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Effects of non-Gaussian waves to the dynamic response of jack-up platforms
Marine Structures 10 (1997) 131-157 PlI:
© 1997 Elsevier ScienceLimited All fights reserved. Printed in Great Britain S0951-8339(96)00021-4 0951-8339/97/$17.00 + .00
ELSEVIER
Effects of Non-Gaussian Waves to the Dynamic Response of Jack-up Platforms
N. Spidsoe & D. K a r u n a k a r a n S I b F r E F Civil a n d E n v i r o n m e n t a l Engineering, N 7034 T r o n d h e i m , N o r w a y
ABSTRACT
The paper discusses the nonlinear response effects to jack-up platfo~ns caused by non-Gaussian waves. The basis for the paper is fieM and model test observations of wave characteristics and simulation of wave processes and dynamic response applying different wave and wave kinematics models. Both normal design sea states and extremely steep sea states are investigated. In the design sea states, effects to the steady state dynamic response are evaluated. For the steep wave situations transient response effects--so called ringing response-are focused. Consequences and implication for design calculations are considered in both cases. © 1997 Elsevier Science Ltd
INTRODUCTION In stochastic dynamic response analysis of offshore structures normal practice is to model the wave process as a Gaussian process. It is well known, however, that real wave processes may be of a different nature, dependent on the sea state characteristics. This may be shown theoretically following the same lines as in the development of the design wave theories, see, for instance, Longuet-Higgins [8] and Martinsen and Winterstein [9]. Furthermore, the presence of such waves in real sea states has also been documented through field observations [12, 15]. Though the phenomenon of non-Gaussian waves is well known, there are several reasons why this has not influenced dynamic response analysis practice. First of all, design practice has been based on design wave analysis using wave models which includes these effects. Next, it has been assumed that the effects of non-Gaussian waves are small compared with dynamic amplification factors which are the main results from stochastic response analysis in a 131
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N. Spidsae, D. Karunakaran
design process. Consequently, practical wave models for such wave processes were not developed. Through the last couple of years, the consequences of non-Gaussian waves for the dynamic response of offshore platforms have caught considerable attention. This is because such waves have been shown to cause higher order wave loads which may cause a significant resonant response of dynamically sensitive, large volume structures, such as deep water gravity platforms and tension leg platforms. In order to overcome these problems, extensive research efforts have been undertaken, involving both model tests and theoretical developments. Response effects from non-Gaussian waves have also been observed for drag-dominated platforms. Examples for quasistatic response of a jacket platform are given by Spidsoe et al. [11] and one example of transient resonant response of a jacket is given in [13]. The consequences of such effects on a dynamically sensitive drag-dominated platform are, however, not investigated. The objective of this paper is therefore to give an introduction to this topic, in order to evaluate if this is a problem of significance or not. This is done by carrying out nonlinear stochastic dynamic response analysis of a traditional jack-up platform, applying a standard dynamic response analysis procedure, but for wave situations which have caused significant problems for large volume platforms and with wave and wave kinematics models developed for the prediction of higher order wave loads on large volume structures. The investigation is divided into two parts. First, normal design sea states and effects to steady state dynamic response are studied. Next, transient responses in sea states with extreme steepness are considered.
THE EXAMPLE P L A T F O R M The example jack-up platform is supported by three lattice legs which rest on a skirted spud can foundations at a water depth of 108m. The top side mass of the structure is 16900tons. The legs are spaced 62.0m apart. The three legs consist each of a triangular truss leg structure (side 16m). The typical member diameters in the leg truss work range from 0.15m to 0.8 m. A three dimensional FEM model is prepared for computer simulation of the platform behaviour, see Fig. 1. The model consists of 31 nodes and 34 elements. Each leg truss work is idealized as a string of beam elements with equivalent stiffness properties to the real leg lattice truss work. The top side mass is lumped at the top nodes. The structural mass and the added mass of the legs are lumped at appropriate nodal points. The first natural period for the platform is calculated from the FEM model to be 5.7sec at the extreme load level condition.
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(-) 91.673 (-)103.975 (-)108.100
Fig. 1. General view and computer model of the platform.
"['HE D Y N A M I C RESPONSE ANALYSIS P R O C E D U R E The nonlinear dynamic response is estimated applying the Finite Element Method (FEM) for structural analysis. The dynamic response analysis is performed in the time domain. The procedure is based on the dynamic equation of motion expressed as: MAJ~ t + C A t t + K A x t = A F t
(1)
where M = mass matrix C = damping matrix K = stiffness matrix AFt = vector of incremental hydrodynamic nodal point forces in line with the wa.ve direction Axt = incremental structural displacement vector Ax't = incremental structural velocity vector Aiit = increments structural acceleration vector. The mass matrix includes deck mass, structural mass and hydrodynamic added mass. Since the submerged part of drag-dominated platforms mainly
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consists of slender elements, the added mass is assumed to be frequency independent. The stiffness matrix contains structural stiffness and foundation stiffness. The structural stiffness includes the P - A effects [10]. Since the vertical loads are dominated by the deck loads, the structural stiffness including p - A effects is assumed to be independent of response level and time. The structural and soil damping is modelled by the Rayleigh damping model by specification of the damping level at the basic modes. The hydrodynamic forces are calculated from Monte Carlo simulated wave kinematics time series using the extended Morison equation based on relative velocities. By this model the hydrodynamic drag damping is included implicitly. The distributed forces are integrated to the simulated or specified instantaneous surface elevation at each time step by numerical integration. The response process samples are obtained by solving eqn (1). A Newmark-fl method with fl=0.25 and 7=0.5 is employed, corresponding to the so-called constant average acceleration algorithm. The time step for time integration in the dynamic response calculation is 0.33sec, ensuring that there is no significant period shift at the first natural period of the structure, caused by the integration procedure itself. The validity of this procedure is discussed by Karunakaran and Spidsoe [7] based on comparison of simulations and model test data.
FIELD OBSERVATIONS OF NON-GAUSSIAN WAVES Based on full scale measurements of waves in the North Sea, Spidsoe and Karunakaran [12] concluded that wave processes characterized by skewness and kurtosis coefficients of the order 0.20 and 3.20, respectively, could be expected in extreme sea states. The non-Gaussian structure of ocean surface waves is later investigated in detail by Vinje and Haver [15], who also developed a parametric model for the coefficient of skewness theoretically by means of a second order Stoke's expansion. Furthermore, the model is shown to fit reasonably well to results obtained by analysing laser measurements from the Gullfaks C platform. The model is: 71 = 3 4 . 4 .
where 71 = coefficient of skewness Hs = significant wave height Te = peak period g = acceleration of gravity.
Hs
gT2P
(2)
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Based on an inspection of data provided by Marthinsen and Winterstein [9], the following correction term for finite depth effects is suggested: A71 = 2.14.10 -6
(3)
where d is the water depth. Concenaing the coefficient of skewness, a second order Stoke's expansion is assumed to yield results of a reasonable accuracy. However, for the kurtosis, a higher order expansion is needed. It is indicated that the contribution to the kurtosis from the third order term is about 75% of the second order contribution. Accounting for this, the coefficient of kurtosis of real seas is assumed to be well modelled by: 72 = 3.0 + 375.
(4)
Equation 4 is found to fit reasonably well to the mean kurtosis level of real seas, at least, as reflected by laser measurements.
