Applied Ocean
Research
E L S E V I E R A p p l i e d d c e a n Research 20 ( 1 9 9 8 ) 3 7 - 5 3 = ^ ^ = = ^ ^ ^ =
5
The extreme dynamic response óf bottom supported structures using an
equivalent quasi-static design wave procedure
J.H. Vugts*, S.L. Doh\ L.A. Harland
Offshore Technology, Faculty of Civil Engineering, Delft University of Teclinology, P.O. Box 5048, 2600 GA Delft, The Netherlands
Abstract W h i l e t h e r e i s g r e a t e x p e r i e n c e w i t h t h e d e t e r m i n i s t i c a n d q u a s i - s t a t i c d e s i g n w a v e m e t h o d f o r t h e d e s i g n o f b o t t o m f o u n d e d o f f s h o r e s t r u c t u r e s , t h e d e t e r m i n a t i o n a n d i n c l u s i o n o f r a n d o m d y n a m i c e f f e c t s i n t h e d e s i g n p r o c e s s i s f a r less d e v e l o p e d a n d o n l y o c c a s i o n a l l y n e e d e d . A n a t t r a c t i v e p r o c e d u r e i s t o d e t e r m i n e a n ' i n e r t i a l l o a d set' t h a t a c c o u n t s f o r t h e e f f e c t s o f t h e d y n a m i c r e s p o n s e a n d t o a d d t h i s t o t h e a p p l i e d l o a d i n g , a f t e r w h i c h t h e s a m e q u a s i - s t a t i c d e s i g n w a v e m e t h o d c a n a g a i n b e u s e d . T h i s e n g i n e e r i n g m e t h o d i s d e s c r i b e d a n d c r i t i c a l l y r e v i e w e d , a f t e r w h i c h i t i s a p p l i e d t o a c o n c e p t u a l d e s i g n o f a c o m p H a n t t o w e r i n 6 0 0 m w a t e r d e p t h w e s t o f t h e S h e t l a n d I s l e s . T h i s is a d e m a n d i n g t e s t f o r t h e m e t h o d a n d t o v e r i f y t h e r e s u l t s t h e s e a r e c o m p a r e d w i t h t h e r e s u l t s o f r a n d o m t i m e d o m a i n s i m u l a t i o n s w h i c h s e r v e as a r e f e r e n c e . T h e ' i n e r t i a l l o a d set' i s d e t e r m i n e d i n t w o d i f f e r e n t v y a y s : {1) b a s e d o n i n c r e a s e s i n t h e g l o b a l r e s p o n s e s ( b a s e s h e a r a n d o v e r t u r n i n g m o m e n t ) a n d ( 2 ) b a s e d o n i n c r e a s e s i n t h e l e v e l s h e a r s at v a r i o u s e l e v a t i o n s o v e r t h e h e i g h t o f t h e s t r u c t u r e . I t i s s h o w n t h a t t h e s e c o n d m e t h o d p r o v i d e s m a r k e d l y b e t t e r r e s u l t s . © 1 9 9 8 E l s e v i e r S c i e n c e L t d . A l l rights r e s e r v e d . Keywords: C o m p l i a n t t o w e r ; D e s i g n ; D e s i g n w a v e m e t h o d ; D y n a m i c s ; E x t r e m e r e s p o n s e ; O f f s h o r e s t r u c t u r e s ; Q u a s i - s t a t i c ; R a n d o m sea 1. Introduction
An effective and reliable dynamic analysis of an offshore stracture is a complex exercise. This paper discusses such a method for fixed or compliant stractures of steel space frame configuration which are piled to the sea floor. The great majority of the world-wide population of offshore stractures belongs to this category. The stractural design of most of them is governed by extreme response to environmental loading in severe storm conditions. With few exceptions these structures exhibit a quasi-static response to the environmental loading. Consequently, exist-ing design methods and experience are largely based on quasi-static and deterministic design wave methods with which a wealth of experience has been gained for design in regard to strength and stability criteria.
When the flexibility of a bottom founded stracture increases, e.g. due to an increase in water depth, dynamic effects can often not be completely ignored.. However, such dynamic effects mostly remain somewhat peripheral to the quasi-static effects that usually continue to dominate strac-tural design. Dynamic effects need to be taken into account
* C o r r e s p o n d i n g author.
' N o w at: A B B L u m m u s G l o b a l B . V . , P.O. B o x 3 2 0 0 3 , 2 3 0 3 D A L e i d e n , T t i e Netherlands.
but these do not fundamentally change the predominantly quasi-static behaviour of the stracture and their quantitative influence rarely exceeds 20% or so. Hence approximate methods usually suffice. The methods used range from the simplistic application of dynamic amplification factors (DAFs) to elaborate and somewhat overdone checks by means of f u l l time domain simulations. There are only a handful of deep water, bottom supported structures in the world where the design has really required detailed dynamic analyses. The subject of this paper is a more rigorously based method to include the effects of dynamic response in an appropriate but still approximate manner, making the best possible use of existing experience and computa-tional tools for the quasi-static design of such stractures. The method would be suitable for deep water fixed strac-tures; for compliant and guyed towers; for jack-ups and for very light and slender stractures in shallower water that are more than usual affected by dynamic response. The applica-tion discussed in this paper concems a conceptual design of a compliant tower west of the Shetland Isles in a water depth of 600 m. A compliant tower forms a very demanding test for the method discussed herein and to verify the results these are compared with the results of random time domain simulations for the same structure which w i l l serve as a reference. It is expressly noted that the aim of the paper is the description and investigation of a method of dynamic 0 1 4 1 - 1 1 8 7 / 9 8 / $ 1 9 . 0 0 © 1 9 9 8 E l s e v i e r S c i e n c e L t d . A l l rights r e s e r v e d
analysis. The choice of the type of structure is solely made with this aim in mind. Details of the analysis may therefore not be judged on their applicability (or lack of it) for the design of a real compliant tower.
2. Dynamic analysis features
The correct determination of the dynamic behaviour of an offshore stracture is a formidable problem. It belongs to the class of muki degree of freedom non-linear systems sub-jected to random excitation. The only known way to solve
such problems is time domain simulation. Höwever, this is a very time consuming and costly exercise, always provided that the available computational tools allow meaningful application in the first place in view of inherent modelling limitations of most of the tools. Even if this is the case, time domain simulations are untransparent and the results are usually difficult to interpret.
