Sensitivity analyses for steel jacket offshore
platforms
S. SHYAM SUNDER and J. J. CONNOR
Department of' Chi! Euqineeriny, Massachusetts' institute of lechnoloqv, Canthridqe, Mass.
02139, USA
INTRODUC1'ION
This paper is concerned with the significant uncertainties and their influence on the predicted response of steel jacket offshore platforms. A simplified numerical model for carrying out sensitivity analyses is outlined and results for two operating platforms with natural periods of 1.15 and 1.73 seconds respectively under rigid foundation conditions are included. The sensitivity analysis is focused on assessing response to: (i) variations in wave height. (ii) uncertainties in wave period to be associated with wave height, (iii) choice of hydrodynamic force coefficients C and C0 particularly in the presence of marine growth. (iv) changes in deck mass and hysteretic structural damping. Detailed discussion of the various uncertainties is pre-sented first, followed by the numerical comparisons.
Spectral representation of wares
Sea aves are generated by several mechanisms. It has long been felt that wind generated waves occur most often, although one of the major findings of the Joint North Sea Wave Project (JONSWAP) experiment was that on average 6O-7O of the observed wave growth can be attributed to non-linear wave-wave interactions'. There is still a diversity of opinion in this respect and research efforts are currently being directed towards understand-ing the wave generation mechanisms more completely.
Owing to the lack of a generally accepted wave
generation model, empirical models which rely on re-corded observations of sea-state characteristics are ap-pliedTypica1lv. one uses records describing the variation of the wave surface elevation with time. Such records show irregular and random behaviour and this has led to a generally accepted view that a seaway can be properly modelled only as a stochastic process2.A study carried out by McClenan and Harris3 ques-tions the validity of such models. They found that aerial photos of coastal regions generally showed complex but well organized wave patterns. The use of the concept of a random sea was, in their view, the result of making most wave observations from too low an elevation, and thus with too limited a field of view. However, they do not preclude the possibility that storm waves, which could not be photographed because of low cloud cover, are more chaotic and less well organized.
Most wave observations, nowadays, consist of con-tinuous records describing completely the wave motion at the point of obser'ation. but with no information on the wave direction. As a conservative estimate, therefore. all the waves are assumed to propagate in the same direction. 1f one assumes that the irregular and random observed records represent this unidirectional wave flow, the use of
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stochastic models is justified. in the following, attention is restricted to the one-d i inensional spectral representation of waves.
Many parametrized wave spectra have been proposed in the literature and it is usual for different spectra to produce diffèrent dynamic structural responses. Typically, these spectra are expressed in terms of short-term sea-state descriptors such as a characteristic wave height. characteristic wave period, etc. The Pierson-Moskowitz spectrum, used in this study. is defined in terms of the significant wave height, H, and zero crossing
period, Tz,4'5:
Gqq(W) = exp( - B/w4) (w? O) (1)
where
B=16z3T'4
(2)The units for H5 are arbitrary. Angelides4 describes some of the other spectra and traces the genealogy of the different representations. Ochi and Bales6 have studied the effect of various spectral formulations in predicting responses of marine vehicles and ocean structures.
Although the use of the sea spectrum in engineering problems is well established2, it is important to recognize its scope and limitations. One of the fundamental bases for the spectral representation is that the phase angles are independent random variables. Any harmonic coupling between them would suggest non-linearity. The JONSWAP study concluded that in no case did a strongly significant coupling occur'. These results immediately question the use of Stokes theory (or other high order theory) for describing the wave particle kinematics since it assumes a coupling of frequencies.
The JONSWAP results appear. however, to conflict with the views of St. Denis who has commented2 on the validity of the different sea-state spectrum representations due to non-linearities of the sea system. St. Denis noted that when the sea-state is light, it can validly be described in principle by the spectral distribution of its energy or variance, but that empirical formulations of the shape of the spectrum are not fully satisfactory although there is a strong convergence towards a definite basic shape. When the sea grows to moderate intensity, the variance spect-rum begins to lOSe validity owing to the appearance of asymmetries which manifest themselves as cusps and haunches and white-caps and breakers, but adjustments can be made so that it remains usable, although with lessened confidence. When the intensity of the sea rises to
014j- I 187181/010013-14S2.00
© 1981 CML Publications Applied Ocean Research, 1981, Vol. 3, No. ¡ 13
E
c'n.itirit analysis Jr steel jacket ojj.shore plaifr;ìis: S. Shvam Sunder and J. J. Connor Ware heiqht and liare period
For carryin out sensitivity analyses using the Pierson-Moskowitz spectrum. the wave environment is characte-rized b two parameters: significant wave height and
mean zero-crossing period. Although the significant wave height is a random variable, generally considered to follow the three-parameter Weibull distribution9, for
2 1 purposes of this study it was decided to select five discrete
i values (4. 12.
20, 28 and 36
ft respectively) which2 1 adequately cover the range of sea states that an offshore
1
1 platform might normally encounter.
2 There is considerable uncertainty in the choice of wave
period to be associated with wave height since the
1 2 relationship between the two is stochastic rather than
1 3 1 1 deterministic. Draper and Squire present a scatter diag-ram (Fig. 1) of significant wave height rersus T for the
3 North Atlantic which clearly demonstrates the stochastic
interdependence of H5 and T71 0 Houmb and Overik have proposed the following two-parameter conditional Weibull distribution11 (based on 3925 samples of in-strumental data from Utsira):
2 1,3 1
1/
1 155
1 2 6,5 9 3 6 2/4,14,,716 4 1/4,4,13 9 10 10 8 8 8 19_14 15\
5 I 2/2 4 131824 22\'6 8 1' 2,/2'2(A'24 JI2 Is iI/f
'5)( 5,' 15 3923 422013 2 7 (24'04''4<2"6 1 4 I711(5 11 11 1
11 22
1 412471
4 3_3 1 211 2 12 1 1 1 7-0 9-0 110 13-0 15-0Zero- crossing period 7 sec)
Figure 1. Scatter diagram of significant ware height and ?ìieail :ero-crossing period for a whole year in the North
,4tlantic'°
such a level that it manifests an appearance of mountains in turmoil, no theory, linear or quasi-linear, is adequate for its description, and clearly, therefore, the variance spectrum cannot be depended upon to describe the sea. Heavy seawavs, in his opinion. can be regarded only as the surface manifestation of turbulence, a phenomenon of considerably greater complexity than that of the
pro-pagation ofseaways of lesser intensity. In such a situation. the prediction of the extreme characteristics of the sea waves will suffer from serious uncertainty.
