1
A b oad CFD-based s udy o he non-linea e ec s on wa e-induced
bending momen s and shea o ces on monohull ships
To mod Lande 1, Ma ine Lasbleis2, Kei Sugimo o3, Guillaume de Hau eclocque2,
Min Xu4, and Hå a d Aus e jo d1
1 IACS – DNV, Oslo, No way
2 IACS – Bu eau Ve i as, Pa is, F ance
3 IACS – Nippon Kaiji Kyokai (ClassNK), Japan
4 IACS – China Class Socie y, Shanghai, China
Abs ac . The In e na ional Associa ion o Classi ica ion Socie ies (IACS) is abou o upda e i s design
equi emen s o s eel ships longe han 90 m, namely Uni ied Requi emen (UR) S11 and S11A, applying o
he majo i y o he wo lds shipping. As a pa o his wo k an ex ensi e CFD-based s udy o wa e loads on 45
monohull ships o a a ie y o ship ypes (oil/o e/gas/con aine ca ie s, Ro-Ro, C uise) has been conduc ed
and is p esen ed in his pape . The goal o he s udy is o p o ide he echnical backg ound o upda ing he
ules o he non-linea e ec s on he ex eme global wa e-induced hull-gi de loads, in pa icula he midship
e ical wa e-induced bending momen (VBM) and he qua e -leng h wa e-induced e ical shea o ces
(VSF) since hese a e among he mos impo an inpu pa ame e s o s uc u al design o la ge sea-going ships.
We s a by showing ha by use o mode n CFD-based ully non-linea me hods, mul iple pa icipan s in
he p ojec a e able o show good ag eemen be ween simula ions and model es s o a igid con aine ship in
i egula wa es. We hen p esen a la ge da abase o ships o a ious ele an ship ypes and show he esul s
o ex ensi e CFD calcula ions o ind ends in how he non-linea ac o s change wi h he main pa ame e s o
he ships in he da abase. Reg ession i s based on ship main pa ame e s a e sugges ed o p edic ing non-linea
wa e loads.
Keywo ds: Wa e Loads; CFD; Non-linea ac o ; VBM; VSF; IACS; CSR; UR S11; UR S11A.
1. In oduc ion
The In e na ional Associa ion o Classi ica ion Socie ies (IACS) has es ablished a p ojec eam o upda e he
hyd odynamic wa e loads applied by he membe socie ies in ule checks o class app o al. Among hese
p esc ip i e ules a e he equi emen s o longi udinal s eng h o all s eel ships longe han 90m. We will he e
p esen a summa y o he wo k pe o med by his p ojec eam which will esul in an upda e o IACS Uni ied
Requi emen s (UR) S11 and S11A. The p e ious upda e o he uni ied equi emen s o longi udinal s eng h was
pe o med in 2015 and was limi ed o con aine ships only [1]. Tha wo k, which lead o UR S11A, was based on
a da abase o non-linea hyd odynamic calcula ions made a ailable by he pa icipa ing class socie ies. A ha
ime he membe s made use o semi-non-linea po en ial- low-based simula ion ools [2]. As no ed in he
p esen a ions om ha p ojec eam he e we e qui e la ge disc epancies be ween he esul s om he a ious ools
and s a is ical me hodologies employed. The esul ing UR S11A o mulas a e used oday in IACS o con aine
essels and he olde UR S11 o mulas a e used o all o he ship ypes.
The IACS ule o mulas o mo ions, accele a ions, and wa e loads such as bending momen s, shea o ces and
wa e- equen ex e nal p essu es a e all o mula ed as p oduc s, 𝑋 = 𝐶𝑊,𝑋 𝑋𝑙𝑖𝑛 𝑓𝑛𝑙−𝑋, whe e 𝐶𝑊,𝑋 is he wa e
pa ame e , 𝑋𝑙𝑖𝑛 is he dimensional linea o mula ion, and 𝑓𝑛𝑙−𝑋 is he non-linea ampli ica ion o educ ion ac o .
