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Verification of methods for simulation of nonlinear dynamic response of jack-up platforms
/
MarineStructures10 (1997) 181-219 © 1997 Elsevier Science Limited All rights re.fred. Printed in Great Britain PII: S0951-8339(96)00023-8 0951-8339/97/$17.00 + .00
ELSEVIER
Verification of Methods for Simulation of Nonlinear Dynamic Response of Jack-up Platforms
D.Karunakaran & N.Spidsoe S 1 N T E F Civil a n d E n v i r o n m e n t a l Engineering, N7034 T r o n d h e i m , N o r w a y
ABSTRACT The validity of mathematical and numerical models commonly applied for simulation of the nonlinear stochastic response of jack-up platforms is discussed. The basis for the discussions is the comparison of simulated response to model test data. The model test data refer to an experiment, which in full scale refers to a typical harsh environment truss leg jack-up. The test data include wave elevation, wave particle kinematics and static and dynamic response. The verification study emphasises wave kinematics, hydrodynamic loading, damping and dynamic response. © 1997 Elsevier Science Ltd
INTRODUCTION Jack-up platforms designed for moderate to deep waters will generally behave as highly dynamic systems because of the structural flexibility. The environmental forces acting on these jack-ups will therefore induce significant dynamic response. Furthermore, the wave and current induced loading is of a highly nonlinear nature due to nonlinear drag force and free surface effects. These nonlinearities in the load process will introduce nonlinear structural response even if the structure acts as a linear system. However, nonlinear behaviour is also introduced by p-A effects due to large structural motions, by nonlinear soil-structure interaction and by the hydrodynamic drag damping arising due to the relative velocity between the structure and the fluid. All these nonlinearities will influence the dynamic response significantly, which therefore demands the application of a time domain analysis, see 181
D. Karunakaran, N. Spidsoe
182
Kjeoy et aL [5], Leira et al. [6], Mommaas and Grundlehner [7], Karunakaran et al. [4] and Spidsoe and Karunakaran [14]. These nonlinearities are modelled by a set of mathematical and numerical models which are combined into a time domain analysis procedure. The objective of this paper is to discuss the validity of some of these models through a systematic comparative study between the numerical simulation and jack-up model test data. The discussions in this paper are limited to selected test cases, and the comparison of simulations with the model test data include: • • • • •
Wave kinematics Hydrodynamic forces Static response Damping Dynamic response.
DATA BASIS A N D VERIFICATION P R O C E D U R E Model test
The model test was carried out by Marine Structures Consultants (MSC) bv, in the offshore wave basin of the Danish Hydraulic Institute (DHI). The model test set-up in the wave basin is shown in Fig. 1. For details of the model test refer to [8] and Grundlehner [1]. The main details of the model test are: • Model scale 1:49 • Model supported by ball joints • 0.4m wide truss legs including racks, with 1.5m leg spacing. The tests were carried out with a full jack-up model and a separate leg model, also shown in Fig. 1. Furthermore, the full jack-up model test is performed both with the jack-up hull fixed so that the static response can be measured and with the hull free which will be similar to the jack-up in operating condition. In the full jack-up model the following responses are measured: • Hull displacement • Forces and moment at top of leg • Forces at the bottom of leg. From the measured leg forces and moments the base shear and overturning moment of the structure are estimated.
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A series of deflection and decay tests were performed for the full jack-up model. From these it is estimated that the natural period is 1.05see and the structural damping is 2.8% (decay test in air). The decay tests in water gave a total damping level of 4.2% which includes the structural damping, hydrodynamic radial damping and potential damping. The separate leg model is used to estimate the hydrodynamic loads at various levels of an individual leg. In the separate leg model the hydrodynamic force on two bays were measured by transducers placed on the leg's aft chords. The separate leg model is constructed such that these transducers will measure the local forces of one bay only. The tests were performed with the sensor bays placed at three different locations, such that it can measure the local forces in a wave crest position, a wave trough position and in a totally submerged position, as shown in Fig. 2.
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M _~---~
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Fig. 2. Arrangement of separate leg test set-up.
In all test cases the wave surface elevation is measured. Both regular and irregular wave cases are considered. In some cases the current is also included. Furthermore, for selected cases, the wave kinematics are measured at one point 0.10m below wave surface and at three fixed positions along the depth, i.e., at el.-0.35m, e l . - 0 . 7 0 m and el.-1.40m.
Applied data The tests were performed for a variety o f wave and current conditions. In this paper, however, only the following three conditions are considered: Regular wave case (Identifier--nc03uni) Irregular wave (Identifier--nc06uni) Irregular wave (Identifier--nc07uni)
H = 0.450m and T = 2.1 sec
Extreme wave
Hs = 0.200m and Tz = 1.5see
Extreme sea state
Hs = 0.062m and Tp = 1.1 sec
Peak period o f sea state corresponding to the natural period o f the model.
All these considered cases are without current.
Methods for simulation of nonlinear dynamic response
185
Simulation procedure and model The simulations are carried out applying the procedure described in Karunakaran 113]. The procedure is based on: • FEM[ modelling of the structure • Morison equation for force calculation • Mea,~ured wave time series as input to wave kinematic calculation. The simulations are carried out in model scale. The key reason for this is that the load coefficients to be applied in the load calculations will be flow dependant, i.e., they must relate to model test conditions and not full scale flow conditions. Both for the separate leg and for the full jack-up, the leg truss work is idealised as single stick models with equivalent diameters, CD and CM as per the Jack-up Industry Recommended Practice (JIRP), Society of Naval Architects and Marine Engineers [11]. The simulation model has a natural period of 1.07 sec. Verification procedure The veritication study is performed in the following five steps: (1) Comparison of measured and simulated wave kinematics. For this comparison the measured wave without the presence of the structure is used. (2) Comparison of measured and simulated hydrodynamic forces from the separate leg model. For this study the wave elevation measured in the presence of the structure is applied. (3) Comparison of static response of the full jack-up model. (4) Comparison of dynamic response of the full jack-up model. (5) Comparison of resonant response in order to estimate the damping level. The con0tparisons are based on time series plots, and estimates of statistical parameters, spectral densities, response functions and probability distributions.
WAVES AND WAVE KINEMATICS The wave kinematics are estimated from the input wave elevation using Fast Fourier Transformation (FFT) using the Wheeler stretched wave model [16]. This comparative study is performed only for two wave conditions, i.e., the extreme regular wave case (nc03uni) and the extreme irregular wave case (nc06uni).
D. Karunakaran, N. Spidsoe
186
The time traces of the measured and simulated wave induced velocities are shown in Fig. 3 for the regular wave case. In the regular wave case, the measured velocities have higher order components seen at the crest-trough transition part. This is probably due to measurement problems caused by vortices as described by Grundlehner [1] and this higher order component is not seen in the measured hydrodynamic forces which will be discussed later. This means that comparison is to be restricted to first and second order components, i.e., for frequencies less than 1.25Hz. The time traces of measured and simulated velocities containing the first and second order components are shown in Figs 4 and 5. As seen from Fig. 4, the measured and simulated velocities are in phase for both first and second order components for velocities at level-0.10. On the other hand, as seen from Fig. 5, at level-0.35, the measured second order component is out-of-phase to the simulated velocity by 180 ° . It was also seen that the measured second order component of velocity is out-of-phase to the measured second order component of wave elevation. However, the measured velocity has full coherence with the measured wave elevation. At the other two levels also, similar out-of-phase second order velocity compoo VelOCity
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nents are observed. There is no explanation for this at this moment and hence in the comparison between measured and simulated velocities for regular waves, only the first order components are considered. Comparison of the first order components shown in Figs 4 and 5 indicate that the simulation overpredicted the velocities by about 20%. In the irregular wave case, the measured surface elevation and measured velocities are in phase with full coherence. Hence, the comparison between measurement and simulation is done without any filtering. The statistical properties of the measured and simulated wave kinematics for the irregular wave are presented in Table 1. For tile irregular wave condition, the simulation underpredicts the kinematics at the wave crests at el.-0.10m. However, at lower levels, the simulated kinematics compare very well with measurements. The underprediction at below wave crest in the surface zone by the Wheeler model is in line with the earlier findings by Gudmestad and Haver [2].