The problems in a dynamic analysis arise mainly from two circumstances, i.e. firstly the non-linearities involved and secondly the requirement for detailed checking of load and stress combinations in members against (code) allowables to determine whether all individual members of the stracture satisfy strength and stability requirements. As regards the last point, the different load (stress) compo-nents in a member vary randomly with time while they are mostly only weakly correlated. Hence any relevant combi-nation of these components needs to be evaluated at each and every instant in time to enable their combination to be compared with a limiting criterion. More practical procedures of adding maxima or statistical measures of the individual random load or stress components will be suiTounded by high uncertainty or are demonstrably unrealistic. Furthermore, the criteria for design against stability and strength as are given in codes, have all been derived on the basis of static conditions. Whether these static criteria are equally applicable to situations where uncorrelated time varying components are at play is an open question.
The system non-linearities include a number of different aspects of the enviroiunental loading (such as Morison drag forces and intermittent loading around the mean still water line between (variable) crest elevations and trough depths) and of the stractural system (such as non-linear foundation stiffness, large displacements and so-called P-ó (effects). Non-linear environmental loading is always present and usually the predominant non-linearity involved. Non-linearities result in the stractural response being non-gaussian, even i f the sea as the source of the random excitation is modelled as a gaussian random process. Extreme responses of general non-gaussian processes are as yet an umesolved topic of continuing research.
For aU these reasons approximate procedures, which are as far as possible founded on a firm theoretical basis but are appfied in a pragmatic manner, are a great asset especially
when they have a direct link with the tools and experience for conventional quasi-static design.
3. Dynamic analysis using an equivalent quasi-static procedure
The equations of motion of a multi degree of freedom dynamic system are:
mxit) + cm + kx(t) = Fit) (1)
where m = (lumped) mass matrix; c = damping matrix; k = stiffness matrix; x = vector of nodal displacements; F = vector of nodal excitation forces.
Eq. (1) may be written in an equivalent static form as:
kx{t) = Fit) - nm{t) - cx{t) = G(t) (2)
In quasi-static design wave methods, Fit) is deterministic and periodic (nearly harmonic). For a harmonic excitation the displacement vector x{t) is also harmonic. Offshore stractures are (like most stractures) lightly damped and the contribution cx{t) in the right-hand side of Eq. (2) is therefore always small and may be neglected compared to the other force components, except in a narrow region around resonance. Vectors varying harmonically i n time may be written in the general form V{t) — Ve'"', where V is a complex number incorporating magnitude and phase of
V{t). The equivalent static excitation force G may thus be
written as:
G = F-mx = F-\-m(j?x (3)
The last term represents an additional load set which sup-plements the trae excitation forces F_ to form the equivalent quasi-static loading vector G; it is termed the inertial load
set. Once the inertial load set has been determined and is
added to the excitation vector F, design calculations can proceed in the normal manner using well-known determi-nistic and quasi-static design wave methods and tools that include all regular design features and code checking facili-ties. The problems associated with practical use lie in: • the determination of an adequate estimate of the
displa-cement vector X due to the forced random excitation F_ for use in Eq. (3); x clearly depends on F_ as well as the properties of the mechanical system involved;
• the basic question of how to represent a stracture's extreme dynamic response under random excitation in a suitably equivalent quasi-static and periodic form as in Eqs. (2) and (3), recognising that F_ and x in Eq. (3) have different phases at one pai'ticular frequency for each degree of freedom x and that these, phases also differ for each component frequency in the random excitation.
4. A'^ritical review of applications
An equivalent quasi-static approach to dynamic problems has been applied with considerable success to fixed
J.H. Vugts et al. / Applied Ocean Research 20 (1998) 37-53 39
Structures of space frame configuration having relatively high natural frequencies: in practical terms for structures with fundamental natural frequencies in excess o f \ 0 . 2 -0.25 Hz. The governing environmental loading condition of such structures continues to be a design storm of high severity with the wave energy mainly concentrated at 'fre-quencies well below the natural frequency of the structure. The extreme response in a random sea is then still predo-minantly quasi-static, but there is a non-negligible dynamic contribution.
To determine the extreme response of such a structure for the purpose of design against strength and stability a quasi-static design wave procedure may be used applying Eqs. (2) and (3). The total applied horizontal loading H^^ is the sum of the (horizontal components) of the environmental load vector F. Similarly, the total horizontal loading //eq in an equivalent quasi-static problem is the sum of the (horizontal components) of the vector G. The difference between these two global load vectors is the sum of the (horizontal components) of the inertial forces in Eq. (3), i.e.
H;=H,
eq (4)
In a general case there will be several phase differences involved, i.e. phase differences:
—between the nodal excitation forces mutually; —between the nodal displacements mutually; —between the loading and response.
The relationship between the global quantities in Eq. (4) may be represented in a vector diagram as shown in Fig. 1. In principle there is a similar but different relationship asso-ciated with each chosen design wave frequency, design wave height and current condition.
In actual applications it is tacitly assumed that all quan-tities are in phase so that Eq. (4) simplifies to an algebraic equation:
— ^eq - ^^ap • hor 'nw X (5)
in which case //eq will always be larger than //gp and / / i > 0.
However, the difference between the equivalent quasi-static and applied loading is actually due to the dynamic response of the structure in its natural environment; quanti-fication of dynamic response due to random excitation requires random analysis techniques using the full dynamic
F i g . 1. D i a g r a m m a t i c representation o f the h o r i z o n t a l vector o f the g l o b a l a p p l i e d l o a d i n g , the equivalent quasi-static e x c i t a t i o n a n d the i n e r t i a l load set.
Eq. (1). An acceptable simplification is though that for the present purpose dynamic calculations of global response using a simplified structural model will normally suffice.
Random dynamic calculations may be performed through time domain simulations or through linearised frequency domain calculations. With regard to linearisation, in most cases it is entirely adequate to consider the structural sys-tem, including its foundation, as linear. The true test of linearisation therefore lies in the environmental loading. However, to capture the additional dynamic response for the case under consideration in this section is not really too much of a problem either. This response occurs in the frequency range around structural resonance and is due to wave components with low to moderate energy content. Drag forces are thus small compared to what they normally are in design waves and the structure is hence mainly excited dynamically by linear wave loads anyway.