M O D E L L I N G OF NON-GAUSSIAN WAVES Gaussian sea surface elevation is normally simulated by Monte Carlo simulation from a model wave spectrum based on linear theory for irregular waves, assuming independent random phase angles uniformly distributed between 0 and 2n, see, for instance, [5]. Models for nonlinear random waves have been proposed by several authors, see, e.g., [8, 1, 4]. So far, however, no complete model which also is applicable for time domain analysis exists. In this study, two methods for simulating non-Gaussian sea surfaces are used: • A Gaussian surface process is transformed such that a particular value of skewness and kurtosis is obtained, by a truncated expansion of a standard Gaussian process in terms of the Hermite polynomials of lowest order. • A non-Gaussian sea surface is simulated by a second order wave model based on Stoke's expansion. In the Hermite expansion method, the non-Gaussian sea surface is derived from a Gaussian wave process by the Hermite expansion proposed by Winterstein [17]. The input for this transformation is the coefficient of skewness and the coefficient of kurtosis of the non-Gaussian surface process. The values of skewness and kurtosis are calculated according to eqn (2) and eqn (4), respectively. The Gaussian sea surface is simulated along the structure using the linear dispersion relationship. Then, at each simulation point, the Gaussian sea surface is transformed by the Hermite transformation.
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This way of simulating surface process is non-physical from a hydrodynamic point of view and it can at best yield an indication of what is likely to happen with the response pattern of a structure exposed to a non-Gaussian surface process. The advantage of the Hermite Expansion Method is its simplicity. However, as an approximation to the physical surface, it lacks a consistent formulation of the dispersion properties. It is not obvious how to maintain a prescribed skewness and kurtosis as the waves move in space. Applying the I r r e g u l a r s e c o n d - o r d e r w a v e m o d e l the non-Gaussian sea surface is simulated by a model based on the second-order Stoke's expansion [8], outlined by Marthinsen and Winterstein [9] for a deep water condition. By this model, a consistent dispersion relation is also obtained. Furthermore, the skewness and kurtosis of the sea surface are implicitly given by the model. Wave kinematics are, in both cases, calculated from the simulated nonGaussian surface elevation time series using the W h e e l e r S t r e t c h i n g m e t h o d . [16] This method is an engineering approximation which lacks a hydrodynamic basis. Comparisons to measurements show, however, that it gives a reasonable representation of the wave flow which is acceptable for most engineering applications [3, 7]. For the Hermite Expansion Method a method of this type has to be used. For the Second Order Wave Model a consistent kinematics model exist [14]. This is, however, somewhat complicated to use and so far it has not been proven to give better results when compared to measurements than the Wheeler method.
RESPONSE EFFECTS IN D E S I G N SEA STATES The objective of this part of the study is to investigate the effects of nonGaussian waves to estimates of extreme dynamic response and dynamic amplification factors in sea states normally considered in design of jack-ups in the North Sea. Five sea states are considered: • Hs = • Hs = • Hs = • Hs = • Hs =
14.0m 14.8m 14.0m 13.0m 12.0m
and Tp = 17.0sec and T e = 16.0sec and T e = 14.0sec and To = 13.0sec and T e = 12.0see.
Each sea state is modelled by a JONSWAP spectrum, assuming longcrested waves. The duration of each sea state is set to 6hours.
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Currem: velocities corresponding to a 10year return period are used along with these: sea states and the current velocities are: • • • •
at mean water level 1.10ms -1 -30m 0.75ms -1 -50m 0.70ms -~ -105m 0.70ms -1.
The hydrodynamic coefficients used in this analysis are: • Mass coefficient CM: 2.0 • Drag coefficient CD: 1.0--In irregular wave analysis • Drag coefficient CD: 0.7--In design wave analysis. The structural damping level in the basic modes is assumed to be 2%. For each sea state 8 samples of waves and response are generated, and the presented results represents averages of the actual parameter from these samples. The complete results from the study are presented and discussed by Karunakaran et al. [6]. Thus, in the following only the main conclusions supported by examples of main findings are given. Simulated sea surface
The average skewness and kurtosis coefficients from the eight independent simulated non-Gaussian sea surfaces by the second order wave model are presented in Table 1. In the same table the input skewness and kurtosis coefficients for the non-Gaussian sea by Hermite expansion are also shown. It is seen that the second order wave model gives slightly lower kurtosis coefficients than was calculated by eqn (4), whereas the skewness coefficients are comparable to what was calculated by eqn (2). Thus, there is a small but clear difference between the two methods, and the difference tends to increase as the steepness increases. Since the Hermite method reflects empirical data, the results indicate that the second order model underestimates the non-Gaussian nature of the waves. Effects on structural response
The response of the platform, simulated using Gaussian and non-Gaussian sea surface using the Hermite Expansion M e t h o d is compared in Table 2. It can be seen that the use of non-Gaussian sea surface increases the standard deviation of the quasistatic response by 5-8% for all response quantities. Concerning extreme response, the use of non-Gaussian sea surface increases the extreme quasistatic response by about 20% for the least steep sea state,
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TABLE 1 Statistical properties of non-Gaussian sea surface elevation
Sea state
H s = 14.0m Te=17s Hs = 14.8m Tp = 16s Hs = 14.0m Tp = 14s Hs = 13.0m Tp= 13s Hs = 12.0m Tp = 12s
Non-Gaussian sea Hermite expansion
Non-Gaussian sea Second order wave model
Skewness
Kurtosis
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Kurtosis
&
0.17
3.09
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&
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0.24
3.10
&
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&
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TABLE 2 Ratio between response estimated using non-Gaussian sea surface (by Hermite expansion) and Gaussian sea surface (Average from eight samples)
Response Quantity
Standard deviation o f response Quasistatic
Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement
Dynamic
Hs = 14.0m & Tp = 1.05 1.06 1.05 Hs = 14.8m & Tp = 1.06 1.06 1.06 1.07 1.06 1.07 Hs = 14.0m & Tp = 1.07 1.07 1.08 1.08 1.08 1.08 Hs = 13.0m & Tp= 1.07 1.07 1.08 1.08 1.08 1.08 Hs = 12.0m & TI, = 1.08 1.08 1.08 1.08 1.08 1.08
Extreme response Quasistatic
Dynamic
1.20 1.23 1.23
1.20 1.23 1.23
1.25 1.29 1.29
1.28 1.30 1.30
1.38 1.44 1.44
1.33 1.36 1.35
1.41 1.47 1.46
1.37 1.42 1.41
1.51 1.57 1.55
1.41 1.43 1.42
17sec
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i.e., H s = 14m & T e = 17sec, while the increase is as high as 57% for the steepest sea state, H s = 12m & Tp= 12sec. The increase in the extreme dynamic response is about 20% for H s = 14m & Tp = 17sec and 43% for H s = 12m & Te = 12sec for base shear and overturning moment. The results obtained by the S e c o n d order wave m o d e l show similar effects as found by the H e r m i t e E x p a n s i o n M e t h o d . The effects are, however, less in this case.
Comparison of design wave analysis and stochastic time domain analysis Stochastic time domain analysis is often used for estimation of dynamic amplification factors which are to be applied to the quasistatic response from the design wave analysis in order to account for the dynamic effects. This requires that consistent wave and wave load parameters and models are used in the two types of analyses. Consistent wave parameters are normally obtained by a long term analysis of wave data. Furthermore, since Morison equation is used in both cases, the drag coefficient is used as a calibration factor for the loading. The results of this calibration factor is obviously influenced by the wave models involved. Normally a Stoke's V order wave model is used in the design wave analysis, while linear, Gaussian waves are used in the stochastic analysis. If non-Gaussian waves are employed in the stochastic analysis the consistency will definitely be improved. However, the results of the calibration will also be changed. In order to check this, a comparison of extreme quasi-static response obtained by a stochastic analysis to design wave analysis results was carried out. The results of this comparison are summarized in Table 3. These results indicate that the choice of wave model in the stochastic analysis has a significant effect on the calibration. Furthermore, they indicate that, along with non-Gaussian waves, a drag coefficient which is comparable to that used in the Stoke's V order design wave analysis should possibly be used. This is based on the assumption that the Stoke's V order design wave analysis gives proper quasistatic load levels.
