PREDICTION METHODS FOR BROACHING AND
THEIR VALIDATION- FINAL REPORT OF SCAPE COMMITTEE (PART 6) Naoya UMEDA*, Hirotada HASHIMOTO*, Atsuo MAKI*, Masatoshi HORI*,
Akihiko MATSUDA** and Tsutomu MOMOKI***Department of Naval Architecture and Ocean Engineering, Osaka University, JAPAN**National Research Institute of Fisheries Engineering, Kamisu, JAPANABSTRACTRegarding broaching associated with surf-riding, thispaper reports experimental, numerical and analyticalstudies conducted by the SCAPE (Strategic ResearchCommittee on Estimation Methods of Capsizing Risk for the
IMO New Generation Stability Criteria) committee,

together with historical review of theoretical progress on
this phenomenon. Here effect of higher-order terms innumerical modelling, extension to novel propulsion
systems and application of global bifurcation theory,
optimal control theory and random process theories are
discussed. Experimental techniques are also investigated.
Based on these outcomes, it also proposes draft stability
criteria consisting of a vulnerability criterion and direct
assessment as a candidate for the new generation
criteria at the IMO (International Maritime
Organization).

KEY WORDS: Broaching, surf-riding, IMO new
generation intact stability criteria, global bifurcation,
broaching probability.

INTRODUCTION
Broaching is one of the three major capsizing scenariosthat the new performance-oriented stability criteria to be
added to the Intact Stability Code at the IMO are requested
to cover (Germany, 2005). This is a phenomenon that a shipcannot keep a constant course despite the maximum steering
effort. If the ship speed is high enough, the centrifugal force
due to this uncontrollable yaw motion could result in
capsizing. This phenomenon often occurs when a ship runs
in following and quartering seas with relatively highforward speed, especially when a ship is surf-ridden. Thus,
this phenomenon is relevant to ships having their Froudenumber of 0.3 or above, such as destroyers, high-speed
RoPax ferries, fishing vessels and so on. Broaching itself

could occur also with slow speed (Oakley et al., 1974) but
such broaching seems not to be a direct threat to shipstability. Thus, this paper focuses on broaching associatedwith surf-riding.

HISTORICAL REVIEWA physical model experiment is definitely suitable forrealising dangerous phenomena in ship stability because fullscale measurement could be too risky. Du Cane (1957)
executed model experiments of broaching in following
waves at a towing tank. Here he added initial disturbance toa ship model running in the straight course and observed theship response. Nicholson (1974) and Fuwa et al. (1982)
conducted runs of manually-controlled models in stern
quartering waves in seakeeping and manoeuvring basins,and identified broaching associated with surf-riding for a
destroyer and a high-speed fishing craft, respectively. Kan
et al. (1990) and Umeda et al. (1999A) executed
free-running model experiments with auto pilots in stern
quartering waves in seakeeping and manoeuvring basins,
and recorded time histories of capsizing due to broachingfor container ships and fishing vessels. In Umeda’s
experiment, it was confirmed that even a ship complying
with the current IMO IS code can capsize as a result of
broaching.
Davidson (1948) investigated directional stability infollowing waves by using a linear sway-yaw coupled model.
Here he estimated wave-induced forces as the sum of the
Froude-Krylov component and the hydrodynamic lift due towave particle velocity. As a result, it was confirmed thateven a directionally stable ship can be directionally unstable
in a wave downslope. Wahab and Swaan (1964) pursued
this approach further but with the Froude Krylov component
on its own. Eda (1972) applied a coupled surge-sway-yaw
model to this linear stability problem. Hamamoto (1973)
and Renilson (1982) investigated the wave effect on linear
manoeuvring coefficients theoretically and experimentally.
Fujino et al. (1983) improved accuracy in prediction ability