For random excitation the influence of dynamics on glo-bal loading can be expressed by a generaUsed form of the Dynamic Amplification Factor (DAF) on the base shear which is defined as the ratio of the most probable maximum (mpm) dynamic base shear S^y^ to the mpm quasi-static base shear rS'qs, both calculated through the same random analysis
teclinique using the same models and the same computa-tional tools:
mpm[Siyn{t)}
(6)
t7ipm{S^,{t)}
Thus, the increase in mpm base shear due to dynamics is:
mpm{Siy,{t)} - mpm{S^,{t)} = {DAF, - iympm{S^,(t)} =H;
(7) The increase may be equated to a force H\ in analogy with Eq. (5). The difference with the single harmonic case of Eq. (5) being that under random conditions, this H\ may in prin-ciple be positive or negative, depending on whether the dynamic response is larger or smaller than the quasi-static response. This will especially be the case when applications are not limited to stmctures with relatively high natural frequencies, as will be discussed in the next section. To enable an equivalent quasi-static design wave analysis in accordance with Eq. (2) we must distribute this total increase in base shear over the height of the stmcture in such a manner that the sum of the (horizontal components) of moP'x in Eq. (5) is equal to the H\ of Eq. (7). This is achieved by determining a set of discrete inertial forces at a number of chosen elevations over the height of the struc-ture (usually at the plan bracing levels) such that:
I
(8)where a = a multiplication factor to make both sides of Eq. (8) equal (m); w; = the fundamental (first) natural frequency of the stmcture (rad/s); m^ = the (summed) lumped mass(es) at elevation k (kg); 0 u = the (normalised) displacement in the first mode of vibration at elevation k (—).
J.H. Vugts et al. / AppUed Ocean Research 20 (1998) 37-53
This set replaces the additional force Hi on the left by a sum of harmonic forces on the right, each acting at a parti-cular elevation and evaluated at the instant of time at which these are a maximum. This is an entirely satisfactory repre-sentation for determining the extreme global response when and only when a number of assumptions underlying Eq. (8) are satisfied:
— the dynamic response in the random environment is dominated by resonant response in a single mode; — the frequency range at which appreciable dynamic
response occurs is well separated from the frequency range at which the predominant quasi-static response occurs;
— the dynamic response is narrow banded and its fre-quency content may be adequately characterised by the first mode natural frequency;
— the horizontal displacements over the height of the struc-ture associated with forced vibration in the fundamental mode under random excitation are for all intents and purposes in phase and thus synchronous;
— the maximum horizontal displacements x{f) associated with forced vibration in the fundamental mode are fully correlated with and occur at the same instant as the excitation F\t) reaches its maximum.
An entirely analogous procedure may of course be fol-lowed using the overturning moment instead of the base shear as a measure of global dynamic structural response, resulting in:
Mi=^co? X ' " i - ^ i A - (9)
k
where: M j = overturning moment about the sea floor (Nm); /3 = a multiplication factor to make both sides of Eq. (9) equal (m); = height of elevation k above the sea floor (m). For the class of fixed structures considered in this section the above assumptions are reasonably satisfied. The quasi-static response is centred around the peak of the wave spec-trum while the dynamic response occurs in the high frequency tail of the wave spectrum, centred around the natural frequency of the (bending) mode. These frequency ranges are well separated and may in approximation be dealt with independently. For resonant dynamic response in the first mode the whole structure moves indeed in phase, while assuming synchronisation of dynamic and quasi-static response is conservative, but not unduly penalising under the circumstances discussed.
To conclude this critical review a further comment should be made. It is usually found that a i= (3. This slight practical problem may be solved by applying the difference between the result of Eq. (9) and the overturning moment which is associated with Eq. (8) as a point moment at an elevation corresponding approximately with the centre of gravity of the inertial loading. Alternatively, the right-hand sides of Eqs. (8) and (9) are sometimes replaced by a linear combi-nation of the first two modes which are associated with the
deformation of the structure in the direction of F(t):
(10)
Mi = TWl S "hMh + X 'nk<l>2khk k k
The two unknown multipfication factors, y and 6 with the dimensions of a length in m, can be found by solving both equations simultaneously. This alternative should not be seen as being more accurate or physically more conect, because the second mode occurs at a still higher frequency than the first and will not noticeably be excited under the applicable random excitation. And even i f it is excited, then the associated forced vibration displacements over the height of the stmcture due to both modes will not be syn-chronous, which is in conflict with the assumptions under-lying the procedure discussed above. However, appunder-lying Eq. (10) for the base of the stmcture does produce the desired outcome of a distributed inertial load set with the correct increase in total base shear and total overturning moment.
5. A more sophisticated method to determine an inertial load set
In addition to applications for fixed stmctures with non-negligible dynamic response as discussed in Section 4, the above method has also been used for the extreme response of jack-ups in deep water. Jack-ups are often much more dynamically sensitive with natural frequencies which lie much closer to the peak of the wave spectmm. The assump-tions underlying Eq. (8) are then less well or clearly not satisfied so that applicability of the method may become questionable. An improved and much more sophisticated engineering method to determine an inertial load set is described i n the Reconunended Practice (RP) for Site Specific Assessment of Mobile Jack-up Units [1], which was developed by Shell International in The Hague. In this work, the method described in the RP for jack-ups has been applied to a compliant tower. With dynamic contribu-tions from two modes on either side of the peak of the wave spectmm the assumptions underlying the method and which are described in Section 4 are definitely not met. Therefore, the results obtained are compared with the outcome of time domain simulations which are used as a reference.
The dynamic response to random loading is quite gener-ally split into two parts:
dynamic response = quasi — static response
\ +'inertial response' (11)
Only the response due to hydrodynamic loading is con-sidered; any quasi-static or dynamic effect as a result of wind loading is to be determined separately. Although the modification of the quasi-static response is termed 'inertial response', this incorporates all differences between
J.H. Vugts et al. / Applied Ocean Research 20 (1998) 37-53 41
4.0E+06
dynamic base shear
quasi-static base shear
O.OE+00
CO 2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Angular Frequency (rad/s)
F i g . 2. T h e d y n a m i c and quasi-static f r e q u e n c y response f u n c t i o n s o f base shear f o r a conceptual C o m p l i a n t T o w e r f o r a l o c a t i o n west o f the Shetland Isles i n 6 0 0 m o f w a t e r ( u s i n g the m e t h o d o f statistical l i n e a r i s a ü o n i n the 100-yr sea state w i t h c u r r e n t ) .
dynamic and quasi-static behaviour and cannot simply be associated with true mass inertia forces. For the circum-stances described in Section 4 the two contributions on the right-hand side are well separated and assumed to be completely independent. However, in general they are intri-cately related in a complex fashion. This is easily demon-strated by comparing dynamic and quasi-static frequency response functions such as shown in Figs 2 and 3 where at each frequency the magnitudes as well as the phases of both responses differ.