Fo mo ions and accele a ions, he non-linea ac o s a e all 𝑓𝑛𝑙−𝑋 = 1.0, while o some global loads and p essu es
i may be highe o lowe han uni y depending on he load mechanism. The de elopmen o he new wa e
pa ame e s and he linea o mula ions a e documen ed sepa a ely [3], we will he e p esen only he wo k
pe o med o upda e he o mula ions o he non-linea ac o s o he wa e-induced e ical bending momen ,
VBM, e ical shea o ce, VSF, and local ex e nal wa e- equen p essu es unde he equi alen design wa es
maximizing momen .
To a oid he la ge sp ead in esul s ound in he de elopmen o UR S11A, i was decided o es ablish new and
highe s anda ds o he applied nume ical and s a is ical me hods, made possible by bo h he inc ease in a ailable
compu a ional powe , and he inc eased a en ion o and de elopmen o ully non-linea me hods o sea-keeping
simula ions. All pa icipan s we e now equi ed o use ully non-linea CFD-based me hods whe e he wo-phase
16 h In e na ional Symposium on P ac ical Design o Ships and O he Floa ing S uc u es PRADS 2025
Ann A bo , MI, USA, Oc obe 19 h – 23 d 2025
2
Na ie -S okes equa ions a e sol ed wi h physically co ec handling o d y and we a eas wi h associa ed ex e nal
p essu es.
The In e na ional Ship and O sho e S uc u es Cong ess, ISSC 2021 Load Commi ee ha e c ea ed a
benchma k including expe imen al model es s o a 6750 TEU con aine essel [4]. The expe imen al s udy
e alua es bending momen s in selec ed ex eme sea s a es wi h a e y igid ship model, allowing he ex ac ion o
he igid body bending momen wi hou he e ec o Whipping. Con aine essels a e known o ha e la ge non-
linea con ibu ion o hull gi de loads. I is assumed ha i pa icipan s in he cu en s udy can ep oduce he
ISSC model es by combining CFD wi h app op ia e s a is ical me hods, hen he same CFD so wa e, mesh,
imes ep, and s a is ical p ocedu e may be used o compu e non-linea ac o s o he essels co e ed by UR S11
and S11A. De ails on he s a is ical and nume ical me hods applied a e shown in sec ion 2. The esul s o he
alida ion done o he ISSC expe imen al benchma k con aine ca ie a e shown in sec ion 3.
Figu e 1. The global hull-gi de e ical bending momen , VBM,
and shea o ce, VSF, wi h posi i e sign di ec ions indica ed.
A la ge numbe o exis ing ocean-going ships we e selec ed o o m he basis o he new CFD-based s udy.
IACS Common S uc u al Rules (CSR) co e Oil anke s and Bulk ca ie s and he CSR has inhe i ed he VBM
and VSF equi emen s om UR S11. These ship ypes a e well ep esen ed in ou da abase o non-linea
calcula ions along wi h he Con aine ca ie s (UR S11A) and he ship ypes co e ed by UR S11 ha a e assumed
o ha e signi ican non-linea e ec s o a endency o be dimensioned by global bending. The included ship ypes
ou side o Oil anke s, Bulk ca ie s, and Con aine essels, a e Ro-Ro ca ie s, Passenge /C uise essels and Gas
ca ie s (LNG/LPG). The essel da abase and he esul ing non-linea ac o s o e ical bending momen s om
CFD a e shown in sec ion 4 along wi h he esul ing eg ession o mulas. Shea o ces associa ed wi h he
maximum momen s a e desc ibed in sec ion 5, and he ex e nal sea p essu e associa ed wi h he ex eme momen s
a e shown in sec ion 6. The wa e-induced non-linea sea p essu e ac o s a e a pa o CSR and no UR S11/S11A.