A second problem arises due to the linearization of the spectral analysis process. since the method is valid only when linear superposition is applicable. The implication is that non-linear loads, such as drag forces, must be small
in comparison to the linear loads such as inertia forces, and that the offshore structufoundation system re-sponds linearly. Hogben8 roughly defines the various loading regimes for the predominance of individual forces such as drag. inertia and lift by relating them to two ratios: D/U and ir K, where D is a member diameter, W is an orbit width parameter. and K is the Keulegan- Carpenter number. The definitions for W and K are:
where H is wave height. h is \vater depth. k is the wave number. U is the fluid particle velocity and T is the zero-crossing period. The regimes are defined as follows:
14 Applied Ocean Research, /98/. Vol. 3, No. I
f T
P(TH5)= I_ex[
T4H5))]
(5) with TH5)= 6.05 exp (0.07H5) (6) y(H5)= 2.35 exp (0.21H5) (H5 in m) (7)Owing to the lack of reliable and adequate wave data, such long-term conditional distributions are not com-monly available and one resorts to the use of a de-terministic relationship between the two variables. Figure 2 shows a relationship between significant wave height and characteristic wave period proposed by Wiegel in 1961 and recently updated'21. An analytical approxi-mation is H5=0.378 D788 (H5 in ft) (8) loo 50 10 o-5 0l
Figure 2. Characteristic itare period versus signUicant D/W it/K >0.2 inertia increasingly important tiare liehj/it 13, North Sea: O, T. :ero crossing; x , T, D,- W it/K <0.6 incipience of lift and drag spectral peak: A, T. :cro crossing. Gulf of Mexico, T,
D/Wir/K <0.2
drag increasingly predominant aig: . hurricane Esther: hurricane A iidreyCl 10 100 N, (feel) 48 40 o o 2: 32 .0 o o > 24 C ci i') 16 H tanh kh (3)
K=T
(4)Table 1. Values of mean :ero-crossing period to be associated with
specified sign ficant waue heights
where T can be selected from the several characteristic periods represented in Fig. 2.
For purposes of sensitivity analyses, five values of T have been selected for each value of significant wave height. These correspond to the upper extreme of wave period for any given H from Draper and Squire's scatter diagram. T5: periods corresponding to a cumulative probability of 0.05 and 0.95, and the average of the mean and median from Houmb and Overik's expression, 7 T095, and Tuo: and finally, the period given by Wiegel's relation, T.,.,. The periods are listed in Table 1.
Estimation offluid loading using Ivíorison's equation In a realistic response assessment of offshore structures it is necessary to identify and improve our understanding of the assumptions and uncertainties that go into the prediction of fluid loading, given the description of the wave and current environment. Hogben et al. emphasize in their state-of-the-art paper that although data on fluid loading are plentiful, there are still many serious un-certainties and gaps in knowledge14.
Severa! studies have shown that the mathematical form
of (he Morison equation, which is
widely used for calculating wave loads, is satisfactory but the difficulty with its application to offshore structures has been the choice, from a wide range of published values, of the empirical coefficients C.,, and CD appropriate to both thestructure and its design sea states'4. The least well
understood wave loading regime, hence the one with the least accurate description, is the regime wherein both drag and inertia forces are important'5. Unfortunately, this is the regime in which Morison's equation is generally assumed to be applicable. As an approach, Hogben et al. recommend using average values of C.,, and CD withfunctional dependence on
Reynolds number and
Keulegan-Carpenter number' °. The Keulegan-Carpenter number represents the relative importance of drag over inertia forces and can be shown to be pro-portional to their ratio.
Ramberg and Niedzwecki point out that there are uncertainties in the estimation of C.,, and CD from
measured force records of experiments and tests'5.
Different estimation techniques will produce different pairs of coefficients for the same force record unless the wave force is identically given by Morison's equation. This identity rarely occurs even in one-dimensional oscillating flow'6. Another uncertainty lies in the ac-curacy with which the phase between the wave force and the wave cycle is known. Incorrect phasing has the effect of trading one force component for another.Effects in the real sea situation, such as orbital motion of the fluid particles, directional properties of the waves and orientation of the orbits with respect to the axis of the
.* &?ML¼ LZVCZV
Sensitit'itv analysis for steel jacket ojTshore plan brins: S. Sh1a,n Sunder and J. J. Comior structural member are not normally accounted for. Sarpkaya's one-dimensional experiments have also poin-ted to the importance of the vortex-generapoin-ted lift or side force which is not included in Morison's equatton and can render the local peak wave force much larger than that predicted by Morison'sequation'7'8. However, the total wave forces obtained by integrating over the member leneths are far more reliable than the local forces'4.
Extensive tests with smooth and rough cylinders sub-jected to flow in wind tunnels and to waves in one of the National Maritime Institute (UK) ship tanks have pro-vided strong evidence that roughness comparable with the expected level for North Sea structures can cause large increases (1500 or more locally) in drag coefficient at high Reynolds number. This results in a loading increase which is
greater than the increment corresponding to the
increased diameter'4. Experimental work also shows that with the rise ir drag coefficient, there is a decrease in the inertia coefficient'9. It should be noted that experimental programs have studied only the effects of hard fouling or rigid marine growths caused by rust, scale, barnacles and mussels. The effects of soft fouling or flexible growths1 caused by seaweed and sea anemones on CD remain unquantifiedThe values of C.,, and C0 as functions of Reynolds number, relative roughness (ratio of roughness height to cylinder diameter) and Keulegan-Carpenter number pre-dicted by Sarpkaya et al's work2° seem to be the most appropriate data for predicting the effect of different heights of surface roughness on C.,, and C0 and the corresponding hydrodynamic force'9. Figure 3 shows the variation of C.,, and CD with Reynolds number and relative roughness for a Keulegan-Carpenter number equal to 30. The fact that Sarpkaya's data were obtained in two-dimensional harmonic flow suggests that such an approach would lead to a conservative design. Sarpk aya does not report the cycle-to-cycle variations in C.,, andC0. At low Reynolds numbers, according to Hogben en al. the
1-9 1-7 1-S 1-3 1-1 0-9 07 0-5 20 '8 16 E 1-4 1-2
App!ied Ocean Research, 1981, Vol. 3, No. 1 15
i I H, (fi) 4 12 20 28 36 T, 3.74 5.87 6.92 7.23 9.20 8.82 11.11 10.66 12.79 12.80 Too5 2.48 4.35 6.52 8.91 11.50 T095 9.46 9.71 10.55 11.88 13.66 TDS 11.25 12.75 12.75 13.75 13.75 10 01 0-5 1 5 8 Re 10
Figure 3. Sarpkaya's curves20 for CM and CD
01 O-5
Sensiriritv analysis for steel jacket offshore platforms: S. Shyam Sunder and J. J. Connor
NN
UN N-1 z 3,'3 _,U2 e1 U1 k57 .-OE9
model aVows for coiphrig foundation impedance(k1.ci,)c iystecetic soil damping
a b
Figure 4. System idealization: (a) idealized structure (b)
real structure
wave force coefficients depart significantly from a mean value over a wave cycle only for KeuleganCarpenter numbers in the region of 15.
The present study utilizes Sarpkaya's results to assess the sensitivity of response of an offshore platform to constantversusflow dependent hydrodynamic force coef-licients and to variations in relative surface roughness. However, one should keep in mind that there remains some doubt as to the direct applicability of Sarpkaya's results to wave flows. Heaf has shown that the modifi-cation of the dynamic response of the offshore structure by the increased mass and added mass arising from marine growth is not so significant as the direct increase in loading due to the same thickness of marine growth'9. These effects have been ignored in this study.