Consider now an arbitrary response variable R(t) with
200
subscript ' d ' for dynamic response, 's' for quasi-static response and ' i ' for 'inertial response'. Next determine the means and standard deviations of the three random responses i ? d ( / ) , R,(t) and R;(t) in the design sea state. This may be done with appropriately simplified structural models and using either time or frequency domain methods. For time domain simulations the 'inertial response' is obtained by direct subtraction of the two random dynamic and quasi-static responses:
^ i ( 0 = ^ d ( 0 - ^ s « (12)
I
-200
quasi-static base shear
Angular frequency (rad/s)
For frequency domain calculations, as used in this work, the frequency response function //Ri(w) of the 'inertial response' is obtained by complex subtraction of the two other frequency response functions to account for phase differences:
(13) Each of the three response functions is subsequently com-bined with the wave spectrum of the design sea state to obtain the means and the standard deviations of the three individual random responses. A least squares (or statistical) linearisation, see Borgman [2,3], should preferably be used; this also allows incorporation of the effect jpf a current on the linearisation. The further procedure is clearly depicted in Figs A 1 - A 3 , which have been copied directly from the jack-up RP [1].
The short-term statistics of the responses in the design sea state are next determined as shown i n the schematic of Fig. A l . This figure is largely self-explanatory; for further details reference is made to the jack-up RP [1]. A very important point is that correlation between the quasi-static and dynamic responses is now accounted for; see block number 7 of Fig. A l . The correlation coefficient PR has a value between - 1 and + 1. The case treated in Section 4 assumes zero correlation (PR = 0) and the split in Eq. (11) then applies to the variances:
'^Rd = 0-Rs + f^Ri (14)
± 1) while for the limiting case of perfect correlation (PR = the split applies to the standard deviations:
«^Rd= IffRs ± (^Ril (15) For all other cases there is no such simple relationship.
Positive values of PR imply that the 'inertial response' acts mainly in the same direction and amplifies the quasi-static response, while negative values mean that the 'inertial response' acts mainly in the opposite direction and de-amplifies the quasi-static response.
The maximum of a short-term random process is itself a random variable. For a gaussian process and a sufficiently large number of response peaks, A^, the ratio between the mpm and the standard deviation of the response is the well-known relationship:
O-R
(16)
As a result of non-linear influences the random response will, however, not be gaussian. Factor C R for an arbitrary non-gaussian response is not known and an engineering postulate is used to account for the effects of non-gaussianity. This is done as follows. The 'inertial response' is assumed to be gaussian; for a standard number of response peaks of N — 1000 the mpm-factor CR; is thus 3.7 (see block 10 of Fig. A l ) . The mpm-factor CRS for the quasi-static response (see block 9) is assumed to be equal to that for Morison-type wave loading per unit length on a
circular cylinder; for this case the statistics have been eval-uated theoretically [4]. The procedure for determining CRS is shown in Fig. A2, while the results are shown in Fig. A3 in the special form of CRS VS. the so-called drag-inertia para-meter K. The parapara-meter K is basically defined as:
magnitude of the drag force magnitude of the inertia force
1
TrpCj:P-c
2cy
•ïïCr^Da (17)
where: p = density of sea water (kg/m );D = diameter (m);
v,a — normal velocity and acceleration acting on the
mem-ber (m/s; m/s^); C^, — drag and inertia coefficients in Morison's equation (—; — ) .
When generalised from periodic waves to random waves and using a least squares linearisation this becomes:
K=\/Ttl% gR(Cm = 0)
aR(Cd = 0) (18)
This relationship and Fig. A3 were first derived by Brouwers and Verbeek [5] and Hagemeijer [4]. However, these authors derived a slightly different ratio CRS for the
mean value of the maximum static response, resulting in a
highest CRS of 8.6. As most work in offshore engineering is based on the mpm the calculations were repeated by Baar [6] resulting in Fig. A3 with a highest CRS of 8.0. Finally, the
CRJ for the dynamic response is determined by a suitable
combination of the quasi-static and 'inertial responses' as shown in block 11 and block 13 of Fig. A l .
To quantify the influence of dynamics a DAF can again be defined. In Fig. A l three different DAFs are stated; see blocks 8, 12 and 16. The first is based on the ratio of the standard deviations as the most basic measure of a random variable, the second on the ratio of the mpm-values in which the effect of the approximate treatment of non-gaussian response has been reflected, while the third is a pragmatic overall measure of the increase in response due to dynamics by taking the ratio of the most probable maximum extreme
(mpme) values also including the mean responses due to current.
An additional load set can now be determined in two different ways. First, by focusing on the global response of the structure. Choose /?(/) to be the base shear 5'(/) and the overturning moment M(t). In accordance with Eq. (7) we have for the increase in base shear and in overturning
moment: (19) H\ = mpme{S^(t)] — mpme{Ss{t)} '=iDAF3s - iympme{S,it)} Mi=mpme{Mdit)} - mpme{M,{t)] =XDAF?,M-l)-mpme{M,(t)} •
J.H. Vugts et al. / Applied Ocean Research 20 (1998) 37-53 4 3
is subsequently determined in accordance with Eq. (10). These forces are to be added to the applied loading vector
Fit) from a deterministic design wave analysis, after which
the common computational tools and code checking fiacili-ties can again be used. The determination of the additional load set is best based on the overall increase in the response due to dynamics. Therefore the mean is included and Eq. (19) uses the mpme-responses and DAF2>^ as shown in blocks 1 4 and 1 6 of Fig. A l . However, for cases where the dynamic mean response is equal to the quasi-static mean response the additional load set can also be deter-mined using Z)A^2R instead of DA/^3R and the mpm-responses instead of the mpme-mpm-responses, achieving identi-cal results (Eq. (20)):
H^=mpm{S^{t)} - mpm[S,{t)} = {DAF2s-iympm[S,it)]
M; = mpmlM^it)] — mpm[M,{t)}
= {DAF2^-l)-mpm[Mm
(20)
The second way is to determine the inertial load at a parti-cular elevation directly from the difference in dynamic and quasi-static shears at that level. The inertial load then
follows from an equation that is analogous to the first one of Eq. (19), i.e.:
Li(z) = mpme[Si{z, t)} - mpme{S,{z, t)}
= {DAF\{z) - l]-mpme{S,{z, t)] (21)
or of Eq. (20) i f the mean responses are equal:
L-Xz) = mpin{Si{z, t)] - mpni{S,{z, f ) }
= {DAF2^{z)-l]-mpm{S,{z,t)} (22)
where the elevation z is an additional parameter and L^z) is the increase in horizontal force at elevation z. When
mpme{Si{z,t)} > mpme{Ss(z,t)} (or mpm[Sj{z,t)} > mpm
{Ss{z,t)]) the force L-^z) is positive and acts in the same direction as the applied loading, when the reverse is true
Li(z) is negative and acts in the opposite direction than the
applied loading at the elevation concerned.