2. Nume ical and s a is ical me hods
One o he pa icipa ing class socie ies is using he comme cial CFD sol e Simcen e STAR-CCM+ om
Siemens wi h he HRIC me hod [5] o wa e-in e ace cap u ing. The wo o he s a e using in-house e sions o
he open-sou ce in e Foam sol e om OpenFOAM wi h he MULES me hod o in e ace cap u ing [6]. The
incoming wa es a e en o ced by Di ichle bounda y condi ions and by o cing-zone me hods o dampen and s op
any dis u bances o he incoming wa e ield c ea ed by he essel om ge ing close o he bounda ies and c ea ing
unwan ed e lec ions. Such o cing zones can signi ican ly educe he compu a ional domain, see e,g. [7] o an
in oduc ion. The incoming wa es a e ep esen ed by i egula long-c es ed wa es desc ibed ei he by linea wa e
kinema ics, o by use o a highe -o de spec al ep esen a ion o he wa e ield [8] [9] (HOSM).
Two o he pa icipan s a e pe o ming he s a is ical assessmen using Response Condi ioned Wa es (RCW,
see e.g. [10] [11]) and one is using a ull Mon e-Ca lo app oach wi h long simula ions in Random I egula Wa es,
RIW. The Response Condi ioned Wa e me hod, in b ie , in ol es combining he linea Response Ampli ude
Ope a o , RAO, and he sea-s a e wa e spec um o compu e he linea esponse spec um and hen compu e he
wa e e en ha is he mos -likely i egula -wa e ime his o y o p oduce a a ge linea esponse le el gi en he
wa e- equency dis ibu ion o he esponse spec um [10]. In his case he a ge esponse is he 25-yea linea
long- e m ex eme om a 3D po en ial heo y panel me hod. One pa icipan is ex ending he s anda d RCW heo y
by compu ing he mos -likely non-linea HOSM wa e p o ile by use o a linea ized esponse model based on he
RAO combined wi h he FFT o he wa e ele a ion ime se ies in a non-linea op imiza ion loop, see [8] o de ails.
VBM
(Hogging posi i e)
VSF
A
Fo e
3
The Mon e-Ca lo RIW app oach is simply o simula e su icien ly long ime se ies o he essel subjec ed o
i egula non-linea HOSM wa es in CFD. As a ough ule o humb, when he expec ed e u n pe iod o he a ge
esponse in he sea s a e is simula ed wen y imes, hen he s a is ics o he co esponding esponse le el can be
conside ed con e ged. This was done by adding andom seed simula ions o 20 minu es each un il he 20- imes-
e u n-pe iod c i e ion was me , mos o en ending up a abou nine hou s o CFD simula ions pe essel and
loading condi ion. In some cases, ex apola ion o he ail o he esul ing empi ical p obabili y dis ibu ion using
a Weibull assump ion was needed owa ds he bow and s e n o he essel whe e he sea s a e selec ed as wo s o
he midship bending momen may no be he op imal choice. The Weibull-assump ion was no needed in he
midship egion whe e he maximum non-linea VBM is ound, and hence his is only ele an o he dis ibu ion
o VBM along he essel, no o he o e all non-linea ac o which is ound om he maximums o he linea
and non-linea dis ibu ions.
Fo bo h he selec ed s a is ical me hods, RCW and RIW, he i s s ep is o selec one o mo e sea s a es whe e
he non-linea esponse should be es ima ed. The ex eme load is expec ed o occu in a sea s a e close o he
s eepness limi o igid-body VBM a i e kno s. I is assumed ha he 𝑓𝑛𝑙 changes wi h he a ge load le el, bu
no e y much wi h he sea s a e pa ame e s 𝐻𝑆 and 𝑇𝑍 wi hin he g oup o sea s a es ha a e likely o cause he
linea maximum load. In his wo k he ollowing p ocedu e is used: he linea 25-yea long- e m CoC (Coe icien
o Con ibu ion, see e.g. [12]) is compu ed o each sea s a e. Fo each conside ed wa e di ec ion he sea s a e
pe iod 𝑇𝑍 wi h he maximum sum o CoC in ha heading is used as he basis o he calcula ions, and he wa e
heigh 𝐻𝑆 is aken as he maximum in he sca e diag am o his 𝑇𝑍, which esul s in a e y s eep sea s a e whe e
he 25-yea ex eme is likely o occu ela i ely equen ly.
