

M t ) = l ^ { P i t f - N { t f } (6) w h i c h depends on introducing an auxiliary, zero-mean Gaussian process P(t), coiTesponding i n a sense to the water particle velocity process U(t). P{t) can be assumed given i n discredzed f o r m as follows [11]

P{t) = Sio)j)Ao}Bj e" (7)

where the two-sided spectral density 5(co) is as yet unknown. Throughout this paper, when the summation index runs f r o m negative to positive values, i t invariably omits zero. 0 :S coj < w j - - - < w„ is a discretization, assumed equidistant f o r simplicity, o f a pertinent part of the positive frequency axis. co_y=—coy and Aoi — o)j^i — ojj. [Bj] is a set of independent, complex Gaussian A'(0,l)-variables w i t h independent, identically distributed real and imaginary parts, and B _j = B* (* sig-nifies complex conjugation). ; = a / - 1. The process N{t) is related to P{f) by being its Hilbert transform[12], that is,

Frequency domain analysis of dynamic offshore structures 253

Fig. 1. Linear plots of the PDFs of FjCf), F^,{t) and $^(0 for m = 1: ,fp^{4>)\ - • (</>); ,hA4>) (<TS = ( 7 f J ; + + + , / * , W K = 1.15(rf^).

A^(0 is defined by the relation

n

N{t)= X Js{coj)Ao>xio^j)Bj e'"'-' (8) ;= -n ^

where x(w) is the transfer f u n c t i o n f o r the Hilbert trans-f o r m [ 1 2 ] , that is, x(w) = - / trans-f o r co > 0, x ( 0 ) = 0, and X(w) = / f o r CO < 0.

I t is seen f r o m eqn (6) that # d ( 0 has a probability density f u n c t i o n (PDF) that is symmetrical around a mean value equal to zero, and that the statistics are given by a quadratic transformation o f a Gaussian process. This clearly agrees w i t h the expression f o r the drag force Fi(t) = fjiV{t)\V(t)\, w h i c h clearly has mean value zero i n the absence o f current. I t is also seen that the drag force is given by a purely quad-ratic expression i n terms o f the Gaussian process V(t) o n each side o f the mean value zero. To complement the dis-cussion above on polynomial approximations to the drag force, we shall briefly describe some recent efforts to go beyond simple linearization. Naess and Y i m [ 5 ] showed that f o r the case o f zero current, no quadratic p o l y n o m i a l expansion can give satisfactory statistical distributions even i f some o f the lower order statistical moments can be approximated f a i r l y w e l l . On the other hand a cubic expan-sion has also been proposed[9,13], w h i c h may be expected to y i e l d much better approximations. For the case o f zero cuiTent, the drag force F^it) is then replaced by an equiva-lent cubic transformation o f the water particle velocity U(t):

F,,{t)^lxV2hr{V(t)+^V{tf} (9)

W i t h this choice o f coefficients, F^^it) and Fi{t) have both mean value zero, and almost the same variance, namely, al^^ = V a r [ f d , ( 0 ] = 2.97/x^ = 0.990^^.

3 S T A T I S T I C S O F P R O P O S E D F O R C E R E P R E S E N T A T I O N S

There are t w o basic conditions that a stochastic model o f the M o r i s o n type force as expressed by eqn (3) has to satisfy. Firstly, the statistical properties should be close to those o f the original force. Secondly, the spectral density o f the model must be i n fair agreement w i t h the original force spectrum. A s a check o f the statistical fitness, the probability distributions as w e l l as the mean level-crossing rates o f the different force formulations w i l l be compared. Agreement between these quantities w i l l give a strong indication o f the appropriateness o f the proposed representation strategies provided the respective spectral densities also agree.

The inertia force term Fi{t) = W(t) is the same f o r all three force formulations. Considered as a random variable, it is independent o f the drag force Fi(t) = ij,V{t)\V(t)\ and also o f the t w o drag force representations $ ^ ( 0 and F^^it). To compare the PDFs o f the different force formulations, i t is therefore sufficient to compare the PDFs o f the three corresponding drag force formulations.

