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Prediction of design wave loads of the ocean structure by equivalent irregular wave approach
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Ocean Engineering 34 (2007) 1422–1430 www.elsevier.com/locate/oceaneng
Prediction of design wave loads of the ocean structure by equivalent irregular wave approach Ming-Chung Fanga,, Chih-Chung Fangb, Chun-Hsien Wuc a
Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan, Taiwan, ROC b Department of Transportation and Navigation Science, National Taiwan Ocean University , Keelung, Taiwan, ROC c United Ship Design and Development Center, Taipei, Taiwan, ROC Received 17 May 2006; accepted 4 October 2006 Available online 19 December 2006
Abstract The present study is employing the equivalent irregular wave approach to predict the wave loads for a ship encountering the worst sea state with respect to the critical dynamic loading parameter. Two different hydrodynamic numerical models, i.e. 3D pulsating source technique and 3D translating pulsating source technique, are applied to calculate the corresponding RAO of the ship moving in waves. Incorporating the RAO of the related physical properties, we can calculate the extreme value for the corresponding ship loading factor, which can be regarded as the worst sea state in the service lifetime of the ship. With the time and period of the occurrence of the corresponding extreme value, we can simulate the time history of the wave load in this period, which is so-called equivalent irregular wave approach. Comparing with the results calculated by the traditional equivalent regular wave approach, we find that the equivalent irregular wave approach can simulate the corresponding wave load more realistic, especially for dynamic pressure. Using the equivalent irregular wave approach can offer the effective and practical base for the ship structural analysis. r 2006 Elsevier Ltd. All rights reserved. Keywords: Design wave load; Equivalent irregular wave; Dynamic load parameters
1. Introduction Generally, the ship design is either based on the classification rules, which are primarily semi-empirical and proven by successful operational experience, or based on direct engineering analysis. The semi-empirical approach used in classification rules has its advantage because it is simple and has some degree of assurance. Therefore it has been the mainstay of ship design practice for some time. Employing the direct engineering analysis is allowed to reduce scantlings based on rules when the sophisticated and rational approach is suitably applied. Many years ago, the American Bureau of Shipping (ABS) offered a new optional class notation called ‘‘Dynamic Load Approach’’, or DLA, which uses the results of the direct engineering analysis to indicate where increasing Corresponding author. Tel.: +886 6 2747018ext.211; fax: +886 6 2080592. E-mail address: [email protected] (M.-C. Fang).
0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.10.006
scantlings should be provided (Liu et al., 1992). This technique has been extensively applied in the ship design recently, especially in ship structural analysis. DLA represents a consistent and rational approach which employs a direct analysis for the particular hull structure being considered. Equivalent regular wave concept was also submitted by ABS early. Combining this concept with the DLA, ABS offered a very effective structural analysis method for ship hull design. In this approach, a set of Dynamic Load Parameters (DLPs) are identified, which are defined as any response process (e.g. bow vertical acceleration, vertical bending moment, lateral acceleration, roll, etc.). The DLP, when maximized, establishes a critical loading for hull structural analysis. Based on the long-term extreme value of DLP, an equivalent regular wave is then defined. The principle of the equivalent regular wave is to use the extreme value to simulate a design regular wave with the equivalent wave amplitude, frequency, and wave incidence for a ship in random waves. Thus it leads to a simple
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sinusoidal wave and a suitable instant of time for extracting a set concurrent load components to be applied to the structural mode. However, the concept still keeps the equivalent wave in ‘‘regular’’ form and cannot correspond to any real or extreme condition at sea. In view of the apparent limitation, the equivalent irregular wave concept was recently introduced by Hutchison et al. (1998) and applied on the analysis of a SWATH ship design. Because the equivalent irregular wave is treated by the time domain analysis, it is more realistic than the equivalent regular wave technique. In both equivalent wave concepts, the long-term extreme values of dynamic loading parameters, DLP, are key factors for hull structural analysis. Both Ochi (1978) and Stiansen and Chen (1982) pointed out that the long-term extreme loading can be predicted by using the short-term statistics analysis on the dynamic loading parameters in the worst sea state case. Before that, Ochi (1973) had already applied the wave spectrum to calculate the extreme values of different dynamic loading parameters of the ship. Later, Buckley (1988) also applied the extreme values of the loading on the ship structural analysis under different sea conditions. In the present study, the vertical bending moment is selected as the critical DLP and its long-term extreme value is determined as the reference of the ship structural analysis. The Bretschneider spectrum is used for simulating the corresponding sea state. The Fourier–Stieljes integral method (St Denis and Pierson, 1953) is applied as the time domain simulation technique for simulating the related ship responses in irregular waves. The results calculated by equivalent regular wave method with respect to the longterm loading extreme values are also shown in the study for comparison. 2. Theoretical formulas 2.1. Short-term prediction While the ship is sailing at sea, the encountered sea state varies with time and the factors influencing the ship response can be regarded as transient and stable. The condition generally lasts for about 1–4 hours. If the probability distribution of the ship response is considered in random sea with such fixed statistic conditions is called short-term distribution. Assume the probability density function of the ship response is Rayleigh distribution, and then it can be expressed as x x2 pðxÞ ¼ exp , (1) m0 2 m0 where x and m0 are the ship response and the mean square value of the related frequency response spectrum, respectively. The probability of the ship response at time t exceeding a ~ i.e. extreme value, can be expressed as the critical value x,
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function of mean square value as below: Z 1 ~ ¼ pðxÞdx Pðx4xÞ x~
Z
x~
pðxÞdx x~ 2 ¼ exp . 2m0 ¼1
0
ð2Þ