Effects on dynamic amplification factors As mentioned above, the dynamic amplification factor (DAF) is often estimated using stochastic time domain analysis and later applied to the quasistatic response from design wave analysis. The influence of non-Gaussian sea surface on the D A F was therefore investigated. The results are given in Table 4 and show that the D A F in extremes, in general, decrease when a non-Gaussian sea surface is used. There are two reasons for this. First, as discussed above, the effect of non-
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TABLE 3 Ratio between quasistatic response from time domain analysis (Hs = 14.8m & Tp = 16 sec) and design wave analysis (H = 27m & T = 14.5 sec) using Stoke wave model
Base shear C D = 1.0 in Gaussian sea surface Non-Gaussain-Hermite expansion Non-Gaussian-Second order wave CD = 0.7 in Gaussian sea surface Non-Gaussian-Hermite expansion Non-Gaussian-Second order wave
Overturning moment
time domain analysis 1.07 1.04 1.33 1.34 1.26 1.33 time domain analysis 0.76 0.74 0.95 0.96 0.92 0.96
Deck displacemen t 1.04 1.34 1.33 0.74 0.96 0.95
TABLE 4 Dynamic amplification factors in extreme response estimated using Gaussian and nonGaussian sea surface
Response Quantity
Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement
Gaussian sea surface Hs = 14.0m 1.13 1.27 1.23 Hs = 14.8m 1.14 1.29 1.25 Hs = 14.0m 1.19 1.37 1.32 Hs = 13.0m 1.18 1.35 1.30 Hs = 12.0m 1.20 1.39 1.33
Non-Gaussian sea surface Hermite expansion Second order wave & Tp= 17sec 1.13 1.26 1.23 & Tp= 16sec 1.16 1.30 1.26 & Tp= 14sec 1.15 1.29 1.24 & Tp = 13sec 1.15 1.30 1.25 & Tp = 12sec 1.12 1.26 1.21
1.07 1.18 1.14 1.13 1.24 1.20 1.07 1.17 1.13 1.11 1.25 1.19 1.07 1.21 1.15
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Gaussian waves is stronger on the quasistatic response than on the dynamic response. Next, it is because the resonant response in the case of non-Gaussian waves includes stronger transient effects than in the Gaussian case and this introduces a phase shift between the maximum quasistatic and dynamic response. This is further discussed in the next section.
D Y N A M I C RESPONSE EFFECTS IN STEEP WAVES Transient resonant response, termed ringing, induced by higher order wave forces in special sea states, may be significant for the extreme response of large volume structures like gravity platforms and tension leg platforms. This has been shown by model tests and computer simulations. Whether ringing is a significant problem for drag-dominated platforms or not is not determined. As mentioned in the introduction, ringing response is observed from full scale measurements, but it is not clear if this has any important effects to the extreme response compared to other nonlinear effects. In the study on effects from non-Gaussian waves, ringing effects were observed. The objective of this study is to investigate further how the applied load models may generate a ringing response, and to investigate the effects of ringing oil the extreme response and the sensitivity of these effects on the load and structural parameters. This is done by using measured surface elevations from experiments on ringing on large volume structures carried out by M A R T I N T E K as an input to simulations. Furthermore, model test data from an experiment with a jack-up platform performed by DHI are studied. Six measured realizations of a sea state are considered for the simulations. The sea state in full scale is characterized by H s = 14.5m and Tp = 16.0sec and each sample is 5.4h long. The skewness and kurtosis coefficients of the samples are typically 0.23 and 3.26, indicating rather strong non-Gaussian nature at a modest steepness. Since the aim of the work is to study the resonant response around the large wave cycles, it is sufficient to use a shorter length of wave time series containing such wave cycles. Hence, from these six realizations, 11 short sea elevation samples of 4859sec (1.35h) long, each having at least one big wave cycle which has been found to generate a ringing response in large volume structure,J, are established. They are here after identified as sample #1, sample #2 etc. Current is not included in this study. The drag and inertia coefficients used in the analysis are 1.0 and 2.0, respectiwfly.
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The investigation is organized in three parts: • a base case study • sensitivity studies • evaluation of model test data. The base case study is performed for a foundation condition, which gives the structure a first natural period equal to 5.7sec. The damping level is assumed to be 2% in the basic modes. This part of the study focuses on the nature of the resonant response and the dynamic amplification of extreme response. The sensitivity study concerns the influence of the ringing response due to variation of the following load and structural parameters: • • • •
Damping level Damping mechanism Natural period Load pattern.
The results are presented only for base shear. The nature of this response quantity is, however, representative for the complete response process. The evaluation of the model test data includes a study of measured dynamic response. Base ease
Figures 2-6 show examples of the wave process, the quasistatic response, the resonant response and the total dynamic response in terms of 200sec long time series window around the extreme wave event in each wave sample. From these plots it is seen that resonant response with typical ringing characteristics is generated by the extreme waves in samples 1, 2 and 11. These examples are representative for the majority of the investigated samples. It is possible that the ringing also occurs in some of the remaining samples. However, in these cases it seems that the resonant response builds up over several wave cycles, and thus does not get the typical characteristics of an instantaneous build up, as shown in samples 3 and 10. It may therefore be concluded that the applied wave and load model generate a ringing response of the platform. From the plots it is seen that the resonant response in the ringing events is significant, in fact in the range of 40--60 percent of the maximum quasistatic response. However, it is also noted that the effect of the ringing response to the extreme total dynamic response generally is small. This is because the extreme response of a dragdominated platform always is dominated by quasistatic response induced by the extreme wave crests and that the ringing response is phase shifted behind this. The effect of the ringing thus depends highly on this phase lag.
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-10 3260
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3340 3360 Time [see]
3380
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Total dynamic response Fig. 6. W a v e h e i g h t a n d quasistatic, d y n a m i c a n d r e s o n a n t r e s p o n s e - s a m p l e #11.
N. Spids~e, D. Karunakaran
148
Sensitivity study The following parameters are included in the sensitivity study: • • • •
Structural damping Hydrodynamic damping Natural period Drag and inertia coefficient.
Structural and soil damping Three different cases are studied; they are: • 1% structural and soil damping • 2% structural and soil damping • 3% structural and soil damping. The results from this study are presented in Table 5. In this presentation only five samples are considered. The results indicate that the damping has only about 1-1.5% effect on the total dynamic response. The damping has about a 5-10% effect on the ringing response. However, the effect is smallest for the condition where the resonant response is of pure ringing type, sample #2 and sample #11, see Table 5. The resonant response of three samples are presented in Fig. 7.
