1
Upda e o he Rule Load Fo mula ions
in he IACS Common S uc u al Rules
Quen in De banne1,
*
, Hå a d Aus e jo d2, Kei Sugimo o3 and Xu Min4
1 Bu eau Ve i as, Nan es, F ance
2 DNV, Oslo, No way
3 Nippon Kaiji Kyokai (ClassNK), Tokyo, Japan
4 China Classi ica ion Socie y, Shanghai, China
Abs ac . Following he upda e o he Recommenda ion n°34, which desc ibes he sea s a e s a is ics
encoun e ed by ships ope a ing in he No h A lan ic, he In e na ional Associa ion o Classi ica ion Socie ies
has unde aken a comp ehensi e e ision o he load o mula ions in he Common S uc u al Rules o Oil
Tanke s and Bulk Ca ie s (CSR). The objec i e was o achie e a good consis ency be ween he ule loads and
he loads es ima ed using a di ec app oach om he sca e diag am. A la ge da abase o mo e han 250
essels, co e ing majo ship ypes (oil anke s, bulk ca ie s, con aine ships, gas ca ie s, o e ca ie s, c uise
ships…), in bo h ull and ballas loading condi ions, was buil . Fo each essel a long- e m s a is ical app oach
was used o compu e he ex eme loads, co esponding o a 25-yea e u n pe iod, as well as he a igue loads
om he 3D Bounda y Elemen s Me hod ans e unc ions. App oxima ely 200 di e en loads quan i ies
(mo ions, accele a ions, hull gi de loads and p essu e along di e en sec ions) ha e been es ima ed o e e y
essel ac oss 12 Equi alen Design Wa es (EDW). This ex ensi e da abase se ed as he e e ence esul s o
calib a e he ule o mula ions o loads en elopes and o p essu es and Load Combina ion Fac o s ac oss
all EDWs. The di e ence be ween he ule o mula ions, which only equi e a ew key ship pa ame e s, and he
e e ence compu a ions we e minimized and ho oughly documen ed. This has esul ed in a signi ican ly mo e
consis en se o ule loads ha will be used in he u u e e ision o he CSR.
Keywo ds: Common S uc u al Rules, Equi alen Design Wa es, Hyd odynamic loads
1. In oduc ion
In 2006, he In e na ional Associa ion o Classi ica ion Socie ies (IACS) issued wo di e en se o ules, o
Oil Tanke s [1] and o Bulk Ca ie s [2]. A ew yea s la e , hese wo se s o ules we e ha monized and me ged
in o a single se , he Common S uc u al Rules o Bulk Ca ie s and Oil Tanke s (CSR) [3]. This se o ules was
submi ed o an audi by he In e na ional Ma i ime O ganiza ion (IMO) o check he compliance wi h he Goal
Based S anda d (GBS) [4]. Acco ding o he GBS, “ships shall be designed in acco dance wi h No h A lan ic
en i onmen al condi ions and ele an long- e m sea s a e sca e diag ams”. IACS explained he No h A lan ic
en i onmen al condi ions a e desc ibed by a sca e diag am in he Recommenda ion n°34 [5], and ha his
ecommenda ion has been used in he ule de elopmen . In 2016, he GBS audi epo wi hin MSC 96/5 includes
he obse a ion IACS/2015/FR1-8/OB/02, saying ha “IACS' Rec. n°34 ha is based on old wa e s a is ics was
las e ised in 2000/2001 and he e is no e idence o moni o ing since i s adop ion”. The audi eam concluded
ha hey had “no ound su icien jus i ica ion ha he wa e da a used in he ules p ope ly ep esen No h
A lan ic condi ions.”
Following his obse a ion IACS has pu in place a dedica ed p ojec eam o upda e he sca e diag am. The
wo k was done be ween 2017 and 2022. By combining mode n hindcas da a wi h ship AIS posi ions, a new sca e
diag am has been issued [6]. Mo e de ails can be ound in [7]. S a ing om his upda ed sca e diag am, ano he
p ojec eam wo ked om 2022 o 2025 o upda e all he load o mula ions om he CSR. The p esen pape shows
he me hodology used o de i e new load o mula ions om di ec analysis esul s using he upda ed sca e
diag am. Only he linea loads o mula ions a e shown in his pape . Fu he non-linea co ec ions based on CFD
calcula ions, o e ical bending momen and shea o ce, a e p esen ed in [8].