Hogben et al. report that the presence of a current seems to influence CD and not C11 Dairymple's con-servative approach for deep water calculates the drag force by adding vectorially the current velocity to the wave particle velocity (for a current approximately in the wave direction) both for the definition of Reynolds number and for use in Morison's equation21. As waves are rarely steeper than H/).=0.1 in the open ocean (... being wave length). slamming due to impulsive pressures caused
b impact between the members and the free surface is
predominantly a vertical force. However, under breaking wave conditions horizontal slamming is significant14. The effects due to current, slamming and impulsive buoyance (sudden application of buoyancy loads as the member is submerged) have also been neglected in the sensitivity analyses.
Finally, it remains to select an appropriate wave theory to describe the fluid particle kinematics. Dean has dis-cussed the relative validities of water wave theories22. It has been found that in deep water, prediction of loading using Airy theory but with integration of the forces up to the actual water surface (not as in strict linear theory to still water level) give results which do not differ greatly from prediction based on Stoke's fifth order theory'4. The use of linear wave theory with integration of forces up to still water level is usual for carrying out frequency domain analysis, and has been adopted in this study.
16 Applied Ocean Research, 1981, Vol. 3, No. I
THEORETICAL MODEL AND COMPUTATIONAL SCHEME
A simplified frequency domain model for the analysis of a
fixed
offshore platform has been developed at the
Massachusetts Institute of Technology423. A brief de-scription of the model outlined in the references is given here, highlighting various simplifications and assump-tions that have been made.
Structural idealization
The platform is modelled as an 'equivalent planar beam' with lumped masses for the dynamic response. The structural geometry and pile configuration are assumed to be doubly symmetric with one of the symmetry axes coinciding with the direction of wave propagation. An approximate two-dimensional model is generated by lumping the masses at the various panel nodes and simulating the stiffness with an equivalent planar beam. Figure 4 illustrates this discretization. There are two
displacement measures per node, the horizontal trans-lation and the rotation of the equivalent beam cross-section. Rotatory inertia is neglected.
The tower, a complex assemblage of tubular elements is first modelled as a pin-jointed truss. This approximation neglects the rotational restraint at the joints. Rigidity coefficients for the equivalent beam are then generated with a complementary energy argument. One applies self-equilibrating force systems to a typical panel, evaluates the complementary energy and then by differentiating with respect to the shear force and bending mbment obtains the flexural and shear rigidities. Equivalent flexural and transverse shear rigidities for X and K bracing systems are listed below (Fig. 5):
X-brace (EI)4 = h2AE 1h2 (GA)eq =1J-2AdEd K-brace (E!)eq=/z2AE
(i\
12 h t 2L3GA)eq = 2h2
AE
+¡ 4AhEh+lh2AdEdThese results are based on perpendicular sides, i.e. the outside verticals are perpendicular to the horizontal
k h f Ad.Ed Ac.Ec a Ah Eh Ad. Ed A. E b
Figure . Bracing systems
piuiiuiir.
members. This approximation has been found to yield acceptable resu Its for structu res normally encountered.
The illustrated derivation of the equivalent rigidities is based on only one frame in planar motion parallel to the flow direction. The total equivalent rigidity coefficients are obtained by summing the contributions of all the frames in the structure. Inherent in this structural model is the assumption that the displacement of all parallel frames at each panel level are identical.
Equations of 1fb! ion
The equations of motion for the system are expressed as
MU + CÛ+ KU = P (13)
Vector U contains the nodal displacements,
1T_1..a ..
..fl)
(14)where u and O. are the horizontal translation and rotation for node i.
The stiffness matrix K is generally complex: the real part, KR, contains the stiffness associated with the struc-ture and the soil, and the imaginary part. K1, contains the hysteretic damping associated with the structure and also with the soil. The soil stiffness is represented by means of
three springs having stiffnesses k and The
hysteretic damping is written as:
K,=1m K=±(2D5K1+K2) (15)
where K1 is the stiffness matrix of the structure, D5 is the hysteretic damping coefficient for the structure, and K, contains the hysteretic damping terms c, c and c for the soil. They form a 2 x 2 submatrix in K5: the remaining terms in this matrix are zeros. The choice of a plus or minus sign is governed by the form of the proposed solution. For hydrodynamic loading, it is convenient to assume a solution which is in phase with and proportional
to the term exp[i(wt+P+)]. Since the hysteretic
damping must be in phase with the velocity and pro-portional to the displacements, a minus sign is required.
The matrix C represents the viscous damping of the structure. This may be expressed by a Rayleigh type damping:
C=1K1+2M+3Ma
(16)where M is the lumped mass matrix of the structure and Ma is the added mass matrix of the structure, to be described later in this section.
Hydrodynarn ic forcing
Hydrodynamic forcing is evaluated on the original three-dimensional structure applying a modified form of Morison's equation which attempts to account for re-lative motion between the fluid and structure. The forces are lumped at the nodes. The equations derived for evaluating the wave forces are based on certain idealiz-ations. Although the velocity of the fluid decays non-linearly with depth (exponentially in the case of deep water conditions), the variation is considered to be step wise; the value at a node level is assumed constant over the tributary segments extending to the mid-points of ad-jacent panels. The force on each member is calculated with
Sc'nsitirity analysis for steel jacket o/fchore platforms: S. Slirani Sunder and J. J. Connor the undisturbed' fluid velocity assuming that there is no interference to the fluid flow due to closeness of the
members. There are no published data on flow
in-terference in waves, although certain thumb rules do exist l4 Only the wave particle motion in the x-direction is considered, assuming no variation in particle motion in the other horizontal direction. Vave forces in thedirection are not considered as they do not excite
horizontal motions. Members that are inclined in the direction of flow present a problem since Morison's equation yields the force only on a vertical member. To get over this problem. the vertical projected length of inclined cylinders are used for calculating the force. Forces on members which lie in the direction of the wave are neglected. since they contribute little to forces in the direction of flow. The net moment created by the forces about a node level is neglected, as it will be relatively small.The wave force for node i is the sum of the forces for the members j of the real structure which lie in the tributary segment associated with the node. Using (ij) to denote such combinations, and applying Morison's equation, the nodal hydrodnamic force, F, is given by:
F =
[PC.i_
I - ii1) + ±- ù1I(r1 -ib)] (17)
where C, CD, are the inertia and drag coefficients which are, in general, dependent upon the flow and structural response characteristics: D.1, l are the diameter and
projected length of the member (iI); i and r are the water particle acceleration and velocity, iand û1 are structural
acceleration and velocity at level i.
The last group of terms in equation (17) is non-linear. A linearization of these terms is required for carrying out spectral analysis which employs a linear superposition over the frequencies describing the sea-state. The assump-tion of staassump-tionarity is implicit in the use of the spectral techniques. Although the sea-state is not stationary, for short-intervals of time lasting one to several hours, the free surface elevation, !7(x.t), from the still water level can be approximated as a stationary process2'26. If the excitation is a zero-mean. Gaussian process and the linearized forcing applies to a linear structure (iteratively linear soil), the output u1 and the relative velocity r= v
-
are also Gaussian with zero mean. Minimizing the error of the linearized form in a least square sense27'28 results in:r1, - ù1I(v - i) = ./8/ir (r - t,c (18)
where o, = root mean square relative velocity.