Hydrodynamic damping The dynamic analysis is also performed using relative velocities in Morison equation. The results are presented in Table 6. The results indicate that the inclusion of hydrodynamic damping decreases the total dynamic response by about 7-10 %, whereas it decreases the resonant response by about 15-30%. In general, hydrodynamic damping reduces the ringing response. TABLE 5 Sensitivity of extreme dynamic response and resonant response to damping--Base shear
Sample identification
Total dynamic response (MN) I°A
damping Sample Sample Sample Sample Sample
#I #2 #3 #10 #11
15.3 16.2 12.9 18.4 13.0
2%
3%
damping damping 15.1 16.1 12.6 17.9 13.0
15.0 16.1 12.3 17.4 12.9
Resonant response ( MN) 1%
2%
3%
damping damping damping 7.66 7.39 5.47 9.51 6.10
7.12 6.95 5.01 8.65 5.78
6.64 6.55 4.73 7.91 5.48
149
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150
iV. Spids~e, D. Karunakaran
TABLE 6 Sensitivity of extreme dynamic response and resonant response to hydrodynamic damping Base shear
Sample identification
Sample #1 Sample #2 Sample #3 Sample #10 Sample #11
Total dynamic response (MN)
Resonant response ( MN)
Without hyd. damping
With hyd. damping
Without hyd. damping
With hyd. damping
15.1 16.1 12.6 17.9 13.0
14.2 15.4 11.7 16.2 12.4
7.12 6.95 5.01 8.65 5.78
5.67 5.57 4.19 6.94 4.87
Drag and inertia coefficients In order to check the importance of drag forces vs inertia forces with respect to ringing, the dynamic response analysis was performed using only the drag loading in one case and only the inertia loading in another case. The resonant response for two samples where the resonance is of pure ringing type is shown in Fig. 8. In this figure the resonant response using both drag and inertia loading is shown, but only the drag loading and only the inertia loading are compared. It is clearly seen that the ringing response is almost totally generated by drag loading.
Natural period The dynamic response analysis is performed for the same structure using two different foundation conditions which gave the first natural periods to be 4.5sec and 7.5sec. The results are shown in Table 7 and in Fig. 9. It is seen that the ringing response and its effects on the total response increases as the natural period decreases. This is because the phase lag between the ringing response and the quasistatic response reduces. It also shows, however, that the ringing response increases as the natural period approaches half of the wave crest period, thus indicating that the integrated drag forces in the wave crest are the main source of ringing for this structure. Evaluation of model test data
The applied model test data refer to an experiment with a dynamic jack-up in model scale 1:49 conducted by MSC at DHI laboratories [2]. This experiment included a variety of sea state conditions. Among these, the one with highest steepness was selected for inspection of ringing events. This sea state
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is characterized by a Pierson-Moskowitz spectrum with significant wave height and peak period equal to 0.20m and 2.2s, respectively. The estimates o f skewness and kurtosis for the measured wave sample are 0.20 and 3.10. The sea state is thus less steep and less non-Gaussian than the waves involved in the base case study. In Figs 10 and 11, observed wave and response situations with a ringing like nature are shown. The ringing nature is definitely not so
N. Spidsoe, D. Karunakaran
152
TABLE 7 Sensitivity of extreme dynamic response and resonant response to natural period, Base shear
Sample identification
Sample Sample Sample Sample Sample
#1 #2 #3 #10 #11
Total dynamic response ( MN)
Resonant response ( MN)
To=4.5sec To=5.Tsec To = 7.Ssec
To=4.5sec To=5.7sec To = 7.5sec
15.6 18.6 12.4 15.8 14.8
15.1 16.1 12.6 17.9 13.0
13.5 15.3 9.77 13.9 12.9
9.51 9.79 7.58 12.1 9.97
7.12 6.95 5.01 8.65 5.78
3.22 3.12 2.39 3.73 2.77
pronounced as in the simulated cases discussed above, probably due to the different sea state nature. However, it is noticed that the quasistatic and transient resonant response combines in the same manner as the simulations, thus indicating that the simulation method produces realistic ringing effects.
CONSEQUENCES FOR D E S I G N CALCULATIONS Design practice for jack-up platforms is based on the design wave method. The extreme quasistatic response is then calculated by deterministic design wave analysis and the dynamic response found by multiplying this with a dynamic amplification factor (DAF) obtained from a stochastic response analysis or from some kind of simplified method. The effects of non-Gaussian waves discussed in this paper concern only the stochastic analysis. The presented results show that if a non-Gaussian wave model is applied, a lower drag coefficient should be used than for Gaussian waves, in order to obtain a consistent quasistatic response compared to the design wave analysis. Furthermore, the results show that if non-Gaussian waves are used, this will give lower DAFs than with Gaussian waves, i.e., a beneficial effect with respect to design. The results presented in the paper also show that significant transient response caused by higher order wave forces must be expected for jackup platforms and that these will influence the dynamic amplification. However, it is also shown that these effects are included in a standard nonlinear stochastic response analysis procedure. This will not be the case if more simplified methods are used for this purpose. Thus, this is another strong argument for using nonlinear stochastic response analysis for estimation of DAFs in the design process of drag-dominated platforms.
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N. Spidsoe, D. Karunakaran
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test.
156
N. Spids~e, D. Karunakaran CONCLUSIONS
The effects of using non-Gaussian waves in the nonlinear stochastic response analysis of jack-up platforms have been discussed. It has been shown that the application of non-Gaussian waves significantly affects both the steady state dynamic response and the transient response, compared to the use of Gaussian waves. The resulting effect on the total response is that the quasistatic response increases while the dynamic amplification reduces. Transient response--so called ringing--has been addressed in particular. It has been shown that commonly applied wave load models for jack-up analysis generate a ringing response due to the higher order components of the drag force, both when Gaussian and non-Gaussian waves are used, but strongest in the latter case. The transient response influences significantly the dynamic amplification. Thus, the results clearly show that a nonlinear stochastic dynamic response analysis procedure should be applied for the estimation of DAFs in the design of jack-ups. Furthermore, the results demonstrate that non-Gaussian waves should be considered in such analyses. The hydrodynamically consistent wave models available for this today are not sufficiently accurate or computer efficient; therefore, improved models for this purpose should be developed. Until then, it is believed that the Hermite Expansion Method applied in this study could be used as a reasonable engineering approximation. However, this method should also be verified, based on model test data and full scale measurements, as soon as possible.
REFERENCES 1. Arhan, M. and Plaisted, R.O., Nonlinear deformation of sea surface profiles in intermediate and shallow water. Oceanologica Acta, 1981, 4, 1-2. 2. Grundlehner, G. J., Systematic model tests on a harsh environment jack-up in elevated condition. In Proceedings of the 5th International Jack-up Conference, London, September 1995. 3. Gudmestad, O.T. and Haver, S., Uncertainties in prediction of wave kinematics in irregular waves. Wave Kinematics and Environmental Forces, 1993, 29, 1-2. 4. Huang, N.E., Long, S.R., Tung, C.C., Yuan, Y. and Bliven, L.F., A nonGaussian model for surface elevation of nonlinear random wave fields. Journal of Geophysical Research, 1983, 88, 1-2. 5. Karunakaran, D., Procedure for nonlinear dynamic response analysis of offshore structures--both for extreme and fatigue response. SINTEF Report STF71 A91016, Trondheim, 1991. 6. Karunakaran, D., Spidsae, N. and Haver, S., Nonlinear dynamic response of jack-up platforms due to non-Gaussian waves. In Proceedings of the 13th lnter-
Non-Gaussian waves
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
157
national Conference on Offshore Mechanics and Arctic Engineering, Houston, April 1994. Karunakaran, D. and Spidsoe, N., Verification of methods for simulation of nonlinear dynamic response of jack-up platforms. In Proceedings of the 5th International Jack-up Conference, London, September 1995. Longuet-Higgins, M.S., The effects of nonlinearities on statistical distributions in the theory of sea waves. Journal of Fluid Mechanics, 1963, 17, 1-2. Marthinsen, T. and Winterstein, S., On the skewness of random surface waves. In Proceedings of the 2nd International Offshore and Polar Engineering Conference, San Fransisco, June 1992. Przenfieniecki, J. S., Theory of Matrix Structural Analysis. McGraw Hill Inc., New York, 1968. Spids~e, N., Brathaug, H.-P. and Skj~stad, O., Nonlinear random wave loading on fixed offshore platforms. In Proceedings of the Offshore Technology Conference, OTC-5101, Houston, May 1986. Spids~e, N. and Karunakaran, D., Statistical and directional properties of measured ocean waves. In Proceedings of the Offshore Technology Conference, Houston, May 1987. Spids~e, N. and Karunakaran, D., Nonlinear dynamic behaviour of jack-up platforms. In Proceedings of the Fourth International Conference on Jack-up Platforms, London, September 1993. Stansberg, C.T., Second-order effects in random wave modelling. In Proceedings of the International Symposium on Waves-Physical and Numerical Modelling, "Vancouver, Canada, 1994. Vinje, T. and Haver, S., On the non-Gaussian structure of ocean waves. Presented at International Conference on the Behaviour of Offshore Structures-BOSS 94, USA, July 1994. Wheeler, J.D., Methods for calculating forces produced on piles in irregular waves. Journal of Petroleum Technology, 1970, 1, 1-2. Winterstein, S.R., Nonlinear vibration models for extremes and fatigue. Journal of Engineering Mechanics, 1988, 114(10), 1772-1790.
