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Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 112 Prediction of Vertical-Plane Wave Loading and Ship Responses in High Seas Zhaohui Wang1 , Jinzhu Xia2 , J.Juncher Jensen1 and Arne Braathen 3 (1Technical University of Denmark, 2The University of Western Australia, 3Det Norske Veritas, Norway) ABSTRACT The non-linearities in wave- and slamming-induced rigid-body motions and structural responses of ships such as heave, pitch and vertical bending moments are consistently investigated based on a rational time-domain strip method (Xia, Wang and Jensen, 1998) A hydrodynamic model for predicting sectional green water force is also outlined for the investigation of the effect of green water loads on the global hull girder bending moment. The computational results based on the non-linear time-domain strip theory are compared with those based on the fully non-linear 3-D panel method SWAN-DNV and other published results. From the rather extensive computations and comparisons, it is found that non-linear effects are significant in head and bow waves in the motion-wave resonant region for both heave and pitch motions, bow accelerations and vertical bending moments for two container ships considered, whereas not significant for a VLCC. The non-linearities in motions and structural loads of conventional monohull ships seem well predicted by the present non-linear strip theory. INTRODUCTION Linear strip theories and 3D linear potential theories have been widely accepted and used by naval architects as the main tools for estimating the performance of a ship in waves due to the relatively small computational effort and the generally satisfactory agreement with experiments. The difficulties come in higher and extreme seas and when trying to establish maximum lifetime loads for structural design. Non-linearities in wave- and slamming-induced structural responses of ships have been observed from full-scale measurements and in model experiments. Strain measurements on ships with fine forms such as warships (Smith, 1966) and container ships (Meek et al, 1972) in moderate and heavy seas have shown that the wave-induced sagging bending moments can be considerably larger than the wave-induced hogging bending moments. The non-linearity in the vertical- plane bending moments has to be taken into account in structural design. To minimise wave-making resistance and enhance seakeeping performance at relatively high speed, fast vessels are usually designed with large length to beam ratio, large bow flare and low block coefficient. These properties put them outside the application range for the rules of the classification societies for hull girder loads calculation. Individual considerations based on direct calculation procedures are therefore required to derive the design loads (Zheng, 1999). Many anpirical non-linear strip methods have been proposed predicting the non-linear wave- and slamming-induced structural loads with reasonably good accuracy (see the proceedings of the International Ship and Offshore Structures Congress (ISSC) and the International Towing Tank Conference (ITTC)). The importance of the non-linearities in heave and pitch motions of ships was not recognised until late 1980's. In the benchmark seakeeping experiments carried out for ITTC on a standard hull form designated the S175 container ship by twenty three organisations, a significant scatter was found in some of the transfer function results for heave and pitch motions in head seas (ITTC, 1987). Later model tests by O'Dea, et al (1992) demonstrated a variation of the heave and pitch transfer functions with wave amplitude, indicating a nonlinear motion behaviour. Recently, Kapsenberg and Brouwer (1998) showed that linear prediction of the heave motion may be insufficient if a ship hull is designed to minimise both resistance and wave-induced motions. Model testing at this stage is still essential in hull form optimisation for seekeeping performance. In order to predict non-linearities consistently in both wave-induced rigid-body motions and structural loads, a rational time-domain strip method was developed by Xia, Wang and Jensen (1998) for the vertical-plane problems. A higher-order ordinary differential equation was used to approximate the hydrodynamic memory effect due to the free surface wave motion. The hydrodynamic and restoring forces were estimated exactly over the instantaneous wetted surface. The ‘momentum slamming' force was automatically obtained in the formulation. The fluid force expression was coupled with the structure represented as a Timoshenko