The selec ed sea s a e is a good ep esen a i e o he sea s a es likely o gi e he ex eme esponse since he
non-linea ac o can be assumed o be independen o he signi ican wa e heigh , hough i is s ongly dependen
on he a ge linea esponse le el. I.e., he 25-yea ex eme VBM will occu mo e o en in ou simula ions by
selec ing he maximum ealis ic 𝐻𝑆, bu he esul ing non-linea ac o should be he same as wha would be ound
o a lowe 𝐻𝑆 o he same 𝑇𝑍, see e.g. [13] and [14]. No e ha he inciden wa e physics would ypically no
include wa e b eaking in he legacy semi-non-linea po en ial heo y p og ams, while he 𝐻𝑆 in he CFD
simula ions should no be inc eased much beyond he s eepness limi as he wa es a e he e much mo e accu a ely
modelled. I mos o he s eep wa es end up b eaking in he CFD wa e simula ion hen he ac ual wa es
encoun e ed by he essels will be la e and longe han he in en ion, and he inc eased-𝐻𝑆 me hod will no gi e
he desi ed accu acy. We ely on he ac ha inc easing he 𝐻𝑆 is jus a scaling o he esponse spec al ampli udes,
while wa e b eaking will also cause a shi in he o e all shape o he esponse spec um since he wa e spec um
ene gy con en changes.
Figu e 2. Illus a ion o he wa e sca e diag am wi h sea-s a e p obabili ies (blue do s),
he linea 25-yea VBM CoC (yellow/ ed), an empi ical sea-s a e s eepness limi (black),
and he sho - e m linea iso- esponse cu es (g een). The blue column has he maximum
in eg a ed Coe icien o Con ibu ion in head seas o his 280m Bulk ca ie in ull load.
Selec ed
sea s a e
4
The p ocedu e applied o ind he sea-s a e o simula e in CFD is illus a ed in Figu e 2. The VBM RAO is
aken om a 280m long Bulk ca ie in he homogenous ully loaded condi ion. The sca e diag am applied is he
No h-A lan ic om IACS Recommenda ion 34 (Re . 2), bu i should be no ed ha he CoC dis ibu ion and
choice o he 𝑇𝑍 o apply in he calcula ion is no e y sensi i e o he exac de ails o he sca e diag am. The
selec ed sea s a e is always picked o be nea he s eepness limi , so he ene gy con en in ano he sca e diag am
(in e ms o CoC as a unc ion o 𝑇𝑍) mus be e y di e en o he sea s a e ha is picked o be e y di e en om
he one we ha e selec ed he e. The linea a ge VBM esponse is on he o he hand qui e impo an , as he non-
linea ac o changes wi h he a ge esponse le el in he sea s a e, so a milde sca e diag am causes non-linea
ac o s ha a e close o uni y, while a ha she sca e diag am (highe 25-yea linea ex eme) will cause he
opposi e.
The applied s a is ical and CFD me hods used by he pa icipa ing class socie ies in he IACS wo king g oup
o wa e loads a e summa ized in Table 1. The CFD sol e s a e un wi hou any u bulence modelling. This
simpli ica ion gi es equi alen esul s in he benchma k o using ei he an Eule sol e o a ull URANS sol e
wi h u bulence modelling, con i ming he negligible in luence o iscosi y on he e ical global loads. All
simula ions a e pe o med a ull scale.
Table 1. Summa y o nume ical and s a is ical me hods employed o compu e non-linea ac o s o VBM and VSF and ypical
mesh size. Fo wo o he pa icipa ing class socie ies, one sea s a e equi es, a minimum, wo CFD simula ions o Response
Condi ioned Wa es (RCW), one o sagging and one o hogging VBM a he linea 25-yea long- e m esponse le el. The las
pa icipan uses a Mon e Ca lo app oach wi h Random I egula Wa es (RIW) wi h a du a ion o 20 imes he linea 25-yea
long- e m esponse e u n pe iod in he selec ed sea s a e. The lis ed du a ion is pe sea s a e (and essel and loading condi ion).