© 1997 Elsevier ScienceLimited All fights reserved. Printed in Great Britain S0951-8339(96)00021-4 0951-8339/97/$17.00 + .00
ELSEVIER
Effects of Non-Gaussian Waves to the Dynamic Response of Jack-up Platforms
N. Spidsoe & D. K a r u n a k a r a n S I b F r E F Civil a n d E n v i r o n m e n t a l Engineering, N 7034 T r o n d h e i m , N o r w a y
ABSTRACT
The paper discusses the nonlinear response effects to jack-up platfo~ns caused by non-Gaussian waves. The basis for the paper is fieM and model test observations of wave characteristics and simulation of wave processes and dynamic response applying different wave and wave kinematics models. Both normal design sea states and extremely steep sea states are investigated. In the design sea states, effects to the steady state dynamic response are evaluated. For the steep wave situations transient response effects--so called ringing response-are focused. Consequences and implication for design calculations are considered in both cases. © 1997 Elsevier Science Ltd
INTRODUCTION In stochastic dynamic response analysis of offshore structures normal practice is to model the wave process as a Gaussian process. It is well known, however, that real wave processes may be of a different nature, dependent on the sea state characteristics. This may be shown theoretically following the same lines as in the development of the design wave theories, see, for instance, Longuet-Higgins [8] and Martinsen and Winterstein [9]. Furthermore, the presence of such waves in real sea states has also been documented through field observations [12, 15]. Though the phenomenon of non-Gaussian waves is well known, there are several reasons why this has not influenced dynamic response analysis practice. First of all, design practice has been based on design wave analysis using wave models which includes these effects. Next, it has been assumed that the effects of non-Gaussian waves are small compared with dynamic amplification factors which are the main results from stochastic response analysis in a 131
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N. Spidsae, D. Karunakaran
design process. Consequently, practical wave models for such wave processes were not developed. Through the last couple of years, the consequences of non-Gaussian waves for the dynamic response of offshore platforms have caught considerable attention. This is because such waves have been shown to cause higher order wave loads which may cause a significant resonant response of dynamically sensitive, large volume structures, such as deep water gravity platforms and tension leg platforms. In order to overcome these problems, extensive research efforts have been undertaken, involving both model tests and theoretical developments. Response effects from non-Gaussian waves have also been observed for drag-dominated platforms. Examples for quasistatic response of a jacket platform are given by Spidsoe et al. [11] and one example of transient resonant response of a jacket is given in [13]. The consequences of such effects on a dynamically sensitive drag-dominated platform are, however, not investigated. The objective of this paper is therefore to give an introduction to this topic, in order to evaluate if this is a problem of significance or not. This is done by carrying out nonlinear stochastic dynamic response analysis of a traditional jack-up platform, applying a standard dynamic response analysis procedure, but for wave situations which have caused significant problems for large volume platforms and with wave and wave kinematics models developed for the prediction of higher order wave loads on large volume structures. The investigation is divided into two parts. First, normal design sea states and effects to steady state dynamic response are studied. Next, transient responses in sea states with extreme steepness are considered.
THE EXAMPLE P L A T F O R M The example jack-up platform is supported by three lattice legs which rest on a skirted spud can foundations at a water depth of 108m. The top side mass of the structure is 16900tons. The legs are spaced 62.0m apart. The three legs consist each of a triangular truss leg structure (side 16m). The typical member diameters in the leg truss work range from 0.15m to 0.8 m. A three dimensional FEM model is prepared for computer simulation of the platform behaviour, see Fig. 1. The model consists of 31 nodes and 34 elements. Each leg truss work is idealized as a string of beam elements with equivalent stiffness properties to the real leg lattice truss work. The top side mass is lumped at the top nodes. The structural mass and the added mass of the legs are lumped at appropriate nodal points. The first natural period for the platform is calculated from the FEM model to be 5.7sec at the extreme load level condition.
133
Non-Gaussian waves
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Fig. 1. General view and computer model of the platform.
"['HE D Y N A M I C RESPONSE ANALYSIS P R O C E D U R E The nonlinear dynamic response is estimated applying the Finite Element Method (FEM) for structural analysis. The dynamic response analysis is performed in the time domain. The procedure is based on the dynamic equation of motion expressed as: MAJ~ t + C A t t + K A x t = A F t
(1)
where M = mass matrix C = damping matrix K = stiffness matrix AFt = vector of incremental hydrodynamic nodal point forces in line with the wa.ve direction Axt = incremental structural displacement vector Ax't = incremental structural velocity vector Aiit = increments structural acceleration vector. The mass matrix includes deck mass, structural mass and hydrodynamic added mass. Since the submerged part of drag-dominated platforms mainly
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consists of slender elements, the added mass is assumed to be frequency independent. The stiffness matrix contains structural stiffness and foundation stiffness. The structural stiffness includes the P - A effects [10]. Since the vertical loads are dominated by the deck loads, the structural stiffness including p - A effects is assumed to be independent of response level and time. The structural and soil damping is modelled by the Rayleigh damping model by specification of the damping level at the basic modes. The hydrodynamic forces are calculated from Monte Carlo simulated wave kinematics time series using the extended Morison equation based on relative velocities. By this model the hydrodynamic drag damping is included implicitly. The distributed forces are integrated to the simulated or specified instantaneous surface elevation at each time step by numerical integration. The response process samples are obtained by solving eqn (1). A Newmark-fl method with fl=0.25 and 7=0.5 is employed, corresponding to the so-called constant average acceleration algorithm. The time step for time integration in the dynamic response calculation is 0.33sec, ensuring that there is no significant period shift at the first natural period of the structure, caused by the integration procedure itself. The validity of this procedure is discussed by Karunakaran and Spidsoe [7] based on comparison of simulations and model test data.
FIELD OBSERVATIONS OF NON-GAUSSIAN WAVES Based on full scale measurements of waves in the North Sea, Spidsoe and Karunakaran [12] concluded that wave processes characterized by skewness and kurtosis coefficients of the order 0.20 and 3.20, respectively, could be expected in extreme sea states. The non-Gaussian structure of ocean surface waves is later investigated in detail by Vinje and Haver [15], who also developed a parametric model for the coefficient of skewness theoretically by means of a second order Stoke's expansion. Furthermore, the model is shown to fit reasonably well to results obtained by analysing laser measurements from the Gullfaks C platform. The model is: 71 = 3 4 . 4 .
where 71 = coefficient of skewness Hs = significant wave height Te = peak period g = acceleration of gravity.