beam to form a hydro the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 113 elasticity theory. By specifying wave amplitudes, non-linear frequency response functions were presented for the S175 Containership in head seas, including the heave and pitch motions, bow acceleration and sagging/hogging bending moments, see Figure 1. Two different bow geometries of the ship were considered to demonstrate the relationship between the bow flare of ships and the non-linearity of the responses, see Figure 2. The predicted results were compared with available experimental data from the elastic model test made by Watanabe, Ueno and Sawada (1989) and the experimental investigation by O'Dea, Powers and Zselecsky (1992). Very good agreements were obtained between the predictions and the measurements for wave-induced rigid-body motions and bending moments. A ship sailing in a heavy sea may experience shipping of water on the fore deck. The green water load may result in severe impact loading on the deck, the superstructure and the equipment mounted on the deck. Prediction of green water loads is especially important for fast ships and for FPSOs as shipping of green water may place severe operational restrictions on these kinds of vessels. Recently, a significant research effort has been initiated to solve the problem. Model tests have been performed on FPSOs in MARIN (Maritime Research Institute Netherlands), and design guidelines are issued addressing the bow shape and the necessary freeboard and breakwater. However, the present numerical methods cannot predict correctly the green water loads due to the very complicated and non-linear water flow around the bow and over the deck. Volume-Of-Fluid (VOF) methods seem to be the most promising, but require significant improvement (Fekken, Veldmann and Buchner, 1999). The extreme sagging wave bending moments in ships are usually determined by taking into account the non-linearities due to momentum slamming and hydrostatic restoring action. These non-linearities are very important to container ships with a large flare, yielding extreme sagging moments twice as high as those obtained by a linear analysis, see Figure 2. However, the effect of green water on deck is seldom included in the calculations of the sectional loads but if it is, the associated vertical forces are often based just on the static water head by which the relative motion exceeds the freeboard. Figure 1: Calculated non-dimensional frequency response functions (FRF) of heave, pitch, bow acceleration (FP) and midship bending moment of the original S175 container ship for different regular wave amplitudes, Fn=0.25 (Xia, Wang and Jensen, 1998). Figure 2: Non-linear sagging (positive) and hogging (negative) bending moments of the original (O) and the modified (M) S175 container ship, moving in regular waves, λ=1.2 L, a=L/60 and Fn=0.25. Comparison is made of the experiment (Watanabe et al., 1989) and the numerical calculation (Xia, Wang and Jensen, 1998). the authoritative version for attribution. Buchner (1994, 1995) has shown by measurements that the actual pressure due to water on deck might be several times larger than the static water head. A much more accurate description of this load was obtained by including a term proportional to the change of the momentum of the water on deck. Later, Wang, Jensen and Xia (1998) proposed a modified formula to account for the forward-speed effect of the ship. The concept of effective relative motion was used and a Smith correction factor was introduced to account for the wave pile-up effect during green water. The present paper outlines several of the recent validations and applications of the non-linear hydroelasticity method for heave and pitch motions, vertical bending moments and other wave-induced responses of ships. A short introduction of the non-linear time-domain strip theory model

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 114 will be given in Section 2. The modelling of the longitudinal distribution of green water loads will be introduced in Section 3. In Section 4, the non-linearities of wave loads and ship responses in head seas will be discussed for the S175 container ships. Comparison will be made of the present predictions with other numerical and experimental results, particularly, the fully 3-D non-linear simulation by SWAN-DNV (Adegeest, Braathen and Vada, 1998). Section 5 of this paper will be devoted to the prediction and validation of wave loads and ship responses in all headings for a panamax container ship and a VLCC. This will also demonstrate the relationship between the non-linearities and the