CLASS
SOCIETY
CFD SOLVER
WAVE
KINEMATICS
STATISTICAL
METHOD
MESH
SIZE
DURATION
A
OpenFOAM
HOSM
RCW
1.2 M cells
~ 2 x 60 sec.
B
OpenFOAM
HOSM
RIW
0.3 M cells
~ 9 hou s
C
S a CCM+
Linea
RCW
1.2 M cells
~ 2 x 60 sec.
3. Benchma king o he CFD and s a is ical me hods
Due o he la ge disc epancies ound be ween di e en nume ical and s a is ical me hods in he wo k leading
up o he publica ion o IACS UR S11A [2], he cu en ule-upda e CFD p ojec s a ed wi h a quali ica ion s ep
o ensu e he me hods employed by he pa icipa ing class socie ies could accu a ely p edic he esul s om an
exis ing model es o igid-body VBM. The ISSC loads commi ee benchma k o global loads on a 6750 TEU
con aine essel [4] was selec ed. Model es s we e pe o med a Ecole Cen ale Nan e, ECN, bo h o egula [15]
and i egula [14] wa es.
The pa icipan s CFD codes we e i s benchma ked by use o he egula wa e expe imen s. He e we ocus
on he i egula sea-s a e loads p edic ion, as his is wha will be done o a la ge se o essels la e o es ablish
he ules-backg ound da abase. Figu e 3 shows he esul s p o ided by he pa icipa ing class socie ies in an
i egula sea s a e wi h 𝐻𝑆=10𝑚 and 𝑇𝑃=14𝑠. ECN has pe o med a ~30-hou ealiza ion in he model basin
in his sea s a e and p o ided ime-se ies o he esul ing midship e ical bending momen o 10 seeds (abou 25-
hou s o usable da a). Each blue do in he igu e ep esen s one posi i e o nega i e momen ex ema om he
model es . Class socie y B has eplica ed he model es by unning 30-hou s coa se-mesh CFD in he same sea
s a e, bu wi h hei own HOSM wa es ha a e no eplica ing he wa es measu ed in he model basin. Class
socie ies A and C ha e used i egula condi ioned ex eme wa es which can be seen a some selec ed p obabili ies
o exceedance.
5
Figu e 3. Wa e-induced dynamic VBM s he p obabili y o exceedance, Q, o he ISSC loads benchma k in i egula
wa es. The 20 mos ex eme poin s a e semi- anspa en as he empi ical alue o Q will be less accu a e o hese.
The disc epancies in he calcula ion o VBM using each class-socie y’s linea po en ial low ool can be seen
o be o he same magni ude as he disc epancies be ween he CFD esul s. I hence seems, om hese esul s, ha
a p ope ly execu ed CFD analysis should no in oduce signi ican ly mo e e o s han a linea po en ial low
seakeeping simula ion. I should be no ed, howe e , ha his is p obably no ue o an inexpe ienced use whose
chances o ob aining e oneous esul s is s ill much highe wi h a complex non-linea CFD sol e .
The disc e ized mesh applied by socie y B is op imized o cap u e he la ges wa es causing he la ges
momen s. I was con i med sepa a ely ha he CFD sagging momen in medium-heigh wa es, whe e he e a e
some di e ences be ween he o ange and blue poin s in Figu e 3, does app oach he model es i he mesh densi y
is inc eased. Since he objec i e o he s udy is o look a he ex emes i was decided ha he mesh densi y was
adequa e.
As a poin o compa ison wi h he s udy pe o med in 2015, class socie y B has p o ided esul s using a quasi-
non-linea po en ial low sol e o he same ype as he ones used in [2]. The only main di e ence is ha HOSM
wa es ha e been applied a his ime ins ead o linea i egula wa es o enable di ec compa ison wi h CFD. As
can be seen his ype o me hod ends o o e p edic he VBM. This is a known issue ac oss many such ools, and
we can hence expec he esul s om he CFD-based s udy documen ed below o be close o eali y bo h o he
con aine essels cu en ly using he UR S11A, and p obably also o he o he ship ypes using UR S11, an olde
ulese which is no based on a da abase o di ec load calcula ions.