For a stationary random process, the probability
distribution of the instantaneous realizations is time-invariant and often symmetric about the mean. In the case of waves, for which the mean is likely to be zero, as assumed, it is convenient to take this distribution as Gaussian. However, as pointed out earlier, asymmetries are introduced under moderate and severe sea-states, and the Gaussian assumption is not valid. Such asymmetries have actually been noted by Kinsman. Longuet-Higgins and others. For practical purposes, one might retain theSensitivity analysis for steel jacket offshore platforms: S. Shvam Sunder and J. J. Connor normality assumption based on the following
justifi-cations offered by Norderistrom: (i) the departures from the normal distribution found by Kinsman and others
were not ver
large, and (ii) the influence of suchdepartures on the response of marine systems is not known26. The latter justilication is really not appropriate. However, results of the North Sea Environmental Study Group also support the Gaussian hypothesis29.
The least squares linearization of the flexible cylinder Morison wave force introduces uncertainty in the repre-sentation of the drag force component. Thelinearization introduces conservatism in the drag component when water particle velocities are less
than \/7
1.6 times the root mean square velocityvalues (which correspond to a88 probability) but underestimates it for higher
velo-cities. The magnitude of the uncertainty depends on the importance of the drag component. the CDemployed in the analysis and increases with
the intensity of the
stationary sea-state in question. In addition, regular drag force linearization methods currently in use like the one above, underestimate high frequency force components on the structure which are associated with structural resonance and therefore important to the fatigue design of the structure30. A typical difference between predicted and observed force spectral density functions is that there are local maxima in the force spectral density function at frequencies twice and three times the peak frequency of the wave spectral density function which are not predicted by the linearized theory. In most cases, the natural period of lixed structures islocated in the range of these
frequency peaks and the structure can be moreseverelyexcited at the higher frequency.
Now, rearranging equation (17) with the objective of separating response dependent and flow dependent terms, and introducing the definitions:
results in
= 1)D,l1
F = - M0ü1 - CÑ1 + F.
F. = p +
18 Applied Ocean Research, 1981, Vol. 3, No. /
P = random phase angle uniformly distributed between 0 and 2ir, gr=gravitational acceleration, h=depth to still water level. :, x,j refer to the coordinate system fixed at the still water level. Note that both B,, and C.,, are independent of time. Phase shift due to the finite width of the structure is accounted for in the terms.
The hydrodynamic drag and inertia coefficients may either be assumed constant for all members or adapted according to Sarpkaya's results. The Reynolds number and KeuleganCarpenter number are defined for wave flows in the following manner:
The terms M and C4 are called
added mass and hydrodynamic drag damping respectively. They lead to two diagonal matrices M and C with the above terms occupying the diagonal positions associated with the translational degrees of freedom.Finally, substituting for F in the system force vector,
P={F, O. F'2, O,..., F, O} (23)
(M+M0)U+(C+Cd)U+KU=F
(24)F=F1, 0, F2, 0,..., FN, 0} (25) Next, the orbital wave velocity and acceleration for member (ij) are expressed as functions of the one-sided wave spectral density function of the free surface elev-ation, Gq(W). with the w axis discretized into M segments, Aw. The resulting expression for force using linear wave
theory is:
-F1=>AF1 sin(wt+tP+yj
(26) where = 2G,,,(w)Aw = nAwF=B+C
tan,Lfl = C,.IBÌB. =[R cos(k,,x1)R
sin (kx1)] C,,, = >[RJ, sin (kx) + R cos (k,,x1j)]R=
R w,=gk tanh k,,h G-
cosh k,,(z + h) sinh k,,h (27), = KeuleganCarpenter number (T0)ZJ (29)where y is the kinematic viscosity and (T0) is the mean zero-crossing period of relative velocity on member (if).
(T0)=2t
(nz) (30)2/i
pCDDJlJ\/8/7r cr,.v1 (22) (Re) = Reynolds number =
(28)
and noting equation (21). the equations of motion are (n13)jJ = (31)
A five parameter expansion was fitted to each of
Sarpkaya's curves. i.e., for the five values of relative roughness indicated in Fig. 3 and Keulegan-Carpenter numbers 20, 30, 40. 60 and 100. Interpolation is used to estimate the parameters corresponding to intermediate values of relative roughness and K. Forward extrapo-lation based on the information at K =60. loo is used forK greater than 100. At the lower end. 15<K<20,
information at K =20, 30 is used for backward extrapo-lation. After the five parameters consistent with the Keulegan-Carpenter number (in the range 15) and roughness are obtained. C, and CD are calculated from the expansion in terms of Reynolds number. Evaluation of C1 and CO3 for K<15 is based on interpolation betweenC,
=2.0, CD=O at K=O and the values of CAt and C0 for the given Reynolds number at K=15.Solution strategy
Since the system is linearized, the nodal displacements can be expressed as a superposition of solutions cor-responding to the M discrete frequencies.
u
Im{AnUne(_'"}
till,,= {CJ,,} ={zt,,e'tln, O,,,e°ln...uNfle,,, ON,,e"Nn} (34)
where tilJ is determined from:
[K - iw,,(C + Cd) - w(M + Ma)] n =
F= {F1,,, 0, F2,,, O...FN,,, 0}
F =
then
F1 = F1,,e''in
Iteration on the hydrodynamic coefficients and li-nearized drag term requires the spectral density function for the relative velocity. Since from linear wave theory
The spectral density functions follow directly from equa-tions (47) and (48).
In the present study the foundation has been assumed rigid. For a discussion of flexible foundation modelling and solution strategy refer to Angelides4 and Arigelides and Connor31.