2.2. Long-term prediction During the design lifetime of the ship or a very long period sailing at sea, the probability distribution of the ship structure stress due to the wave loading will vary with the sea states and the other relating factors. Therefore we cannot assume it is transient and stable, and the probability distribution of the ship response under such condition is then called long-term distribution. The long-term distribution can be regarded as the combination of infinite short-term distributions. Generally the short-term probability distribution is assumed to be Rayleigh distribution and its energy spectrum or mean square value is also different from each other. If the lifetime of the ship is set to be 20 years and total annual duration actually spent at sea is 300 days, then the long-term distribution is composed of 105 short-term distribution. If the average period of the encounter wave is assumed to be 10 s, the total numbers of the ship response are about 107 108 . The long-term distribution response is the summation of the related short-term distribution response, therefore the probability of occurrence of the ship response with longterm distribution, q(x), is the summation of that of each short-term distribution component, pm(x). However, the probability of occurrence for the mean square value of each short-term distribution is different in the long-term distribution. Therefore the probability of occurrence of the ship response x for each short-term distribution component is mðxÞ pm0 ðm0 Þ in the long-term distribution where m0 ðm0 Þ is the probability of occurrence of the short-term distribution mean square value in the long-term distribution, then Z 1 qðxÞ ¼ pm ðxÞpm0 ðm0 Þ dm0 . (3) 0
For the long-term distribution of the ship response, the probability exceeding the critical response x~ is written as Z x~ ~ ¼1 Qðx4xÞ qðxÞdx 0 Z 1 ~ m0 ðm0 Þdm0 ¼ Pm ðx4xÞp 0 Z 1 x~ 2 ¼ exp ð4Þ p dm0 . 2m0 m0 0 From the above formulas, we can obtain the related statistic value for ship response in the long-term distribution. Since the long-term extreme is defined as that value
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which is expected to be exceeded only once in the lifetime of the ship, the corresponding long-term extreme in the ~ can then be determined. lifetime of the ship, x, In Eq. (4), the probability of occurrence of the shortterm distribution mean square value in the long-term distribution, Pm0 ðm0 Þ, is rather difficult to be predicted. In order to obtain the accurate Pm0 ðm0 Þ, we must in advance understand the probability of each sea state encountered, ship heading, ship speed etc. At present, the main ocean data in the world have been established quite well, e.g. North Atlantic by Walden (1964), ‘‘OWS’’ and ‘‘GWS’’ by Hogben (1988). Using these related ocean statistic data and wave spectra, the long-term probability of the response value x exceeding the level of x~ in all sea states is written as Pm Pm P P P ~ ¼ Pðx4xÞ
i¼1
j¼1
wave crest position relative to the ship. Generally, it is applied to simulate the irregular wave defined by the wave spectrum. The response amplitude operator (RAO) corresponding to the DLP is first computed for a range of frequencies and headings. The RAO of a DLP is the complex amplitude of the DLP per unit wave amplitude for a given frequency and heading. The frequency and heading combination which maximizes is then identified. This defines the frequency and heading of the equivalent regular wave. With the long-term extreme value of the correspond~ predicted from Eq. (4), the amplitude of ing response, x, the equivalent regular wave can be obtained by aw ¼
x~ . RAO
(6)
2
~ =2m0ijV yL ÞÞ L n Pij PV Py PL exp ððx Pmy P m P P P V y L n Pij PV Py PL i¼1 j¼1
V
(5) where Pij is the probability of occurrence of a sea state described by significant wave heights, Hi, and modal wave periods, Tj; and PV, Py and PL are the probabilities of ship speed, heading and loading condition respectively. n* is the number of responses in a given sea state and ship condition. 2.3. Equivalent regular wave approach According to the ABS dynamic loading approach, we can define an equivalent regular wave for each DLP, which is characterized by an amplitude, frequency, heading and
2.4. Equivalent irregular wave approach The above approach is design to accomplish, in a simple manner with the regular wave pattern, a transfer from the frequency and probabilistic domains to the time domain. The present concept of the equivalent irregular wave is a different approach which can accomplish the same transformation but is more realistic. Similar to the algorism of Hutchison et al. (1998), the present equivalent irregular wave approach is accomplished by the following procedure which is also described by using the flow chart in Fig. 1: Step 1: The choice of the wave spectrum and scatter diagram for the encountered sea environment.
Scatter Diagram & spectrum generation 3D Program
USDDC/Long-term Program
Wave spectrum RAO: motions or loads
RH-Table generation& Section Extreme Value Probability at 10-8
Irregular response simulation: k
Σ {F (k / χ) cos[kt + (k / χ) + (k)]√2sw(k)d}
RH = 3.72√m2
k=1
Judgement Random Phase, (k)
Extreme Value
Generation Return
Fig. 1. The proceeding flow of equivalent irregular wave approach.
Export T0 & (k)
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Step 2: Calculate the RAO of the corresponding DLP which is selected for design consideration. Step 3: Make the long-term extreme value analysis. In the present study, the probability of occurrence of the DLP is set to be smaller than 108 to determine the extreme value. (i.e. Eq. (4)). Step 4: Select the suitable sea state. Calculate the average of the 1/1000 highest values of the corresponding DLP with respect to the each sea state from the scatter pffiffiffiffiffiffi diagram, i.e. significant value RH ¼ 3.72 m0 , where m0 is the area under the DLP spectrum, and record the sea state with the significant value near the long-term extreme value. Step 5: Select a random phase set, fðok Þ, for the wave by using the random process. Step 6: Use the Fourier–Stieljes integral method to transfer the RAO of the corresponding DLP from frequency domain to time domain in random waves. Step 7: Find the time, t0, for the occurrence of the worst sea condition that has the amplitude greater than the extreme value calculated in step 3. If the time, t0, exists, then we can assure the worst sea condition indeed occurs at this time and proceed the corresponding loading analysis in irregular waves. If the time, t0, does not exist, then we repeat the steps 5 to 7 until the occurrence time, t0, of the worst sea condition is found. 2.5. The Fourier–Stieljes integral method The application of the Fourier–Stieljes integral method on the transformation of the RAO with respect to the corresponding DLP from the frequency domain to the time domain can be expressed by the following formula, Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðt=wÞ ¼ F ðo=wÞj cosfot þ bðo=wÞ þ fðoÞg 2Sw ðoÞdo 0
(7) and it can also be expressed as the following finite discrete summation form, f ðt=wÞ ¼
K X
F ðok =wÞj cosfok t þ bðok =wÞ þ fðok Þg
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sw ðok Þdok ,
k¼1
(8) where f ðt=wÞ is the time-varied DLP, w is the wave heading, and F ðok =wÞ is the response amplitude operator pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (RAO) of the DLP. 2S w ðok Þdok and ok are the kth wave amplitude and wave frequency with respect to the corresponding wave spectrum, respectively. t is the time, f (ok) is the random phase for the kth wave (02p), and b (ok/w) is the phase angle of the DLP with respect to the kth wave. Sw(ok) is the wave spectrum. The following Bretschneider spectrum is selected in the present study: A B S w ðoÞ ¼ 5 exp 4 (9) ðm2 sÞ o o
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where A ¼ 0:3125H 21=3 o4m ; B ¼ 1:25o4m ; om ¼ 2p/T0 (modal wave frequency, rad/s); T0 the modal wave period, s; o the rad/s. If the RAO is expressed in complex form, Eq. (8) can be written by f ðt=wÞ ¼
K X