Hs
gT2P
(2)
Non-Gaussian waves
135
Based on an inspection of data provided by Marthinsen and Winterstein [9], the following correction term for finite depth effects is suggested: A71 = 2.14.10 -6
(3)
where d is the water depth. Concenaing the coefficient of skewness, a second order Stoke's expansion is assumed to yield results of a reasonable accuracy. However, for the kurtosis, a higher order expansion is needed. It is indicated that the contribution to the kurtosis from the third order term is about 75% of the second order contribution. Accounting for this, the coefficient of kurtosis of real seas is assumed to be well modelled by: 72 = 3.0 + 375.
(4)
Equation 4 is found to fit reasonably well to the mean kurtosis level of real seas, at least, as reflected by laser measurements.
M O D E L L I N G OF NON-GAUSSIAN WAVES Gaussian sea surface elevation is normally simulated by Monte Carlo simulation from a model wave spectrum based on linear theory for irregular waves, assuming independent random phase angles uniformly distributed between 0 and 2n, see, for instance, [5]. Models for nonlinear random waves have been proposed by several authors, see, e.g., [8, 1, 4]. So far, however, no complete model which also is applicable for time domain analysis exists. In this study, two methods for simulating non-Gaussian sea surfaces are used: • A Gaussian surface process is transformed such that a particular value of skewness and kurtosis is obtained, by a truncated expansion of a standard Gaussian process in terms of the Hermite polynomials of lowest order. • A non-Gaussian sea surface is simulated by a second order wave model based on Stoke's expansion. In the Hermite expansion method, the non-Gaussian sea surface is derived from a Gaussian wave process by the Hermite expansion proposed by Winterstein [17]. The input for this transformation is the coefficient of skewness and the coefficient of kurtosis of the non-Gaussian surface process. The values of skewness and kurtosis are calculated according to eqn (2) and eqn (4), respectively. The Gaussian sea surface is simulated along the structure using the linear dispersion relationship. Then, at each simulation point, the Gaussian sea surface is transformed by the Hermite transformation.
N. Spidsoe, D. Karunakaran
136
This way of simulating surface process is non-physical from a hydrodynamic point of view and it can at best yield an indication of what is likely to happen with the response pattern of a structure exposed to a non-Gaussian surface process. The advantage of the Hermite Expansion Method is its simplicity. However, as an approximation to the physical surface, it lacks a consistent formulation of the dispersion properties. It is not obvious how to maintain a prescribed skewness and kurtosis as the waves move in space. Applying the I r r e g u l a r s e c o n d - o r d e r w a v e m o d e l the non-Gaussian sea surface is simulated by a model based on the second-order Stoke's expansion [8], outlined by Marthinsen and Winterstein [9] for a deep water condition. By this model, a consistent dispersion relation is also obtained. Furthermore, the skewness and kurtosis of the sea surface are implicitly given by the model. Wave kinematics are, in both cases, calculated from the simulated nonGaussian surface elevation time series using the W h e e l e r S t r e t c h i n g m e t h o d . [16] This method is an engineering approximation which lacks a hydrodynamic basis. Comparisons to measurements show, however, that it gives a reasonable representation of the wave flow which is acceptable for most engineering applications [3, 7]. For the Hermite Expansion Method a method of this type has to be used. For the Second Order Wave Model a consistent kinematics model exist [14]. This is, however, somewhat complicated to use and so far it has not been proven to give better results when compared to measurements than the Wheeler method.
RESPONSE EFFECTS IN D E S I G N SEA STATES The objective of this part of the study is to investigate the effects of nonGaussian waves to estimates of extreme dynamic response and dynamic amplification factors in sea states normally considered in design of jack-ups in the North Sea. Five sea states are considered: • Hs = • Hs = • Hs = • Hs = • Hs =
14.0m 14.8m 14.0m 13.0m 12.0m
and Tp = 17.0sec and T e = 16.0sec and T e = 14.0sec and To = 13.0sec and T e = 12.0see.
Each sea state is modelled by a JONSWAP spectrum, assuming longcrested waves. The duration of each sea state is set to 6hours.
Non-Gaussian waves
137
Currem: velocities corresponding to a 10year return period are used along with these: sea states and the current velocities are: • • • •
at mean water level 1.10ms -1 -30m 0.75ms -1 -50m 0.70ms -~ -105m 0.70ms -1.
The hydrodynamic coefficients used in this analysis are: • Mass coefficient CM: 2.0 • Drag coefficient CD: 1.0--In irregular wave analysis • Drag coefficient CD: 0.7--In design wave analysis. The structural damping level in the basic modes is assumed to be 2%. For each sea state 8 samples of waves and response are generated, and the presented results represents averages of the actual parameter from these samples. The complete results from the study are presented and discussed by Karunakaran et al. [6]. Thus, in the following only the main conclusions supported by examples of main findings are given. Simulated sea surface
The average skewness and kurtosis coefficients from the eight independent simulated non-Gaussian sea surfaces by the second order wave model are presented in Table 1. In the same table the input skewness and kurtosis coefficients for the non-Gaussian sea by Hermite expansion are also shown. It is seen that the second order wave model gives slightly lower kurtosis coefficients than was calculated by eqn (4), whereas the skewness coefficients are comparable to what was calculated by eqn (2). Thus, there is a small but clear difference between the two methods, and the difference tends to increase as the steepness increases. Since the Hermite method reflects empirical data, the results indicate that the second order model underestimates the non-Gaussian nature of the waves. Effects on structural response
The response of the platform, simulated using Gaussian and non-Gaussian sea surface using the Hermite Expansion M e t h o d is compared in Table 2. It can be seen that the use of non-Gaussian sea surface increases the standard deviation of the quasistatic response by 5-8% for all response quantities. Concerning extreme response, the use of non-Gaussian sea surface increases the extreme quasistatic response by about 20% for the least steep sea state,
N. Spidsoe, D. Karunakaran
138
TABLE 1 Statistical properties of non-Gaussian sea surface elevation
Sea state
H s = 14.0m Te=17s Hs = 14.8m Tp = 16s Hs = 14.0m Tp = 14s Hs = 13.0m Tp= 13s Hs = 12.0m Tp = 12s
Non-Gaussian sea Hermite expansion
Non-Gaussian sea Second order wave model
Skewness
Kurtosis
Skewness
Kurtosis
&
0.17
3.09
0.18
3.06
&
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3.12
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3.08
&
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3.19
0.24
3.10
&
0.27
3.22
0.25
3.10
&
0.29
3.26
0.27
3.12
TABLE 2 Ratio between response estimated using non-Gaussian sea surface (by Hermite expansion) and Gaussian sea surface (Average from eight samples)
Response Quantity
Standard deviation o f response Quasistatic
Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement
Dynamic
Hs = 14.0m & Tp = 1.05 1.06 1.05 Hs = 14.8m & Tp = 1.06 1.06 1.06 1.07 1.06 1.07 Hs = 14.0m & Tp = 1.07 1.07 1.08 1.08 1.08 1.08 Hs = 13.0m & Tp= 1.07 1.07 1.08 1.08 1.08 1.08 Hs = 12.0m & TI, = 1.08 1.08 1.08 1.08 1.08 1.08
Extreme response Quasistatic
Dynamic
1.20 1.23 1.23
1.20 1.23 1.23
1.25 1.29 1.29
1.28 1.30 1.30
1.38 1.44 1.44
1.33 1.36 1.35
1.41 1.47 1.46
1.37 1.42 1.41
1.51 1.57 1.55
1.41 1.43 1.42
17sec
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13sec
12sec
Non-Gaussian waves
139
i.e., H s = 14m & T e = 17sec, while the increase is as high as 57% for the steepest sea state, H s = 12m & Tp= 12sec. The increase in the extreme dynamic response is about 20% for H s = 14m & Tp = 17sec and 43% for H s = 12m & Te = 12sec for base shear and overturning moment. The results obtained by the S e c o n d order wave m o d e l show similar effects as found by the H e r m i t e E x p a n s i o n M e t h o d . The effects are, however, less in this case.