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The time traces of the measured and simulated wave induced velocities are shown in Fig. 6. As seen from this figure the measured and simulated velocities compare very well, except for the wave crests at el.-0.10m. TABLE 1 Comparison Between the Measured and Simulated Velocities Irregular Wave (nc06uni) Standard deviation
Skewness
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Minimum
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(m/s}
(re~s) At el. -0.10 At el. -0.35 At el. -0.70
Meas.
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Meas.
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0.154 0.091 0.060
0.152 0.102 0.067
0.06 -.25 -.27
0.20 -.27 -.23
Meas. Sire.
3.80 3.61 3.24
3.18 3.09 3.05
Meas. Sire.
Meas.
Sim.
0.66 0.29 0.20
-0.43 -0.38 -0.23
-0.48 -0.41 0.24
0.58 0.28 0.21
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H Y D R O D Y N A M I C FORCES Separate leg model For this study both the two extreme wave conditions are used (nc03uni and nc06uni). As described earlier, the leg truss work is idealised as a single stick model in the simulations. The hydrodynamic forces are measured at two levels representing the force from two bays in the leg truss work. Furthermore, the tests were repeated three times by keeping the sensor bays at different elevations, thereby the wave forces at different elevations are measured, see Fig. 2. The different positions and their corresponding levels are: Upper Lower Upper Lower Upper Lower
bay--Upper level bay--Upper level bay--Intermediate level bay--Intermediate level bay--Lower level bay--Lower level
--Elevation --Elevation --Elevation --Elevation --Elevation --Elevation
around + 0.17 around + 0.00 around-0.18 around-0.35 around-0.87 around- 1.04.
190
D. Karunakaran, N. Spidsoe
Regular wave--ncO3uni For the regular wave case, the force measurements are available for all sensor bays at all levels. The measured and simulated wave forces along the leg are shown in Fig. 7. It is seen that the simulations significantly overpredict the forces in the surface zone. The overprediction is the highest at the top level and reduces systematically at lower levels. As seen from Figs 3, 4 and 5, there is no indication that this is due to the difference in the wave particle velocity. Hence, the reason for the significant difference in the forces is from the force mechanisms. Alternatively, a reduced Cn in the surface zone should be used. This is in accordance with the results from an earlier study where the forces on a single pile were measured almost in similar flow conditions. In that study, the finding was that the force coefficients were lower in the splash zone than for the submerged part of the structure [12]. I~orce
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For the parts of the leg which are totally submerged (bays at lower level), the simulated forces are only slightly larger than measurements. For the bay which is just below the splash zone (Lower bay at intermediate level), the simulated forces are about 50% higher than the measured forces. The elevation of this bay is approximately - 0 . 3 5 m , a location at which the measured velocity is available, see Fig. 5. Part of the difference in forces attributes to the difference in the wave kinematics. Furthermore, the measured forces and the simulated forces have different frequency components. In order to study this, the measured and simulated forces are band-pass filtered such that the forces around the wave loading frequency, and multiples of the wave frequencies are compared. This comparison is found in Fig. 8. It is seen that the first order forces are about 20% higher in the simulation than the measurement. As discussed in Section 3, the first order
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component of simulated wave particle velocity is 20-25% larger than the measurement. This should have given a 50% increase in forces if the assumed load coefficients are correct. This implies that the load coefficients used are lower than what should have been, indicating that the Co should be higher than 1.0. The Reynolds number (Re) calculated based on the velocities close to the surface is about 1×104 and the Keulegan Carpenter number (Kc) is about 60. From all the measured data summarized in Sarpkaya and Isaacson [9] and further discussed in Spidsoe [12], the Cn in such flow conditions are higher than 1.0. The second and higher order forces in the simulation are significantly higher than the measurements. Possible explanations may be: (1) The second order forces are caused by the direct excitation caused by second order wave component. The Re for waves at the higher frequencies will be around 4000 and the Kc will be about 10. This indicates that the CM around these frequencies will be around 1.4 [10, 15]. A lower C ~ at these frequencies will reduce the second order forces. This will have significant effect on the dynamic response, since the wave frequency corresponding to the second order forces match the natural frequency of the model. (2) The third order component has both direct excitation and the superharmonic excitation arising from the Morison equation. However, in the measurements the third order forces are very low compared to the simulations. This is because in the measurements, the third order forces are cancelled due to the phase effects between the chords of the leg. The cancellation frequency is exactly three times the wave frequency. In the simulation model, the leg is idealised by a stick model which do not have any cancellation.
Irregular wave--ncO6uni For the irregular wave case, the force measurements are available for all sensor bays at upper and intermediate levels only. The comparison between the measured and simulated wave forces along the leg are shown in Table 2 and in Fig. 9. In this case also, it is seen that the simulations overpredict the forces in the surface zone as in the regular wave case, due to the same reasons discussed for regular wave case. At lower down in the leg the simulated forces compare reasonably well with the measured forces. At lower bay--intermediate level, the simulated forces are systematically smaller than measurements. As discussed in Section 3, for the irregular wave case, the measured and simulated kinematics compared very well, which indicates that the smaller simulated force is due to a
195
Methods for simulation of nonlinear dynamic response
TABLE 2 Comparison Betweenthe Measured and Simulated Forces--Irregular Wave
Up. bay-Up. level Low. bay-Up. level Up. bay-Int. level Low. bay-Int. level
St. Devation (N)
Skewness
Meas. Sim.
Meas. Sire.
Kurtosis Meas.
0.086 0.082 7.77 12.35 96.6 0.465 0.535
3.13
Maximum (iV)
Sim.
Meas. Sire.
195
1.563 2.013
Minimum (iV) Meas.
Sim.
3.63 16.87 22.60 4.273 5.476 -0.589-1.005
0.564 0.510 0.43 0.35 4.38
3.75 2.307 2.128-1.896-1.537
0.459 0.433-0.79-0.80
6.40 1.474 1.530-2.603-3.013
5.68
lower CD in simulations. This is again in line with the discussion from the regular wave case.
Full jack-up model The hydrodynamic load on the full jack-up model is measured by fixing the hull. In this comparative study, two wave conditions are considered, the extreme regular wave case (nc03uni) and the extreme irregular wave case (nc06uni). Regular wave-ncO3uni
The measured and simulated base shear and overturning moments are compared in Fig. 10. It is clearly seen that the simulations overpredict both the base :shear and overturning moments. The difference is higher for overturning moments than for base shear. If a reduced CD is used in the wave surface zone as indicated in earlier sections, both base shear and overturning m o m e n t will be reduced and the reduction will be higher for overturning moment. This is, however, not the only source of difference between the measured and simulated forces. As seen from Fig. 10, there is a difference in the frequency components. This is highlighted in Fig. 11, where the band-pass filtered base shear is shown. The figure indicates that the simulated base shear is slightly lower than measurements at the wave frequency and is higher than measurements at frequencies in multiples of wave frequency. As discussed previously, if an increased Cz) is used, the force a r o u n d wave frequency can be matched. The main reasons for large
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Methods for simulation of nonlinear dynamic response
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difference at high frequency are the superharmonic excitation at second and third harmonics and also the reduced C u at high frequencies as discussed earlier and the big difference at three times wave frequency is also due to the wave cancellations seen in the model test which are not modelled in the simulations.