*
Co espondence o: [email p o ec ed]
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
2. Rule upda e p inciples
The upda es ollow he main p inciples o cu en CSR. Fo ex eme loads and a igue loads, he ules gi e
en elope alues o mo ions, accele a ions and hull gi de loads. The new en elope o mula ions a e de ailed in 3.
The Equi alen Design Wa e (EDW) concep wi h egula design wa es and load combina ion ac o s (LCF)
emain. Fo each EDW, he ins an aneous alues o mo ions, accele a ions and hull gi de loads a e gi en by a
LCF mul iplied wi h he en elope alue (see 4), while he wa e p essu e is di ec ly gi en by ule o mula ions
(see 5). All hese ule o mula ions a e calib a ed e sus di ec compu a ions.
2.1. E alua ed lee
2.1.1. Type o essels
A e y la ge hyd odynamic da abase o mo e han 250 essels has been used o compu e ex eme and a igue
loads, and o i he ule o mula ions. In p inciple he upda e o he CSR equi es compu ing all ypes o loads
(mo ions, accele a ions, hull gi de loads, p essu e) o only bulk ca ie s and oil anke s. I was howe e decided
o ha e a b oade da abase, including mo e ship ypes. I has been checked ha including non-CSR essels in he
eg essions o ule o mula ions do no signi ican ly impac he accu acy o CSR essels. On he opposi e, i
p e en s us om doing o e - i ing (as some pa ame e s, such as 𝐶𝐵 and 𝐶𝑊, a e e y co ela ed o CSR essels),
and i b ings mo e obus ness o he applica ion o he ules o essels wi h abno mal shapes. Table 1 shows he
numbe o essels in he da abase, pe ship ype and loading condi ion.
Table 1. Numbe o essels by ship ype in he da abase
Ship ype
Full load
Ballas
CSR
essels
Bulk ca ie
47
43
Oil anke
48
43
Non-CSR
essels
Con aine ship
52
41
Ca go ca ie
30
30
Gas ca ie
30
20
Ro- o ship
18
13
Passenge ship
11
6
O sho e essel
21
21
O he
7
8
To al
264
225
In he ollowing g aphs and ables, he essels will be spli in o ou g oups: CSR essels in ull o in ballas
loading condi ion, non-CSR essels in ull o ballas loading condi ion.
2.1.2. Main cha ac e is ics
Figu e 1. Main essels cha ac e is ics.
3
The di e en essels o he da abase ha e been selec ed o co e a wide ange o dimensions and shapes. A
single essel (and loading condi ion) is cha ac e ized by he ollowing quan i ies: ule leng h 𝐿 (m), b ead h 𝐵 (m),
a e age d augh a he co esponding loading condi ion 𝑇 (m), block coe icien a he co esponding loading
condi ion 𝐶𝐵=∆/𝐿𝐵𝑇 (∆ being he displacemen ), wa e plane coe icien a he co esponding loading condi ion
𝐶𝑊𝑃=𝐴𝑊/𝐿𝐵 (𝐴𝑊 being he wa e plane a ea), and oll pe iod 𝑇𝜃 (s). Figu e 1 shows he alues o hese quan i ies
o all he essels o he da abase.
2.2. Compu a ion o he linea long- e m hyd odynamic loads
2.2.1. Loads ans e unc ions
Fo each essel and each loading condi ion in he da abase, loads ans e unc ions (RAOs) a e compu ed
using linea po en ial low sol e s Hyd os a , Wasim, 3-DPM.L o WALCS. RAOs a e compu ed o mo ions ( oll
and pi ch), accele a ions a he cen e o g a i y (su ge, sway, hea e, oll and pi ch), hull gi de loads a 21 sec ions
along he ship ( e ical and ho izon al bending momen , e ical shea o ce, o sion) and p essu es a 105 di e en
poin s along he we ed hull. RAOs a e compu ed o wo speeds: 5 kno s and 75% o he design speed. This is in
o al abou 400 RAOs pe essel and loading condi ion.