It is obtained that E[Fi{t)] = £ [ $ d ( 0 ] = 0, aj^ = V a r [ F d ( 0 ] = 3^i^ and

4,

VCT'*,

Fig. 2. Logarithmic plots of the PDFs of FjCO, Fde(/) and *d(0 for ft = L K = 1.15a.).

= leads to the relation CT= ( 3 / 4 ) " ' ' « 0.93. The expression f o r the PDF fp^{-) o f F^it) is reported i n the literature[2,14]

- 1/2

exp

2ft (10)

Naess and Y i m [ 5 ] have calculated the P D F o f $^(0- K is given as f o l l o w s

1

3 ^ 0

101

(11) where /iro(-) denotes a Bessel f u n c t i o n o f imaginary argu-ment[15].

The P D F o f the equivalent c u b i c i z a t i ö n force F^^it) can be shown to be Here V- =v-{ti>)={ q 1/3 (12) (13) and q = ^2-K2,(i>l{An)

The three PDFs have been plotted i n Figs. 1 and 2 under the t w o conditions a^^=ap^, and a = ay — \.Q, w h i c h implies CT$^ = l.lScTf^j. The reason f o r imposing the last con-dition w i l l be clear f r o m the discussion below about the crossing rates. I t is seen f r o m Figs 1 and 2 that, f r o m a

pracdcal point o f view, the agreement between the PDFs o f Fi{t) and $ d ( f ) is quite good under both conditions, even i n the tail region. The P D F o f fjeCO also shows reasonably good agreement w i t h that o f F^it).

A n important motivation i n studying the proposed drag force representation is the search f o r an efficient tool f o r calculating long-term fatigue i n drag-dominated offshore structures. A statistical quantity o f particular importance f o r fatigue calculations is the mean level upcrossing rate. Hence i t must be required that the proposed force repre-sentation also has reasonably accurate mean level crossing rates.

The mean level upcrossing rate of the drag force F^it) is given by the expression[3,16]

4_ Oy

2m (14)

where Oy = Vai[_V{t)].

B y f o l l o w i n g the derivation given by Naess[17], i t can be shown that the upcrossing rate ï'$j(</>) o f ^(t) is given to good approximation by

TTO (15)

Here p = E[P(t)NitMaó), where o^ = V a r [ P ( f ) ] . Since P(t) is a zero-mean stationary Gaussian process, the mean zero upcrossing rate vp (0) o f P{t) is given by the relation

The corresponding mean level upcrossing rate o f the equivalent c u b i c i z a t i ö n force Fi^it) can be calculated by i n v o k i n g a result f r o m Naess[3], w h i c h leads to the f o l l o w -ing equation

g e x p { - < ^ : ± ^ } ,16, where v~ is given b y eqn (13).

To illustrate the agreement between the upcrossing rates o f the three alternative formulations o f the drag force, results f r o m the next section w i l l be used. For a class o f wave spectra characterized b y a J O N S W A P type spectrum, it is shown that good agreement can be obtained between the spectral densities o f F^it) and § ( 0 . There is, however, one particular point that has to be decided on before a compar-ison o f crossing rates can be done. This relates to the fact that the J O N S W A P spectrum behaves as w " ^ f o r large co, w h i c h implies that crossing rates o f water particle velocities cannot be calculated without introducing a truncation o f the spectrum at a finite frequency. The truncation frequency coir used here is, somewhat arbitrarily, set to be cotr = 5cOp. This is believed to cover the frequency range o f interest i n most cases o f practical concern. I t should also be noted that the relative accuracy o f the crossing rates to be compared is not sensitive to the choice o f cotp

Figs. 3 and 4 show the three different crossing rates f o r the chosen class o f J O N S W A P spectra w i t h the particular choice o f peak period Tp = 16 s. I t is seen that under the condition CT$^ = jf^, v}^ (</>) underestimates Vp^ ((f)) to some

extent, w h i l e very good agreement is obtained by requiring that ff = c7v,= 1.0, i.e. ais,^ = l.l5<jf^. VF^^{4>) also shows good agreement w i t h vp^'t') over the interval o f levels shown.