Irregular wave--ncO6uni The measured and simulated base shear is shown in Fig. 12. The figure indicates that the simulation underestimates the response slightly at large wave cycles. This is also noticed from the hydrodynamic response function, which is tlhe ratio between the response spectrum and wave spectrum, shown in Fig. 13, where there is significant differences in the frequency contents. The simulations underestimate the forces at and around wave peak frequency and overestimates at higher frequencies. This can also be seen from the filtered response shown in Fig. 14. The reason for this is discussed in the previous section and they are: (1) The Re calculated based on velocities simulated close to the surface from the large wave cycle is around 9000. However, the average Re willt be much lower at the surface and will decrease further down. The same is the case with Kc which is about 50 for the large wave cycle. From the measured data summarized in Sarpkaya and Isaacson [9] and further discussed in Spidsoe [12], the CD in such cases are higher than 1.0. This is also seen from an earlier study with force measuremeats from a single pile [12]. (2) At frequencies around 1Hz, the Re and Kc will be of the order 3000 and 10, respectively. For such low Kc values, the C u will be of the order 1.4 (instead of 1.8 used in the simulations). (3) A reduced CD at the surface zone as discussed previously. To investigate the possible effects, the simulation is carried out with the following variable coefficients: (1) CD in the surface zone is reduced to 0.7. (2) Cn is increased to 1.3 for elements below surface zone. (3) CM is reduced to 1.4 for frequencies above 0.75Hz. The hydrodynamic response functions of base shear and overturning moments obtained using variable load coefficients are shown in Fig. 15. Comparing Figs 13 and 15, it can be seen that the use of variable coefficients has improved the simulated response function. This exercise also indeed indicates that the load coefficients are flow dependant.
D. Karunakaran, N. Spids~e
202
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DYNAMIC RESPONSE The dynalmc response is performed using the full jack-up model. The structural damping applied in the dynamic analysis is 2.8% as given by the decay tests in air. The simulations are performed with P-A effects and with relative velocities :in the Morison equation. For this comparison, two wave conditions are used, they are the extreme irregular wave case (nc06uni) and the resonant sea state (nc07uni). Extreme sea state (neO6uni)
The simulations are first carded out with Cn and C ~ as per JIRP. The frequency response function of measured and simulated base shear is compared in Fig. 16. It is clearly seen that around the wave peak frequency, there is an underestimation by the simulation and around the resonant frequency, there is a significant overestimation. As discussed in the previous section, if' an increased Cn is applied, the underestimation of loads around wave peak frequency, can be avoided. However, the overestimation of resonant response is due to reduced C~r as well as the level of damping. It is found that 5% linear damping is needed in order to get a good comparison with measurements when reduced CM is used. This linear damping level is slightly higher than the measured damping level of 4.2% in the decay test in water. The frequency response functions of simulated base shear using variable load coefficients and 5% linear damping are compared with measurements in Fig. 17. As seen from this figure, simulations with variable load coefficients and 5% linear damping gives the best fit to measurements. This is also confirmed by the comparison of statistical properties of measured and simulated response presented in Table 3. The next important step is the validation of the use of relative velocities in Morison equation. To verify that, the dynamic response is performed with varying load coefficients, but with absolute velocities in Morison equation. In this case, the linear damping level is increased to substitute the hydrodynamic drag damping. 3% increase in linear damping level (total 8% linear damping) gave a good match between simulated and measured response standard deviation. This increase in damping level, 3%, is also the same level recommended by the JIRP to account for the hydrodynamic drag damping. The statistical properties of this simulated response is also compared with measurements in Table 3. It is seen that the standard deviation of response is comparable to the measurements, whereas, the kurtosis coefficient of the simulated response is significantly higher when absolute velocity is used in the Morison equation. The response simulated using absolute velocities are more non-Gaussian than
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D. Karunakaran, N. Spidsoe
TABLE 3 Comparison of Measured and Simulated Response Statistics Irregular Wave--Extreme Sea State (nc06uni) Base shear
Overturning moment
Standard deviation (iV)
Skewness
Kurtosis
Standard deviation (Nm)
Skewness
Kurtosis
7.69
-0.04
3.93
19.73
0.03
3.95
Simulated-CD = 1 & 2.8% damping
8.26
-0.05
3.37
24.22
0.02
3.24
Sim.-Vari. Co and CM & 5% damp. + Rel. velocity
7.41
-0.06
4.01
20.83
0.01
3.78
Sim.--Vari. Co and CM & 5% damp. + Abs. velocity
7.48
-0.09
4.37
20.93
0.02
4.14
4.68
0.0
3.08
14.29
0.0
3.13
Simulated-42D = 1 & 2.8% damping
6.58
0.0
2.95
20.52
0.0
2.96
Sim.--Vari. Co and CM & 5% damp. + Rel. velocity
4.94
0.0
3.04
15.40
0.0
3.07
Sim.--Vari. CD and damp. + Abs. velocity
4.95
0.0
3.29
15.34
0.0
3.32
Total response Measured
Resonant response Measured
C M ~£ 5 %
the measured response, which will give larger extreme response t h a n the measurements. This is also d e m o n s t r a t e d by the comparison o f m a x i m a probability distributions shown in Fig. 18.
Resonance sea state (nc07uni) F o r this sea state, the dynamic analysis is performed using variable load coefficients. F u r t h e r m o r e , the analysis is performed with 5% linear d a m p i n g with relative velocities. The statistical properties o f the measured and simulated responses are c o m p a r e d in Table 4. It can be seen t h a t the s t a n d a r d deviation responses compare very well, however, the kurtosis coefficient is too small and the m a x i m a response is underestimated, see
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212
D. Kanmakaran, N. Spidsoe
TABLE 4 Comparison of Measured and Simulated Response Statistics Irregular Wave--Resonant Sea State (nc07uni) Base shear
Overturning moment
Standard deviation (N)
Skewness
Kurtosis
Standard deviation (Nm)
Skewness
Kurtosis
Measured
2.51
0.0
3.66
7.56
0.0
3.80
Sim.--Vari. Co and CM & 5% damp. + Rel. velocity
2.44
0.0
3.17
7.61
0.0
3.17
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3.12
0.0
3.11
8.60
0.0
3.12
Sim.--Vari. CD and CM & 5% damp. + Abs. velocity
2.60
0.0
3.28
8.11
0.0
3.29
maxima, see Fig. 20. This indicates that for this low sea state the relative velocity formulation is questionable.
MODELLING OF DAMPING It is evident from the discussions in the previous section that modelling of damping, both linear damping and hydrodynamic damping, is very important. The decay tests of the model in air indicated that the structural damping is of the order 2.8%. The decay tests of the model in water indicated the total damping to be about 4.2%, which includes the structural damping and hydrodynamic damping due to wave radiation and hydrodynamic potential damping. As seen from the previous section, for both extreme and resonant sea states, the linear damping level needed to have a good match with the measurements was about 5%. The structural damping from the decay tests is about 2.8% and the rest is due to hydrodynamic damping other than the drag damping. It is also seen from the comparisons that, for extreme sea states, the hydrodynamic drag damping is important and should be modelled by relative velocity formulation, whereas, for low sea state, the hydrodynamic drag damping is not present. Furthermore, the probabilistic nature of measured static and dynamic
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response confirm that there is Normalisation due to the linear and nonlinear damping mechanism [13]. This is exemplified in Fig. 21 where the maxima probability distribution of static, dynamic and resonant response of base shear at the extreme sea state is presented. It can be seen that the measured static response is highly non-Gaussian, whereas the measured dynamic response is less non-Gaussian than the static response and the resonant response is totally Gaussian.
CONCLUSIONS A comparison is made between the jack-up model test data and nonlinear time domain simulations applying measured wave elevations. The comparison is made in model scale in order to have proper reflection of flow conditions. The comparisons include: • • • •
Waw~ kinematics Hydrodynamic forces Static response Dynamic response and damping.