2.2.2. Long- e m compu a ion
The loads RAOs a e combined wi h he sca e diag am o compu e he ex eme and a igue loads. The
assump ions o his compu a ion a e he ones gi en in Rec.34 e .2. The sea s a es s a is ics a e gi en by he sca e
diag am. The heading dis ibu ion is uni o m. The sea s a es a e modeled by a JONSWAP spec um wi h 𝛾=1.5
and a di ec ional sp eading in cos3. Ex eme loads a e compu ed a 5 kno s and co espond o a 25-yea e u n
pe iod. They a e de ined by he ollowing equa ion, whe e 𝑝𝑖 is he p obabili y o encoun e a gi en sea s a e and
heading, 𝑇𝑧𝑖 is he mean esponse pe iod on his sea s a e (in seconds), and 𝑃𝑖(𝑥) is he sho - e m non-exceedance
p obabili y gi en by a Rayleigh dis ibu ion:
∑𝑝𝑖
𝑇𝑧𝑖(1−𝑃𝑖(𝑋25))
𝑖=1
25∗365.25∗24∗3600
(1)
Fa igue loads a e compu ed a 75% o design speed and co espond o a 10-2 p obabili y o exceedance. They
a e de ined by he ollowing equa ion:
∑𝑝𝑖
𝑇𝑧𝑖(1−𝑃𝑖(𝑋10−2))
𝑖=10−2∑𝑝𝑖
𝑇𝑧𝑖
𝑖
(2)
These ex eme and a igue loads a e called ‘en elope’ loads. They a e no concomi an . To ob ain a ealis ic
combina ion o loads, we mus use he EDWs.
2.2.3. Equi alen design wa es
The EDWs a e he same as he ones al eady used in he CSR. We ha e 7 EDWs o ex eme loads and 5 o
a igue. Each EDW is a egula wa e ha is ailo ed o each he en elope alue o i s go e ning load. Fo ins ance,
HSM is maximizing he e ical bending momen a midship in head sea and BSP is maximizing he wa e p essu e
a midship wa e line in beam sea. The exac de ini ion o all he EDWs can be ound di ec ly in he CSR, bu a
summa y is gi en in Table 2.
Table 2. Equi alen Design Wa es de ini ion
Name
Heading
Go e ning load
HSM
180°
Ve ical Bending Momen a midship
Ex eme and Fa igue
HSA
180°
Ve ical accele a ion a Fo e Pe pendicula
Ex eme only
FSM
0°
Ve ical Bending Momen a midship
Ex eme and Fa igue
BSR
90° / 270°
Roll angle
Ex eme and Fa igue
BSP
90° / 270°
P essu e a midship wa e line
Ex eme and Fa igue
OST
60° / 300°
To sion a 0.25L and baseline
Ex eme and Fa igue
OSA
120° / 240°
Pi ch accele a ion
Ex eme only
4
Using he calcula ed en elope alue o he go e ning loads, and using he RAOs o all he o he loads, we
we e able o compu e he ins an aneous alue o all he loads (mo ions, accele a ions, hull gi de loads and
p essu e) unde each EDW. I means ha we end up wi h a huge da abase o esul s whe e o each ship and loading
condi ion, we ha e he alue o 200 loads componen s unde he 7 ex eme EDWs and 5 a igue EDWs.
2.3. Me hodology o de eloping he ule o mula ions
2.3.1. E o quan i ica ion
Fo each essel, and each loading condi ion, he ule alue is compa ed o he compu ed alue. The e o is
de ined as:
𝜀=𝐶𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒−𝑅𝑢𝑙𝑒 𝑣𝑎𝑙𝑢𝑒
𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑖𝑛𝑔 𝑡𝑒𝑟𝑚
(3)
The no malizing e m can be de ined in di e en ways, depending on he ype o load we wan o compa e:
• Fo en elope alues o mo ions and accele a ions which a e, by de ini ion, always s ic ly posi i e,
he no malizing e m is chosen o be he ule alue. Hence he e o is exp essed as a ac ion o he
ule alue.