4 A P P R O X I M A T I O N O F T H E D R A G F O R C E S P E C T R U M

A remaining issue is to c l a r i f y to what extent i t is possible to approximate the spectral density o f the drag f o r c e Fi{t) by that o f ^i{t). This problem has already been discussed at some length by Naess and Y i m [ 5 ] , w h o demonstrated that an experimentally determined drag force spectrum could be fitted accurately by the spectral density o f $ d ( f )

-I n the f o l l o w i n g , a superscript -|- on the s y m b o l repre-senting a spectral density w i l l s i g n i f y a one-sided density. A c c o r d i n g l y , let S$^(co) denote the (one-sided) spectral density o f $ d ( 0 - It can be shown that[5]

5 + ( c o ) = 4 f i ^ _ ^ 5 + ( c o ' ) 5 + ( c o - c o ' ) d c o '

= 4/x^5+ *5+(co) (17) where (co) denotes the spectral density o f the auxiliary

process P{t). The asterisk denotes convolution.

The mathematical problem o f determining a spectral density iS'"'"(co) so that S^^iui) given b y eqn (17) is equal to Sp^iui), does not have an exact solution i n general[5].

10

Fig. 4. Logarithmic plots of the mean upcrossing rates of Fi{t), Fi^{t) and ^^{f) for ft = \: , VF^{4>)\ , (4>)

However, to show that a good approximation can be obtained, Naess[4] (see also Naess and Y i m [ 5 ] ) used a simple method to demonstrate the feasibility o f obtaining a practical solution. I n particular, i t was assumed that 5'*' (w) could be approximated as a sum o f Gaussian densities i n the f o l l o w i n g manner

27ra;

(co-co,-r

2a? (18)

I n v o k i n g w e l l - k n o w n properties o f the Gaussian density, eqn (17) leads to the f o l l o w i n g result

4/x a^aj

= 1 1

exp<

-[co - (O); - I - COy)]

2(a? + o f ) (19)

The accuracy o f this approach w i l l be illustrated b y application to the drag force spectrum determined by a class o f J O N S W A P type wave spectra, w h i c h is used extensively on the Norwegian Continental Shelf. W r i t t e n in terms o f non-dimensional frequency vc^co/cop, where the peak frequency cop = 2 i r / r p , the (one-sided) J O N S W A P spectrum, denoted by S^(w), is given as

f o l l o w s [ l ]

5 + (w) = 0.3125(1 - 0.287 In y)

{w-\Y

2^2 (20)

where 7 denotes the peakedness p a r a m e t e r [ l ] , and s — 0.07 w h e n w < \ and s = 0.09 when w > 1. A representative value o f Y f o r calculation o f long-term fatigue could be 2.0, and this value w i l l be adopted i n what f o l l o w s .

T o derive the drag f o r c e spectrum associated w i t h the specified class o f wave spectra, Borgman's approach[18] w i l l be used. For the drag force Fi{t) = jxV{t)\V{f)\, i t is then obtained that[18]

SFAW)--8f.-'

5 y ( v v ) + ^ [ 5 v ( v v ) ] * ^ - h , (21) where the superscript *3 denotes triple convolution, i.e.

Sy{ii)Syiy — u)S(\v — v) du dv.

A l s o note that eqn (21) is written i n terms o f two-sided spectra, indicated by the missing superscript -|- . For our puiposes, sufficient accuracy is obtained by neglecting the higher order terms beyond the first two appearing on the right-hand side o f eqn (21).