The study is limited to three wave conditions only and the following conclusions may be drawn from this limited study:
Wave kinematics The applied wave model, Wheeler stretching model, gives good comparison to measured wave kinematics below wave surface zone. However, this model slightly underestimates the kinematics below the wave crests at the surface zone.
Hydrodynamic forces The measured forces indicate that the load coefficients are flow dependant and it is important to apply force coefficients that are valid for the flow conditions in the model test.
Dynamic' response and damping The measured dynamic response clearly indicate that the dynamic response is less non.-Gaussian than the static response, thereby validating the Normalisation effects.
218
D. Karunakaran, N. Spidsoe
For extreme sea states, the relative velocity formulation seems valid. Modelling of the hydrodynamic drag damping by equivalent linear damping will not give a dynamic response having a similar probabilistic nature as the measured dynamic response. For small sea states, the relative velocity formulation should not be used and there is no hydrodynamic drag damping at the small sea states. The over all linear damping level for this structure is about 5%.
ACKNOWLEDGEMENTS The authors are grateful to Prof. Knut-Aril Farnes, Division of Marine Structures, Norwegian Institute of Technology, for the valuable discussions and suggestions during this study.
REFERENCES 1. Grundlehner, G. J., Systematic model tests on a harsh environment jack-up in elevated condition. In Proc. 5th Int. Jack-up Conf., London, September 1995. 2. Gudmestad, O. T. and Haver, S., Uncertainties in prediction of wave kinematics in irregular waves. In Wave Kinematics and Environmental Forces, vol. 29, Society for Underwater Technology, 1993. 3. Karunakaran, D., Procedure for nonlinear dynamic response analysis of offshore structures--both for extreme and fatigue response. SINTEF Report STF71 A91016, Trondheim, 1991. 4. Karunakaran, D., Gudmestad, O. T. and Spidsee, N., Nonlinear dynamic response analysis of dynamically sensitive slender offshore structures. In Proc. 11th Int. Conf. Offshore Mech. and Arctic Engng, Calgary, June 1992. 5. Kjeey, H., Bee, N. G. and Hysing, T., Extreme-response analysis of jack-up platforms. Marine Structures, 1990, 2, 125-154. 6. Leira, B.J., Karunakaran, D. and Nordal, H., Estimation of fatigue damage and extreme response for a jack-up platform. Marine Structures, 1990, 3, 461493. 7. Mommaas, C. J. and Grundlehner, G. J., Application of a dedicated stochastic nonlinear dynamic time domain analysis program in design and assessment of jack-ups. In Proc. Third Int. Conf. Jack-up drilling platform, London, September 1991. 8. MSC bv., Jack-up model tests at DHI-model test report and measurement results. MSC Report, Sehiedam, Nederland, 1995. 9. Sarpkaya, T. and Isaacson, M., Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, 1981.
Methodsfor simulation of nonlinear dynamic response
219
10. Sarpkaya, T., Discussion of quasi-2D forces on vertical cylinders in waves. Journal of Waterways, Port, Coastal & Ocean Engineering, 1984, 110, 1-2. 11. Recommended Practice for Site Specific Assessment of Mobile Jack-up Units, SNAME, 1994. 12. Spidsa~e, N., Wave force models for time domain dynamic analysis of dragdominated platforms: summary of a literature survey. SINTEF Report STF A89024, Trondheim, 1990. 13. Spids~e, N., Karanakaran, D. and Gudmestad, O., Nonlinear effects of damping to dynamic amplification factors for drag-dominated offshore platforms. In Proc. llth Int. Conf. on Offshore Mech. and Arctic Engng, Calgary, June 1992. 14. Spids~e, N. and Karunakaran, D., Nonlinear dynamic behaviour of jack-up platforms. Marine Structures, 1996, 9, 71-100. 15. Stansby, P.K., Bullock, G.N. and Shorr, I., Quasi-2D forces on a vertical cylinder in waves. Journal of Waterways, Port, Coastal & Ocean Engineerinng, 1983, 109, 1-2. 16. Wheeler, J.D., Methods for calculating forces produced on piles in irregular waves. Journal of Petroleum Technology, 1970, 1, 1-2.
MarineStructures10 (1997) 181-219 © 1997 Elsevier Science Limited All rights re.fred. Printed in Great Britain PII: S0951-8339(96)00023-8 0951-8339/97/$17.00 + .00
ELSEVIER
Verification of Methods for Simulation of Nonlinear Dynamic Response of Jack-up Platforms
D.Karunakaran & N.Spidsoe S 1 N T E F Civil a n d E n v i r o n m e n t a l Engineering, N7034 T r o n d h e i m , N o r w a y
ABSTRACT The validity of mathematical and numerical models commonly applied for simulation of the nonlinear stochastic response of jack-up platforms is discussed. The basis for the discussions is the comparison of simulated response to model test data. The model test data refer to an experiment, which in full scale refers to a typical harsh environment truss leg jack-up. The test data include wave elevation, wave particle kinematics and static and dynamic response. The verification study emphasises wave kinematics, hydrodynamic loading, damping and dynamic response. © 1997 Elsevier Science Ltd
INTRODUCTION Jack-up platforms designed for moderate to deep waters will generally behave as highly dynamic systems because of the structural flexibility. The environmental forces acting on these jack-ups will therefore induce significant dynamic response. Furthermore, the wave and current induced loading is of a highly nonlinear nature due to nonlinear drag force and free surface effects. These nonlinearities in the load process will introduce nonlinear structural response even if the structure acts as a linear system. However, nonlinear behaviour is also introduced by p-A effects due to large structural motions, by nonlinear soil-structure interaction and by the hydrodynamic drag damping arising due to the relative velocity between the structure and the fluid. All these nonlinearities will influence the dynamic response significantly, which therefore demands the application of a time domain analysis, see 181
D. Karunakaran, N. Spidsoe
182
Kjeoy et aL [5], Leira et al. [6], Mommaas and Grundlehner [7], Karunakaran et al. [4] and Spidsoe and Karunakaran [14]. These nonlinearities are modelled by a set of mathematical and numerical models which are combined into a time domain analysis procedure. The objective of this paper is to discuss the validity of some of these models through a systematic comparative study between the numerical simulation and jack-up model test data. The discussions in this paper are limited to selected test cases, and the comparison of simulations with the model test data include: • • • • •
Wave kinematics Hydrodynamic forces Static response Damping Dynamic response.
DATA BASIS A N D VERIFICATION P R O C E D U R E Model test
The model test was carried out by Marine Structures Consultants (MSC) bv, in the offshore wave basin of the Danish Hydraulic Institute (DHI). The model test set-up in the wave basin is shown in Fig. 1. For details of the model test refer to [8] and Grundlehner [1]. The main details of the model test are: • Model scale 1:49 • Model supported by ball joints • 0.4m wide truss legs including racks, with 1.5m leg spacing. The tests were carried out with a full jack-up model and a separate leg model, also shown in Fig. 1. Furthermore, the full jack-up model test is performed both with the jack-up hull fixed so that the static response can be measured and with the hull free which will be similar to the jack-up in operating condition. In the full jack-up model the following responses are measured: • Hull displacement • Forces and moment at top of leg • Forces at the bottom of leg. From the measured leg forces and moments the base shear and overturning moment of the structure are estimated.
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A series of deflection and decay tests were performed for the full jack-up model. From these it is estimated that the natural period is 1.05see and the structural damping is 2.8% (decay test in air). The decay tests in water gave a total damping level of 4.2% which includes the structural damping, hydrodynamic radial damping and potential damping. The separate leg model is used to estimate the hydrodynamic loads at various levels of an individual leg. In the separate leg model the hydrodynamic force on two bays were measured by transducers placed on the leg's aft chords. The separate leg model is constructed such that these transducers will measure the local forces of one bay only. The tests were performed with the sensor bays placed at three different locations, such that it can measure the local forces in a wave crest position, a wave trough position and in a totally submerged position, as shown in Fig. 2.
