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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
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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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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)
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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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PREDICTION OF VERTICAL-PLANE WAVE LOADING AND SHIP RESPONSES IN HIGH SEAS 115
(6)
where he(x, t)=−ze(x, t)−Df(x) with Df(x), the freeboard; B e(x) is an effective breadth of the green water. In the present
study B e(x) is taken to be half the sectional breadth Bd(x) of the deck, i.e., Be (x)=0.5 Bd(x).
Due to the dominant positive component in the second term of Equation 4, the force fgw will be directed upward just at
the moment when the water enters or leaves the deck. 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
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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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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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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
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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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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.
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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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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
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moments.

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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
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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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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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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
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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.
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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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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.