hull forms. THE TIME-DOMAIN STRIP THEORY According to Xia, Wang and Jensen (1998), the non-linear time-domain hydrodynamic force F(x, t) at the longitudinal positions x on the hull maybe expressed by (1) where I in represents both the impulsive and memory effects in the hydrodynamic momentum; D/Dt is the total with U being the forward speed of the ship; derivative with respect to time and are the so-called frequency-independent hydrodynamic coefficients derived by a rational approximation from the frequency dependent added-mass and damping coefficients. Furthermore, the relative where w(x, t) is the vertical motion of the hull and motion is the wave elevation with Smith correction. are taken as functions of only x, i.e. If the frequency-independent hydrodynamic coefficients and the change of wetted body surface is neglected, Equation (1) represents a time-domain counterpart of the linear strip theories, for example, Salvesen, Tuck and Faltinsen (1970) Generally, J=3 suffices for most sectional shapes for symmetric ship motion problems. By integration of the higher order differential equation in Equation (1) and by incorporation of the hydrostatic buoyancy force fb under the instantaneous wave surface and the green water force fgw, the total non-linear external fluid force Z(x, t) acting on a ship section can be expressed as (2) where is the added mass of the ship section when the oscillating frequency tends to infinity; accounts for the ‘memorial' hydrodynamic effect with qJ governed by the following set of differential equations (3) The third term of Z(x, t) in Equation (2) is the momentum slamming force. It is assumed to be zero when the ship section exits water. The still-water response of the ship due to the difference of the distribution of the weight and the buoyancy forces is ignored in the calculations. MODELING OF GREEN WATER LOADS A brief introduction to the formulation of the green water sectional force fgw (in Equation 2) is given below, whereas a detailed derivation can be found in Wang, Jensen and Xia (1998) and Wang (2000). The vertical load fgw per unit length due to green water on deck in a longitudinal position x and at a time t is taken to be (4) the authoritative version for attribution. directed positively upwards. Here ze(x, t) is defined as the effective relative motion and mgw denotes the instantaneous mass per unit length of green water. The effective relative motion ze(x, t) is taken to be a function of the nominal relative motion zn(x, t)=w(x, t)−ζ(x, t) based on the undisturbed wave elevation ζ(x, t), (5) Here Cs is the Smith correction factor for the instantaneous wetted body sections. The green water mass is taken to be proportional to the effective water height he(x, t) on the deck:

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This unphysical behaviour is excluded by assuming fgw=0 if fgw > 0. Equation 4 is of the same form as suggested by Buchner (1994, 1995), except that the forward speed effect is included in the definition of velocities and accelerations. The present approach simply treats the green water load in the same way as the added mass of water for a submerged section. The change in wave profile due to the bow and the flow of water on the deck is accounted for by using ze(x, t), instead of zn(x, t). The Smith correction is introduced because it has been successfully used in the strip theories to account for the diffraction effect of the incident waves, and it gives a plausible variation of ze with the geometry of the submerged part of the section, the wave elevation and the frequency. For instance, an increasing bow flare will increase Cs and thus decrease ze. When the wavelength is long or the wave frequency is small, the effective relative motion is close to the nominal relative motion, which indicates a physically rational asymptotic behaviour of the dynamic wave deformation, see Figure 3. The relative motion amplitude za in Figure 3 is shown as a function of the wave frequency ω for the S175 ship sailing in a head sea with a Froude number of 0.25. (Hereafter, subscript a represents the amplitude of the full and a is defined as wave amplitude. The relative motion amplitude za is defined as half the peak-to-peak value). The location considered is FP and the wave amplitude a of the regular waves is L/60, where L is the length (175 m) of the ship. At this wave amplitude significant non-linearities are present in the responses, see Figure 2. Figure 3 illustrates that Equation 5 yields a rather good agreement with the measurements over the whole frequency range, whereas the nominal relative motions zn deviate quite