184
D. Karunakaran, N. Spidsoe
M _~---~
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In all test cases the wave surface elevation is measured. Both regular and irregular wave cases are considered. In some cases the current is also included. Furthermore, for selected cases, the wave kinematics are measured at one point 0.10m below wave surface and at three fixed positions along the depth, i.e., at el.-0.35m, e l . - 0 . 7 0 m and el.-1.40m.
Applied data The tests were performed for a variety o f wave and current conditions. In this paper, however, only the following three conditions are considered: Regular wave case (Identifier--nc03uni) Irregular wave (Identifier--nc06uni) Irregular wave (Identifier--nc07uni)
H = 0.450m and T = 2.1 sec
Extreme wave
Hs = 0.200m and Tz = 1.5see
Extreme sea state
Hs = 0.062m and Tp = 1.1 sec
Peak period o f sea state corresponding to the natural period o f the model.
All these considered cases are without current.
Methods for simulation of nonlinear dynamic response
185
Simulation procedure and model The simulations are carried out applying the procedure described in Karunakaran 113]. The procedure is based on: • FEM[ modelling of the structure • Morison equation for force calculation • Mea,~ured wave time series as input to wave kinematic calculation. The simulations are carried out in model scale. The key reason for this is that the load coefficients to be applied in the load calculations will be flow dependant, i.e., they must relate to model test conditions and not full scale flow conditions. Both for the separate leg and for the full jack-up, the leg truss work is idealised as single stick models with equivalent diameters, CD and CM as per the Jack-up Industry Recommended Practice (JIRP), Society of Naval Architects and Marine Engineers [11]. The simulation model has a natural period of 1.07 sec. Verification procedure The veritication study is performed in the following five steps: (1) Comparison of measured and simulated wave kinematics. For this comparison the measured wave without the presence of the structure is used. (2) Comparison of measured and simulated hydrodynamic forces from the separate leg model. For this study the wave elevation measured in the presence of the structure is applied. (3) Comparison of static response of the full jack-up model. (4) Comparison of dynamic response of the full jack-up model. (5) Comparison of resonant response in order to estimate the damping level. The con0tparisons are based on time series plots, and estimates of statistical parameters, spectral densities, response functions and probability distributions.
WAVES AND WAVE KINEMATICS The wave kinematics are estimated from the input wave elevation using Fast Fourier Transformation (FFT) using the Wheeler stretched wave model [16]. This comparative study is performed only for two wave conditions, i.e., the extreme regular wave case (nc03uni) and the extreme irregular wave case (nc06uni).
D. Karunakaran, N. Spidsoe
186
The time traces of the measured and simulated wave induced velocities are shown in Fig. 3 for the regular wave case. In the regular wave case, the measured velocities have higher order components seen at the crest-trough transition part. This is probably due to measurement problems caused by vortices as described by Grundlehner [1] and this higher order component is not seen in the measured hydrodynamic forces which will be discussed later. This means that comparison is to be restricted to first and second order components, i.e., for frequencies less than 1.25Hz. The time traces of measured and simulated velocities containing the first and second order components are shown in Figs 4 and 5. As seen from Fig. 4, the measured and simulated velocities are in phase for both first and second order components for velocities at level-0.10. On the other hand, as seen from Fig. 5, at level-0.35, the measured second order component is out-of-phase to the simulated velocity by 180 ° . It was also seen that the measured second order component of velocity is out-of-phase to the measured second order component of wave elevation. However, the measured velocity has full coherence with the measured wave elevation. At the other two levels also, similar out-of-phase second order velocity compoo VelOCity
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nents are observed. There is no explanation for this at this moment and hence in the comparison between measured and simulated velocities for regular waves, only the first order components are considered. Comparison of the first order components shown in Figs 4 and 5 indicate that the simulation overpredicted the velocities by about 20%. In the irregular wave case, the measured surface elevation and measured velocities are in phase with full coherence. Hence, the comparison between measurement and simulation is done without any filtering. The statistical properties of the measured and simulated wave kinematics for the irregular wave are presented in Table 1. For tile irregular wave condition, the simulation underpredicts the kinematics at the wave crests at el.-0.10m. However, at lower levels, the simulated kinematics compare very well with measurements. The underprediction at below wave crest in the surface zone by the Wheeler model is in line with the earlier findings by Gudmestad and Haver [2].
188
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The time traces of the measured and simulated wave induced velocities are shown in Fig. 6. As seen from this figure the measured and simulated velocities compare very well, except for the wave crests at el.-0.10m. TABLE 1 Comparison Between the Measured and Simulated Velocities Irregular Wave (nc06uni) Standard deviation
Skewness
Kurtosis
Maximum
Minimum
(m/s)
(m/s}
(re~s) At el. -0.10 At el. -0.35 At el. -0.70
Meas.
Sim.
Meas.
Sim.
0.154 0.091 0.060
0.152 0.102 0.067
0.06 -.25 -.27
0.20 -.27 -.23
Meas. Sire.
3.80 3.61 3.24
3.18 3.09 3.05
Meas. Sire.
Meas.
Sim.
0.66 0.29 0.20
-0.43 -0.38 -0.23
-0.48 -0.41 0.24
0.58 0.28 0.21
Methods for simulation of nonlinear dynamic response Velocity
at
-
0 . 1 0
rn
.
W h e e l e r
*
189
ncO6unl
o e
M I~UW h i¢~U tit iO~
........
h4 r=L u red S*muballon
........
J -o.I *o2
I 3'L6
~0 4 ~ I 0
Velocity
I
I
32O
at
-
0 . 3 5
m
-
W h e e l e r
-
ncO6uni
0 2
-o
I
-o.2
~0 I ~ '1 o
3"ll
~2o
325
33o
Nil. 6. Comparison of measured and simulated vel0cities-Irregular wave--ncO6uni.
H Y D R O D Y N A M I C FORCES Separate leg model For this study both the two extreme wave conditions are used (nc03uni and nc06uni). As described earlier, the leg truss work is idealised as a single stick model in the simulations. The hydrodynamic forces are measured at two levels representing the force from two bays in the leg truss work. Furthermore, the tests were repeated three times by keeping the sensor bays at different elevations, thereby the wave forces at different elevations are measured, see Fig. 2. The different positions and their corresponding levels are: Upper Lower Upper Lower Upper Lower
bay--Upper level bay--Upper level bay--Intermediate level bay--Intermediate level bay--Lower level bay--Lower level
--Elevation --Elevation --Elevation --Elevation --Elevation --Elevation
around + 0.17 around + 0.00 around-0.18 around-0.35 around-0.87 around- 1.04.
190
D. Karunakaran, N. Spidsoe
Regular wave--ncO3uni For the regular wave case, the force measurements are available for all sensor bays at all levels. The measured and simulated wave forces along the leg are shown in Fig. 7. It is seen that the simulations significantly overpredict the forces in the surface zone. The overprediction is the highest at the top level and reduces systematically at lower levels. As seen from Figs 3, 4 and 5, there is no indication that this is due to the difference in the wave particle velocity. Hence, the reason for the significant difference in the forces is from the force mechanisms. Alternatively, a reduced Cn in the surface zone should be used. This is in accordance with the results from an earlier study where the forces on a single pile were measured almost in similar flow conditions. In that study, the finding was that the force coefficients were lower in the splash zone than for the submerged part of the structure [12]. I~orce
In
u p p e r
Dilly
.
U D D e r
l e v e l
-
n o ~ l ~ u n l
=m
0
=D,
Tile
F o r c e
in
lower
:;t=0
I~ay
-
U p p e r
~
level
~6
-
=Dm
n c O 3 u n l
"le .jI 0
F o r c e
In
u p D e r ,
b a y
-
Interrl-)e(31ate
level
-
n o O 3 u n l
o
.e
Fig. 7. Comparison of simulated and measured forces--Regular wav~--nc03uni.