The additional load set derived through Eq. (19)(Eq. (20)) or Eq. (21)(Eq. (22)) may again be referred to as an 'inertial load set'. However, it should be recognised that there is only a loose connection with true mass inertial forces. The additional load set incorporates all the differences between dynamic and static behaviour of the structure, including effects of phase differences between contributions from several modes of vibration, phase differences over the
O m
—
-201m
F i g . 4. 3 - D m o d e l o f the conceptual design o f a C o m p l i a n t T o w e r f o r 6 0 0 m
J.H. Vugts et al. / Applied Ocean Researcli 20 (1998) 37-53
height of the structure and (in case global forces are used) the resulting phase differences between S and M. Conse-quently, the net result cannot simply be related to mass inertial forces associated with a dominant mode.
6. Compliant tower application considered
The two different applications of the method outlined in Section 5 have been applied to a conceptual design of a Compliant Tower (CT) for a water depth of approximately 600 m in the area west of the Shetland Isles, the details of which were made available for this project by Shell Inter-national Exploration and Production B V; see Fig. 4. The CT is a slender, tubular steel structure with tensioned risers inside the tower. The sub-structure is an X-braced space frame which is subdivided into 9.5 bay levels with constant bay width and height of 67 m. The conceptual design has a total sub-structure height of 637 m. Five piles are located around each of the four legs. The piles range from approxi-mately - 1 5 0 m below the mudline to 400 m above the mudline. A t the top the piles are rigidly connected to the space frame; between their top and the mudline the piles are connected to the space frame via pile sleeves leaving them free in the axial direction.
-402 +
-469 - - i
-536 -- I
-603 -I 1 1 1
O.OB-l-00 5.0E-I-06 1.0E-^07 1.5E-f07
M a s s ( k g )
F i g . 6. D i s t i i b u t i o n o f structural and added mass as a f u n c t i o n o f e l e v a t i o n .
In this work two software programs were used to analyse the responses of the CT, i.e. SESAM (Det Norske Veritas [7]) and NIRWANA (Karunakaran et al. [8]). SESAM was used for the (equivalent) quasi-static design wave procedure while NIRWANA was used for the verification based on time domain analyses. For the purpose of this study the complex 3-D,structure of Fig. 4 was replaced by a 2-D model as shown in Fig. 5. The five piles are now modelled by a composite vertical spring at their top. The horizontal forces are transferred into the seabed by a lateral spring at the base of the tower. Furthermore, the part of the tower between the pile-frame connection and the seabed is sim-plified by means of stick elements. The mass and stiffness as well as the hydrodynamic diameter of the stick elements vary over the height and are modelled such that they corre-spond to the original design. The distribution of the mass over the height of the tower (including structural and added mass) is given in Fig. 6. The mass distribution is necessary to determine mode shapes and natural frequencies as well as the 'inertial load set'.
In Table 1 and Fig. 7, the results of a free vibration
d e t e r m i n e d using SESAIvl 0 • d e t e r m i n e d using N I R W A N A / 1 -67 - / f / /' / / ' -134''^^ - / ' b e n d i n g mode / ' / Ao\
-<
/ /
If / r -268 - 1 fl / i i I -335 • 1 i i i i 1 -402 •1 ^
\ \
\
\ -469 • sway m o d e \ V \ \ \ \ -536 • \ \ N \ 1 \ 4303 i — = H -0.5 0 0.5 1 \ Normalised mode shapeF i g . 7. T h e m o d e shapes o f the m o d e l o f F i g . 5, d e t e r m i n e d u s i n g S E S A M and N I R W A N A .
J.H. Vugts et al. / Applied Ocean Research 20 (1998) 37-53 4 5
T a b l e 1
N a t u r a l periods o f the first and the second m o d e
N a t u r a l p e r i o d S E S A M N I R W A N A
T, (s) 30.3 3 0 . 1
5.96 5.48
analysis of the model are presented using both SESAM and NIRWANA. The differences for the first mode are insignif-icant, but the differences for the second mode are remark-able as both programs have used exactly the same model. This difference could not be simply explained and was not pursued further for the purpose of this study.
Damping was modelled as proportional damping using the first two modes with 10% of critical damping for the first mode and 5% of critical damping for the second mode. The design environmental conditions for the structure were taken as:
100-yr sea state: Pierson-Moskowitz spectrum with
H, = 18.0 m T,= 15.19 s
Design water depth: d = 603 m
Current profile: surface = 1.0 m/s seabed = 1 . 0 m/s
Design wave: wave height = 32.6 m wave period = 21.4 s
It should expressly be noted that these conditions fully served the purpose of this study and are reasonably repre-sentative of west of Shedand conditions, but should not be interpreted as authoritative design conditions for this geo-graphical area or for this type of structure. In particular, for a compliant tower with its very low natural frequency, the choice of a 100-yr design sea state may be unconservadve when compared with normal design practice for other structures.
7. Determination of the 'inertial load set' through global responses and results obtained
Following Secdon 5 for global responses, the base shear S{t) and overturning moment about the sea floor Mit) were calculated. From these two quantities all the variables in Eq. (19) were determined using the procedure shown in Figs. A 1 - A 3 . It should be noted that the overturning moment at the base of the tower is 'theoretically' zero as the global overturning moment is taken out by axial forces in the vertical piles. The moment provided by these axial pile forces remains constant between the pile to frame con-nection and the seabed: It is this moment from the axial
forces in the piles that is used as the global overturning moment for this particular CT-application.