strongly from the measurements. In a stochastic sea an average value Cs is applied with the individual wave amplitudes as weight factors. Numerical results for the relative motion in stochastic seaways and comparison with experimental results can be found in (Wang et al., 1998). Figure 3: Relative motion amplitude za at the FP for the S175 container ship sailing in regular head waves. Wave amplitude a=L/60, Fn=0.25 (Wang et al, 1998). Measured results from Watanabe et al. (1989). THE S175 CONTAINER SHIP IN HEAD WAVES Prediction of the non-linear motions and wave loads of the S175 container ship in head waves is presented below with comparisons with 3D non-linear results by SWAN-DNV (Adegeest, Braathen and Vada, 1998) and other available publications. The body-plan and the main particulars of the ship are identical to those used in Watanabe et al. (1989). Figures 4 and 5 compare the non-linear calculation by the present method with the experimental data for the first harmonics of heave and pitch of the original S175 container ship. The presentation is made for three wavelengths, λ/L =1.0, 1.2, 1.4, and two Froude numbers, Fn=0.2, 0.275, for both amplitude and phases (defined as leads relative to the wave elevation at the LCG, with wave and heave defined as positive upward and pitch defined as positive bow down). The predicted non-dimensional amplitudes of the heave and pitch as functions of the wave steepness ka seem to agree very well with the experimental results. It is seen that as the wave amplitude increases, the non-dimensional heave and pitch monotonously decrease. Also presented in Figures 4 and 5 are the numerical results obtained by the 3-D non-linear time- domain codes SWAN-DNV and LAMP-4 (Lin et al., 1994), the partly non-linear simulation (Xia and Wang, 1997) where only non-linearity in hydrostatic and ‘momentum slamming' forcing is considered. Other results in Fig. 4 and 5 are (obtained by various non-linear strip theory formulations (ISSC, 2000). The discrepancy between the partly non-linear the authoritative version for attribution. simulation and the non-linear and experimental results indicates that non-linearities in the hydrodynamic forcing due to variation of added mass and damping are important. The present nonlinear theory prediction seems to agree very well with the experimental results, particularly for the variation of heave and pitch amplitudes with wave sloop. It is noted that green water on deck is not detected in the computations for the nonlinear strip theory results in Figures 4 to 6.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 116 The bow acceleration is a sensitive indication of higher harmonics since in regular waves acceleration corresponds to multiplying the displacement by ω2, 4ω2, 9ω 2 for the first, second and third harmonics, respectively. If the response is properly represented by a Volterra functional series, these harmonics would be expected to vary as the square of the wave amplitude for the second harmonic, cube for the third harmonic, etc. The comparison of the bow acceleration of the second and the third harmonics (non-dimensionalised by the acceleration of the gravity g) for the wavelength λ= L and the Froude numbers Fn=0.2 is presented Figure 6. The present results agree well for the third harmonics, whereas they under-predict the second harmonics when the wave steepness ka≥0.08, i.e. a≥2.2m. The second harmonics results (dot line) obtained by the quadratic strip theory (Jensen and Pedersen, 1979) and the results by various non-linear strip theory formulations (ISSC, 2000) are also included in Figure 6. Figure 4: Comparison of the calculated and the experimental (hollow square points, O'Dea et al., 1992) magnitudes and phases of the heave (left) and pitch (right) of the original S175 container ship with respect to wave steepness, Fn=0.2. Solid lines for the present method, solid circle points for SWAN-DNV, dash lines for the partly non-linear simulation (Xia and Wang, 1997) and other strip theories (ISSC, 2000). the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 117 Figure 5: Comparison of the calculated and the experimental (hollow square points, O'Dea et al., 1992) magnitudes and phases of the heave (left) and pitch (right) of the original S175 container ship with respect to wave steepness, Fn=0.275. Solid lines for the present method, solid circle points for SWAN-DNV, solid square points for LAMP-4, dash lines for the partly non- linear simulation (Xia and Wang, 1997) and other strip theories (ISSC, 2000). The load due to green water on deck might influence the maximum wave-induced sagging bending