I t can be shown that ^v^'(w) = (cop/ffy)^^^^^ (vi^) and a?, = 0.108(cOpH,)' (ffu standard deviation o f water

Frequency domain analysis of dynamic offshore structures 257 1.5r

particle velocity U{t)). W i t h the choice y = 2.0, i t then f o l l o w s that the (one-sided) spectral density o f the non-dimensional particle velocity Vit) is given by

2.32

^ - e x p ^ - F (In 2) exp

2s^ (22) This expression is independent o f significant wave height

and peak period T^, w h i c h is then also the case f o r eqn (21).

The drag force spectrum Sp^{w) corresponding to eqns (21) and (22) is shown i n F i g . 5 together w i t h the spectral density St (vr) obtained f r o m the f o l l o w i n g representation

27r^,--exp {w - vt^,)' 2s} (23) 0.063, A5 0.034), where (0;,)?=! = ( 2 . 1 1 1 , 0.549, 0.211, (vi^/)f=i = ( 0 . 6 , 0.7, 1.2, 1.8, 2.5) and (s,);'^ 1 = (0.075, 0.35, 0.30, 0.30, 2.00). These constants have been obtained by a simple trial and error procedure, w h i c h is quite easy to p e r f o r m .

I t is seen that excellent agreement is obtained between the t w o drag force spectra. Hence, accepting the adopted para-metrization o f the J O N S W A P spectrum given by eqn (20), the problem o f representing the drag force has been solved f o r all sea states specified by a J O N S W A P wave spectrum w i t h a peakedness parameter 7 = 2.0. The dimensional force spectra required f o r a long-term response analysis are n o w all obtained by simple reseating o f the spectra given above.

Concerning the spectral density o f Fde(0. it can be shown by direct calculation that this is given by the first t w o terms on the right-hand side o f eqn (21). I t is therefore exact w i t h i n the adopted approximation.

5 T R A N S F O R M A T I O N O F F O R C E S

The drag force representation given above relates to a specified location, e.g. the origin o f a chosen coordinate system. The desire to calculate the response o f an extensive structure like a jack-up p l a t f o r m necessitates a procedure f o r calculating the force field over the entire structure f r o m the force related to a reference point. For the (normahzed) inertia force W{t), this is simply a linear transformation given exactiy as f o r the water particle acceleration. A l s o , f o r F{t) and F^{f)^W{t) + Fi^{t), w h i c h are given expH-citiy i n terms o f the normalized water particle velocity field V{t) = V{t;x), the transformation o f the force f r o m one point to any other can be done i n a straightforward manner by i n v o k i n g the linear transfer f u n c t i o n f o r V{t) w h i c h specifies the change i n the velocity f r o m one p o i n t to another o f the wave field. However, by this method the transfer mechanism f o r the forces becomes h i g h l y n o n linear, w h i c h makes i t d i f f i c u l t to calculate statistical i n f o r -mation beyond moments, as demonstrated i n the paper b y Quek et al.[\9}. Our approach to this problem is to assume that the transfer o f point forces over the wave field can be calculated by using the linearized drag force. This means that the transfer mechanism f o r drag forces is approximated

II

Fig. 6. Sketch of model structure and nodal representation.

by a linear transfer f u n c t i o n . L e t the auxiliary processes P{i) and N(i) be referred to the o r i g i n x = 0 o f the chosen coor-dinate system, see F i g . 6. The corresponding drag force representation $ d ( f ) = x = 0) can then be written (see eqns ( 6 ) - ( 8 ) )

n n

X = 0) = M X S ^('^Z' - e''"' ~

- « ; =

where

^(co,-, - CO,) = v'5(co,)5(co;)Aco(l - x(co,-)x^-)*

(24)

(25) W i t h i n the approximation o f a linear transfer f u n c t i o n f o r the drag force, ^^{t; x) at the point x = (,\-, y, z) can then be written as

$ d ( f ; x ) = M(x) X X n w , - c o ; ; x )

X i?(co,-, - co,)fi,i?; e (26) where f(co;x) denotes the transfer f u n c t i o n associated w i t h the water particle velocity field. Note that jli(x) — ixy{x)avix), where 7(x) is a geometric factor (7(x = 0) = 1.0) and ay{x) is the standard deviation o f V{t;x) (crv(x = 0) = 1.0). To s i m p l i f y the analysis, we shall assume that the long-crested waves propagate i n the positive x-direction. For linear wave theory, w h i c h is assumed here, i t can be shown t h a t [ l ]

^ cosh[k{z + d)]

s i n h ( M ) ' (27) where d denotes the water depth, and k = k(ui) is deter-mined by the dispersion relation

The same r(co;x) applies f o r transformation o f the inertial force W{t).