n o Re F ðok =wÞeiðok tþfðok ÞÞ
k¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sw ðok Þdok ,
ð10Þ
where F ðok =wÞ the RAO of the corresponding DLP in complex form. While processing the equivalent irregular wave approach, it may not be convergent if the random phase is not suitably selected, which will make the approach unrealistic. However, it usually takes long time and not sure to find the suitable random phase. Therefore, a fast searching formula is developed here to solve this problem and shown as below: fðok Þ ¼ f2np ½ok t0 þ bðok =wÞgð1 þ dk Þ; ðn ¼ 1; 2; . . .Þ, (11) where dk is a random variable from 0.5 to 0.5. 2.6. Equations of motions In the study, the motion response is assumed to be small and the RAO of the motion displacement can be obtained from the following the equations of motions: 6 X
½o2 ðM jk þ Ajk Þ ioBjk þ C jk x¯ k =z0
k¼1
¼ FW j =z0 ðj ¼ 1; 2; 3; 4; 5; 6Þ,
ð12Þ
where x¯ k =z0 is the RAO of the kth mode of motion displacement. Mjk is the mass matrix. Ajk, Bjk and Cjk are the added mass, damping coefficient, and restoring force, respectively. z0 is wave amplitude and FjW is the wave exciting force. The subscripts j and k represents the direction of the force acting on and the motion mode, respectively (j or k ¼ 1, 2, 3, 4, 5, 6 for surge, sway, heave, roll, pitch, and yaw, respectively). The related hydrodynamic forces in Eq. (12) are calculated by using the three dimensional source distribution method, either with pulsating source (3DP) or with translating pulsating source (3DT) (Chan, 1990). While doing the ship structral analysis by finite element method, the hydrodynamic pressure distribution on ship hull must be well pre-calculated as the input. Based on the threedimensional source distribution methods, the corresponding resultant hydrodynamic pressure pT ðtÞ on the instantaneous wetted body surface can be calculated by the following formulas: 1 1 2 pT ðtÞ ¼ r Ft þ rF rF U þ gz , (13) 2 2
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where r is the water density, F is the resultant velocity potential and U is the ship speed. Expanding this pressure in a Taylor series about the undisturbed position of the hull and neglecting higher-order terms and static pressure terms, the RAO of the resultant hydrodynamic pressure, ¯ T , can be split into four parts: The Froude–Krylov P ¯ FK ðtÞ, diffraction pressure, P ¯ D ðtÞ, radiation pressure, P ¯ pressure, PR ðtÞ, and dynamic restoring pressure, p¯ qs ðtÞ, as follows: ¯ D ðtÞ þ P¯ R ðtÞ þ P ¯ qs ðtÞ. ¯ T ðtÞ ¼ P¯ FK ðtÞ þ P P
(14)
In Eq. (14), the linearized hydrodynamic pressures for radiation and diffraction over the body surface can be written in the form ¯ R ðtÞ ¼ r io þ U q fk x¯ k eiot =z0 ; k ¼ 1; 2; . . . ; 6, P qx (15) ¯ D ðtÞ ¼ r io þ U q f7 eiot =z0 , (16) P qx where fk and f7 are the radiation potential and diffraction potential, respectively. Generally, the above resultant pressure is nonlinear because the draft of the ship hull is time varied due to motions. Here, for simplicity, only the Froude–Krylov ¯ FK ðtÞ and dynamic restoring pressure p¯ qs ðtÞ are pressure P considered to be nonlinear. The nonlinear Froude–Krylov pressure is expressed as below ¯ FK ðtÞ ¼ rgekz eikðx cos wþy sin wÞ eiot ; P
zp0,
(17)
where z is the relative depth with respect to wave and k is the wave number. However, when 0pzpz0 , the linear interplation is applied between the following two values: ¯ FK ðtÞ ¼ rgeikðx cos wþy sin wÞ eiot ; P ¯ FK ðtÞ ¼ 0; ðz ¼ z0 Þ. P
shape are shown in Table 1 and Fig. 2, respectively. The RAO of the corresponding DLP is calculated by using the three-dimensional source distribution methods, i.e. pulsating source (3DP) and translating pulsating source (3DT). The long-term motion and wave loading are calculated using the USDDC/long term computer program. The extreme values for different DLP mentioned here are obtained by using the Eq. (5) and the two-parameter Bretschneider spectrum based on the long term ocean statistic data in Taiwan. The results are shown in Table 2 and the limit of the probability of occurrence is set to be 108. Generally the extreme value obtained by 3DP is larger than that of 3DT, which means the 3DP method is more conservative. With the aid of the fast searching formula developed here, i.e. Eq. (11), we can easily and quickly find all the corresponding sea states with the possibility exceeding the extreme value. Because the 3DT source distribution method is more realistic, therefore is applied here to calculate the following wave loadings and hydrodynamic pressure. Figs. 3–6 show some results of the
Table 1 The principle dimension of RD-200 Length of overall Length of perpendiculars Breadth Design draft Displacement Vertical center of gravity (from Keel) Longitudinal center of gravity (from AP) Matercentric height (from Keel) Roll radius of gyration Pitch radius of gyration Roll natural period
98.0 m 90.0 m 12.29 m 3.84 m 2055 m3 5.094 m 42.64 m 1.485 m 4.43 m 22.5 m 7.4 s
ðz ¼ 0Þ, ð18Þ
The nonlinear dynamic restoring pressure due to the varied draft at the relative depth h(t) can be written as. ¯ qs ðtÞ ¼ rgðhðtÞÞ, P
(19)
where hðtÞ ¼ x3 ðtÞ þ yi ðt DtÞx4 ðtÞ xi ðt DtÞx5 ðtÞ
(20)
If the nonlinear terms in Eqs. (18) and (19) are considered in the Eq. (14), we call it quasi-nonlinear hydrodynamic pressure. 3. Calculation results and discussion In order to investigate the application of the present equivalent irregular wave approach, a high-speed displacement craft, RD-200, moving with 24 knots is selected as the calculation model and its bow vertical acceleration, vertical shear force, torsion moment, and vertical bending moment are taken as DLPs for calculation. The wave heading to ship is set to be w ¼ 170 . The principle dimension and ship
Fig. 2. Body shape of RD-200.
Table 2 Long-term response prediction program spectrum scatter diagram motion method F.P. Acc.(A3) m/s^2 V.S.F.(F3) N T.M.(F4) N m V.B.M.(F5) N m
USDDC/Longterm Program BRETSCHNEIDER Annual ocean statistics in Taiwan 3DP 2.92E+01 5.68E+06 9.07E+06 1.81E+08
3DT 2.27E+01 6.71E+06 8.46E+06 1.60E+08
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simulation of different DLP of RD-200 moving with 24 knots at different sea state. From these simulations, we can find the time, t0, exceeding the extreme value of the corresponding DLP. Figs. 3 and 4 show the simulations of the torsion moment and the bow vertical acceleration at sea state with T0 ¼ 10 s and H1/3 ¼ 5.75 m, respectively. The extreme values for both DLPs are 8.46 106 N M and 22.7 m/s2, and the corresponding time of exceeding the extreme value are at t0 ¼ 1894.8.and 641.3 s, respectively. Fig. 5 is the simulation of the vertical shear force at sea state with T0 ¼ 8.0 s and H1/3 ¼ 5.75 m. The extreme value is 6.71 106 N and the time of exceeding the extreme value is at t0 ¼ 2154.5 s. Fig. 6 is the simulation of the vertical bending moment at sea state with T0 ¼ 10.0 s and H1/3 ¼ 5.25 m. The extreme value is 1.6. 108 N m and the time of exceeding the extreme value is at t0 ¼ 4092.5 s. From the above cases we can see that the time of exceeding the corresponding extreme values of DLP is different even with the same sea state. Therefore which DLP is suitable to
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Fig. 5. The time simulation history of vertical shear force (T0 ¼ 8.0 s, H1/3 ¼ 5.75 m).
Fig. 6. The time simulation history of vertical bending moment (T0 ¼ 10.0 s, H1/3 ¼ 5.25 m). Fig. 3. The time simulation history of torsion moment (T 0 ¼ 10:0 s, H1/3 ¼ 5.75 m).