Methods for simulation of nonlinear dynamic response I~oroll)
A
I1~1 I o w l l r
l[:~,lLy *
Intermediate
ImVCDI
191
-- r l ~ ' ( ) 3 u n l
s
o
*l
4'=o E
]
I~orc~
It1 Upper
bay
-
L o w e r
14bvOI
-
not~3unl
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In
I~ilLy
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I_owor
lilIVLII
.
noO3urll
08
..o •
°
R
o
Iowor
oi
~s I
3
Fig. 7. Continued.
For the parts of the leg which are totally submerged (bays at lower level), the simulated forces are only slightly larger than measurements. For the bay which is just below the splash zone (Lower bay at intermediate level), the simulated forces are about 50% higher than the measured forces. The elevation of this bay is approximately - 0 . 3 5 m , a location at which the measured velocity is available, see Fig. 5. Part of the difference in forces attributes to the difference in the wave kinematics. Furthermore, the measured forces and the simulated forces have different frequency components. In order to study this, the measured and simulated forces are band-pass filtered such that the forces around the wave loading frequency, and multiples of the wave frequencies are compared. This comparison is found in Fig. 8. It is seen that the first order forces are about 20% higher in the simulation than the measurement. As discussed in Section 3, the first order
-
First
2,
orcler
-%',
"I"oTiBLI
force
force
In
2=
in
2=
lower
lower
bay
Intermediate
1...~]
"3
-
-%'~
Intermediate
"r , m ~
..
bay
-
-%.'
2.
level
-
-
25
~'5
M eIL,1~u Wheelo~ .-
ncO3unl
noO3unl
S0mu|m.on.
level
........
"
.
Fig. 8. Comparison of simulated and measured forces--Regular wave---nc03uni.Lower bay--intermediate level--filtered forces.
-'-%o
-2
!o
2
-%
d
"IS-% 0
-4
-3
-2
-1
0
2
_%
4
t~
Methods for simulation of nonlinear dynamic response
193
o
o
o
o
o
?
0
o
o,
o
,;
o
~
o
o
eJ
?
~
o,
•
?
194
D. Karunakaran.N. Spids~e
component of simulated wave particle velocity is 20-25% larger than the measurement. This should have given a 50% increase in forces if the assumed load coefficients are correct. This implies that the load coefficients used are lower than what should have been, indicating that the Co should be higher than 1.0. The Reynolds number (Re) calculated based on the velocities close to the surface is about 1×104 and the Keulegan Carpenter number (Kc) is about 60. From all the measured data summarized in Sarpkaya and Isaacson [9] and further discussed in Spidsoe [12], the Cn in such flow conditions are higher than 1.0. The second and higher order forces in the simulation are significantly higher than the measurements. Possible explanations may be: (1) The second order forces are caused by the direct excitation caused by second order wave component. The Re for waves at the higher frequencies will be around 4000 and the Kc will be about 10. This indicates that the CM around these frequencies will be around 1.4 [10, 15]. A lower C ~ at these frequencies will reduce the second order forces. This will have significant effect on the dynamic response, since the wave frequency corresponding to the second order forces match the natural frequency of the model. (2) The third order component has both direct excitation and the superharmonic excitation arising from the Morison equation. However, in the measurements the third order forces are very low compared to the simulations. This is because in the measurements, the third order forces are cancelled due to the phase effects between the chords of the leg. The cancellation frequency is exactly three times the wave frequency. In the simulation model, the leg is idealised by a stick model which do not have any cancellation.
Irregular wave--ncO6uni For the irregular wave case, the force measurements are available for all sensor bays at upper and intermediate levels only. The comparison between the measured and simulated wave forces along the leg are shown in Table 2 and in Fig. 9. In this case also, it is seen that the simulations overpredict the forces in the surface zone as in the regular wave case, due to the same reasons discussed for regular wave case. At lower down in the leg the simulated forces compare reasonably well with the measured forces. At lower bay--intermediate level, the simulated forces are systematically smaller than measurements. As discussed in Section 3, for the irregular wave case, the measured and simulated kinematics compared very well, which indicates that the smaller simulated force is due to a
195
Methods for simulation of nonlinear dynamic response
TABLE 2 Comparison Betweenthe Measured and Simulated Forces--Irregular Wave
Up. bay-Up. level Low. bay-Up. level Up. bay-Int. level Low. bay-Int. level
St. Devation (N)
Skewness
Meas. Sim.
Meas. Sire.
Kurtosis Meas.
0.086 0.082 7.77 12.35 96.6 0.465 0.535
3.13
Maximum (iV)
Sim.
Meas. Sire.
195
1.563 2.013
Minimum (iV) Meas.
Sim.
3.63 16.87 22.60 4.273 5.476 -0.589-1.005
0.564 0.510 0.43 0.35 4.38
3.75 2.307 2.128-1.896-1.537
0.459 0.433-0.79-0.80
6.40 1.474 1.530-2.603-3.013
5.68
lower CD in simulations. This is again in line with the discussion from the regular wave case.
Full jack-up model The hydrodynamic load on the full jack-up model is measured by fixing the hull. In this comparative study, two wave conditions are considered, the extreme regular wave case (nc03uni) and the extreme irregular wave case (nc06uni). Regular wave-ncO3uni
The measured and simulated base shear and overturning moments are compared in Fig. 10. It is clearly seen that the simulations overpredict both the base :shear and overturning moments. The difference is higher for overturning moments than for base shear. If a reduced CD is used in the wave surface zone as indicated in earlier sections, both base shear and overturning m o m e n t will be reduced and the reduction will be higher for overturning moment. This is, however, not the only source of difference between the measured and simulated forces. As seen from Fig. 10, there is a difference in the frequency components. This is highlighted in Fig. 11, where the band-pass filtered base shear is shown. The figure indicates that the simulated base shear is slightly lower than measurements at the wave frequency and is higher than measurements at frequencies in multiples of wave frequency. As discussed previously, if an increased Cz) is used, the force a r o u n d wave frequency can be matched. The main reasons for large
a
E
z
0
2
4
05
0
0.5
1
2
force
force
in
In
, 323
lower
a 32~
upper
T*rr~
bay
-
! 3~0
U p p e r
i 330
U p p e r
lid.c]
-
Ibm)
Tan'm
bay
level
level
ncO6unl
i 313S
iomubufo(l . Wh~llr
i 335
ncO6unl
$imUllllOfl
-
-
........
Fig. 9. Comparison of simulated and measured forces--Irregular wave--nc06uni.
320
Total
l 320
Total
340
340
0~
J
z
J
m
-2"315
-2
-I .5
-1
-05
0
0.5
I
1.5
2
~1%1 =
-1
Total
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force
320
=
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i
force
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^
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I
325
bay
i 325
bay
I~I
Intermediate
T=rne
-
330
=
'
i 330
Intermediate
T , r~,I ll.e¢|
-
Pig. 9. Continued.
lower
upper
-
-
335
I
~ o i s l ~ u t e=~ WhNI~F ........
noO6uni
| 33:5
F
- Wh~lor
ncOGunl
SimUlilion-
level
S6mul~on
level
340
3.=0
..,.,I
;=
r~
o
' °%o
o
-4O3o
-3o
-2o
-lo
,io
~.o
;,
'
Total
base
"
2~
~'=
-
ncO3unl
.k"
m o m e n t
sheer-
O v e r t u r n i n g
Total
ncO3unl
-
~"
. . . .
Static
......
" s,il
Static
~'~
M ~ a u' L u r ~
-':':,7~.,~=
........
Fig. 1O. Comparison of simulated and measured static response--Regular wave----nc03uni.