6 D Y N A M I C R E S P O N S E O F E X T E N S I V E

S T R U C T U R E S

Fig. 6 depicts a jack-up p l a t f o r m , w h i c h could represent the structure to be investigated. The adopted dynamic model is a lumped parameter model o f the f o l l o w i n g type

M X - ( - C X - f K X = W - F $ d (29) Here M , C and K are suitable matrices. X = ( X i , ...,Xj^) where Xi^=Xi^{t), k= I, ...,N, denote the displacement response i n the wave direction at node no. k. W =

iW{t;x,),...,Wit;xN)f and $d = (^>d(«;xi),...,«•(<;X;^))^

where x<. denotes the coordinates o f nodal point no. k. Each o f the W(t;x{) and $d(/;xjr) must now be interpreted as locally integrated point forces to be consistent w i t h the present dynamic model.

For the calculations o f this paper, and f o r the sole purpose o f illustration, the horizontal m o t i o n o f the p l a t f o r m is approximated b y the dominant mode shape assumed to be given by the expression

Hz)=l-cosMz + d)/2L] (30)

where L is the length o f each p l a t f o r m leg, see F i g . 6. Further simplifications are introduced by assuming that the p l a t f o r m is symmetrical about the A^z-plane, and that the phase difference between forces at the fore and aft legs can be neglected, w h i c h implies that the only spatial variable to appear i n the subsequent expressions is z- A g a i n , these simplifications are introduced just to reduce complexity o f expressions, and have nothing to do w i t h a h m i t a t i o n o f the theory. Introducing the generalized coordinate Z = Z(t), that is, Xk=Xi,{t)=Zit)\!^{zk), we obtain the f o l l o w i n g S D O F equation o f motion

Z + 2^w,Z + o)iZ = ni-\W + ^ j ) (31) where in, ^ and coe denote modal mass (including added mass c o n ü i b u t i o n s ) , modal damping ratio and undamped modal frequency, respectively. $d = ^^(t) is given b y the equation

*d(0= X *da;z*)\^(Zi) (32)

The expression f o r W = W{t) is entirely similar. The f o l l o w i n g (quadratic) representation is obtained

*d(0= X Ê G , B , B ; e ' ( " - " ^ ) ' where N k= 1 (33) (34) CO = gk tmh{kd) (28)

A g a i n , an analogous (linear) representation can be written d o w n f o r W{t).

Frequency domain analysis of dynamic offshore structures 259 3000 2500 2000 1500 1000 500 0.2 0.4 0.6 0.8

Fig. 7. Drag force spectrum derived from simulated force time series and the corresponding force spectrum for the proposed drag force representation.

The linear transfer f u n c t i o n L(co) of the dynamic model o f eqn (31) is given as f o l l o w s

L ( w ) = [ - co^ + 2/^WeW + We] " (35) Let the modal response process Z ( r ) = Z i ( r ) -|-Zj(?), where Z i ( r ) is the response component due to m^^W^t), and Z2(r) the one due to m " ' * d ( 0 - Z i ( / ) and Ziit) are independent stochastic processes. I t has been shown[20] that the modal response component Zjit) can then be represented by an expression very similar to that o f eqn (33), namely

Z2(0= X S e,yB,B;e''("'-'->)'

where

Qij = m 'L(cO, - W ; ) G y

(36)

(37) A corresponding expression can be written down f o r Z i ( r ) , w h i c h is a Gaussian process.