Fig. 4. The time simulation history of vertical acceleration (T0 ¼ 10.0 s, H1/3 ¼ 5.75 m).
be selected as the critical load parameter is very important. The determination of the critical load parameter generally depends on the problem we concerned, for example, if the structural analysis problem is concerned, we will choose the transverse side force and prying moment in beam seas as the critical DLP for SWATH ships and the vertical bending moment in head seas for high- speed mono-hull ships. One of the most important applications of the equivalent irregular wave approach is the structural analysis by finite element method and the hydrodynamic pressure distribution is the key input for the direct analysis. Therefore, in the following part of the paper, we concentrate on the calculation of the hydrodynamic pressure distribution with respect to the critical DLP using the equivalent irregular wave approach. The equivalent regular wave approach is also used for comparison. Since the mono-hull ship sailing with high speed in bow waves is studied in the present study, the vertical bending moment is selected as the critical loading parameter. Assume the RD-200 sails in bow waves with w ¼ 170 , T0 ¼ 8.0 s and H1/3 ¼ 9 m and the
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corresponding time history simulation of the vertical bending moment is shown in Fig. 7. The long term extreme value and the time of exceeding the extreme value are found to be 1.6 108 N m and 1507.6 s, respectively. According to the equivalent regular wave approach, the equivalent wave height is at 13.66 m and period is at 8.97 s with respect to the extreme value, 1.6 108 N m. The time of the occurrence of the extreme value using the equivalent regular wave simulation is found to be 2.48 s that is the time for us to pick up the pressure distribution results to compare those obtained by the equivalent irregular wave approach at 1507.6 s in the next discussions. Figs. 8 and 9 are the results using the equivalent regular wave approach with the equivalent wave height at 13.66 m and period at 8.97 s for the linear and quasi-nonlinear instantaneous resultant total pressure distribution at 2.48 s, respectively. Fig. 8 shows that the bow submerges into the water and forms a high-pressure region around bow and stern, this is a sagging condition for the RD-200. However, the pressure distribution on ship hull above the calm water
Fig. 7. The time simulation history of vertical bending moment (T0 ¼ 8.0 s, H1/3 ¼ 9.0 m).
Fig. 8. The linear resultant pressure distribution by equivalent regular wave (T0 ¼ 8.97 s, Hequivalent ¼ 13.66 m; w ¼ 1701).
Fig. 9. The quasi-nonlinear resultant pressure distribution by equivalent regular wave (T0 ¼ 8.97 s, Hequivalent ¼ 13.66 m; w ¼ 1701).
cannot be obtained because of the limitation of linear assumption. In order to provide more realistic pressure distribution for the direct ship structural analysis, the quasi-nonlinear method is applied in Fig. 9, which includes the time-varied buoyancy force and Froude–Krylov force, i.e. Eqs. (17)–(19), and the nonlinear dynamic pressure distributions along the exact water line can be obtained. The results show that there still exists a high-pressure region around the bottom of bow but the value is about 12% smaller than that obtained from the linear technique. Generally, the nonlinear resultant pressure distributions around calm water line are similar to those with linear assumption in the mid-ship region but very different around bow and stern. Using the same conditions as in Figs. 8 and 9, we apply the equivalent irregular approach to calculate the resultant pressure distribution in Figs. 10 and 11 with linear and quasi-nonlinear assumptions, respectively. Both results are selected from the pressure distribution simulation history at the time exceeding the extreme value of the corresponding vertical bending moment, i.e. t0 ¼ 1507.6 s. The instantaneous total pressure distributions along the ship hull are calculated from 40 sets of different wave frequencies followed by the equivalent irregular approach as shown in Figs. 10 and 11. The nonlinear dynamic pressure at high-pressure region around bow is about 30% smaller than that obtained from the linear theory. The difference is more significant than those obtained from the equivalent regular model. Besides, the results in Figs. 8–11 also reveal that there is no significant difference between two approaches if the linear total pressure distributions are considered. However, if the quasi-nonlinear method is applied, the local highpressure values predicted by equivalent irregular wave approach are generally 25% lower than those predicted by equivalent regular wave approach at the same long-term extreme structural bending load condition.
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Fig. 12. The linear hydrodynamic pressure distribution by equivalent regular wave (T0 ¼ 8.97 s, Hequivalent ¼ 13.66 m; w ¼ 1701). Fig. 10. The linear resultant pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s, H1/3 ¼ 9.0 m; w ¼ 1701).
Fig. 11. The quasi-nonlinear resultant pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s, H1/3 ¼ 9.0 m; w ¼ 1701).
In order to investigate the advantage of the equivalent irregular wave approach, we further consider the hydrodynamic pressure only and the comparisons with the equivalent regular wave approach are shown in Figs. 12–15. From the results, it is surprising to find that significant difference appears between two approaches either using linear assumption (i.e. Figs. 12 and 14) or quasi-nonlinear assumption (i.e. Figs. 11 and 15). These differences are believed to come from the sum effects of multi-frequencies of irregular waves. This finding indicates that using the equivalent irregular wave approach can show the hydrodynamic pressure variations in a more realistic way since it can consider both ocean environment and wave load characteristics of vessel traveling in seas rather than a simple large sinusoidal wave used in the equivalent regular wave approach.
Fig. 13. The quasi-nonlinear hydrodynamic pressure distribution by equivalent regular wave (T0 ¼ 8.97 s; Hequivalent ¼ 13.66 m; w ¼ 1701).
Fig. 14. The linear hydrodynamic pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s; H1/3 ¼ 9.0 m; w ¼ 1701).
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4. The number of frequencies for simulating the irregular procedure is very important. It had better be large enough to have a sufficient long record before the signal repeats itself. The present paper adopts 40 sets of frequencies because of the limitation of computing time for the pressure distribution mesh required by finite element analysis. Generally, form the present analysis, we strongly suggest to apply the equivalent irregular wave approach to make the ship structural analysis instead of the equivalent regular one if the realistic simulation is needed. Acknowledgements Fig. 15. The quasi-nonlinear hydrodynamic pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s; H1/3 ¼ 9.0 m; w ¼ 1701).
4. Conclusion A simple and realistic equivalent irregular wave approach is developed and applied in the present study. A series of calculations for a high-speed mono-hull ship incorporating the long term prediction of corresponding DLP have been done and the equivalent regular wave approach is also adopted for comparison. Some important conclusions can be drawn as below: 1. From the present comparisons, the wave loads and hydrodynamic pressure distribution on ship hull predicted by the equivalent irregular approach is indeed more realistic and reliable. 2. Because of the large amount hydrostatic pressure effect, there is only slight difference between two equivalent wave approaches if the resultant total pressure distribution is considered either with the linear or with quasinonlinear assumptions. 3. A fast formula for searching the random phase has been developed in the present study which indeed reduces the searching time to find the time of occurrence to reach the extreme values of the corresponding DLP.