moment if the two events are in phase. Figure 7 illustrates the predicted force and response time histories of the S175 container ship in regular waves with two different wavelengths λ= L and λ=1.2L. The time histories include the input wave elevation ζ, the effective relative ze , the buoyancy force fb, the momentum slamming force fsl , the green water force fgw and the hydrostatic part of the green water force fs, all at the FP and the midship wave bending moment Mt. The incident waves are regular head-sea waves with the wave amplitude a=L/60 and Fn=0.25. The result for the bending moment on the assumption of no green water is included for comparison. Here the hull is taken to be fairly rigid to suppress hydroelastic effects. For λ=1.2L it is seen that the green water force fgw is larger than the momentum slamming force fsl at the FP, but contrary to the momentum slamming force fsl , fgw appears slightly after the peak in the midship bending moment. Hence, the green water load only marginally influences the peaks of the midship bending moment. The magnitude of the green the authoritative version for attribution. water force is seen to be about twice the hydrostatic value fs given by the first term in Equation 4. If the nominal relative motion zn is used instead of ze, no green water on deck appears, see Figure 3, for this wave amplitude. In Buchner (1994) the average value of the ratio between the local pres

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 118 sure in the centre line at FP and the static water head is found to be about 3.5 from measurements on a frigate at 20 knots. As the pressure must be expected to vary both in magnitude and phase over the breadth of the deck and with the highest value in the centre line, the presently predicted value of fgw is believed to be of the correct magnitude. The results for λ=L show only marginally water on deck. Hence, fgw is small but the phase nearly coincides with the peak of the midship bending moment. A small reduction in the peak sagging moment is therefore seen in Figure 7. Figure 6: Comparison of the calculated and the experimental (cross and square points, O'Dea et al., 1992) bow acceleration (15%L aft of FP) of the second and third harmonics of the original S175 container ship with respect to wave steepness, λ/L=1.0, Fn=0.2. Solid line for the present method, dot line for the quadratic strip theory and other lines, ISSC, 2000. Figure 7: Time histories of the wave elevation ζ , the effective relative motion ze, the buoyancy force fb, the momentum slamming force fsl, the green water force fgw and the hydrostatic part of the green water force fs, all at the FP, together with the midship bending moment Mt for the S175 container ship in regular head waves. Wave amplitude a=L/60, Fn= 0.25 (Xia, Wang and Jensen, 1998). SHIP RESPONSES IN ALL HEADINGS For short-term and long-term predictions, it is important to investigate the motions and loads in all wave directions. This is because the highest stresses may be expected in 30 and 60 degrees wave directions, either approaching from the bow or the stern due to the superposition of the vertical and lateral bending moments and torsional moments. In this section comparison with the model experiments of a panamax container ship (Tan, 1972) and a VLCC (very large crude carrier) (Tanizawa et al., 1993) will be made for all headings. the authoritative version for attribution. A Panamax Container Ship Towards the end of 1969, collaborative research was performed in the Netherlands on ship behaviour at sea, particularly for third generation container ships, designed for trade on the Far East with low block coefficient. Model experiments were carried out to investigate the effect of wave direction, length, height, and ship speed on the bending and torsional moments and shear forces. A detailed description of the experiments including the body plan and the main particulars of the ship can be found in Tan (1972). The model tests were conducted in seven wave directions: β=25, 45, 65, 180, 205, 225 and 245 degrees, (180 degrees denote head waves). Seven wavelength to ship length ratios are used: λ/L=0.35, 0.5, 0.6, 0.7, 0.9, 1.1 and 1.4. The wave height was kept constant at L/60. A range of ship