• Fo en elope alues o hull gi de loads, he no malizing e m is chosen o be he maximum alue
along he ship o he ule en elope. Indeed, a o e and a end o he essel, whe e he hull gi de
loads a e negligible, i makes no sense o exp ess he e o as a ac ion o a e y small quan i y. Hence
he e o , a any loca ion, is exp essed as a ac ion o he maximum ule en elope alue.
• Fo ins an aneous loads unde a gi en EDW, he no malizing e m is chosen o be he en elope ule
alue o mo ions and accele a ions, and he maximum alue along he ship o he en elope ule alue
o hull gi de loads. Hence he e o is exp essed as a ac ion o he maximum en elope alue.
• Fo p essu e, he no malizing e m is equal o 1. The e o is hen he di e ence o p essu e (in kN/m²).
In any case, a posi i e e o means ha he compu ed alue is bigge han he ule alue, and a nega i e e o
means ha he compu ed alue is smalle han he ule alue. Once he e o is compu ed o each essel, each
loading condi ion (and each longi udinal loca ion o hull gi de loads and p essu es), he ollowing global
quan i ies a e de ined:
Mean E o (ME)
𝑀𝐸=1𝑁∑𝜀𝑖
𝑁
𝑖=1
(4)
Roo Mean Squa ed E o (RMSE)
𝑅𝑀𝑆𝐸=√1𝑁∑𝜀𝑖2
𝑁
𝑖=1
(5)
2.3.2. Rule o mula ions
Rule o mula ions should gi e a simple bu accu a e es ima ion o he compu ed hyd odynamic loads, using
only a ew pa ame e s: 𝐿, 𝐵, 𝑇, 𝐶𝐵, 𝐶𝑊, 𝑇𝜃 and 𝑉. They should, as much as possible, be consis en om a uni
poin o iew. Fo each o mula ion, he e o is compu ed o e e y essel and loading condi ion, and he ME and
RMSE a e compu ed. A good o mula ion should ha e 𝑀𝐸=0, and a RMSE as small as possible.
5
3. En elopes o ex eme loads and a igue loads
3.1. Rule o mula ion empla e
3.1.1. Gene alized wa e pa ame e
A gene alized wa e pa ame e , 𝐶𝑤, is used in he de ini ion o all he loads. This wa e pa ame e akes a simila
o m as he one al eady de ined in UR S11A [9]. I is compu ed om he ule leng h 𝐿 and a e e ence leng h 𝐿𝑟𝑒𝑓
which is de ined o each load.
𝐶𝑤=
{
1−1.3(1−√𝐿
𝐿𝑟𝑒𝑓)1.8𝑖𝑓 𝐿<𝐿𝑟𝑒𝑓
1−0.9(√𝐿
𝐿𝑟𝑒𝑓−1)1.8𝑖𝑓 𝐿≥𝐿𝑟𝑒𝑓
(6)
The wa e pa ame e is conside ed o ha e a uni in me e . A de ailed de ini ion o wha he wa e pa ame e
ep esen s can be ound in [10]. This me hodology has al eady been used o de ine he wa e pa ame e in UR
S11A, as explained in [11]. In a nu shell, i ep esen s he in e ac ion be ween he wa e sca e diag am and he
load RAO o a gi en essel. As illus a ed in Figu e 2, he RAOs o longe essels ha e hei peak a longe pe iods
han sho e essels. Hence he wo s sea s a e in he sca e diag am o longe essels is no he same as he wo s
sea s a e o sho e essels. The wa e pa ame e is somehow p opo ional o his wo s wa e heigh o a gi en
essel. The quan i y 𝐿𝑟𝑒𝑓, de ined in me e s, gi es he ship leng h o which he wa e pa ame e eaches i s
maximum alue. When he ship leng h 𝐿 is close o 𝐿𝑟𝑒𝑓, i means ha he highes sea s a es in he sca e diag am
a e he wo s sea s a e o his essel and his load pa ame e . 𝐿𝑟𝑒𝑓 is de ined o each load (mo ions, accele a ions,
hull gi de loads). I e lec s he ac ha o he same essel, he peak o he di e en loads RAOs a e a di e en
equencies, and ha he wo s sea s a e o e ical bending momen is no he wo s sea s a e o ho izon al
bending momen o p essu e. I is impo an o unde s and ha , up o a mul iplica i e cons an , he wa e pa ame e
can be compu ed o a gi en essel and a gi en load, om he co esponding RAO and he sca e diag am.