The authors wish to thank the National Science Council, Republic of China and USDDC for their financial support under Grant No. NSC 91-2611-E006-016 and Contract No. USDDC-221-T572(92), respectively. References Buckley, W.H., 1988. Extreme and climatic wave spectra for use in structural design of ship. Naval Engineers Journal 100 (5), 36–58. Chan, H.S., 1990. A three-dimensional technique for predicting first and second order hydrodynamic forces on a marine vehicle advancing in waves. Ph.D. Thesis, University of Glasgow, UK. Hogben, N., 1988. Experience from compilation of global wave statistics’. Ocean Engineering 15 (1), 1–31. Hutchison, B.L., Mathai, T., Morgan, J.M., 1998. Hydrodynamic loads on SWATH hulls. In: Proceeding of High Speed Craft Motion and Maneuverability, London, UK. Liu, D., Spencer, J., Itoh, T., Kawachi, S., Shigematsu, K., 1992. Dynamic load approach in tanker design. SNAME Transactions 100, 143–172. Ochi, M.K., 1973. On prediction of extreme values. Journal of Ship Research 17 (1), 29–37. Ochi, M.K., 1978. Wave statistics for the design of ship and ocean structures. SNAME Transactions 86, 47–76. St Denis, M., Pierson, W.J., 1953. On the motion of ships in confused seas. SNAME Transactions 61, 280–357. Stiansen, S.G., Chen, H.H., 1982. Application of probabilistic design methods to ship structures analysis. SNAME T&R Bulletin, 2–27. Walden, H., 1964. Die Eigenschaften der Meerswellen im Nordatlantischen Ozean, Deutscher Wetterdienst, No.41, Hamburg.
Ocean Engineering 34 (2007) 1422–1430 www.elsevier.com/locate/oceaneng
Prediction of design wave loads of the ocean structure by equivalent irregular wave approach Ming-Chung Fanga,, Chih-Chung Fangb, Chun-Hsien Wuc a
Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan, Taiwan, ROC b Department of Transportation and Navigation Science, National Taiwan Ocean University , Keelung, Taiwan, ROC c United Ship Design and Development Center, Taipei, Taiwan, ROC Received 17 May 2006; accepted 4 October 2006 Available online 19 December 2006
Abstract The present study is employing the equivalent irregular wave approach to predict the wave loads for a ship encountering the worst sea state with respect to the critical dynamic loading parameter. Two different hydrodynamic numerical models, i.e. 3D pulsating source technique and 3D translating pulsating source technique, are applied to calculate the corresponding RAO of the ship moving in waves. Incorporating the RAO of the related physical properties, we can calculate the extreme value for the corresponding ship loading factor, which can be regarded as the worst sea state in the service lifetime of the ship. With the time and period of the occurrence of the corresponding extreme value, we can simulate the time history of the wave load in this period, which is so-called equivalent irregular wave approach. Comparing with the results calculated by the traditional equivalent regular wave approach, we find that the equivalent irregular wave approach can simulate the corresponding wave load more realistic, especially for dynamic pressure. Using the equivalent irregular wave approach can offer the effective and practical base for the ship structural analysis. r 2006 Elsevier Ltd. All rights reserved. Keywords: Design wave load; Equivalent irregular wave; Dynamic load parameters
1. Introduction Generally, the ship design is either based on the classification rules, which are primarily semi-empirical and proven by successful operational experience, or based on direct engineering analysis. The semi-empirical approach used in classification rules has its advantage because it is simple and has some degree of assurance. Therefore it has been the mainstay of ship design practice for some time. Employing the direct engineering analysis is allowed to reduce scantlings based on rules when the sophisticated and rational approach is suitably applied. Many years ago, the American Bureau of Shipping (ABS) offered a new optional class notation called ‘‘Dynamic Load Approach’’, or DLA, which uses the results of the direct engineering analysis to indicate where increasing Corresponding author. Tel.: +886 6 2747018ext.211; fax: +886 6 2080592. E-mail address: [email protected] (M.-C. Fang).
0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.10.006
scantlings should be provided (Liu et al., 1992). This technique has been extensively applied in the ship design recently, especially in ship structural analysis. DLA represents a consistent and rational approach which employs a direct analysis for the particular hull structure being considered. Equivalent regular wave concept was also submitted by ABS early. Combining this concept with the DLA, ABS offered a very effective structural analysis method for ship hull design. In this approach, a set of Dynamic Load Parameters (DLPs) are identified, which are defined as any response process (e.g. bow vertical acceleration, vertical bending moment, lateral acceleration, roll, etc.). The DLP, when maximized, establishes a critical loading for hull structural analysis. Based on the long-term extreme value of DLP, an equivalent regular wave is then defined. The principle of the equivalent regular wave is to use the extreme value to simulate a design regular wave with the equivalent wave amplitude, frequency, and wave incidence for a ship in random waves. Thus it leads to a simple
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sinusoidal wave and a suitable instant of time for extracting a set concurrent load components to be applied to the structural mode. However, the concept still keeps the equivalent wave in ‘‘regular’’ form and cannot correspond to any real or extreme condition at sea. In view of the apparent limitation, the equivalent irregular wave concept was recently introduced by Hutchison et al. (1998) and applied on the analysis of a SWATH ship design. Because the equivalent irregular wave is treated by the time domain analysis, it is more realistic than the equivalent regular wave technique. In both equivalent wave concepts, the long-term extreme values of dynamic loading parameters, DLP, are key factors for hull structural analysis. Both Ochi (1978) and Stiansen and Chen (1982) pointed out that the long-term extreme loading can be predicted by using the short-term statistics analysis on the dynamic loading parameters in the worst sea state case. Before that, Ochi (1973) had already applied the wave spectrum to calculate the extreme values of different dynamic loading parameters of the ship. Later, Buckley (1988) also applied the extreme values of the loading on the ship structural analysis under different sea conditions. In the present study, the vertical bending moment is selected as the critical DLP and its long-term extreme value is determined as the reference of the ship structural analysis. The Bretschneider spectrum is used for simulating the corresponding sea state. The Fourier–Stieljes integral method (St Denis and Pierson, 1953) is applied as the time domain simulation technique for simulating the related ship responses in irregular waves. The results calculated by equivalent regular wave method with respect to the longterm loading extreme values are also shown in the study for comparison. 2. Theoretical formulas 2.1. Short-term prediction While the ship is sailing at sea, the encountered sea state varies with time and the factors influencing the ship response can be regarded as transient and stable. The condition generally lasts for about 1–4 hours. If the probability distribution of the ship response is considered in random sea with such fixed statistic conditions is called short-term distribution. Assume the probability density function of the ship response is Rayleigh distribution, and then it can be expressed as x x2 pðxÞ ¼ exp , (1) m0 2 m0 where x and m0 are the ship response and the mean square value of the related frequency response spectrum, respectively. The probability of the ship response at time t exceeding a ~ i.e. extreme value, can be expressed as the critical value x,