E
tR
!
:lE
30
40
60
•so
O0
-4o
-~o
o
First
order
comp.
base
of
shear-
lt~c]
ncO3unl
"r , r y ~ I ~ c ]
base
Time
A
shear-
-
15
~..-,...
Static M o ; I M b u r ~¢1 Simull|ton Jer . . . . . . . .
!
..........
Static
ncO3unl
-
~e
Fig. 11. Comparison of simulated and measured static base shear. Regular wave--nc03uni--Filtered forces.
~
~0
30
40
50
"40~0
-20
A
Total
',0
;=
¢<
200
D. Karunakaran, N. Spidsoe
Methods for simulation of nonlinear dynamic response
201
difference at high frequency are the superharmonic excitation at second and third harmonics and also the reduced C u at high frequencies as discussed earlier and the big difference at three times wave frequency is also due to the wave cancellations seen in the model test which are not modelled in the simulations.
Irregular wave--ncO6uni The measured and simulated base shear is shown in Fig. 12. The figure indicates that the simulation underestimates the response slightly at large wave cycles. This is also noticed from the hydrodynamic response function, which is tlhe ratio between the response spectrum and wave spectrum, shown in Fig. 13, where there is significant differences in the frequency contents. The simulations underestimate the forces at and around wave peak frequency and overestimates at higher frequencies. This can also be seen from the filtered response shown in Fig. 14. The reason for this is discussed in the previous section and they are: (1) The Re calculated based on velocities simulated close to the surface from the large wave cycle is around 9000. However, the average Re willt be much lower at the surface and will decrease further down. The same is the case with Kc which is about 50 for the large wave cycle. From the measured data summarized in Sarpkaya and Isaacson [9] and further discussed in Spidsoe [12], the CD in such cases are higher than 1.0. This is also seen from an earlier study with force measuremeats from a single pile [12]. (2) At frequencies around 1Hz, the Re and Kc will be of the order 3000 and 10, respectively. For such low Kc values, the C u will be of the order 1.4 (instead of 1.8 used in the simulations). (3) A reduced CD at the surface zone as discussed previously. To investigate the possible effects, the simulation is carried out with the following variable coefficients: (1) CD in the surface zone is reduced to 0.7. (2) Cn is increased to 1.3 for elements below surface zone. (3) CM is reduced to 1.4 for frequencies above 0.75Hz. The hydrodynamic response functions of base shear and overturning moments obtained using variable load coefficients are shown in Fig. 15. Comparing Figs 13 and 15, it can be seen that the use of variable coefficients has improved the simulated response function. This exercise also indeed indicates that the load coefficients are flow dependant.
D. Karunakaran, N. Spids~e
202
0 I
I
I
I
i
i
I
I
.~.
¢J
~2 ¢a
8 0 |
o~
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0
0
e4 0
~
0
~
0
~dRo~~
•
0
•
0 ~
4o
6o
8o
12o
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01.2
Hydrodynamic
0 •
0 6
response
Ff~uertCy
function
(HZ]
0 8
-
Base
1
shear
1.2
~vt o m J r . u r ~ Slmulglton
- ncO6uni
1.4
........
Fig. 13. Comparison of measured and simulated hydrodynamic response functions. Irregular wave---ncO6uni.
~
_s
~
"160
1 80
200
=1
R e s p o n s e
I 320
I 320
a r o u n d
Tot~ll
w a v e
TJrn~
fre.
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shear-
p e a k
I
~ 325
b a s e
1~4,¢1
-
[r.~c)
I 330
b a s s
|
n c O 6 u n l llon-
I
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A 338
~4eaLuo l~l * Wha~141r
-
Wh~le!
n c O 6 u n l
scroll,
Static
s h e a r -
-
........
........
340
Fig. 14. Comparison of simulated and measured static base shear. Regular wave---nc03uni--Filtered forces.
-~0
-15
-I0
16
318
o
-6
°
$
10
16
20
25
-20
-15
-10
-5
0
6
10
g
z
i
z
16
~6
Methodsfor simulation of nonlinear dynamic response
205
t~ ~~~ 1
0
,
I
I
_
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I
I
I
'"
"
-~
I
0
I
_
_
~
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~
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I
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20 0
40
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Hydrodynamic
014
016
response
F=r4~lUOncy [HZ}
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function
- Base
;
shear
' 1.2
1 ,' 4
- ncO6uni ~loaM~uro(:l Simulation ........
Fig. 15. Comparison of measured and simulated hydrodynamic response functions. Variable load coefficients--Irregular wave---nc06uni.
~
i
160
180
200
=;
I'O
Methodsfor simulation of nonlinear dynamic response
207
DYNAMIC RESPONSE The dynalmc response is performed using the full jack-up model. The structural damping applied in the dynamic analysis is 2.8% as given by the decay tests in air. The simulations are performed with P-A effects and with relative velocities :in the Morison equation. For this comparison, two wave conditions are used, they are the extreme irregular wave case (nc06uni) and the resonant sea state (nc07uni). Extreme sea state (neO6uni)
The simulations are first carded out with Cn and C ~ as per JIRP. The frequency response function of measured and simulated base shear is compared in Fig. 16. It is clearly seen that around the wave peak frequency, there is an underestimation by the simulation and around the resonant frequency, there is a significant overestimation. As discussed in the previous section, if' an increased Cn is applied, the underestimation of loads around wave peak frequency, can be avoided. However, the overestimation of resonant response is due to reduced C~r as well as the level of damping. It is found that 5% linear damping is needed in order to get a good comparison with measurements when reduced CM is used. This linear damping level is slightly higher than the measured damping level of 4.2% in the decay test in water. The frequency response functions of simulated base shear using variable load coefficients and 5% linear damping are compared with measurements in Fig. 17. As seen from this figure, simulations with variable load coefficients and 5% linear damping gives the best fit to measurements. This is also confirmed by the comparison of statistical properties of measured and simulated response presented in Table 3. The next important step is the validation of the use of relative velocities in Morison equation. To verify that, the dynamic response is performed with varying load coefficients, but with absolute velocities in Morison equation. In this case, the linear damping level is increased to substitute the hydrodynamic drag damping. 3% increase in linear damping level (total 8% linear damping) gave a good match between simulated and measured response standard deviation. This increase in damping level, 3%, is also the same level recommended by the JIRP to account for the hydrodynamic drag damping. The statistical properties of this simulated response is also compared with measurements in Table 3. It is seen that the standard deviation of response is comparable to the measurements, whereas, the kurtosis coefficient of the simulated response is significantly higher when absolute velocity is used in the Morison equation. The response simulated using absolute velocities are more non-Gaussian than
J
,.=
-I
=¢
0
IO0
200
3O0
400
SO0
600
7OO
0 1
/\
~t
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-
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1.5
function sire.
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2 .8
2
* y,
shear ¢~mp.
,
2.5
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Fig. 16. Comparison of measured and simulated frequency response functions.
O.S
Frequency - - .......
O0
loo
200
500
o
o
0.5
Frequency
fi
.'~
I
".~
response
~,
-
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-
ncO6uni
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hey
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function
_~Jt~,~.
- - .......
Fig. 17. Comparison of measured and simulated frequency response functions. Simulations with variable load coefficients and 5% structural damping.