Figu e 2. Illus a ion o he in e ac ion be ween he sca e diag am and he essel RAOs
3.1.2. P obabili y ac o
As in he cu en CSR he a igue loads en elopes a e de ined om he ex eme load en elopes mul iplied by
a p obabili y ac o 𝑓𝑝. Hence his coe icien conside s he speed e ec ( om 5 kno s o 75% o design speed) and
he p obabili y le el ( om 25 yea s o 10-2). Fo ex eme loads we ha e 𝑓𝑝=1.
6
3.1.3. En elope empla e and i ing me hodology
The ule o mula ions o en elopes alues, which by de ini ion a e always s ic ly posi i e, a e made using
he ollowing empla e:
{
𝑋=𝑎0 𝑓𝑝𝐶𝑤 𝐿𝑛 ∏𝑃𝑖𝑎𝑖𝑓(𝑥′)
𝑓𝑝=𝑏0 𝐿𝑏7 ∏𝑃𝑖𝑏𝑖
𝑥′=𝑥𝐿
⁄−𝑥𝑎
𝑥𝑓−𝑥𝑎
𝑥𝑎=𝑐0+∑𝑐𝑖𝑃𝑖
𝑥𝑓=𝑑0+∑𝑑𝑖𝑃𝑖
𝑃𝑖∈[𝐵𝐿,𝑇𝐿,𝐶𝐵,𝐶𝑊𝑃,𝐹𝑟=𝑉
√𝑔𝐿,𝑇𝑅=𝑇𝜃√𝑔𝐿]
(7)
𝑛 is he F oude dimension o he ans e unc ion o he conside ed load: -1 o linea accele a ion and angula
mo ions, -2 o angula accele a ions, +2 o shea o ces, and +3 o momen s. 𝑃𝑖 a e a se o dimensionless
pa ame e s, 𝑎𝑖, 𝑏𝑖,𝑐𝑖 and 𝑑𝑖 a e numbe s o be uned. 𝑓(𝑥′) is he dis ibu ion unc ion (used only o hull gi de
loads). To conside he di e en beha io o deep and shallow d a essels, CSR and non-CSR essels, he
dis ibu ion unc ions a e de ined as a unc ion o 𝑥′. I means ha he dis ibu ion unc ion will s a a 𝑥𝑎𝐿 (no
necessa y 𝑥=0) and end a 𝑥𝑓𝐿 (no necessa y 𝑥=𝐿).
The bes o mula ion is ound by minimizing he RMSE, ensu ing a ME equal o ze o, and ying o use he
smalles numbe o pa ame e s 𝑃𝑖: no need o ha e a e y complex o mula ion o a small gain o RMSE. I needed,
physical cons an such as 𝜌 and 𝑔 a e included in he pa ame e 𝑎0 o ha e a consis en o mula ion in e ms o
uni s, assuming he wa e pa ame e is in me e . Fo ins ance, linea accele a ions will be p opo ional o 𝑔𝐶𝑤𝐿−1,
and momen s will be p opo ional o 𝜌𝑔𝐶𝑤𝐿3.
The i s s ep is o i he ex eme en elope, by uning he pa ame e s 𝑎𝑖. Fo hull gi de loads, his i is done
on he maximum bending momen along he essel, and he maximum shea o ce and o sion momen in he a
pa o in he o e pa . In a second s ep he o mula ions o a igue loads a e adjus ed by uning he pa ame e s
𝑏𝑖 o he p obabili y ac o . In a las s ep, o hull gi de loads only, he dis ibu ion unc ions a e i ed by uning
he coe icien s 𝑐𝑖 and 𝑑𝑖 and adjus ing he dis ibu ion unc ion.