Surf-riding is regarded as a prerequisite of broaching.
Thus it is very important to estimate the surf-riding
threshold. Grim (1951) explained that surf-riding boundary
coincides with the case that a trajectory from an unstableequilibrium of an uncoupled surge model on a wave isconnected to another unstable equilibrium. This is aheteroclinic bifurcation in nonlinear dynamical systemtheory. Makov (1969) demonstrated the validity of Grim’s
statement by using a phase plane analysis. Ananiev (1966)
obtained an analytically approximated solution by applying
a perturbation technique. Umeda and Renilson (1992)
extended the nonlinear dynamical system approach from an
uncoupled surge model to a coupled surge-sway-yaw model.
Spyrou (1995, 1996, 1997) and Umeda (1999) numericallyobtained the heteroclinic bifurcation for an uncoupled surge
model and the coupled surge-sway-yaw-roll model with a
PD autopilot, respectively. Umeda and Vassalos (1996)
obtained the region of unstable periodic motion in stern
quartering waves by applying an averaging method. Spyrou(2001) presented an exact analytical solution of heteroclinic
bifurcation of uncoupled surge model.
The above researches deal with broaching in regularwaves only. Although Rutgerssen and Ottosson (1987) andMotora et al. (1982) executed model experiments inirregular waves in a seakeeping and manoeuvring basin andfull scale measurement at actual sea, respectively, no
probabilistic study on broaching had been reported. Umeda(1990), however, proposed a theoretical method for
calculating the surf-riding probability in irregular waves.
Here the surf-riding probability is calculated with thedeterministic surf-riding threshold and the joint probabilityof local wave height and wave period.

for manoeuvring coefficient in waves with free surface
effects taken into account.
Motora et al. (1982) and Renilson (1982) numericallyintegrated nonlinear equation of surge-sway-yaw motion
and concluded that necessary condition of broaching is thatwave-induced yaw moment exceeds the maximum yaw
moment due to rudder. Umeda and Renilson (1992 and1994) developed a 4 DOF mathematical model based on a
manoeuvring model with a linear wave forces. Assumingthat wave steepness and manoeuvring motions are small, allhigher-order terms, such as interactions due to manoeuvring
and waves, are consistently ignored. Umeda and Hashimoto(2002) applied this model to explain the free-running modelexperiments (Umeda et al., 1999A) so that qualitative
agreement was confirmed. The ITTC specialist committee
on extreme motions and capsizing executed a benchmark
testing study of numerical models with the free-runningmodel test data by Umeda et al. (1999A), and concluded
that some numerical codes can predict the occurrence of
broaching qualitatively (Umeda and Renilson, 2001).
For realizing quantitative prediction, it is necessary to
improve the mathematical model by adding higher orderterms. Umeda et al. (2003) and Hashimoto et al. (2004A)
developed a mathematical model with second-order terms
taken into account, and reported that quantitative prediction
is realized. Here hydrodynamic forces due to interaction
between manoeuvring and waves, hydrodynamic forces due
to large roll angle, nonlinear wave and manoeuvring forces
and so on are estimated with potential theories or captivemodel experiments. Hashimoto et al. (2004B) utilised
fully-captive model experiments in waves with apurpose-built apparatus.

original
model
first
order
model

present model
0.5

nominal Froude number

nominal Froude number

0.5
0.4
0.3
0.2
0.1
0

15

30

45

0.4
0.3
0.2
0.1

auto pilot course (degrees)capsizing due to broaching
capsizing without broachingbroaching without capsizing

stable surf-riding
periodic motion
not identified

0

15

30

45

auto pilot course (degrees)
EXP.(capsizing due to broaching)
EXP.(capsizing without broaching)
EXP.(near stable surf-riding)
EXP.(periodic motion)

Figure 1 Comparison between the experimental results and numericalresults (first order model and present model) for the ITTC Ship A-2 withwave steepness of 0.1 and wavelength to ship length ratio of 1.637(Umeda and Hashimoto, 2006C).

-0.30Roll Moment (kg.f.m)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-0.35

CFD
EFD

-0.40

-0.45

-0.50

Fn

Figure 2

Comparison between the physical model experiment (EFD) and the RANS solution (CFD) in roll moment
acting on the ONR tumblehome model running in still water with the heel angle of 10 degrees (Hashimoto andStern, 2007).

0.50
capsize without broaching

Nominal Froude Number

0.45
0.40

stable surf-riding

0.35
periodic motion
0.30
surfriding with oscillation

0.25
0.20

not identified

0.15
by heteroclinic bifurcation

0.10
0

5

10

15

20

25

30

35

40

Auto Pilot Course (degrees)

Figure 3

Comparison in motions of the ITTC Ship A-2 running in following waves between numerical simulation
results and the heteroclinic bifurcation (Umeda et al., 2006A).

To avoid broaching, it is effective to increase ship dragby some means. Renilson (1986) investigated the Sea Brake,
which is a kind of sea anchor to add drag to ship resistance.
Umeda et al. (1999B) invented the Anti-Broaching Steering
System (ABSS), in which the rudder deflection limit isallowed to be a large angle, such as 70 degrees, so that therudder drag can deteriorate surf-riding equilibrium under theyaw deviation from the desired course, and successfully
validated it with free-running model experiments. Further,
Umeda et al. (2002) optimised the design of the ABSS byusing numerical simulations.