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 119 speeds between Fn=0.22 and Fn=0.27 was investigated. Here comparisons are only made for Fn=0.245, since the responses very slightly depend on the speed in the considered speed range (Flokstra, 1974). The comparison of the frequency response functions of amplitude of heave za /a, pitch θa/(ka), vertical acceleration at the FP, midship bending moment Ma/(ρgaBL 2) at the seven headings is shown in Figures 8 to 11. More detailed comparisons including phase and structural responses at other sections are given in Wang (2000). The agreement is generally good for the frequency response functions, defined as half to peak-to-peak value. It is seen that the non-linear calculation of motions (Figures 8 to 10) agrees well with the experiment for head, bow and beam waves, whereas the linear calculation over-predicts the motions as also found by Flokstra (1974) and Wahab and Vink (1975). Figure 8: Frequency response functions of the heave za/a of the container ship. Fn=0.245. Figure 9: Frequency response functions of the pitch θa/(ka) of the container ship. Fn=0.245. The non-linearity of motions is obvious for head and bow waves in the motion-wave resonant region, but small for quartering waves. The difference between the amplitudes of linear and non-linear calculations of vertical bending moment is small, but the small difference does not mean small nonlinear effect of the structural responses. As seen in Figures 11, the increase due to non-linear effect of the sagging moments is about the same as the corresponding decrease of the hogging the authoritative version for attribution. moments.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 120 Figure 10: Frequency response functions of the vertical acceleration at the FP of the container ship. Fn=0.245. Figure 11: Frequency response functions of the midship sagging and hogging bending moment Ma/(ρgaBL2) of the container ship. Fn=0.245. A Very Large Crude Carrier Here a VLCC is used to validate the program and to investigate the nonlinearities of ship motion and responses as another kind of ship different from container ships. VLCCs have more vertical sidewalls and larger block coefficients than container ships. The non-linear effect of the VLCCs may be expected to be small. A free-run experiment of a VLCC model ship was carried out in the Ship Research Institute of Japan (Tanizawa et al., 1993). The model tests were conducted in seven wave directions: β=0, 30, 60, 90, 120, 150 and 180 degrees. (180 degrees denoted head waves). Ten wavelength to ship length ratios were used: λ/L=0.2, 0.3, 0.4, 0.5, 0.625, 0.75, 0.875, 1.0, 1.25 and 1.5. The wave height H was kept constant at L/64. The Froude number was 0.131. The ship motion, the vertical and lateral bending moments amidships, the relative water level and the wave pressure were measured. The comparison of the frequency response functions of heave za/a, pitch θa /(ka) and midship bending moment Ma/ (ρgaBL2) amidships at the seven headings is shown in Figures 12 to 14. Generally, the calculated results agree reasonably well with the experiments and, as expected, the differences between the linear and the non-linear results are very small. Calculations have also been performed for greater wave heights, but the frequency response functions change little. It is seen from Figures 12 that the ship experiences larger heave motion in bow waves than in head and quartering the authoritative version for attribution. waves. The predicted amplification of the heave motion for bow waves in the heave-wave resonant region is not as large as measured in the experiment. The vertical bending moment curves in head and bow waves in Figures 14 show multiple peaks. The occurrence of the two larger peaks for the head sea

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. the inertial forces Figure 12: Frequency response functions of the heave za/a of the VLCC. Fn=0.131. Figure 13: Frequency response functions of the pitch θa/(ka) of the VLCC. Fn=0.131. PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS Figure 14: Frequency response functions of the midship bending moment Ma/(ρgaBL2 ) of the VLCC. Fn=0.131. 121 case can be explained by considering the bending moment as the sum of moments due to the hydrodynamic forces and to

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 122 CONCLUSION In this paper, we have outlined several recent validations and applications of a non-linear time-domain strip theory method for heave and pitch motions, vertical bending moments and other wave-induced responses of ships. From the extensive computations and comparisons, it is found that the non-linear theory significantly improves the overall accuracy of prediction of both heave, pitch motions and structural loads of monohull ships in high seas. More specifically, the following conclusions maybe drawn: • - The non-linear effects are significant in the motions and structural responses of the two container ships considered, whereas