4. Ve ical wa e-induced bending momen
The s udy ini ially con ained 103 essels in homogenous ully loaded and ballas condi ions. Passenge and
C uise essels a e only included in he ully loaded condi ion since he ballas and ully loaded condi ions a e
ypically e y close in e ms o displacemen and im o such essels. The o he essel ypes a e included wi h
wo loading condi ions mean o span he ull ange o possible ope a ional d augh s.
The ini ial CFD simula ions o all essels we e pe o med in head sea wa es o sea s a es nea he b eaking
limi . The sea s a es we e selec ed as explained in sec ion 2. F om he esul ing ime se ies and ideos i was ound
ha se e al o he essels go signi ican amoun s o g een wa e on he o edeck du ing he maximum sagging
wa e encoun e s. The wa e-induced sagging momen is in pa due o he p essu e om a wa e c es a he bow,
pushing he bow upwa ds. The p essu e om he g een wa e on he o edeck is hence coun e ac ing he wa e
c es p essu e and educing he sagging momen . One al e na i e o how o deal wi h his p oblem is o con inue
he bulwa k geome y e ically un il he e is no g een wa e obse ed (o li he whole o edeck o b eakwa e
depending on he ship ype and design). Fo some essels his was ound o wo k wi hou esul ing in designs ha
a e ob iously di e en om eal ships in ope a ion [16], bu o o he s he essel d awings indica ed ha he
eeboa d in he CFD models was easonable and ha he wa es would ealis ically b eak o e he bow in he 25-
yea ex eme wa e encoun e s. As seen in Figu e 2 he encoun e ed wa es in he wo s s o ms in he No h-A lan ic
Ocean can be e y la ge, especially conside ing ha he maximum wa e heigh encoun e ed can be oughly wo
imes he signi ican wa e heigh , HS.
6
I should be no ed ha he RIW simula ions also showed ha he mos ex eme sagging momen s had wa e on
he o edeck. I is hence no he case ha sligh ly lowe wa es would be mo e c i ical since no g een wa e would
be p esen o hose. In he long i egula wa e simula ions hose sligh ly lowe wa es a e p esen in he
encoun e ed i egula seas, and hey we e no gi ing VBM exceeding he wa e encoun e s ha ga e wa e on deck
so we can conclude hey we e no mo e c i ical in e ms o e ical wa e-induced bending momen s.
A e unning some ini ial es s wi h ollowing sea wa es in CFD i was decided ha all essels included in ou
CFD da abase, c ea ed explici ly o he pu pose o i ing new ule o mulas, should ha e he non-linea bending
momen calcula ed also in ollowing sea wa es by he exac same p ocedu e as was used o head sea wa es. Fo
many o he included essels he linea long- e m s a is ics indica e a e y sligh educ ion in VBM when swi ching
om head o ollowing sea wa es, he ma ginal CoC in ollowing sea wa es is jus below he ma ginal CoC o
head sea wa es ( o some essels i is e en sligh ly highe ). This is due o he low o wa d speed used as ule basis
in ex eme condi ions, a maximum o 5 kno s is assumed in he mos ex eme sea s a es in he No h-A lan ic [17].
Ou inal da abase hence consis s o 45 essels whe e head and ollowing sea esul s a e a ailable. This is
un o una ely smalle han he ~100 essels we aimed o bu is s ill a sizable da abase co e ing a la ge ange o
ele an essel designs and sizes. A e y ough es ima e is ha 40 000 CPU-days ac oss h ee di e en mode n
high-pe o mance compu e clus e s we e used o gene a e he p esen ed esul s. The main pa ame e s o he essels
in he CFD da abase a e lis ed in Table 2 and shown in Figu e 4.
Table 2 Main pa ame e s o he essels included in he CFD da abase wi h esul s o head- and ollowing-sea wa es.