With SESAM, the transfer functions of the quasi-staüc and dynamic base shear and overturning moment were determined using the statistical linearisation technique for the design sea state. The results for base shear are given in Fig. 2. This shows several cancellation points, where (« — 0.5) wavelengths correspond to the distance between the legs. At these cancellation points the phase angle jumps by 180°, as can be seen in Fig. 3. Fig. 3 also shows phase transitions for the dynamic base shear at reso-nance in the first mode (at 0.21 rad/s) and the second mode (at 1.06 rad/s). Fig. 3 further clearly shows that the dynamic base shear is broadly in coiinterphase with the quasi-static base shear (and hence also with the applied environmental loading) in the frequency range between 0.2 and 1.0 rad/s, i.e. between the two resonances, and in phase in the fre-quency range above 1.2 rad/s, i.e. beyond the second mode. Additionally there is a gradual change of phase with fre-quency for both cases as one would expect. Altogether this figure clearly demonstrates that phase relationships vary in a complex manner with frequency. The ultimate base shear and overturning moment experienced in a random sea are the result of a weighted summation over all frequencies. Hence it is to be expected that simplifications of these phase relationships can have a significant impact on abso-lute as well as comparative results. The procedure described in Section 5 takes the phase relationships fully into account. A further demonstration of the importance of phase rela-tions is given in the vector diagrams of Fig. 8 showing the quasi-static, the 'inertial' and the dynamic base shear vec-tors at a few selected frequencies. A final point to note is that resonance in the second mode at 1.06 rad/s is unfortunately close to a cancellation point in the base shear frequency response functions (Fig. 2). Consequently, in view of the proximity of resonance and the cancellation point the (rela-tively small) differences between SESAM and N I R W A N A for the second mode may influence the quantitative com-parisons to some extent.
The results for the global base shear and moment (as defined in the first paragraph of this section) are shown i n Table 2. The table presents the results for all parameters featuring in the schematic of Fig. A l . The p.- and a-values were determined through linearised frequency domain cal-culations using the SESAM-program. The two correlation coefficients clearly suggest that the static and dynamic global shear and moment are rather strongly correlated, while the fact that the coefficients are negative indicates that they are largely in counterphase. The 'inertial response' thus acts in the opposite direction to the applied loading. Whether this opposition represents de-amplification or
amplification with respect to the quasi-static base shear
and moment depends entirely on the magnitude and phase of the 'inertial response' over frequency (cf. Fig. 8). The D A F 2 R and DAF3^ are larger than 1.0 so that there is actu-ally amplification for both global shear and moment.
D e t e r m i n i n g the ' i n e r t i a l l o a d set' t h r o u g h the schematics o f F i g . A l and F i g . A 2 u s i n g f r e q u e n c y d o m a i n analyses f o r l e v e l shears and g l o b a l responses
2 /^Rs A^Rd IT R(l "Si PR DAFIR C R , C R I mpm Us mpm^i mpm Ri DAFIR CM mpme^ mpme^ Increase in mpme*
Inertia] load set
Corresponding block number in Fig. A I 7 8 10 11 12 13 IA 15 16
Corresponding block number in Fig. A 2 9e 9d
Global responses (MN or MNm) BS 8.25 8.25 2.28 4.06 5.72 - 0.82 1.78 4.63 4.20 3.7 9.56 14.40 21.15 1.51 3.54 17.81 22.65 1.27 4.84 O T M 120 120 923 1640 2282 - 0.80 1.7S 5.06 4.74 3.7 4373 5601 8443 1.28 3.41 4493 5721 1.27 1227 Level shears (MN) L l 0 0.07 1.76 1.74 0.33 24.02 6.11 6.00 3.7 0.44 6.58 6.42 14.97 3.74 0.44 6.58 14.97 6.14 6.14 L2 - 67 0.73 1.53 1.91 - 0.66 2.10 7.18 7.16 3.7 5.21 5.34 7.06 1,02 3.49 5,21 5.34 1.02 0,13 - 6.02 L3 - 134 1.01 1.56 2.10
-
0.71 1.55 6.97 6.94 3.7 7.01 5.68 7.79 0,81 3.64 7.01 5.68 0.81 - 1,33 - 1.45 L4 - 2 0 1 1.13 1.86 2.52-
0.73 1.65 5.63 5.44 3.7 6.15 6.39 9.32 1.04 3.44 6.15 6.39 1.04 0.24 1.57 L5 - 2 3 5 " " 1.38 2.15 3.02 - 0.77 1.56 5.31 5.06 3,7 6.99 7.34 11.19 1.05 3.41 6.99 7.34 1.05 0.35 0.11 L6 - 2 6 8 1.58 2.44 3.49 - 0.79 1.55 5.11 4.80 3.7 7.59 8.35 12.90 1,10 3.42 7.59 8.35 1.10 0.76 0.41 L7 - 302 1.73 2.73 3.91-
0.80 1,58 4.98 4.64 3,7 8.03 9.38 14.46 1.17 3.43 8.03 9.38 1.17 1.36 0.59 L8 - 336 1.85 3.01 4.29 - 0.81 1.63 4.89 4.53 3.7 8.36 10.39 15.88 1.24 3.45 8.36 10.39 1.24 2.03 0.67 1.9 - 3 6 9 1.94 3.27 4.64-
O.Sl 1.6S 4.82 4.44 3,7 8.63 11.35 17.16 1.32 3.47 8.63 11.35 1.32 2.72 0.69 L l O - 4 0 3 2.02 3.49 4.94-
0.81 1.73 4.77 4.38 3,7 8.84 12.20 18.26 1.38 3.49 8.84 12.20 i.38 3.36 0.64 L l l - 4 3 6 2.08 3.69 5.19 - 0.82 1,77 4.73 4.33 3.7 9.01 12.93 19.19 1.44 3.50 9.01 12.93 1.44 3.92 0.56 L12 - 470 2.13 3.85 5.39 - 0.82 1.80 4.70 4.29 3,7 9.15 13.52 19.94 1,48 3.52 9.15 13.52 1.48 4.38 0.46 L13 - 503 2.17 3.9T 5.54 - 0.82 1.82 4.68 4.26 3.7 9.27 13.98 20.51 1,51 3.52 9.27 13.98 1.51 4.71 0.33 L14 - 537 2.21 4.04 5.65 - 0,82 1.83 4.66 4.24 3.7 9.37 14.27 20.91 1.52 3.53 9.37 14.27 1.52 4.90 0.19 L15 - 570 2.25 4.08 5.71 - 0.82 1.82 4,64 4.22 3.7 9.47 14.43 21.12 1.52 3.54 9.47 14.43 1.52 4.96 0.06 L16 - 603 2.28 4.06 5.72-
0,82 1.78 4.63 4.20 3.7 9.56 14.40 21.15 1.51 3.54 9.56 14.40 1.51 4.84 0.46J.H. Vugts et al. / Applied Ocean Research 20 (1998) 37-53 4 7 'inertial' 2.01*10' R e quasi-static 0.75* 10« 'inertial' 4.00*10'
F i g . 8. Phase diagrams s h o w i n g the quasi-static, ' i n e r t i a l ' a n d d y n a m i c base shear vectors i n N / m at w = 0.25 rad/s, o> = 0.9 rad/s and co = 1.2 rad/s, r e s p e c t i v e l y .