tan G.,, sin k,,x + zi,, sinß,,
- G,, cos k,,x, + u1,, cosß,, (42) Noting that P,, is random and uniformly distributed, the spectral density function reduces to:
G,(w,,)= (43)
The base shear and moment are evaluated with the stiffness coefficients for the equivalent beam. The general expressions are:
F =Jm{(l +2Di)[SF(u, -"2) + S,O, +SFß2]}
M'= ¡m[(l+2D5i)[S,(u1 - u2)+ S,,01 + SO2]}
where D is the hysteretic structural damping coefficient and Se.,, S represent the equivalent beam stiffness
coefficients which are taken here as: 6E1*
SF,=SF,= 2
12E1*
12/EI \
a=T4-A)
6E1* Ei* EI*
(33) SA,=
-2
S,,=(4+a)
¡S=(2-a)
(46) Il+a
Substituting for the complex nodal translation and ro-tation measures defined by equation (33), and taking the imaginary part of the resulting expressions, one obtains the general form:
Computational procedure
Initially a zero response is assumed and(T,
and (l), are
calculated with the relations presented already with the terms associated with the response dropped. Next, the coefficients C,,. C0 (if Sarpkaya's curves are used), the hydrodynamic drag damping Cd and the forcing vector are determined. The equations of motion are then solved M times and the response is generated. Updated values of
(T0), (now functions of the response) are used to
etermine new estimates for Cs,, C0 and the damping coefficients. In each iteration the new value of a, is checked for convergence. Iteration is terminated when the following criterion is satisfied:
Applied Ocean Research, 198/, Vol. 3, No. 1 19
i
v4t)
= cos( -w,,t+
one obtains using equation (33):
+ P,,) (39)
-
= A,,w,,H,, cos( - wt + ± P,,) where(40)
H,2,,, =G,, + ut,, + 2G,,u,, cos (k,,x1 ß,,) (41)
The spectral density function for displacement at node ni F = A,,F8,, sin( -w,,t+ P,, ± y,,) (47) is G(w,,)=(A,,u,,,,)2/2Aw. If the force F., equation (26), is
written in complex exponential form as:
M
=
ni
A,,MB, sin( - w,,t + 'I',, + ji,,) (48) 1IIu uy ucuiiwuii(32) (T, =(tu0)
Sunder and J. J. Connor Sc'nsitirity analysis for steel jacket off thore platforms: S. Shvani
Sensitirity analJsis for steel jacket offshore plarforns: S. Shyan Sunder and J. J. Connor 50-6 202 5 ? 3 729
:kVÁ
VA
(a,)2(a)2
42 5 bFigure 6. General characteristics of platform A. (i) Frames A and D are identical. Similarly frames B and C are identical. (ii) Dimensions on the left in (b) correspond fo frames A and D. whereas those on the right correspond fo frames B and C. Deck mnass= 19,118 kips
z=max all
.120 Applied Ocean Research, 1981, Vol. 3, No. /
(49)
where the prime denotes the current value. Although iteration is required, the process converges rapidly.
PRESENTATION AND DISCUSSION OF RESULTS FROM SENSITIVITY ANALYSIS
Two steel jacket platforms, A and B. have been analysed using the computer program POSEIDON developed for the model outlined in the earlier section. Platform A isa
North Sea type structure of 404 ft height and lias a fundamental period of 1.73 sec under rigid foundation conditions. The water depth is 319.5 ft. Platform B is a
smaller structure of 145 ft height and has a period of 1.15 sec. The water depth for this structure is 104 ft. The salient features of both structures are indicated in Figs. ô and 7. In the following, results of the sensitivity analyses carried out for the cases indicated in the introductory discussion are presented. A brief discussion of the results for each case is also included.
Influence of ware height on response
Platform A is subjected to the selected significant wave heights with zero crossing periods being given by Wiegel's relation. i.e., T (Table 1). Constant values of C.1= 2 and C13= 1.4 are used in the analysis. The response quantities of interest include the top node displacement, the base shear and moment along with their average zero-crossing period, and the root-mean-square relative velocity. Table 2 gives a summary of the results.
A number of interesting conclusions may he arrived at. First, the rms dispacements are ver small in all cases. including the case where H5 = 36 ft which corresponds to
an extreme wave of 72 ft. Figures 8 and 9 show the Ta spectrum and the displacement spectrum for the two extreme cases. H5 = 4 and H9 = 36 ft. The location of the natural frequency of the platform on the frequency axis has also been indicated. It is clear from the response spectral density function that there is no dynamic amplifi-cation and that, in fact, a quasi-static analysis procedure would have yielded identical results. The multiple peak response spectral density of Fig. 8 requires some expla-nation. The zeros (or valleys) in the spectrum are due to the occurrence of out-of-phase harmonics in the load effects. While the overall (global) displacement response prediction is always accurate, in some cases, the true wave-induced stress state may be underestimated. This corresponds to the condition where certain force har-monics are associated with wave lengths which are whole number fractions of the distance separating column planes of the platform. This effect is a function of the
o
,154 J,,
kA
L.. 165
1
Figure 7. General characteristics of platform B. Platform is square in plan. doubly symmetric and lias similar properties about both symmetry axes. Deck mass = 1434 kips
Table 2. Influence of ware height on response for platform A H9 (ft)
Top node dis-placement. c (in) Base shear, (kips) Base moment, 0M kip-ItX
i0)
Base force period, i, (sec) RMS rei, ve!. just above SWI (mis) RMS rei, el just below SWL (injs) 4 12 20 28 36 0.008 0.060 0.118 0.183 0.266 12.0 I 11.5 242.3 394.8 591.4 3.5 27.3 55.2 86.9 127.4 4.7 9.1 II.! 12.8 14.4 0.0 IO 0.040 0.064 0.087 0.112 3.26 15.12 24.84 33.34 41.48200-0
0-16
800
40-0
0-0
Figure 8. Excitation and response spectral densitvfunctions for platform A (H = 4ft, L = 3.74 sec). , Response;
Sensitivity analysis for steel jacket offs/wre platforms: S. Shyam Sunder and J. J. Connor
S, FurrdarnentaJ natural
s'
s, frequency of Platt or A
shear and base moment for a given structure. Figure 10 plots h = aS,'aF, versus , where Ii is a measure of the location of centre of force. Such relations are useful in developing simple models for assessing stiffness de-gradation of soil, etc.431. The base force period listed in Table 2 is the average of mean zero-crossing periods computed from the base shear and base moment spectral density functions. In all cases, the base force period is larger than the correspondng wave zero-crossing periods, T1 This occurs because the high frequency components of excitation, and hence response. attenuate faster with depth. In addition, the high frequency wave energy was observed lo contribute little to force.
Third. the motion of the structure is negligible com-pared to the fluid motion, and this reduces the uncertainty associated with the use of equation (17). This is discernible from the last two lines of Table 2. The RMS relative velocities listed in the Table correspond to nodes just above and below the still water level and to a position at the middle of the structure along the wave propagation direction. It was observed that the variation in RMS velocities for different positions along the wave pro-pagation diretion is negligible. It should be noted that the RMS relative velocity for the node above still water
level
is equal to the RMS structural node velocity.
However, the conclusion arrived at here does not preclude the possibility that, for more flexible platforms, structural motion might not be negligible.
Effect of uncertainty in wave period ro be associated with wave height
Platform A has been subjected to the different H, T combinations indicated in Table I. Table 3 lists the ratio of period used to Wiegel's period, TH-, for different values of significant wave height.
300
275
250
C
225
00 O-20 O-40 O-60 080 1-00
Freqoerrc-y (Hz)
Figure 9. Excitation and response spectral densityfunctions
for platform A (H=36 ft. T= 12.79
sec).Response; - - -, wave spectrum
distance between major column planes of the structure and the bandwidth of the wave spectral density function. Therefore, it is more prominent in the response to the lover to moderate sea -states, which are associated with smaller wave lengths. In order to accurately account for the out-of-phase load effects, a more complex structural model should be used, such that the behaviour of the major column planes can be studied separately. Similar conclusions apply to the base shear and base moment spectral density functions.