Ca go Type
Coun
Leng h [m]
B ead h [m]
CB [-]
CWP [-]
Con aine
11
120 – 380
20 – 51
0.54 – 0.72
0.66 – 0.91
Bulk
8
140 – 280
24 – 45
0.74 – 0.88
0.81 – 0.97
Gas
8
110 – 270
19 – 49
0.65 – 0.78
0.74 – 0.89
Oil
8
160 – 260
24 – 49
0.69 – 0.86
0.78 – 0.94
Passenge
6
120 – 330
22 – 47
0.59 – 0.72
0.82 – 0.93
Ro–Ro
4
160 – 250
28 – 32
0.57 – 0.69
0.78 – 0.92
To al
45
110 – 380
19 – 51
0.54 – 0.88
0.66 – 0.97
Figu e 4. Main pa ame e s o he essels included in he CFD da abase wi h esul s o head- and ollowing-sea wa es.
Ship dimensions: L=leng h [m], T=d augh [m], B=b ead h [m], CB=block coe icien [-], CWP=wa e -plane coe icien [-]
The esul ing non-linea ac o s o VBM in he IACS Head Sea Momen (HSM) wa e condi ions can be seen
in Figu e 5, and he ollowing-sea wa e esul s (FSM) can be seen in Figu e 6. The o mulas used o he x-axis
a e explained u he below. The non-linea ac o is compu ed as he maximum VBM om CFD along he leng h
o he essel di ided by he maximum VBM om linea po en ial low along he leng h o he essel, see
Equa ion 1. These longi udinal VBM dis ibu ions a e ex ac ed a he same 25-yea -ex eme long- e m p obabili y
le el. I is impo an o no e ha he CFD and linea momen dis ibu ions may no ha e hei maximums a he
same longi udinal loca ion, so he epo ed non-linea ac o s o VBM a e global pa ame e s, independen o
loca ion.
7
𝒇𝒏𝒍,𝑽𝑩𝑴 =𝐦𝐚𝐱
𝒙 ∈ [𝟎,𝑳] 𝑽𝑩𝑴𝐂𝐅𝐃(𝒙)
𝐦𝐚𝐱
𝒙 ∈ [𝟎,𝑳] 𝑽𝑩𝑴𝐥𝐢𝐧.(𝒙)
(1)
Figu e 5. The non-linea ac o s o VBM om he head-sea wa e simula ions.
The 𝑓𝑛𝑙 om CFD is compu ed acco ding o equa ion 1.
Figu e 6. The non-linea ac o s o VBM om he ollowing-sea wa e simula ions.
The 𝑓𝑛𝑙 om CFD is compu ed acco ding o equa ion 1.
A compa ison be ween he esul ing non-linea ac o s om he HSM and FSM simula ions can be seen in
Figu e 7. In gene al, he ollowing-sea wa e simula ions gi es highe sagging momen s o he ully loaded
condi ion, while i is he head-sea wa es ha gi e he highes sagging momen s in ballas . This seems easonable
since g een wa e should be much less likely o essels in ballas condi ion, when hey ypically ha e a qui e high
eeboa d a he bow. Fo hogging we see ha he occu ence o he mos ex eme non-linea i ies (𝑓𝑛𝑙 below 0.7)
a e educed when ollowing-sea wa es a e included. Fo essels ha a e app oxima ely equally likely o see he
highes e ical wa e hogging momen in head o ollowing seas a low o wa d speed (based on linea heo y), i
may hence be ad isable o include ollowing-sea wa e simula ions when unning CFD since he non-linea ac o
may hen be conse a i e compa ed o he non-linea ac o in head-sea wa es. Please no e ha when including
luid-s uc u e in e ac ion e ec s such as Whipping i is no expec ed ha he same is ue since he e ical
whipping bending momen is ypically highes a highe o wa d speed in mo e benign sea s a es when signi ican
bow slamming occu s. Please also no e ha he e ec o whipping and sp inging a e no included in he IACS 𝑓𝑛𝑙
ac o , i co e s only igid-body non-linea i ies. Hull lexibili y is no a pa o he CFD-based ule upda e s udy
documen ed he ein.
8
Figu e 7. Compa ison o esul ing 𝑓𝑛𝑙 o VBM om head and ollowing sea wa es.
The 𝑓𝑛𝑙 om CFD is compu ed acco ding o equa ion 1.