A key result facilitating the statistical analysis o f the response process is the f o l l o w i n g equation[20]

(38)

a=\

where the are the positive eigenvalues o f the matrix Q = iQij)- For each positive eigenvalue there w i l l be a cor-responding negative eigenvalue o f equal magnitude. The P „ ( r ) are real A^(0,I) Gaussian processes defined as f o l l o w s

(39)

where the v „ = (v„_ _ „ , v „ ^ „ ) are the orthonormal eigenvectors corresponding to the Aq,. I t also turns out

that Nait) is the Hilbert transform o f ƒ'„(?) f o r every a [ 2 0 ] . Note the strong siinilarity between the response process Z2(0 as given by eqn (38) and the drag force repre-sentation given b y eqn (6). The probabilistic properties o f the response w i l l therefore to a certain extent resemble those o f the force. H o w close this resemblance w i l l be i n a particular case depends on the eigenvalues. I f there are many eigenvalues o f comparable size, the response tends to become more Gaussian than the force because o f the central l i m i t theorem.

It appears that no closed f o r m expression f o r the P D F o f Z ( f ) is k n o w n . However, an accurate numerical method f o r calculating this P D F has been developed[20]. I t consists o f calculating the characteristic f u n c t i o n o f the response v a r i -able given by eqn (38), and then using a saddle point method to calculate the Fourier transform o f the obtained character-istic f u n c t i o n to produce the PDF. A method f o r calculating the upcrossing rates is also available[21]. This w i h be exem-p l i f i e d i n the next section by calculation o f statistical moments and P D F o f the dynamic response o f a jack-up type p l a t f o r m structure subjected to random waves.

7 N U M E R I C A L E X A M P L E

I n agreement w i t h the simplification introduced i n the pre-vious section, the jack-up structure o f F i g . 6 is assumed without lateral extension. The p l a t f o r m is assumed to be placed on location i n 80 m water depth. The structure is represented vertically by 10 nodal points. 60% o f the total mass o f the p l a t f o r m is allocated to the nodal point at the mean free surface, w h i l e the remaining 4 0 % is evenly dis-tributed over the nine other nodes. I t is emphasized that the

sole purpose o f this simplified stmctural representation is to reduce non-essential complexity o f the expression. A much more complex structure could have been analysed w i t h little increase i n required C P U time.

To illustrate the results o f a dynamic analysis, we shall calculate the statistics o f the displacement response o f the model stmcture assumed to be drag dominated, that is, inertia forces are neglected, and subjected to random waves. Specifically, it w i l l be assumed that the drag force is determined by the J O N S W A P spectral density f o r the wave elevation as specified above. The significant wave height H^— 10 m and the peak wave period Tp = 16 s are chosen f o r the example, w h i c h , o f course, is somewhat exaggerated f o r a water depth o f 80 m . For calculation o f the drag force, the relation = 0.5pCdA is used. Here, p = density o f sea water = 1026 k g m " ^ Cd = drag coefficient and A = exposed area perpendicular to the direction o f wave propagation. I t is assumed that C j = 1.0, and that the sub-surface stmcture has an exposed area equivalent to 5 m^ per m along the legs (z-direction). Further, i t is assumed that the dynamics o f the stmcture can be determined by the modal representation given by eqn (24) w i t h the f o l l o w i n g para-meters: modal structural and added mass, M = 8.8 X 10^ k g ; modal damping ratio, ^ — 0.10; natural period,

= 5 s.

To v e r i f y the calculated response statistics obtained by the proposed method, a Monte Carlo simulation o f the response o f the extended structure was carried out based on the standard drag force f o r m u l a t i o n . T o estiinate the drag force spectrum needed f o r the proposed drag force representation, simulated time series f o r the drag force

were used. The estimated drag force spectrum at the mean free surface derived f r o m such time series is shown i n Fig. 7, together w i t h a corresponding spectrum (w) f o r the drag force representation $d(0- It should be mentioned that this numerical example was calculated before the results above on the spectral representation o f the drag force f o r JONS-W A P type wave spectra were available. This explains the sHght deviation i n shape between the 5^^(w) o f F i g . 7 and that o f Fig. 5. However, this has negligible consequences f o r the results to be presented here.