Second, there is a definite relationship between base
200
-0 100 200 300 400
crF )KIPS)
Figure 10. Plot of h versus crfor platform A
500 600
Table 3. Mean zero-crossing period as a ratio to Wiegets raiue
700
Applied Ocean Research, 1981, Vol. 3, No. ¡ 21
t
---,
wave spectrum 480-0 O-30 400 O-25 -320-0 -0 20 o E E 6 240-0 -0-15 V o 160-0 - O-10 80-0 - - O-05 Fundarr,errtal natural A O-O - 00 H (ft) Period used 4 12 20 28 36 7-w 1.00 1.00 1.00 1.00 ¡.00 .T10 1.57 ¡.05 0.96 0.96 1.00 T005 0.66 0.63 0.71 0.80 - 0.90 T095 153 1.40 1.15 ¡.07 1.07 T05 3.01 1.84 1.39 L24 1.08 0 80 00 0-20 0-40 0-60 Frequency (Hz) 0-24 0-20 0-12 E 0-08 0-04 00I
1 LSensitirit)' anal)'sis for st eel jacket offshore platforms: S. Shyam Sunder and J. J. Connor
Table 4. RMS top node displacement for tarious H. T pairs as a ratio ro that predicted by Wiegers period
H, (ft) T. 1.00 1.00 1.00 1.00 1.00 THO 2.02 1.03 1.00 1.00 1.00 T005 0.71 0.53 0.86 0.99 1.00 T095 2.50 1.09 1.00 1.00 0.99 T 2.42 1.02 0.97 0.97 0.99
Table 5. RMS base shear for carious H, T, pairs as a ratio to that
predicted by WiegePs period
H,(It) Period used 4 12 20 28 36 1.00 1.00 1.00 1.00 1.00 2.45 1.05 0.98 0.98 1.00 T005 0.67 0.43 0.75 0.91 0.96 T095 3.76 1.26 1.05 1.03 1.01 T 3.90 1.32 1.10 1.05 1.01
Table 6. RMS base moment for tarious H,, T, pairs as a ratio to that
predicted by Wirgers period
Tables 4-7 list the corresponding response ratios. The actual values of response for Wiegel's period have already been given in Table 2. Figures 11 and 12 represent the same information in plot form.
Perusal of the results show that uncertainty in T is large for small H5, and small for large H5. In addition,
un-certainty in response is positively correlated to the uncertainty in T. One notes that, in general, at large H,,
uncertainty in response is less than uncertainty in 7. At lowH5, uncertainty in top node response is almost equal to the uncertainty in period, whereas uncertainty in base shear and moment is larger. It is significant that the uncertainty in base force zero-crossing period, TFu is also
positively correlated to the uncertainty in L.
Therefore, for extreme sea states, period uncertainty does not appear to be critical. However, for such sea states it is the adequacy of the wave spectrum in representing the sea surface that is questionable. For lower sea states, which are potentially important for fatigue assessment, the uncertainty in force is appreciable. An increase in force is partially compensated for by a reduction in effective number of loading cycles since the base force zero-crossing period increases at the same time. It would be interesting at this stage to look at the conclusions of a study on the fatigue behaviour for North Sea conditions of a fictitious steel jacket platform with period 3.85 sec32. An extract from the report follows:
22 Applied Ocean Research, 1981. Vol. 3, No. ¡
-.,. .,..'I..
most of the cumulative fatisue damage comes from the higher sea states. The highest contribution corresponds to the largest slope (of the cumulative fatigue damage curve), which occurs at about a 23 feet significant wave height. This result contradicts the common idea that fatigue in offshore structuresisprimarily due to lower to moderate sea states. As
a matter of fact, the lowest six sea states considered, for which the degree of dynamic response is significant, do not contri-bute to the fatigue in the joint. Even though the probability of their occurrenceissmall, the higher sea states play the major
role in inducing the fatigue process at the structural joints.' It thus appears that uncertainty in period, L is unlikely to significantly influence design or fatigue life assessment. The response spectral density function forH5= 4 ft and T005= 2.48 sec requires some explanation for two reasons: (i) there is no second peak due to dynamic amplification even though there is significant wave energy at the natural frequency of the platform, and (ii) the quasi-static re-sponse is negligible above a frequency of
-0.4 Hz,
although there is significant wave energy in that region (Fig. 13). Equations (26) and (27) are the key to under-standing this phenomenon. In equation (26), the term Fi,, is essentially the force transfer function. It is observed that this transfer function becomes very small for f>0.4 Hz and thus there is no response whether quasi-static or dynamic.Influence of hydrodvnarnic force coefficients and marine
growth on response
In the first instance platform A was analysed for
constant cersus flow dependent hydrodynarnic
co-efficients using the various H,.T pairs identified through Wiegel's relation and a relative roughness of 0.02. The worst case, represented by relative roughness equal to
Table 7. Arerp ge base force :ero-crossing period for various H,, T, pairs as a ratio to that predicted by Wiegers period
2-3
o
Figure Il. Plot ofTiT,, versus signifIcant ware height. I,
Wiege!. T,,.; 2, Houmb and Orerik. T1i0; 3, Houmnb ((mid
Orerik, T005;4, Hounth a,i1 Oterik, 1095; ®,Draper and Squire,T1)5 H,(It) Period used 4 12 20 28 36 T, 1.00 1.00 1.00 1.00 1.00 T 2.17 1.03 0.99 1.00 1.00 T005 0.69 0.50 0.83 0.97 0.99 T095 2.85 1.13 1.01 1.01 0.99 T05 2.81 1.10 1.00 0.99 0.99 H, (fi) Period used 4 12 20 28 36 1.00 1.00 1.00 1.00 1.00 1.72 1.03 0.97 0.97 1.00 T005 0.77 0.64 0.79 0.85 0.92 T095 2.40 1.26 1.10 1.05 1.05 T05 2.69 1.52 1.27 1.17 1.05 Period used 4 12 20 28 36 io 20 30 H, (feet)
Figure 12. Response ratios for platform A. (a) RMS top node displacement: (b) RMS base shear; (e) RMS base moment: (d) average base force :ero-crossing period
012 Tablc 8. Response of platform A to variable hvdrodynamic coefficients with relative roughness 002
120-O 100.0 40-O 20-0 0-0 0-02 ',. Fundamental natural frequency of Iatforrn A
Sensitivity anal vsis for steel jacket offshore platforms: S. Shya,n Sunder and J. J. Connor
b O O-10 O-089 E 0-06
t
O-04 00 H, (ft)Top node dis-placement. o (in) Base shear, a (kips) Base moment, o (kip-ft x l0-) Base force period, (sec) RMS reL vel. just above SWL (mis) RMS rel. vel. just below SWL(in/s)
Reynolds number. C.,, and C0 in a convenient and systematic manner, the structure has been divided into four types of members by diameter:
GROUP 1 120">DIA>60" GROUP 2 60">DIA>40" GROUP 3 40">DIA>20" GROUP 4
20">DIA> 0"
and further divided into horizontal zones (counted from bottom) corresponding to each vertical bay i.e., the region including and between adjacent nodes. For each group and zone, an average member diameter has been calcu-lated and is given in Table 9. Table 10 lists the average values of K,Re.C,., and CD for each of the groups in all zones, for the case H5 = 36 ft. L = 12.79 sec. When H5 = 4 ft and T = 3.74 sec, the KeuleganCarpenter numbers are all
Applied Ocean Research, 1981, Vol. 3, No. 1 23
4 12 20 28 36 0.008 0.055 0.095 0.142 0.240 11.9 103.1 203.6 309.4 498.0 3.4 24.9 44.9 67.4 112.4 4.7 9.1 ILl 12.7 14_4 0010 0.036 0.052 0.068 0.100 3.26 15.12 24.84 33.33 41.46 O-20 040 0-60 O-80 1-00 Frequency (liz)
Figure 13. Excitation and response spectral density
functions for platform A (H5=4
fi;L=2.48 sec).