~"
-
•. f
e;O0
700
210
D. Karunakaran, N. Spidsoe
TABLE 3 Comparison of Measured and Simulated Response Statistics Irregular Wave--Extreme Sea State (nc06uni) Base shear
Overturning moment
Standard deviation (iV)
Skewness
Kurtosis
Standard deviation (Nm)
Skewness
Kurtosis
7.69
-0.04
3.93
19.73
0.03
3.95
Simulated-CD = 1 & 2.8% damping
8.26
-0.05
3.37
24.22
0.02
3.24
Sim.-Vari. Co and CM & 5% damp. + Rel. velocity
7.41
-0.06
4.01
20.83
0.01
3.78
Sim.--Vari. Co and CM & 5% damp. + Abs. velocity
7.48
-0.09
4.37
20.93
0.02
4.14
4.68
0.0
3.08
14.29
0.0
3.13
Simulated-42D = 1 & 2.8% damping
6.58
0.0
2.95
20.52
0.0
2.96
Sim.--Vari. Co and CM & 5% damp. + Rel. velocity
4.94
0.0
3.04
15.40
0.0
3.07
Sim.--Vari. CD and damp. + Abs. velocity
4.95
0.0
3.29
15.34
0.0
3.32
Total response Measured
Resonant response Measured
C M ~£ 5 %
the measured response, which will give larger extreme response t h a n the measurements. This is also d e m o n s t r a t e d by the comparison o f m a x i m a probability distributions shown in Fig. 18.
Resonance sea state (nc07uni) F o r this sea state, the dynamic analysis is performed using variable load coefficients. F u r t h e r m o r e , the analysis is performed with 5% linear d a m p i n g with relative velocities. The statistical properties o f the measured and simulated responses are c o m p a r e d in Table 4. It can be seen t h a t the s t a n d a r d deviation responses compare very well, however, the kurtosis coefficient is too small and the m a x i m a response is underestimated, see
Methods for simulation of nonlinear dynamic response I
0.9999
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S T . DEV. OF PARENT SERIES BOSQ she~" MeosuPed rssponse
..... I XXXXMo 0,9999-
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Fig. 18. Comparison of maxima probability distribution. Base shear--Irregular wave (nc06uni). Fig. 19. If the linear damping level is decreased, it increases the standard deviation and the kurtosis coefficient decreases, see Table 4, and the maxima level has no significant change. This is due to the Normalisation effect (i.e., the dynamic response is less non-Gaussian than the static response) due to the linear damping and the quadratic drag damping mechanism, as explained in Spidsee et al. [13]. It is commonly assumed that for low sea states such as this one, the application of relative velocity formulation is questionable. The dynamic response analysis is therefore also performed using 5% linear damping, but with ab,~olute velocities in the Morison equation. The comparison of measured and statistical properties are also shown in Table 4. The kurtosis coefficient of the simulated responses is still lower than for the measured response, but the simulated response maxima compare well with measured
212
D. Kanmakaran, N. Spidsoe
TABLE 4 Comparison of Measured and Simulated Response Statistics Irregular Wave--Resonant Sea State (nc07uni) Base shear
Overturning moment
Standard deviation (N)
Skewness
Kurtosis
Standard deviation (Nm)
Skewness
Kurtosis
Measured
2.51
0.0
3.66
7.56
0.0
3.80
Sim.--Vari. Co and CM & 5% damp. + Rel. velocity
2.44
0.0
3.17
7.61
0.0
3.17
Sim.--Vari. Co and CM & 4% damp. + Rel. velocity
3.12
0.0
3.11
8.60
0.0
3.12
Sim.--Vari. CD and CM & 5% damp. + Abs. velocity
2.60
0.0
3.28
8.11
0.0
3.29
maxima, see Fig. 20. This indicates that for this low sea state the relative velocity formulation is questionable.
MODELLING OF DAMPING It is evident from the discussions in the previous section that modelling of damping, both linear damping and hydrodynamic damping, is very important. The decay tests of the model in air indicated that the structural damping is of the order 2.8%. The decay tests of the model in water indicated the total damping to be about 4.2%, which includes the structural damping and hydrodynamic damping due to wave radiation and hydrodynamic potential damping. As seen from the previous section, for both extreme and resonant sea states, the linear damping level needed to have a good match with the measurements was about 5%. The structural damping from the decay tests is about 2.8% and the rest is due to hydrodynamic damping other than the drag damping. It is also seen from the comparisons that, for extreme sea states, the hydrodynamic drag damping is important and should be modelled by relative velocity formulation, whereas, for low sea state, the hydrodynamic drag damping is not present. Furthermore, the probabilistic nature of measured static and dynamic
Methodsfor simulation of nonlinear dynamic response
8 ÷
!
ii 0 r~
0
0
0
0
0
0
0
Or
o
213
113(
-40
-30
-20
-10
0
1135
moment
- nc07uni -
5%
damp.
÷
Abs.
61mul~llon
Vel. i~lluf - Wh
¢1 NI*¢
1 ~0
Fig. 20. Comparison of measured and simulated overturning moment. Simulation with absolute velocities--Resonant sea state.
zo
10
2O
3O
Overturning
I
X
_
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S
DEV. OF PARENT SERIES Bose s h s o r Stottc response
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Methodsfor simulationof nonlineardynamicresponse
217
response confirm that there is Normalisation due to the linear and nonlinear damping mechanism [13]. This is exemplified in Fig. 21 where the maxima probability distribution of static, dynamic and resonant response of base shear at the extreme sea state is presented. It can be seen that the measured static response is highly non-Gaussian, whereas the measured dynamic response is less non-Gaussian than the static response and the resonant response is totally Gaussian.
CONCLUSIONS A comparison is made between the jack-up model test data and nonlinear time domain simulations applying measured wave elevations. The comparison is made in model scale in order to have proper reflection of flow conditions. The comparisons include: • • • •
Waw~ kinematics Hydrodynamic forces Static response Dynamic response and damping.
The study is limited to three wave conditions only and the following conclusions may be drawn from this limited study:
Wave kinematics The applied wave model, Wheeler stretching model, gives good comparison to measured wave kinematics below wave surface zone. However, this model slightly underestimates the kinematics below the wave crests at the surface zone.
Hydrodynamic forces The measured forces indicate that the load coefficients are flow dependant and it is important to apply force coefficients that are valid for the flow conditions in the model test.
Dynamic' response and damping The measured dynamic response clearly indicate that the dynamic response is less non.-Gaussian than the static response, thereby validating the Normalisation effects.
218
D. Karunakaran, N. Spidsoe
For extreme sea states, the relative velocity formulation seems valid. Modelling of the hydrodynamic drag damping by equivalent linear damping will not give a dynamic response having a similar probabilistic nature as the measured dynamic response. For small sea states, the relative velocity formulation should not be used and there is no hydrodynamic drag damping at the small sea states. The over all linear damping level for this structure is about 5%.
ACKNOWLEDGEMENTS The authors are grateful to Prof. Knut-Aril Farnes, Division of Marine Structures, Norwegian Institute of Technology, for the valuable discussions and suggestions during this study.
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10. Sarpkaya, T., Discussion of quasi-2D forces on vertical cylinders in waves. Journal of Waterways, Port, Coastal & Ocean Engineering, 1984, 110, 1-2. 11. Recommended Practice for Site Specific Assessment of Mobile Jack-up Units, SNAME, 1994. 12. Spidsa~e, N., Wave force models for time domain dynamic analysis of dragdominated platforms: summary of a literature survey. SINTEF Report STF A89024, Trondheim, 1990. 13. Spids~e, N., Karanakaran, D. and Gudmestad, O., Nonlinear effects of damping to dynamic amplification factors for drag-dominated offshore platforms. In Proc. llth Int. Conf. on Offshore Mech. and Arctic Engng, Calgary, June 1992. 14. Spids~e, N. and Karunakaran, D., Nonlinear dynamic behaviour of jack-up platforms. Marine Structures, 1996, 9, 71-100. 15. Stansby, P.K., Bullock, G.N. and Shorr, I., Quasi-2D forces on a vertical cylinder in waves. Journal of Waterways, Port, Coastal & Ocean Engineerinng, 1983, 109, 1-2. 16. Wheeler, J.D., Methods for calculating forces produced on piles in irregular waves. Journal of Petroleum Technology, 1970, 1, 1-2.