not significant for the VLCC. • - For the two container ships, the non-linearity of the motions seems stronger in head and bow waves in motion- wave resonant region, but less important in quartering waves. • - In the ship-wave resonant region for head and bow waves for the two container ships investigated, heave, pitch, bow acceleration and hogging bending moment monotonously decrease while the sagging bending moment increases with increasing wave steepness. REFERENCES Adegeest, L, Braathen, A. and Vada, T., 1998, “Evaluation of methods for estimation of extreme non-linear ship responses based on numerical simulations and model tests”, Proceedings: 22nd Symposium on Naval Hydrodynamics, Vol. 1, pp.70–84. Buchner, B., 1994, “On the effect of green water impacts on ship safety (a pilot study)”, Proc. NAV'94, Vol. 1, Session IX, Rome, Italy. Buchner, B., 1995, “On the impact of green water loading on Ship and Offshore Unit Design”, the Sixth International Symposium on Practical Design of Ships and Mobile Units, Seoul, 1.430–1.443. Fekken, G., Veldmann, A.E.P. and Buchner, B., 1999, “Simulation of Green Water Loading Using the Navier-Stokes Equations”, Proc. 7th Int. Conf. Numerical Ship Hydrodynamics, Nantes. Flokstra, C., 1974, “Comparison of ship motion theories with experiments for a container ship”, International Shipbuilding Progress, Vol. 21, No. 238, 168–189. ISSC, 2000, “Committee VI.1: on Extreme Hull Girder Loading”, Proc. of the 14th International Ship & Offshore Structures Congress, Vol. 2. ITTC, 1987, “Report of the Seakeeping Committee”, Proc. of the 18th International Towing Tank Conference, Vol. 1, pp. 401–468. Jensen, J.J. and Pedersen, P.T., 1979, “Wave-Induced Bending Moments in Ships—a Quadratic Theory”, Trans. RINA, Vol. 121, pp. 151–165. Kapsenberg, G.K. and Brouwer, R, 1998, “Hydrodynamic development for a frigate for the 21 century”, Proc. of PRADS'98, The Netherlands, pp. 555– 566. Lin, W.-M., Meinhold, M., Salvesen, N. and Yue, D.K.P., 1994, “Large-amplitude motions and wave loads for ship design”, Proceedings of 20th Symposium on Naval Hydrodynamics. Meek, M. et al., 1972, “The structural design of the O.C.L. container shops”, Trans. RINA, Vol. 114, pp. 241–292. O'Dea, J., Powers, E. and Zselecsky, J., 1992, “Experimental determination of nonlinearities in vertical plane ship motions”, Proceedings: 19th Symposium on Naval Hydrodynamics. Salvesen, N., Tuck, E.O. and Faltinsen, O., 1970, “Ship motions and sea loads”, Trans SNAME, 78, pp. 250–287. Smith, C.S., 1966, “Measurement of service stresses in warships”, Conference on Stresses in Service, Inst. of Civ. Engrs., March. Tan, S.G., 1972, “Wave load measurements on a model of a large container ship”, Netherlands Ship Research Center, Report 173S. Tanizawa, K., Taguchi, H., Saruta, T. and Watanabe, I., 1993, “Experimental study of wave pressure on VLCC running in short waves” (in Japanese), JSNAJ, Vol. 174. Wahab, R. and Vink, J.H., 1975, “Wave induced motions and loads on ships in oblique waves”, International Shipbuilding Progress, Vol. 22, No. 249, 151–184. Wang, Z., 2000, “Hydroelastic analysis of high speed ships”, Ph.D. Thesis, Department of Naval Architecture and Offshore Engineering, Technical University of Denmark the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 123 Wang, Z., Jensen, J.J. and Xia, J., 1998, “On the Effect of Green Water on Deck on the Wave Bending Moment”, Proc. 7th International Symposium on Practical Design of Ships and Mobile Units, PRADS'98, The Netherlands, pp. 239–245. Watanabe, I., Ueno, M. and Sawada, H., 1989, “Effects of bow flare shape to the wave loads of a container ship, J. Soc Naval Architects of Japan, 166, pp. 259–266. Xia, J. and Wang, Z., 1997, “Time-Domain Hydroelasticity Theory of Ships Responding to Waves”, J. of Ship Research, Vol. 41, No. 4. Xia, J., Wang, Z. and Jensen, J.J., 1998, ‘Nonlinear Wave Loads and Ship Responses by a Time-Domain Strip Theory”, Marine Structures, Vol. 11, No. 3, pp. 101–123. Zheng, X., 1999, “Global Loads for Structural Design of Large Slender Monohulls”, FAST'99, Seatle, USA, pp. 719–729. the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 124 DISCUSSION L.J.Doctors University of New South Whales Australia The reviewer would like to congratulate the four authors on a most practical and interesting piece of research. It is particularly heartening to see the many plotted examples of nonlinearity, exemplified by a comparison of response curves corresponding to differing wave amplitudes. Could the authors first clarify precisely what nonlinear effects are included in their theory? The text in the section “The