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function of mean square value as below: Z 1 ~ ¼ pðxÞdx Pðx4xÞ x~
Z
x~
pðxÞdx x~ 2 ¼ exp . 2m0 ¼1
0
ð2Þ
2.2. Long-term prediction During the design lifetime of the ship or a very long period sailing at sea, the probability distribution of the ship structure stress due to the wave loading will vary with the sea states and the other relating factors. Therefore we cannot assume it is transient and stable, and the probability distribution of the ship response under such condition is then called long-term distribution. The long-term distribution can be regarded as the combination of infinite short-term distributions. Generally the short-term probability distribution is assumed to be Rayleigh distribution and its energy spectrum or mean square value is also different from each other. If the lifetime of the ship is set to be 20 years and total annual duration actually spent at sea is 300 days, then the long-term distribution is composed of 105 short-term distribution. If the average period of the encounter wave is assumed to be 10 s, the total numbers of the ship response are about 107 108 . The long-term distribution response is the summation of the related short-term distribution response, therefore the probability of occurrence of the ship response with longterm distribution, q(x), is the summation of that of each short-term distribution component, pm(x). However, the probability of occurrence for the mean square value of each short-term distribution is different in the long-term distribution. Therefore the probability of occurrence of the ship response x for each short-term distribution component is mðxÞ pm0 ðm0 Þ in the long-term distribution where m0 ðm0 Þ is the probability of occurrence of the short-term distribution mean square value in the long-term distribution, then Z 1 qðxÞ ¼ pm ðxÞpm0 ðm0 Þ dm0 . (3) 0
For the long-term distribution of the ship response, the probability exceeding the critical response x~ is written as Z x~ ~ ¼1 Qðx4xÞ qðxÞdx 0 Z 1 ~ m0 ðm0 Þdm0 ¼ Pm ðx4xÞp 0 Z 1 x~ 2 ¼ exp ð4Þ p dm0 . 2m0 m0 0 From the above formulas, we can obtain the related statistic value for ship response in the long-term distribution. Since the long-term extreme is defined as that value
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which is expected to be exceeded only once in the lifetime of the ship, the corresponding long-term extreme in the ~ can then be determined. lifetime of the ship, x, In Eq. (4), the probability of occurrence of the shortterm distribution mean square value in the long-term distribution, Pm0 ðm0 Þ, is rather difficult to be predicted. In order to obtain the accurate Pm0 ðm0 Þ, we must in advance understand the probability of each sea state encountered, ship heading, ship speed etc. At present, the main ocean data in the world have been established quite well, e.g. North Atlantic by Walden (1964), ‘‘OWS’’ and ‘‘GWS’’ by Hogben (1988). Using these related ocean statistic data and wave spectra, the long-term probability of the response value x exceeding the level of x~ in all sea states is written as Pm Pm P P P ~ ¼ Pðx4xÞ
i¼1
j¼1
wave crest position relative to the ship. Generally, it is applied to simulate the irregular wave defined by the wave spectrum. The response amplitude operator (RAO) corresponding to the DLP is first computed for a range of frequencies and headings. The RAO of a DLP is the complex amplitude of the DLP per unit wave amplitude for a given frequency and heading. The frequency and heading combination which maximizes is then identified. This defines the frequency and heading of the equivalent regular wave. With the long-term extreme value of the correspond~ predicted from Eq. (4), the amplitude of ing response, x, the equivalent regular wave can be obtained by aw ¼
x~ . RAO
(6)
2
~ =2m0ijV yL ÞÞ L n Pij PV Py PL exp ððx Pmy P m P P P V y L n Pij PV Py PL i¼1 j¼1
V
(5) where Pij is the probability of occurrence of a sea state described by significant wave heights, Hi, and modal wave periods, Tj; and PV, Py and PL are the probabilities of ship speed, heading and loading condition respectively. n* is the number of responses in a given sea state and ship condition. 2.3. Equivalent regular wave approach According to the ABS dynamic loading approach, we can define an equivalent regular wave for each DLP, which is characterized by an amplitude, frequency, heading and
2.4. Equivalent irregular wave approach The above approach is design to accomplish, in a simple manner with the regular wave pattern, a transfer from the frequency and probabilistic domains to the time domain. The present concept of the equivalent irregular wave is a different approach which can accomplish the same transformation but is more realistic. Similar to the algorism of Hutchison et al. (1998), the present equivalent irregular wave approach is accomplished by the following procedure which is also described by using the flow chart in Fig. 1: Step 1: The choice of the wave spectrum and scatter diagram for the encountered sea environment.
Scatter Diagram & spectrum generation 3D Program
USDDC/Long-term Program
Wave spectrum RAO: motions or loads
RH-Table generation& Section Extreme Value Probability at 10-8
Irregular response simulation: k
Σ {F (k / χ) cos[kt + (k / χ) + (k)]√2sw(k)d}
RH = 3.72√m2
k=1
Judgement Random Phase, (k)
Extreme Value
Generation Return
Fig. 1. The proceeding flow of equivalent irregular wave approach.
Export T0 & (k)
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Step 2: Calculate the RAO of the corresponding DLP which is selected for design consideration. Step 3: Make the long-term extreme value analysis. In the present study, the probability of occurrence of the DLP is set to be smaller than 108 to determine the extreme value. (i.e. Eq. (4)). Step 4: Select the suitable sea state. Calculate the average of the 1/1000 highest values of the corresponding DLP with respect to the each sea state from the scatter pffiffiffiffiffiffi diagram, i.e. significant value RH ¼ 3.72 m0 , where m0 is the area under the DLP spectrum, and record the sea state with the significant value near the long-term extreme value. Step 5: Select a random phase set, fðok Þ, for the wave by using the random process. Step 6: Use the Fourier–Stieljes integral method to transfer the RAO of the corresponding DLP from frequency domain to time domain in random waves. Step 7: Find the time, t0, for the occurrence of the worst sea condition that has the amplitude greater than the extreme value calculated in step 3. If the time, t0, exists, then we can assure the worst sea condition indeed occurs at this time and proceed the corresponding loading analysis in irregular waves. If the time, t0, does not exist, then we repeat the steps 5 to 7 until the occurrence time, t0, of the worst sea condition is found. 2.5. The Fourier–Stieljes integral method The application of the Fourier–Stieljes integral method on the transformation of the RAO with respect to the corresponding DLP from the frequency domain to the time domain can be expressed by the following formula, Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðt=wÞ ¼ F ðo=wÞj cosfot þ bðo=wÞ þ fðoÞg 2Sw ðoÞdo 0
(7) and it can also be expressed as the following finite discrete summation form, f ðt=wÞ ¼
K X
F ðok =wÞj cosfok t þ bðok =wÞ þ fðok Þg
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sw ðok Þdok ,
k¼1
(8) where f ðt=wÞ is the time-varied DLP, w is the wave heading, and F ðok =wÞ is the response amplitude operator pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (RAO) of the DLP. 2S w ðok Þdok and ok are the kth wave amplitude and wave frequency with respect to the corresponding wave spectrum, respectively. t is the time, f (ok) is the random phase for the kth wave (02p), and b (ok/w) is the phase angle of the DLP with respect to the kth wave. Sw(ok) is the wave spectrum. The following Bretschneider spectrum is selected in the present study: A B S w ðoÞ ¼ 5 exp 4 (9) ðm2 sÞ o o