However, this is due to an 'overshoot' of the 'inertial response' in the opposite direction compared to the quasi-static response, rather than to inertial forces simply adding to the excitation forces as was the case in the more common application discussed in Section 4.
With Eq. (19) the mpme increase in base shear Hi and overturning moment Mj can now be determined. These are then used to determine the 'inertial load set' through Eq. (10), which gave the values y = 1.810 m and Ö = - 0.063 m. In Fig. 9 the distributed 'inertial load set'
Hi(k) over the height of the CT is shown; this load set should
be added to the quasi-static design wave loading. From Fig. 9 i t is seen that the additional 'inertial forces' Hi(k) at each level are all positive. Therefore, they act in the same direction as the excitation forces F(k). The environmental force F is time-dependent with non-zero mean. To deter-mine the extreme responses from an equivalent quasi-static design wave analysis Hi(k) and F(k) are added at the instants in time at which the global applied force F is a maximum
and a minimum. The maximum absolute response is the desired extreme response; this will usually be when F is a maximum in the direction of wave propagation.
The benefit of the equivalent quasi-static design wave method would be its ability to use routine computational tools, including regular code checking facilities, in a normal design process for checking individual members. Therefore the results for local response variables need to be verified. The local response variables chosen were the shear force and bending moment in the cross-sections along the height of the stracture, the so-called level shear forces and level bend-ing moments. When these are predicted correctly, individual member forces may also be expected to be well represented. The design wave analysis was performed using a 5* order Stokes wave with a wave height of 32.6 m and an associated period of 21.4 s. The cumulative distribution of the applied design wave and current loading over the height of the stracture is shown in Fig. 10a. When the 'inertial load set' of Fig. 9 is added the distribution in Fig. 10b is obtained.
-603
- 2 - 1 0 1
'Inertial load' per level ( M N )
48 J.H. Vugts et al. / Applied Ocean Research 20 (1998) 37-53 * " equivalent quasi-static design w a v e procedure using g l o b a l responses o r level shears 5 10 15 20 mpme quasi-static level shear (MN)
25 ( b ) o t -67 -134 -201 -268 -335 -402 -469 -536 t -603 equivalent quasi-static design w a v e procedure using g l o b a l responses X - equivalent quasi-static design wave procedure using level shears
O - time domain simulations
-+-10 15 20 mpme dynamic level shear (MN)
25
F i g . 10. (a) m p m e quasi-static l e v e l shears f r o m t i m e d o m a i n analysis c o m p a r e d w i t h the a p p l i e d d e s i g n w a v e and current l o a d i n g , (b) m p m e d y n a m i c l e v e l shears f r o m t i m e d o m a i n analysis c o m p a r e d w i t h the a p p l i e d design w a v e a n d c u r r e n t l o a d i n g p l u s the ' i n e r t i a l l o a d set'.
The corresponding distributions of the bending moments acting in the cross sections, which are due to F alone and
XoF+ the 'inertial load set' above the elevation z, are shown
in Fig. I l a and Fig. l i b , respectively.
8. Determination of the 'inertial load set' through local responses in the form of level shears and results obtained
The results for the shear forces at 16 different elevations are also shown in Table 2. The inertial load set L^z) is determined from Eq, (22), which was used in preference to Eq. (21) since it proved difficult to abstract the mean
values for level shears from the SESAM frequency domain output.
As can be seen from this table, the level shear at z = 0 is nearly entirely dynamic {mpm^, = 0.44 M N , mpm-m = 6.58 M N , mpm^i = 6.42 M N , DAF2R = 14.97). Further-more, the correlation coefficient at z = 0 is positive and relatively small. This is in contrast to all elevations lower down, where quasi-static and dynamic level shears are of the same order of magnitude, while the correlation coefficients arè all negative and increase gradually i n magnitude towards the base. It is further noteworthy that the inertial contribution mpniRi is the largest of all at all elevations.
J.H. Vugts et al. / AppUed Ocean Research 20 (1998) 37-53 4 9 ( a ) -4000 (b) -4000 -2000 equivalent quasi-static design wave procedure using g l o b a l responses or level shears
-2000 0 2000 4 0 0 0 6000
mpme quasi-static level moment (MNm)
8000 10000
X - equivalent quasi-static
/ J l
-design wave procedure using level shears
0 - t i m e d o m a i n simulations \ . . " ' -201 ' -268 • 335 -+ - equivalent quasi-static design w a v e procedure using g l o b a l responses 402 469 536 -1 -1 =m-< 1 1 1 1 1 0 2000 4 0 0 0 6000
mpme dynamic level moment (MNm)
8000 1000
F i g . 1 1 . (a) m p m e quasi-static l e v e l m o m e n t s f r o m t i m e d o m a i n analysis c o m p a r e d w i t h the m o m e n t s due t o the a p p l i e d design w a v e and c u r r e n t l o a d i n g , ( b ) m p m e d y n a m i c l e v e l m o m e n t s f r o m t i m e d o m a i n analysis c o m p a r e d w i t h the m o m e n t s due to the a p p l i e d design w a v e and current l o a d i n g p l u s die ' i n e r t i a l l o a d set'.
z = Otoz = - 1 3 4 m and then increase gradually from z — — 134 m towards a nearly constant value over the lower part of the structure. Due to the different influences of non-gaussian effects on the quasi-static and, the dynamic responses, the DAF2R-value is actually smaller than 1.0 at elevation z= — 134 m, resulting in a negative value of Z,i( - 134) = - 1.33 M N . The distributed 'inertial load set' over the height of the structure is obtained from Li(A: -1- 1) - Li(^) and shown in Fig. 9. The 'inertial load sets' determined through global shear/moment and local shear parameters are seen to be markedly different. The
results for the level shears and moments using die equivalent quasi-static design wave procedure and an 'inertial load set' determined through local shear forces are also plotted in Figs. 10b and l i b .