Response; ---, wave spectrum
0.02, was chosen to see if f here was any effect at all of flow dependent hydrodynamic coefficients on the structure. The usual upper bound design values C.,, = 2 and CD = 1.4 were used for the constant coefficients case. Table 2 gives a summary of results for the constant coefficients case and Table 8 gives results for the variable coefficients case. In order to present results for KeuleganCarpenter number,
8.0-O
E
Thhle 9. Analysis of arcraqt' member diameters in platform A Average diameter of Distribution of troups
members (in) within zones (") Group Group Group Group Group Group Group Group
24 Applied Ocean Research, 1981, Vol. 3, No. /
Table 10. Flow parameters andforce coefficients for platform A. Zone 6 is outside water. (H=36 ft. T= 12.79 sec, relatire roughness=0.02)
close to zero yielding CM2 and CDO for all members. The results show that the usual upper bound values of C.0=2 and C0= 1.4 give an upper bound response for all sea
states. Table 10 shows that
for this structure Keulegan-Carpenter numbers are all below 20, the lowest value for which Sarpkaya's data are available. The mean zero-crossing period of the wave, from the relative velocity spectrum, was found to increase with depth. This confirms the earlier finding that the high frequency components of excitation attenuate faster with depth from still water level.An interesting feature of the results is the variation of the hydrodynamic force coefficients with wave height, wave period and depth below surface. It is reported that. there is a trend for C0 to decrease with increasing wave
height and period at all depths below the surface33. Pierson and Holmes suggest that variation in wave force coetTicients with depth below the surface is significant, although results from the Gulf of Mexico data suggest that C0 is more or less constant, with possibly smaller values near the surface: C5, appears to increase signi-ficantly towards the surface34. Results from platform A, which is inertia dominant (i.e.. large member diameters), shows: (i) C0 increases and C decreases with increasing wave height and period,(ii)t here is significant variation in wave force coefficients with depth below the surface as pointed out by Pierson and Holmes, (iii) C0 varies from 0.55 to 1.90. with larger values near the surface. and (iv) C.0 varies from 1.16 to 1.76. with smaller values near the surface. Thus, these results are at direct variance with the findings of Evans from the Gulf of Mexico data33.
Further investigation was carried out on platform B, a drag dominant structure, to assess effects of constant
Sensitiuity analysis for steel jacket offshore plat fortns: S. Sh yam Sunder and J. J. Connor
versus flow dependent hydrodynamic coefficients and marine growth. Significant wave heights and T. identified ytrough Wiegel's relation and a range of relative rough-ness values from 0.00125 to 0.02 were considered. Table 11 gives a summary of results for the constant coelticicnts case and Tables
12 and 13
list results for relative roughness 0.00125 and 0.02 for the variable coefficients case. Average member diameters within different groupsTable Il. Response of platform B to Constant hyd rod vnamic coefficients
Table 12. Response of platform B to rariahle hvd rod ynatnic coefficients and relative roughness =0.00125
Table 13. Response of platform B to variable hvd rod vnamic coefficients and re/otite roughness = 0.02
Zone Group K Re( x 10-6) C C0
1 1 4.23 0.9803 1.76 0.55 2 5.61 0.6405 1.68 0.73 3 9.01 0.4074 1.49 1.17 2 1 4.63 1.0660 1.74 0.60 2 6.61 0.6981 1.62 0.86 3 9.88 0.4629 1.43 1.28 3 I 6.02 1.1953 1.66 0.79 2 8.46 0.7998 1.52 1.10 3 11.51 0.5789 1.34 1.50 4 1 7.81 1.5225 1.55 1.02 2 11.53 1.0312 1.34 1.50 3 13.23 0.7681 1.24 1.72 5 1 8.49 1.7850 1.51 1.10 2 12.53 1.2090 1.28 1.63 3 15.20 1.0017 1.16 1.90 H, ft) 4 12 20 28 36
Top node dis-placement. a,, (in)
0.009 0.030 0.081 0.176 0.259 Base shear, 2.7 10.5 30.3 68.7 103.9 C (kips) Base moment, ou (kip-ft X 10) 0.22 0.76 2.09 4.59 6.79 Base force period. T5, (sec) 5.0 7.7 10.2 12.1 13.2 RMS rel. el. just above 0.011 0.024 0.049 0.090 0.120 S\VL (mss) RMS rei, vet. just below 7.0 22.4 36.0 48.4 58.4 SWL (mis) H, (ft) 4 12 20 28 36
Top node dis-placement, a, (in) 0.009 0.028 0.063 0.132 0.194 Base shear, 2.7 10.0 23.8 51.4 77.8 0F(kips) Base moment, 0.22 0.72 1.63 3.45 5.09 0, (kip-ft x 10) Base force period, T (sec) 5.0 7.6 10.0 11.9 13.1 RMS rel. vel. just above 0.011 0.023 0.039 0.068 0.091 SWL (ints) RMS rel. veL just below 7.0 22.4 36.0 48.4 58.4 SWL (ints) Zone 1 2 3 4 I 2 3 4 1 2 3 4 5 6 77.71 76.25 67.25 62.00 62.00 54.00 50.57 47.14 42.00 42.00 48.00 33.00 33.00 33.00 33.43 34.20 34.20 41.2 23.5 23.5 215 23.5 23.5 41.2 41.2 35.3 17.6 41.2 35.3 35.3 35.3 41.2 58.8 58.8 H, (ft) 4 12 20 28 36
Top node dis-placement, a, (in)
0.009 0.037 0.080 0.142 Ö.206
Base shear, 2.8 13.2 30.4 56.0 82.7
0F (kips) Base moment, a., (kip-ft x 10-i)
0.23 0.95 2.08 3.71 5.40 Base force period, TFM (sec) 5.0 7.7 10.0 11.8 13.0 RMS rel. vel. just above 0.011 0.030 0.050 0.074 0.097 SWL (ints) RMS rel. vet. just below 7.0 22.4 36.1 48.4 58.4 SWL (ints)
Group Group Group Group Group Group Group Group
hle 14. Analrsis of averaqe member diameters in ph/rn B
Table15. Flow parameters andforce coefficients for platform B. Zone 5 is outside water. (H,=36 ft. T= 12.79 sec, reiatire roughness=0.02)
Table16. Fundamental natural period of platform Afor rarious ralues of deck mass
are listed in Table 14. and average values of K.Re,C and C0 for the case H,=36 It, L= 12.79 sec and the relative roughness 0.02 are given in Table 15. For the same excitation but relative roughness 0.00125, 1.30 and C0 1.40 for members in Group 3, and Ci 1.72 and CD 1.30 for members ¡n Group 4. There was almost no variation with depth. As before. C.2 and C00 when H5=4 ft and L =3.74 sec. irrespective of relative rough-ness values. lt was observed that significant variation of the force coefficients with depth occurred for moderate sea states, and in these cases the trend was similar to that observed in platform A. For the higher sea states there was only a mild variation in the coefficients with depth, as obsêrvable in Table 15. This minimal variation unlike that for platform A, is similar to results reported by Evans on the basis of the Gulf of Mexico data33.