A l l the required i n f o r m a t i o n to determine the matrix Q = {Qij) is now available. A standard library routine is used to calculate the positive eigenvalues o f the obtained matrix. The first f o u r statistical moments o f the displace-ment response X j o at nodal point no. 10 (free suiface) can then be calculated directly f o r m the eigenvalues[l 1,20], and are f o u n d to be h i , = E[XiQ] = 0, ni2 = Var[X,o] = 3.14 X 1 0 ^ 2 m ^ « 3 = £ [ ( X i o ) ^ ] = 0 and nu = E[{Xiof] = 7.92 X 1 0 " ^ m ' ' . This leads to akurtosis Ta = ' " 4 / ( ' " 2 ) ^ = 80. i n d i -cating that the response statistics w i l l deviate significantly f r o m the Gaussian case, w h i c h has 72 = 3.0.

The corresponding results obtained by M o n t e Carlo simu-lation o f the response to pure drag forces are «i2 = 3.06 X 1 0 " ^ m^ and m4 = 7.77 X 1 0 " ^ m'', w h i c h are i n good agreement w i t h those f o u n d by using the new drag force representation. The corresponding value f o r the kurtosis is 72 = 8.3.

The P D F o f the displacement response obtained by the proposed method is shown i n Figs. 8 and 9 together w i t h the PDF estimated f r o m the simulation results. It is seen that the overall agreement between the two PDFs is indeed very

good. For compai-ison, the corresponding Gaussian P D F w i t h the same mean value and standard deviation is also shown. A s expected, the Gaussian response assumption leads to a substantial underestimation o f large responses.

8 C O N C L U D I N G R E M A R K S

A new stochastic representation o f the drag forces on an offshore structure subjected to random waves is described. It is demonstrated that the proposed force representation to a large extent has the same statistical properties as the original drag force. I t is also pointed out that the spectral density o f the new force representation can be made to accurately f i t that o f the drag force. The advantage o f the proposed force representation is that i t makes i t possible to carry out a dynamic analysis o f a linear structure i n the frequency domain using available techniques, w h i c h includes accurate calculation o f PDFs and corresponding statistical descrip-tors o f various kinds.

B y comparison w i t h the response P D F o f a jack-up struc-ture estimated f r o m M o n t e Carlo simulations based on the standard drag force formulation, it is shown that f o r the specific case study presented, the proposed stochastic drag force model leads to very good response predictions.

I t is also shown that the drag force representation pro-posed here obtains the same level o f accuracy as a recently proposed c u b i c i z a t i ö n approach. However, the c u b i c i z a t i ö n approach is i n practice l i m i t e d to providing statistical moments up to fourth order, w h i c h means that only esti-mates o f skewness and kurtosis can be given. B y assuming

additionally that the response process can be approximated by an algebraic p o l y n o m i a l , o f degree three, o f a standard Gaussian process, generally i n terms o f Hermite p o l y -nomials[22], it is possible to derive more detailed response statistics.

The inherent limitation o f zero current o f the proposed method is j u d g e d to be o f minor importance f o r the cal-culation o f long-term fatigue damage o f a drag-dominated offshore structure. Since the proposed method also presup-poses that the water particle velocity process is stationary and Gaussian, splash zone effects cannot directly be talcen into account.

A C K N O W L E D G E M E N T S

Part o f this w o r k was carried out w h i l e the second author was a visiting researcher at the Norwegian University o f Science and Technology ( N T N U ) under the H u m a n Capital and M o b i h t y network 'Stochastic Mechanics i n Structural and Mechanical Engineering'. The authors gratefully acknowledge the financial support f r o m the E U Commission to this network.

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