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where A ¼ 0:3125H 21=3 o4m ; B ¼ 1:25o4m ; om ¼ 2p/T0 (modal wave frequency, rad/s); T0 the modal wave period, s; o the rad/s. If the RAO is expressed in complex form, Eq. (8) can be written by f ðt=wÞ ¼
K X
n o Re F ðok =wÞeiðok tþfðok ÞÞ
k¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sw ðok Þdok ,
ð10Þ
where F ðok =wÞ the RAO of the corresponding DLP in complex form. While processing the equivalent irregular wave approach, it may not be convergent if the random phase is not suitably selected, which will make the approach unrealistic. However, it usually takes long time and not sure to find the suitable random phase. Therefore, a fast searching formula is developed here to solve this problem and shown as below: fðok Þ ¼ f2np ½ok t0 þ bðok =wÞgð1 þ dk Þ; ðn ¼ 1; 2; . . .Þ, (11) where dk is a random variable from 0.5 to 0.5. 2.6. Equations of motions In the study, the motion response is assumed to be small and the RAO of the motion displacement can be obtained from the following the equations of motions: 6 X
½o2 ðM jk þ Ajk Þ ioBjk þ C jk x¯ k =z0
k¼1
¼ FW j =z0 ðj ¼ 1; 2; 3; 4; 5; 6Þ,
ð12Þ
where x¯ k =z0 is the RAO of the kth mode of motion displacement. Mjk is the mass matrix. Ajk, Bjk and Cjk are the added mass, damping coefficient, and restoring force, respectively. z0 is wave amplitude and FjW is the wave exciting force. The subscripts j and k represents the direction of the force acting on and the motion mode, respectively (j or k ¼ 1, 2, 3, 4, 5, 6 for surge, sway, heave, roll, pitch, and yaw, respectively). The related hydrodynamic forces in Eq. (12) are calculated by using the three dimensional source distribution method, either with pulsating source (3DP) or with translating pulsating source (3DT) (Chan, 1990). While doing the ship structral analysis by finite element method, the hydrodynamic pressure distribution on ship hull must be well pre-calculated as the input. Based on the threedimensional source distribution methods, the corresponding resultant hydrodynamic pressure pT ðtÞ on the instantaneous wetted body surface can be calculated by the following formulas: 1 1 2 pT ðtÞ ¼ r Ft þ rF rF U þ gz , (13) 2 2
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where r is the water density, F is the resultant velocity potential and U is the ship speed. Expanding this pressure in a Taylor series about the undisturbed position of the hull and neglecting higher-order terms and static pressure terms, the RAO of the resultant hydrodynamic pressure, ¯ T , can be split into four parts: The Froude–Krylov P ¯ FK ðtÞ, diffraction pressure, P ¯ D ðtÞ, radiation pressure, P ¯ pressure, PR ðtÞ, and dynamic restoring pressure, p¯ qs ðtÞ, as follows: ¯ D ðtÞ þ P¯ R ðtÞ þ P ¯ qs ðtÞ. ¯ T ðtÞ ¼ P¯ FK ðtÞ þ P P
(14)
In Eq. (14), the linearized hydrodynamic pressures for radiation and diffraction over the body surface can be written in the form ¯ R ðtÞ ¼ r io þ U q fk x¯ k eiot =z0 ; k ¼ 1; 2; . . . ; 6, P qx (15) ¯ D ðtÞ ¼ r io þ U q f7 eiot =z0 , (16) P qx where fk and f7 are the radiation potential and diffraction potential, respectively. Generally, the above resultant pressure is nonlinear because the draft of the ship hull is time varied due to motions. Here, for simplicity, only the Froude–Krylov ¯ FK ðtÞ and dynamic restoring pressure p¯ qs ðtÞ are pressure P considered to be nonlinear. The nonlinear Froude–Krylov pressure is expressed as below ¯ FK ðtÞ ¼ rgekz eikðx cos wþy sin wÞ eiot ; P
zp0,
(17)
where z is the relative depth with respect to wave and k is the wave number. However, when 0pzpz0 , the linear interplation is applied between the following two values: ¯ FK ðtÞ ¼ rgeikðx cos wþy sin wÞ eiot ; P ¯ FK ðtÞ ¼ 0; ðz ¼ z0 Þ. P
shape are shown in Table 1 and Fig. 2, respectively. The RAO of the corresponding DLP is calculated by using the three-dimensional source distribution methods, i.e. pulsating source (3DP) and translating pulsating source (3DT). The long-term motion and wave loading are calculated using the USDDC/long term computer program. The extreme values for different DLP mentioned here are obtained by using the Eq. (5) and the two-parameter Bretschneider spectrum based on the long term ocean statistic data in Taiwan. The results are shown in Table 2 and the limit of the probability of occurrence is set to be 108. Generally the extreme value obtained by 3DP is larger than that of 3DT, which means the 3DP method is more conservative. With the aid of the fast searching formula developed here, i.e. Eq. (11), we can easily and quickly find all the corresponding sea states with the possibility exceeding the extreme value. Because the 3DT source distribution method is more realistic, therefore is applied here to calculate the following wave loadings and hydrodynamic pressure. Figs. 3–6 show some results of the
Table 1 The principle dimension of RD-200 Length of overall Length of perpendiculars Breadth Design draft Displacement Vertical center of gravity (from Keel) Longitudinal center of gravity (from AP) Matercentric height (from Keel) Roll radius of gyration Pitch radius of gyration Roll natural period
98.0 m 90.0 m 12.29 m 3.84 m 2055 m3 5.094 m 42.64 m 1.485 m 4.43 m 22.5 m 7.4 s
ðz ¼ 0Þ, ð18Þ
The nonlinear dynamic restoring pressure due to the varied draft at the relative depth h(t) can be written as. ¯ qs ðtÞ ¼ rgðhðtÞÞ, P
(19)
where hðtÞ ¼ x3 ðtÞ þ yi ðt DtÞx4 ðtÞ xi ðt DtÞx5 ðtÞ
(20)
If the nonlinear terms in Eqs. (18) and (19) are considered in the Eq. (14), we call it quasi-nonlinear hydrodynamic pressure. 3. Calculation results and discussion In order to investigate the application of the present equivalent irregular wave approach, a high-speed displacement craft, RD-200, moving with 24 knots is selected as the calculation model and its bow vertical acceleration, vertical shear force, torsion moment, and vertical bending moment are taken as DLPs for calculation. The wave heading to ship is set to be w ¼ 170 . The principle dimension and ship
Fig. 2. Body shape of RD-200.
Table 2 Long-term response prediction program spectrum scatter diagram motion method F.P. Acc.(A3) m/s^2 V.S.F.(F3) N T.M.(F4) N m V.B.M.(F5) N m
USDDC/Longterm Program BRETSCHNEIDER Annual ocean statistics in Taiwan 3DP 2.92E+01 5.68E+06 9.07E+06 1.81E+08
3DT 2.27E+01 6.71E+06 8.46E+06 1.60E+08
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simulation of different DLP of RD-200 moving with 24 knots at different sea state. From these simulations, we can find the time, t0, exceeding the extreme value of the corresponding DLP. Figs. 3 and 4 show the simulations of the torsion moment and the bow vertical acceleration at sea state with T0 ¼ 10 s and H1/3 ¼ 5.75 m, respectively. The extreme values for both DLPs are 8.46 106 N M and 22.7 m/s2, and the corresponding time of exceeding the extreme value are at t0 ¼ 1894.8.and 641.3 s, respectively. Fig. 5 is the simulation of the vertical shear force at sea state with T0 ¼ 8.0 s and H1/3 ¼ 5.75 m. The extreme value is 6.71 106 N and the time of exceeding the extreme value is at t0 ¼ 2154.5 s. Fig. 6 is the simulation of the vertical bending moment at sea state with T0 ¼ 10.0 s and H1/3 ¼ 5.25 m. The extreme value is 1.6. 108 N m and the time of exceeding the extreme value is at t0 ¼ 4092.5 s. From the above cases we can see that the time of exceeding the corresponding extreme values of DLP is different even with the same sea state. Therefore which DLP is suitable to
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Fig. 5. The time simulation history of vertical shear force (T0 ¼ 8.0 s, H1/3 ¼ 5.75 m).