OUTCOMES FROM SCAPE ACTIVITIESThe two-year activities of the SCAPE committee output
the following fruits for broaching.
Nonlinear modelling
Although Hashimoto et al. (2004A) realised quantitative
prediction by adding many higher-order terms, it isimportant to provide guidelines for simpler but accurateenough modelling. For this purpose Hashimoto and Umeda(2005), Umeda and Hashimoto (2006C) recommended to
add the following three terms to their first-order model as aresult of systematically degrading study.
- nonlinearity of wave-induced surge force,- nonlinearity of sway-roll coupling,- nonlinearity of roll-induced moment in calm water.
Here captive tests in a towing tank are required but not in aseakeeping and manoeuvring basin. Agreement between therecommended mathematical model and the free-runningmodel experiment by Umeda et al. (1999A) is satisfactory as
shown in Figure 1
The above mathematical modelling is developed forsingle-screw and single-rudder ships, while other ships arealso susceptible for broaching. They include twin-screw andtwin-rudder ships as shown in model experiments andwater-jet propelled craft. For twin-screw and twin-rudder
ships Umeda et al. (2006B) proposed a mathematical model
as an extension of the above model and then validated with
Lloyd’s experiment (Renilson, 1982). Furthermore, for a
craft propelled by water-jet Umeda et al. (2005A) provides amathematical model.

For making captive tests easier to obtain some higher
order terms, Matsuda et al. (2007) developed a new captive
testing technique, which allows heave and pitch motions. To
replace such model experiment with numerical calculation,
Hashimoto and Stern (2007) applied a RANS-based CFD
code to heel-induced hydrodynamic forces and showed agood agreement with the model experiment, as shown in
Figure 2.
Bifurcation analysis
It was already established that the critical condition forsurf-riding in regular following and quartering waves is aheteroclinic (or homoclinic) bifurcation. Next step is to
directly evaluate such bifurcation point from mathematicalmodels, especially a 4 DOF model. Hori and Umeda. (2005)
and Umeda et al. (2006A) proposed a theoreticalformulation of this bifurcation problem with the coupledsurge-sway-yaw-roll mathematical model and well validatedwith the direct initial-value simulation as shown in Figure 3.
Here the critical Froude number for surf-riding is determined
for requesting the unstable invariant manifold from a saddle
to reach another saddle within the framework of the Newton
method. Thus the repetition of numerical simulation formany different initial and control values can be avoided.
Above this surf-riding threshold, capsizing due to broachingoccurs if wave steepness is very high. Maki et al. (2007)
further improved this methodology, focusing to an
uncoupled surge model in regular following waves, fromtheoretical and numerical viewpoints so that deterministic
surf-riding threshold hyperplane could be efficiently
estimated.

validated with the Monte Carlo simulation as shown in
Figure 4.
Optimal control theory
In the above theoretical and experimental studies, steering
is modelled as a PD auto pilot. The maximum steering effort,
however, could be different from a PD auto pilot. Therefore,
Maki and Umeda (2006) proposed the application of optimal
control theory. When it is applied, a kind of the bang-bang
control could be obtained as maximum steering effort toprevent broaching.
Prevention device
Other than the Sea brake and the ABSS, a new device forpreventing broaching was proposed by Hashimoto et al.
(2007B). It is a pair of wings attached to bow above a stillwater plane. Once broaching is initiated, a ship deviatedfrom the desired course. This turning results in outward roll.
The wing in the side of roll will be submerged so that yaw
moment towards desired course and the roll moment towards
upright position can be generated. As a result, broaching can
be avoided. The effectiveness of this device was validatedwith a free-running model experiment.

RECOMMENDATION FOR IMO NEW GENERATIONCRITERIA
The IMO requests both the vulnerability criteria and the
direct assessment for broaching, following the framework
proposed by Japan, the Netherlands and the United States(2007). Responding to this requirement, the SCAPE
committee recommends the following draft criteria.