Comparison of design wave analysis and stochastic time domain analysis Stochastic time domain analysis is often used for estimation of dynamic amplification factors which are to be applied to the quasistatic response from the design wave analysis in order to account for the dynamic effects. This requires that consistent wave and wave load parameters and models are used in the two types of analyses. Consistent wave parameters are normally obtained by a long term analysis of wave data. Furthermore, since Morison equation is used in both cases, the drag coefficient is used as a calibration factor for the loading. The results of this calibration factor is obviously influenced by the wave models involved. Normally a Stoke's V order wave model is used in the design wave analysis, while linear, Gaussian waves are used in the stochastic analysis. If non-Gaussian waves are employed in the stochastic analysis the consistency will definitely be improved. However, the results of the calibration will also be changed. In order to check this, a comparison of extreme quasi-static response obtained by a stochastic analysis to design wave analysis results was carried out. The results of this comparison are summarized in Table 3. These results indicate that the choice of wave model in the stochastic analysis has a significant effect on the calibration. Furthermore, they indicate that, along with non-Gaussian waves, a drag coefficient which is comparable to that used in the Stoke's V order design wave analysis should possibly be used. This is based on the assumption that the Stoke's V order design wave analysis gives proper quasistatic load levels.
Effects on dynamic amplification factors As mentioned above, the dynamic amplification factor (DAF) is often estimated using stochastic time domain analysis and later applied to the quasistatic response from design wave analysis. The influence of non-Gaussian sea surface on the D A F was therefore investigated. The results are given in Table 4 and show that the D A F in extremes, in general, decrease when a non-Gaussian sea surface is used. There are two reasons for this. First, as discussed above, the effect of non-
N. Spidsoe, D. Karunakaran
140
TABLE 3 Ratio between quasistatic response from time domain analysis (Hs = 14.8m & Tp = 16 sec) and design wave analysis (H = 27m & T = 14.5 sec) using Stoke wave model
Base shear C D = 1.0 in Gaussian sea surface Non-Gaussain-Hermite expansion Non-Gaussian-Second order wave CD = 0.7 in Gaussian sea surface Non-Gaussian-Hermite expansion Non-Gaussian-Second order wave
Overturning moment
time domain analysis 1.07 1.04 1.33 1.34 1.26 1.33 time domain analysis 0.76 0.74 0.95 0.96 0.92 0.96
Deck displacemen t 1.04 1.34 1.33 0.74 0.96 0.95
TABLE 4 Dynamic amplification factors in extreme response estimated using Gaussian and nonGaussian sea surface
Response Quantity
Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement Base shear Overturning moment Deck displacement
Gaussian sea surface Hs = 14.0m 1.13 1.27 1.23 Hs = 14.8m 1.14 1.29 1.25 Hs = 14.0m 1.19 1.37 1.32 Hs = 13.0m 1.18 1.35 1.30 Hs = 12.0m 1.20 1.39 1.33
Non-Gaussian sea surface Hermite expansion Second order wave & Tp= 17sec 1.13 1.26 1.23 & Tp= 16sec 1.16 1.30 1.26 & Tp= 14sec 1.15 1.29 1.24 & Tp = 13sec 1.15 1.30 1.25 & Tp = 12sec 1.12 1.26 1.21
1.07 1.18 1.14 1.13 1.24 1.20 1.07 1.17 1.13 1.11 1.25 1.19 1.07 1.21 1.15
Non-Gaussian waves
141
Gaussian waves is stronger on the quasistatic response than on the dynamic response. Next, it is because the resonant response in the case of non-Gaussian waves includes stronger transient effects than in the Gaussian case and this introduces a phase shift between the maximum quasistatic and dynamic response. This is further discussed in the next section.
D Y N A M I C RESPONSE EFFECTS IN STEEP WAVES Transient resonant response, termed ringing, induced by higher order wave forces in special sea states, may be significant for the extreme response of large volume structures like gravity platforms and tension leg platforms. This has been shown by model tests and computer simulations. Whether ringing is a significant problem for drag-dominated platforms or not is not determined. As mentioned in the introduction, ringing response is observed from full scale measurements, but it is not clear if this has any important effects to the extreme response compared to other nonlinear effects. In the study on effects from non-Gaussian waves, ringing effects were observed. The objective of this study is to investigate further how the applied load models may generate a ringing response, and to investigate the effects of ringing oil the extreme response and the sensitivity of these effects on the load and structural parameters. This is done by using measured surface elevations from experiments on ringing on large volume structures carried out by M A R T I N T E K as an input to simulations. Furthermore, model test data from an experiment with a jack-up platform performed by DHI are studied. Six measured realizations of a sea state are considered for the simulations. The sea state in full scale is characterized by H s = 14.5m and Tp = 16.0sec and each sample is 5.4h long. The skewness and kurtosis coefficients of the samples are typically 0.23 and 3.26, indicating rather strong non-Gaussian nature at a modest steepness. Since the aim of the work is to study the resonant response around the large wave cycles, it is sufficient to use a shorter length of wave time series containing such wave cycles. Hence, from these six realizations, 11 short sea elevation samples of 4859sec (1.35h) long, each having at least one big wave cycle which has been found to generate a ringing response in large volume structure,J, are established. They are here after identified as sample #1, sample #2 etc. Current is not included in this study. The drag and inertia coefficients used in the analysis are 1.0 and 2.0, respectiwfly.
N. Spidsge, D. Karunakaran
142
The investigation is organized in three parts: • a base case study • sensitivity studies • evaluation of model test data. The base case study is performed for a foundation condition, which gives the structure a first natural period equal to 5.7sec. The damping level is assumed to be 2% in the basic modes. This part of the study focuses on the nature of the resonant response and the dynamic amplification of extreme response. The sensitivity study concerns the influence of the ringing response due to variation of the following load and structural parameters: • • • •
Damping level Damping mechanism Natural period Load pattern.
The results are presented only for base shear. The nature of this response quantity is, however, representative for the complete response process. The evaluation of the model test data includes a study of measured dynamic response. Base ease
Figures 2-6 show examples of the wave process, the quasistatic response, the resonant response and the total dynamic response in terms of 200sec long time series window around the extreme wave event in each wave sample. From these plots it is seen that resonant response with typical ringing characteristics is generated by the extreme waves in samples 1, 2 and 11. These examples are representative for the majority of the investigated samples. It is possible that the ringing also occurs in some of the remaining samples. However, in these cases it seems that the resonant response builds up over several wave cycles, and thus does not get the typical characteristics of an instantaneous build up, as shown in samples 3 and 10. It may therefore be concluded that the applied wave and load model generate a ringing response of the platform. From the plots it is seen that the resonant response in the ringing events is significant, in fact in the range of 40--60 percent of the maximum quasistatic response. However, it is also noted that the effect of the ringing response to the extreme total dynamic response generally is small. This is because the extreme response of a dragdominated platform always is dominated by quasistatic response induced by the extreme wave crests and that the ringing response is phase shifted behind this. The effect of the ringing thus depends highly on this phase lag.
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N. Spidsoe, D. Karunakaran
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N. Spids~e, D. Karunakaran
148
Sensitivity study The following parameters are included in the sensitivity study: • • • •
Structural damping Hydrodynamic damping Natural period Drag and inertia coefficient.