4.1. Resul en elopes o VBM and eg ession o mulas
The maximum non-linea ac o ound be ween he head-sea and ollowing-sea wa e simula ions is used o
de ine an en elope non-linea ac o which is o be applied o he bending momen in bo h he HSM and FSM
equi alen design wa es. The maximum o he wo is shown in Figu e 8. He e he s a is ical me hod is highligh ed
ins ead o ship ype, and we emind he eade ha he esul s om he RCW and RIW me hods a e compu ed
sepa a ely by non-collabo a ing CFD expe s each in hei sepa a e class socie ies. The essel geome y and mass
models a e picked om ha class socie y’s da abase o eal essel designs. I is hence a good quali y assu ance o
see ha he esul s all oughly in line.
Figu e 8. The non-linea ac o s o VBM, aking he maximum om head-sea and ollowing-sea wa es.
The RMSRE eg ession i ing e o s a e 5.6% (hogging) and 7.4% (sagging).
The 𝑓𝑛𝑙 om CFD is compu ed acco ding o equa ion 1.
Ou sugges ed o mula ion o he non-linea ac o s o VBM in head and ollowing sea wa es ha e been used
as he x-axis alues in he abo e igu es. Fo he e ical wa e-induced hogging momen he p oposed non-linea
ac o ound om a eg ession i is
𝒇𝒏𝒍−𝒗𝒉 = 𝟏.𝟎𝟔(𝑩𝑻
𝑳)𝟎.𝟏𝟓 𝑪𝑩
𝟎.𝟔,
(2)
9
and he p oposed non-linea ac o eg ession i o e ical wa e-induced sagging momen is
𝒇𝒏𝒍−𝒗𝒔 = 𝟎.𝟗𝟖 (𝑪𝑩𝑪𝑾𝑷)−𝟎.𝟔.
(3)
The pa ame e s T (d augh amidships), 𝐶𝐵 (block coe icien ), and 𝐶𝑊𝑃 (wa e -plane coe icien ) a e aken
om he ac ual loading condi ions in he abo e plo s (whe e hese o mulas a e used on he x-axis) and in he
eg ession i ing p ocess used o ind he p oposed p esc ip i e ule o mulas. The IACS ule o mulas may in
some cases use he alues om scan ling d augh ins ead when compu ing he non-linea ac o o apply. The ship
leng h is L, and b ead h is B. The leng h pa ame e s (L, B, T) a e o be aken in me e s, and he o he coe icien s
a e dimensionless.
5. Ve ical wa e-induced shea o ce
The wa e-induced hull-gi de Ve ical Shea Fo ce, VSF, has been ex ac ed om he same CFD wa e
simula ions as we e used o he VBM. The e ical shea o ces in head and ollowing seas a e assumed o occu
simul aneously wi h he maximum e ical bending momen s in he Equi alen Design Wa e app oach used by
he cu en IACS ules. We do no ha e easons o modi y his, so he Head and Following Sea Momen design
wa es, HSM and FSM, will be used also o p oduce he non-linea ac o o VSF. This ac o mus be ound bo h
a and o wa d o midship due o he wo-peaked VSF dis ibu ion along he hull. Fo each o he essel hal es,
he shea o ce du ing bo h he ex eme wa e-induced sagging and hogging bending momen s need o be
es ablished, leading o ou o mula ions o he non-linea ac o in VSF. This is equi alen o he cu en IACS
p ocedu e. Also equi alen is ou sugges ed app oach o using he non-linea ac o o VBM o es ima e he non-
linea ac o s o VSF.
The esul s om he CFD analyses a e shown in Figu e 9. The x-axes in he sub- igu es show he calcula ed
ule alue om he sugges ed eg ession i s, which a e shown in he axis labels. The y-axes show he esul om
he CFD calcula ions. Jus like o VBM, he CFD non-linea ac o s a e calcula ed om he ex emum o he non-
linea dis ibu ions on each hal o he essel di ided by he ex emum o he linea dis ibu ions on he same
hal es o he essel. The non-linea and linea dis ibu ions may no ha e hei ex emum a he exac same
longi udinal posi ion.