1broaching probability

0.9

theory

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

numerical
exp(no.1)

numerical
exp(no.2)

numerical
exp(no.3)

numerical
exp(no.4)

numerical
exp(no.5)

Figure 4 Comparison in broaching probability between
numerical experiments and theory for ahigh-speed fishing craft in irregular stern
quartering waves (Umeda et al., 2007).
Probabilistic assessment
To evaluate danger of broaching in actual seaways, it isessential to evaluate broaching probability or probability ofcapsizing due to broaching in irregular waves. Umeda et al.
(2007) proposed a theoretical calculation method forbroaching probability as an extension of Umeda’s theory forsurf-riding probability (Umeda, 1990). Here the broachingprobability is calculated by integrating the joint probability
density of local wave height and wave period, which was
proposed by Longuet-Higgins (1983), within thedeterministic broaching region. The present theory was well

Vulnerability criterion
It is important that a vulnerability criterion can be easily
used, can guarantee conservative safety level and should bebased on non-empirical approach. The surf-riding thresholdin regular following waves can be substituted to that ofcapsizing due to broaching because surf-riding is aprerequisite to broaching and following waves are mostdangerous for surf-riding. To determine the surf-riding
threshold in regular following waves, an analytical solution
is obviously most suitable because it is easily evaluated but
is still based on theoretical background. Spyrou (2001)
proposed an analytical solution by assuming the balance ofresistance and propeller thrust to be proportional to square ofship velocity. Maki and Umeda (2008) utilised a piece-wiselinear approximation of wave-induced surge force and thendifferent analytical formula for surf-riding threshold in
regular following waves was obtained. Here the balance ofresistance and thrust is approximated to have a linear
relationship with ship velocity. In this procedure, ship
resistance in calm water, propeller thrust and distribution ofsectional hull area are required. They executed comparisonsamong the two analytical formulae, numerically-obtained
bifurcation points and a free-running model experiment sothat they concluded that Maki and Umeda’s analyticalsolution is more accurate than Spyrou’s and almost identicalto the numerically-obtained bifurcation points and theexperiment, as shown in Figure 5.

Direct assessment
If a ship design fails to comply with the vulnerability
criterion, it is expected to apply direct stability assessment tothe subject ship design. It is preferably to evaluate risk levelof the ship design. For this purpose, it is necessary toevaluate probability of capsizing due to broaching. The
SCAPE committee already developed a theoreticalprocedure for broaching probability in irregular stern
quartering waves (Umeda et al., 2007) and an extension ofthis method to probability due to broaching isstraight-forward. First, we should identify deterministic
thresholds for capsizing due to broaching as a function ofwave height and period by heteroclinic bifurcation analysisof the coupled surge-sway-yaw-roll model with an auto pilot
or the repetition of numerical simulation with the samemodel. Second, the probability of capsizing due to broachingis calculated by integrating the joint probability density of
local wave height and wave period, which was proposed by
Longuet-Higgins (1983), within the deterministic capsizing
region mentioned before. In this procedure, ship resistancein calm water, propeller thrust, hull form offset, calm-water
manoeuvring coefficients, roll damping coefficient andrestoring moment are required at least.

0.5
0.45
0.4
0.35

Fn

0.3
0.25

Numerical Bifurcation Analysis
Spyrou's Formula

0.2

Present Method

0.15
0.1
0.05
0
1

1.2

1.4

1.6

1.8

2

λ/L

Figure 5 The surf-riding thresholds in following waves obtained by model experiments, numerical bifurcation analysis andanalytical bifurcation analysis for the ITTC Ship A-2 with the wave steepness of 0.08 (Maki and Umeda, 2008).

CONCLUDING REMARKS
The activity of the SCAPE committee for broachingprovides the following concluding remarks:
1. For quantitative prediction in regular waves, anenhanced system-based model is proposed andvalidated with the free-running model experiments.
2. For estimating the deterministic threshold of surf-riding
in stern quartering waves as a heteroclinic bifurcation,an equation set was theoretically formulated and wasvalidated with the direct numerical simulation.
3. For calculating broaching probability in irregularwaves, a theory was formulated and was validated with
the Monte Carlo simulation.
4. A device for preventing broaching was proposed andwas validated with model experiments.

ACKNOWLEDGEMENTS
This work was supported by Grant-in Aids for
Scientific Research of the Japan Society for Promotion of
Science (Nos. 18360415, 18360426 and 18760619) and the
US Office of Naval Research contract No. 0014-06-1-0646
under the administration of Dr. Patrick Purtell. The work

described here was partly carried out as a research activity of
SPL project of Japan Ship Technology Research Association,funded by the Nippon Foundation. The authors express theirsincere gratitude to the above organisations.
The authors thank Prof. F. Stern, Messrs. G. Sakamoto,
M. Shuto and S. Sanya for their contribution to the works
described in this paper.

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