Time-Domain Strip Theory” seems to suggest that the nonlinearity is included with respect to the buoyancy force, green-water force, and the slamming force. On the other hand, has the linearized free-surface condition been used for the other components of force? It is noted that the influence of nonlinearity is generally that increasing sea wave amplitude leads to a reduction in the response-amplitude operator (RAO), at least as far as heave, pitch, and bow acceleration are concerned. However, the opposite appears to be the case for midship bending moment. This is seen in Figure 2, for example, most clearly near the resonance. Is this a generally true result for other vessels as well? The much improved correlation when utlizing the nonlinear theory is seen in Figure 8 through Figure 11. On the other hand, the case of a 45-degree heading in Figure 8 shows poorer correlation between the experiments and both the linear and nonlinear theory. This seems to suggest a difficulty with the experiments rather than with the theory. Could this point be elaborated upon? Once again, I would like to express my appreciation to the authors for a most informative paper. AUTHOR'S REPLY As expressed in Equation 2, the non-linearities are included in the theory with respect to the buoyancy force, green water loads, slamming action and the other hydrodynamic effects such as inertia and damping terms calculated to the instantaneous wet surface of the body. On the other hand, the linearized free-surface condition has been used for the hydrodynamic calculations at the instantaneous submersion. Based on our calculations, the trend of non-linearity shown in Figure 1 is generally confirmed for container ships and frigates. No opposite trend has yet been found for other ship types. The poorer correlation of heave in the 45-degree heading in Figure 8 might be due to the difficulties in the heading control of the self-propelled model tests in this heading. Another reason might be the non-linear coupling between the heave and roll motions, which is not included in the calculation but could be significant in the model testing for the 45- degree heading. DISCUSSION H.Kagemoto University of Tokyo, Japan The authors claim that the nonlinear characteristics of motion responses observed at resonant region, where the motion responses per unit wave height are reduced as the wave height becomes larger, can be accounted for by the potential-based nonlinear theory. However, I think and I understand that it is generally agreed that such nonlinearities in the resonant range are caused due to the nonlinear characteristics of viscous damping forces. AUTHOR'S REPLY We agree with Professor Kagemoto that viscosity can have an effect on vertical ship motions. For example, Beukelman (1983) experimentally studied a ship-like model with rectangular cross-sections in regular head sea waves of different wavelengths and found that viscous effect, particularly flow separation around sharp corners, may significantly contribute to heave and pitch damping. However, viscous effect is not likely to be important for heave and pitch motions of fine ships such as frigates and container ships (Faltinsen and Svensen, 1990). And it has been common practice to include viscous effect in the prediction of roll motion but neglect the effect of viscosity on heave and pitch motions of ships. The characteristics of motion responses observed in the present analysis may be interpreted as a non-linear potential flow effect, including the non-linearity in the potential-flow-based hydrodynamic damping. Similar motion characteristics have been observed based on use of non-linear potential flow theory and 3-D Rankine source method as reviewed by Beck and Reed (2000) at this symposium. A recent discussion on viscous effects on wave-induced ship motions may be found in the proceedings of ISSC (2000, pp. 268–269). REFERENCES: the authoritative version for attribution. Beck, R.F. and Reed, A.M., 2000, Modern seakeeping computations, 23rd Symposium on Naval Hydrodynamics, France.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 125 Beukelman, W., 1983, Vertical motions and added resistance of a rectangular and triangular cylinder in waves, Report No. 594, Ship Hydrodynamics Laboratory, Delft University of Technology. Faltinsen, O. and Svensen, T., 1990, Incorporation of seakeeping theories in CAD, Intl Symposium on CFD and CAD in Ship Design, Wageningen, The Netherlands. ISSC, 2000, “Committee VI.1: on Extreme Hull Girder Loading”, Proc. of the 14th International Ship & Offshore Structures Congress, Vol. 2. the authoritative version for attribution.