Perusal of the results for platform B indicates that for relative roughness 0.02 and H,=36 ft. 71=12.79 sec, the variable coefficients option yields RMS response that are larger by - 25 over the constant coefficients case. In cases where variable C0 and C0 yielded larger reponse, it was found that the Keulegan-Carpenter number fell within those for which Sarpkaya's results are available. For relative roughness 0.02. the variable coefficients case response is larger for H,>20 ft. As relative roughness reduces, this limit for H, increases to above 20 ft. This reult is consistent with the fact that as relative roughness increases C0 increases, and that as H, increases drag force starts dominating. For sea states below limiting sea states (i.e., 20 ft
for 0.02). the constant C,,C0 values are
conservative, and the Keulegan-Carpenter number tends to fall outside Sarpkaya's published curves. However, even in those cases where constant C and C0 yieldSensitivity analysis for steel jacket offshore plaijornis: S. Shva,n Sunder and J. J. Connor conservative results, the results show that the choice ofC0
= 1.4 is not necessarily an upper bound, although C51= 2 most certainly is.
For drag dominant structures such as platform B. it is important to interpret results keeping in mind the simpli-fication introduced by the linearization of the drag force term.
Influence of deck mass and hysteretic structural dam ping on response
Sensitivity of fundamental natural period to changes in deck mass, M of platform A is shown in Table 16 and Fig. 14. For a single-degree-of-freedom system the fundamen-tal natural period should be proportional to square root of mass, and this is borne out by the plot in Fig. 14 which shows period versus square root of mass. The plot is and should be a straight line for the above to be true. The proportionality constant yields the stiffness of the equiva-lent single-degree-of-freedom system. Results showed that even a mass of l.4.I. which makes the natural frequency move closer to regions of larger wave energy, yielded no dynamic amplification. Thus, large variations in deck mass do not affect lateral motions of the platform.
To check whether structural hysteretic damping. D5 influenced platform response, structure A was analysed for the larger mass i.e., 14M, H,=4 ft and T=3.74 sec with D5 = 0.05 and 0.01. Results are presented in Table 17. None of the results are different more than 0.8%. In addition, there is imperceptible dynamic amplification in both cases. Therefore, for the North Sea structure con-sidered, quasi-static analysis procedures would suffice for both short and long-term response predictions as has already been pointed out.
CONCLUSIONS
One of the major objectives of this study has been to highlight the numerous uncertainties that go into the characterization of the enviroment and the offshore system, and the effect that these uncertainties are likely to have on short and long-term response. Thus, results of any
2'O 18 C o 14 06M 0-8M 10M Deck mass
Figure 14. Variation of natural period with deck mass for platform A
Table 17. Response of platform A to two djfferent talues of hysteretic structural dwnpinq
D, o (in) F (kips) o (kip-ft x I0) T1.,1 (sec)
Appli.ed Ocean Research, 198 1, t'bl. 3, No. 1 25
h
-- J
Zone Group K Re)x 10-6) C4 C0
1 3 17.72 0.9902 1.16 1.93 4 41.56 0.4201 1.30 1.77 2 3 18.68 1.0765 1)7 1.92 4 44.92 0.4362 1.32 1.76 3 3 20.06 1.2610 1.19 1.90 4 51.91 0.4725 1.37 1.74 4 3 20.70 1.3780 1.19 1.89 4 53.24 0.5370 1.37 1.73 Zone 1 2 3 4 I 2 3 4
-
34.00 14.67 25.0 75.0 1-
34.00 14.00 25.0 75.0 -.5 34.00 13.17 25.0 75.0 -4 34.00 I3.I7 25.0 75.0 -s 34.00 13.17 25.0 75.0 Mass 0.6M 08M M 12M 14MFund, period (sec) 1.42 1.58 1.73 1.87 1.98
0.05 0.00885 12.374 3.632 4.62
0.01 0.00892 12.402 3,642 4.61
Average diameter of Distribution of groups
members (in) within zones (°)
cnsitirify anal ysis for steel jacket offchore platforms: S. Shyani Sunder and J. J. Connor analysis should be accepted only after field verification and
calibration of the model.
Some of the major conclusions of the sensitivity analyses are summarized below:
(il For the platforms considered. RMS displacement even under extreme sea conditions, are small. There is neglig-ible or imperceptneglig-ible dynamic amplification irrespective of the hysteretic structural damping. Quasi-static analyses procedures are therefore likely to be adequate.
A definite functional relationship was observed between RMS base shear and RMS base moment.
The zero-crossing period of the relative velocity spectrum increases with depth below still water level. The zero-crossing period of base shear and moment are larger than the wave zero-crossing period.
Structural motion is negligible compared to fluid motion for the structures considered. However, for flex-ible structures this might not be so.
(y) Uncertainty in zero-crossing period to be associated with significant wave height is unlikely to influence design or fatigue life assessment.
Constant hydrodynamic force coefficients equal to C51=2 and C= 1.4 yield conservative results for inertia dominant structures. For drag dominant sfructures. the flow dependent C., and CD from Sarpkaya's experiments are conservative for large values of relative roughness and significant wave height.
It was observed that Cci decreases and CD increases with increasing wave height and period. C. and CDvary significantly with depth below still water level, especially for moderate sea states, with smaller C5, and 1arier CD near the water surface in these cases.
Large variations in deck mass do not affect lateral motion of the platform since thechange in period does not induce perceptible dynamic amplification.
This investigation has ignored sensitivity of response of finite foundation impedances and changes in hydrody-namic drag damping. The phasing effect, which causes multiple peaking of the response spectral densities, leads to the neglecting of the breathing mode which may be of importance in fatigue life assessment, especially of hori-zontal members. For correcting the phasing problema more complex structural model is warranted.
ACKNOWLEDGEMENTS
This work was funded by the Instituto Tecnologico Venezolano del Petroleo. Caracas, Venezuela. We are most appreciative of their support. The contribution of Dr. D. C. Angelides in the development of the com-putational strategy for the computer code POSEIDON is most gratefully acknowledged.
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I 1 Houmb. O. G. and Overik. T. Parametrization of wave spectra
and long termjoint distribution ofwave height and period, Proc. ¡nr. Conf Behariour ofOffhore Structures, Trondheim, 1976
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