Structural and soil damping Three different cases are studied; they are: • 1% structural and soil damping • 2% structural and soil damping • 3% structural and soil damping. The results from this study are presented in Table 5. In this presentation only five samples are considered. The results indicate that the damping has only about 1-1.5% effect on the total dynamic response. The damping has about a 5-10% effect on the ringing response. However, the effect is smallest for the condition where the resonant response is of pure ringing type, sample #2 and sample #11, see Table 5. The resonant response of three samples are presented in Fig. 7.
Hydrodynamic damping The dynamic analysis is also performed using relative velocities in Morison equation. The results are presented in Table 6. The results indicate that the inclusion of hydrodynamic damping decreases the total dynamic response by about 7-10 %, whereas it decreases the resonant response by about 15-30%. In general, hydrodynamic damping reduces the ringing response. TABLE 5 Sensitivity of extreme dynamic response and resonant response to damping--Base shear
Sample identification
Total dynamic response (MN) I°A
damping Sample Sample Sample Sample Sample
#I #2 #3 #10 #11
15.3 16.2 12.9 18.4 13.0
2%
3%
damping damping 15.1 16.1 12.6 17.9 13.0
15.0 16.1 12.3 17.4 12.9
Resonant response ( MN) 1%
2%
3%
damping damping damping 7.66 7.39 5.47 9.51 6.10
7.12 6.95 5.01 8.65 5.78
6.64 6.55 4.73 7.91 5.48
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iV. Spids~e, D. Karunakaran
TABLE 6 Sensitivity of extreme dynamic response and resonant response to hydrodynamic damping Base shear
Sample identification
Sample #1 Sample #2 Sample #3 Sample #10 Sample #11
Total dynamic response (MN)
Resonant response ( MN)
Without hyd. damping
With hyd. damping
Without hyd. damping
With hyd. damping
15.1 16.1 12.6 17.9 13.0
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7.12 6.95 5.01 8.65 5.78
5.67 5.57 4.19 6.94 4.87
Drag and inertia coefficients In order to check the importance of drag forces vs inertia forces with respect to ringing, the dynamic response analysis was performed using only the drag loading in one case and only the inertia loading in another case. The resonant response for two samples where the resonance is of pure ringing type is shown in Fig. 8. In this figure the resonant response using both drag and inertia loading is shown, but only the drag loading and only the inertia loading are compared. It is clearly seen that the ringing response is almost totally generated by drag loading.
Natural period The dynamic response analysis is performed for the same structure using two different foundation conditions which gave the first natural periods to be 4.5sec and 7.5sec. The results are shown in Table 7 and in Fig. 9. It is seen that the ringing response and its effects on the total response increases as the natural period decreases. This is because the phase lag between the ringing response and the quasistatic response reduces. It also shows, however, that the ringing response increases as the natural period approaches half of the wave crest period, thus indicating that the integrated drag forces in the wave crest are the main source of ringing for this structure. Evaluation of model test data
The applied model test data refer to an experiment with a dynamic jack-up in model scale 1:49 conducted by MSC at DHI laboratories [2]. This experiment included a variety of sea state conditions. Among these, the one with highest steepness was selected for inspection of ringing events. This sea state
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is characterized by a Pierson-Moskowitz spectrum with significant wave height and peak period equal to 0.20m and 2.2s, respectively. The estimates o f skewness and kurtosis for the measured wave sample are 0.20 and 3.10. The sea state is thus less steep and less non-Gaussian than the waves involved in the base case study. In Figs 10 and 11, observed wave and response situations with a ringing like nature are shown. The ringing nature is definitely not so
N. Spidsoe, D. Karunakaran
152
TABLE 7 Sensitivity of extreme dynamic response and resonant response to natural period, Base shear
Sample identification
Sample Sample Sample Sample Sample
#1 #2 #3 #10 #11
Total dynamic response ( MN)
Resonant response ( MN)
To=4.5sec To=5.Tsec To = 7.Ssec
To=4.5sec To=5.7sec To = 7.5sec
15.6 18.6 12.4 15.8 14.8
15.1 16.1 12.6 17.9 13.0
13.5 15.3 9.77 13.9 12.9
9.51 9.79 7.58 12.1 9.97
7.12 6.95 5.01 8.65 5.78
3.22 3.12 2.39 3.73 2.77
pronounced as in the simulated cases discussed above, probably due to the different sea state nature. However, it is noticed that the quasistatic and transient resonant response combines in the same manner as the simulations, thus indicating that the simulation method produces realistic ringing effects.
CONSEQUENCES FOR D E S I G N CALCULATIONS Design practice for jack-up platforms is based on the design wave method. The extreme quasistatic response is then calculated by deterministic design wave analysis and the dynamic response found by multiplying this with a dynamic amplification factor (DAF) obtained from a stochastic response analysis or from some kind of simplified method. The effects of non-Gaussian waves discussed in this paper concern only the stochastic analysis. The presented results show that if a non-Gaussian wave model is applied, a lower drag coefficient should be used than for Gaussian waves, in order to obtain a consistent quasistatic response compared to the design wave analysis. Furthermore, the results show that if non-Gaussian waves are used, this will give lower DAFs than with Gaussian waves, i.e., a beneficial effect with respect to design. The results presented in the paper also show that significant transient response caused by higher order wave forces must be expected for jackup platforms and that these will influence the dynamic amplification. However, it is also shown that these effects are included in a standard nonlinear stochastic response analysis procedure. This will not be the case if more simplified methods are used for this purpose. Thus, this is another strong argument for using nonlinear stochastic response analysis for estimation of DAFs in the design process of drag-dominated platforms.
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Non-Gaussian waves
E
Surface e l e v a t i o n - ncO6unlid
0.25 0.20 0.15 0.10 0.05
o as
0 -0.05 -0.10 -0.15 410
415
420
425
430
T i m e [see] 25 20 15 10 5 0 -5 -10 -15 -20 410
Q u a s i s t a t i c - B a s e shear - n c 0 6 u n l i d
4]5
410
4½5
430
T i m e [sec] R e s o n a n t r e s p o n s e - b a s e shear - n c 0 6 u n l i d 20 15
z
10 5 0 -5 -10 -15 -20 410
4 !5
420
425
430
425
430
T i m e [sec] Total b a s e shear - n c 0 6 u n l i d
40 30 20 10 0 -10 -20 -30 -40 410
415
420 T i m e [see]
F i g . 11. W a v e h e i g h t a n d q u a s i s t a t i c , d y n a m i c a n d r e s o n a n t r e s p o n s e - - - m o d e l
test.
156
N. Spids~e, D. Karunakaran CONCLUSIONS
The effects of using non-Gaussian waves in the nonlinear stochastic response analysis of jack-up platforms have been discussed. It has been shown that the application of non-Gaussian waves significantly affects both the steady state dynamic response and the transient response, compared to the use of Gaussian waves. The resulting effect on the total response is that the quasistatic response increases while the dynamic amplification reduces. Transient response--so called ringing--has been addressed in particular. It has been shown that commonly applied wave load models for jack-up analysis generate a ringing response due to the higher order components of the drag force, both when Gaussian and non-Gaussian waves are used, but strongest in the latter case. The transient response influences significantly the dynamic amplification. Thus, the results clearly show that a nonlinear stochastic dynamic response analysis procedure should be applied for the estimation of DAFs in the design of jack-ups. Furthermore, the results demonstrate that non-Gaussian waves should be considered in such analyses. The hydrodynamically consistent wave models available for this today are not sufficiently accurate or computer efficient; therefore, improved models for this purpose should be developed. Until then, it is believed that the Hermite Expansion Method applied in this study could be used as a reasonable engineering approximation. However, this method should also be verified, based on model test data and full scale measurements, as soon as possible.
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Non-Gaussian waves
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
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