9. Comparison of results with random time domain simulations
The extreme quasi-static and dynamic global responses have also been determined by time domain simulations
50
T a b l e 3
m p m e values o f the base shear at the sea bed a n d the o v e r t u m i n g m o m e n t f r o m a x i a l forces i n the v e r t i c a l piles
E q u i v a l e n t quasi-static d e s i g n w a v e E q u i v a l e n t quasi-static d e s i g n w a v e procedure u s i n g g l o b a l responses procedure u s i n g l e v e l shears
C o n s t r a i n e d r a n d o m t i m e d o m a i n s i m u l a t i o n s
Base shear at seabed m p m e 5qua ( M N ) m p m e Siy„ ( M N )
Moment from axial forces in the piles m p m e Afq„j ( G N m ) m p m e A/d,„ ( G N m ) 23.7 28.5 9.31 10.5 23.7 28.5 9.31 10.8 21.3 28.4 8.11 10.2
using the NIRWANA program [8]. A recently developed technique based on constrained random time domain simu-lations has been adopted to estimate the extreme stmctural response in a given sea state with relatively small simulation effort. This technique [9] is transparent and provides the whole statistical structure of the extreme response (being quasi-static or dynamic). It should be noted that this technique is not based on the extrapolation of individual maxima of response time series and provides complete statistical information rather than an estimate of one measure of central tendency only.
Wave kinematics were determined using linear wave the-ory in association with the well-known Wheeler stretching model [10]. In Table 3, the numerical results for the mpme base shear and moment are presented using the three approaches, both statically and dynamically.
Table 3 demonstrates that the mpme of the global responses can indeed be estimated using the equivalent design wave procedure in the manner described in Section 5. The relatively small differences in the estimates are the result of differences in the modelling of the wave kinematics (Stokes theory vs. Linear wave theory plus Wheeler stretch-ing) in the two programs. This good resemblance is of course not really a surprise since the whole procedure is based on supplementing the quasi-static design wave values of base shear and overtuming moment with the additional 'inertial response' reflecting the effects of dynamics. However, the good agreement is an indication that for cases like this linearised frequency domain calcula-tions to determine the 'inertial load set' are not a severe limitation to the application of the method, even though the deterministic design wave calculations are highly drag dominated.
The cumulative distribution of the applied design wave and current loading from time domain simulations is com-paied to the applied loading from the equivalent quasi-static design wave procedure in Fig. 10a. The moments are com-pared in Fig. I l a . As can be seen there is a good match between the two. The differences which can be observed are a direct consequence of the different wave kinematics formulations in SESAM (periodic design wave) and NIR-W A N A (random time domain simulations), as already referred to above. These differences manifest themselves
in the region above - 200 m below M W L ; the applied forces at lower elevations are dominated by current drag and virtually equal in the two procedures. The conclusion from Fig. 10a and Fig. 1 l a is therefore that the design wave loading correctiy represents the mpme applied loading due to wave and current in a random sea.
The results for the dynamic shears and moments at each elevation, as presented in Fig. 10b and Fig. 1 lb, are mark-edly different in the top part of the stracture when the 'iner-tial load set' is determined using either global responses or level shears. This is of course a reflection of the differences shown in Fig. 9. Comparing these results with the reference case of time domain simulations, also shown in Fig. 10b and Fig. l i b , it is clear that only an inertial load set based on level shears is able to represent the rather complex dynamic behaviour ofthe CT correctly, especially in the upper part of the stmcture above the pile to frame connection. Below this point the results for the dynamic shears and moments result-ing from the two different 'inertial load sets' are in good agreement. The remaining differences here are for the most part due to the same differences in applied loading from the two programs that were discussed earlier.
It is thus apparent that more complex dynamic behaviour over the height of a stmcture cannot generally be repre-sented by an 'inertial load set' derived from global base shear and moment. To capture such dynamic behaviour adequately in an equivalent quasi-static design wave method the procedure of Section 5 should be applied direcdy to the level shears at a sufficientiy large number of elevations over the height of the structure. While the latter application goes more into detail than the first, the calculations are simply a repeat of the calculations that are required for the base shear anyway. In addition to the base shear the level shear forces are to be abstracted from the stractural analysis for the design sea state per-formed by SESAM in the frequency domain. The level shear frequency response functions are subsequently pro-cessed by spectral analysis in the same manner as the base shear frequency response functions. Finally, the results are evaluated in the spreadsheet manner of Table 2. Hence this more detailed application pf Section 5 does not really require a great deal more effort and provides clearly superior results.
J.H. Vugts et al. / Applied Ocean Research 20 (1998) 37-53 5 1
10. Conclusions and recommendations
70.7. Conclusions , (i) Load calculations using a deterministic design wave
correctly represent the magnitude and distribution of the applied wave and current loading on space frame structures in a random sea of design severity, as shown by comparison with time domain simulations.
(ii) Frequency domain calculations using a statistical linearisation technique and a simplified structural model can adequately determine the additional global base shear and overturning moment or, alternatively, the additional level shear forces resulting from dynamic response. This is so even i f the applied loading in deterministic design wave conditions is highly drag dominated. From the addi-tional global shear and moment or the addiaddi-tional level shear forces, an 'inertial load set' can be determined for use in an equivalent quasi-static design wave procedure.
(iii) Application of the equivalent quasi-static design wave procedure to a compliant tower is a demanding test for this method. Representation of dynamic effects by an 'inertial load set' based on global base shear and overturn-ing moment should not generally be expected to give satis-factory results for individual member checking over the full height of the structure. To achieve this latter goal the 'iner-tial load set' should be based on dynamic additions to the level shear forces.
10.2. Recommendation
To get a fuller understanding of the adequacy and limita-tions of the equivalent quasi-static design wave procedure and the two ways to determine the 'inertial load set', similar studies to that described in this paper should be performed for fixed structures with progressively increasing fundamen-tal natural periods.
Acknowledgements
The authors wish to thank Shell Intemational Exploration and Production BV for kindly making the conceptual design of the compliant tower available to them for this project. They also gratefully acknowledge SINTEF, Trondheim, for permission to use the program NIRWANA at the offshore group in Delft.
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