Fig. 6. The time simulation history of vertical bending moment (T0 ¼ 10.0 s, H1/3 ¼ 5.25 m). Fig. 3. The time simulation history of torsion moment (T 0 ¼ 10:0 s, H1/3 ¼ 5.75 m).
Fig. 4. The time simulation history of vertical acceleration (T0 ¼ 10.0 s, H1/3 ¼ 5.75 m).
be selected as the critical load parameter is very important. The determination of the critical load parameter generally depends on the problem we concerned, for example, if the structural analysis problem is concerned, we will choose the transverse side force and prying moment in beam seas as the critical DLP for SWATH ships and the vertical bending moment in head seas for high- speed mono-hull ships. One of the most important applications of the equivalent irregular wave approach is the structural analysis by finite element method and the hydrodynamic pressure distribution is the key input for the direct analysis. Therefore, in the following part of the paper, we concentrate on the calculation of the hydrodynamic pressure distribution with respect to the critical DLP using the equivalent irregular wave approach. The equivalent regular wave approach is also used for comparison. Since the mono-hull ship sailing with high speed in bow waves is studied in the present study, the vertical bending moment is selected as the critical loading parameter. Assume the RD-200 sails in bow waves with w ¼ 170 , T0 ¼ 8.0 s and H1/3 ¼ 9 m and the
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corresponding time history simulation of the vertical bending moment is shown in Fig. 7. The long term extreme value and the time of exceeding the extreme value are found to be 1.6 108 N m and 1507.6 s, respectively. According to the equivalent regular wave approach, the equivalent wave height is at 13.66 m and period is at 8.97 s with respect to the extreme value, 1.6 108 N m. The time of the occurrence of the extreme value using the equivalent regular wave simulation is found to be 2.48 s that is the time for us to pick up the pressure distribution results to compare those obtained by the equivalent irregular wave approach at 1507.6 s in the next discussions. Figs. 8 and 9 are the results using the equivalent regular wave approach with the equivalent wave height at 13.66 m and period at 8.97 s for the linear and quasi-nonlinear instantaneous resultant total pressure distribution at 2.48 s, respectively. Fig. 8 shows that the bow submerges into the water and forms a high-pressure region around bow and stern, this is a sagging condition for the RD-200. However, the pressure distribution on ship hull above the calm water
Fig. 7. The time simulation history of vertical bending moment (T0 ¼ 8.0 s, H1/3 ¼ 9.0 m).
Fig. 8. The linear resultant pressure distribution by equivalent regular wave (T0 ¼ 8.97 s, Hequivalent ¼ 13.66 m; w ¼ 1701).
Fig. 9. The quasi-nonlinear resultant pressure distribution by equivalent regular wave (T0 ¼ 8.97 s, Hequivalent ¼ 13.66 m; w ¼ 1701).
cannot be obtained because of the limitation of linear assumption. In order to provide more realistic pressure distribution for the direct ship structural analysis, the quasi-nonlinear method is applied in Fig. 9, which includes the time-varied buoyancy force and Froude–Krylov force, i.e. Eqs. (17)–(19), and the nonlinear dynamic pressure distributions along the exact water line can be obtained. The results show that there still exists a high-pressure region around the bottom of bow but the value is about 12% smaller than that obtained from the linear technique. Generally, the nonlinear resultant pressure distributions around calm water line are similar to those with linear assumption in the mid-ship region but very different around bow and stern. Using the same conditions as in Figs. 8 and 9, we apply the equivalent irregular approach to calculate the resultant pressure distribution in Figs. 10 and 11 with linear and quasi-nonlinear assumptions, respectively. Both results are selected from the pressure distribution simulation history at the time exceeding the extreme value of the corresponding vertical bending moment, i.e. t0 ¼ 1507.6 s. The instantaneous total pressure distributions along the ship hull are calculated from 40 sets of different wave frequencies followed by the equivalent irregular approach as shown in Figs. 10 and 11. The nonlinear dynamic pressure at high-pressure region around bow is about 30% smaller than that obtained from the linear theory. The difference is more significant than those obtained from the equivalent regular model. Besides, the results in Figs. 8–11 also reveal that there is no significant difference between two approaches if the linear total pressure distributions are considered. However, if the quasi-nonlinear method is applied, the local highpressure values predicted by equivalent irregular wave approach are generally 25% lower than those predicted by equivalent regular wave approach at the same long-term extreme structural bending load condition.
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Fig. 12. The linear hydrodynamic pressure distribution by equivalent regular wave (T0 ¼ 8.97 s, Hequivalent ¼ 13.66 m; w ¼ 1701). Fig. 10. The linear resultant pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s, H1/3 ¼ 9.0 m; w ¼ 1701).
Fig. 11. The quasi-nonlinear resultant pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s, H1/3 ¼ 9.0 m; w ¼ 1701).
In order to investigate the advantage of the equivalent irregular wave approach, we further consider the hydrodynamic pressure only and the comparisons with the equivalent regular wave approach are shown in Figs. 12–15. From the results, it is surprising to find that significant difference appears between two approaches either using linear assumption (i.e. Figs. 12 and 14) or quasi-nonlinear assumption (i.e. Figs. 11 and 15). These differences are believed to come from the sum effects of multi-frequencies of irregular waves. This finding indicates that using the equivalent irregular wave approach can show the hydrodynamic pressure variations in a more realistic way since it can consider both ocean environment and wave load characteristics of vessel traveling in seas rather than a simple large sinusoidal wave used in the equivalent regular wave approach.
Fig. 13. The quasi-nonlinear hydrodynamic pressure distribution by equivalent regular wave (T0 ¼ 8.97 s; Hequivalent ¼ 13.66 m; w ¼ 1701).
Fig. 14. The linear hydrodynamic pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s; H1/3 ¼ 9.0 m; w ¼ 1701).
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4. The number of frequencies for simulating the irregular procedure is very important. It had better be large enough to have a sufficient long record before the signal repeats itself. The present paper adopts 40 sets of frequencies because of the limitation of computing time for the pressure distribution mesh required by finite element analysis. Generally, form the present analysis, we strongly suggest to apply the equivalent irregular wave approach to make the ship structural analysis instead of the equivalent regular one if the realistic simulation is needed. Acknowledgements Fig. 15. The quasi-nonlinear hydrodynamic pressure distribution by equivalent irregular wave (T0 ¼ 8.0 s; H1/3 ¼ 9.0 m; w ¼ 1701).
4. Conclusion A simple and realistic equivalent irregular wave approach is developed and applied in the present study. A series of calculations for a high-speed mono-hull ship incorporating the long term prediction of corresponding DLP have been done and the equivalent regular wave approach is also adopted for comparison. Some important conclusions can be drawn as below: 1. From the present comparisons, the wave loads and hydrodynamic pressure distribution on ship hull predicted by the equivalent irregular approach is indeed more realistic and reliable. 2. Because of the large amount hydrostatic pressure effect, there is only slight difference between two equivalent wave approaches if the resultant total pressure distribution is considered either with the linear or with quasinonlinear assumptions. 3. A fast formula for searching the random phase has been developed in the present study which indeed reduces the searching time to find the time of occurrence to reach the extreme values of the corresponding DLP.
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