3.2. Mo ions and accele a ions
The o mula ions in Table 3 a e p oposed o he mo ions and accele a ions ex eme alues. The d a 𝑇, he
block coe icien 𝐶𝐵, he wa e plane coe icien 𝐶𝑊𝑃 and he oll pe iod 𝑇𝜃 a e depending on he loading condi ion
( ull o ballas ). The F oude numbe 𝐹𝑟 should be compu ed a a speed o 5 kno s.
Table 3. Ship mo ions and accele a ions: new o mula ions o ex eme and a igue en elopes, mean and oo mean squa ed
e o s
Load
Rule o mula ion
𝐿𝑟𝑒𝑓
𝑓𝑝
Ex eme
Fa igue
ME
RMSE
ME
RMSE
Su ge
acc.
12.1 𝑔𝑓𝑝𝐶𝑤𝐿−0.85𝑇−0.15𝐶𝐵−0.6
203/𝐶𝑊𝑃
0.9
1.03 𝐿−0.2𝐵−0.15𝑇−0.1
0.4%
2.4%
0.7%
10.3%
Sway
acc.
11.1 𝑔𝑓𝑝𝐶𝑤𝐿−0.35𝐵−0.5𝑇−0.15
132(𝐿𝐵)0.65
0.55 𝐿−0.08𝐵−0.15𝐶𝐵−0.1
-0.1%
3.1%
-0.5%
4.7%
Hea e
acc.
56 𝑔𝑓𝑝𝐶𝑤𝐿−0.8𝐵−0.2𝐶𝑊𝑃
−0.2
192(𝐿𝑇)0.35
1.87 𝐿−0.4𝐶𝐵−0.5
-0.4%
3.6%
0.1%
9.7%
Roll
mo ion
𝑓𝑝3050(1−𝑇𝜃/54)
𝐵+51
2530
𝑇𝑅2
0.805 𝐿−0.2𝑇𝑅−0.2𝐶𝐵−1.2
0.6%
31.2%
0.1%
23.0%
Roll
acc.
𝑓𝑝136(1−𝑇𝜃/53)
𝑇𝜃1.2𝐶𝑊𝑃1.3(𝐵+16)
2530
𝑇𝑅2
5.45 𝐿−0.45𝑇𝑅−0.45
-0.1%
20.8%
1.9%
27.0%
Pi ch
mo ion
3650 𝑓𝑝𝐶𝑤𝐿−1𝐶𝑊𝑃
−0.75𝐹𝑟025
309
0.86 𝐿−0.15𝐵−0.2𝐶𝑊𝑃
−0.2
0.0%
4.7%
-0.5%
6.0%
Pi ch
acc.
182 𝑓𝑝𝑔𝐶𝑤𝐿−1.4𝐵−0.6𝐶𝑊𝑃
1.4𝐹𝑟0.3
132(𝐿𝐵)0.8
0.72 𝐵−0.3𝐶𝑊𝑃
−0.6
0.2%
4.9%
-0.2%
9.1%
7
Figu e 3 shows he e o o e e y essel. All he do s ep esen indi idual essels, spli in o 4 g oups (CSR
essels in ull load o in ballas condi ion, non-CSR essels in ull load o ballas condi ion). La ge diamonds wi h
he ho izon al e o ba ep esen he mean e o o each g oup, plus o minus he s anda d de ia ion o he e o .
The ag eemen is e y good: he mean e o is close o ze o o all he loads (small di e ences a e due o he
ounding o he nume ical coe icien 𝑎0 in he o mula ion), and he a iabili y is qui e low: RMSE is less han
5% o ex eme loads and less han 10% o a igue loads (excep o oll mo ions and accele a ion). Fu he mo e,
he a iabili y is he same o all he 4 ca ego ies, showing ha he e ec s o loading condi ion and ship shape
ha e been p ope ly conside ed in he ule o mula ion.
A signi ican sca e exis s howe e o he oll mo ion and he oll accele a ion. Roll mo ion is indeed e y
di icul o compu e, and e en di ec compu a ion may no ep esen accu a ely he eali y. Fu he mo e, i was
di icul o i a good o mula ion o he compu ed da a. I was decided o keep a o mula ion like he cu en oll
o mula ion in he CSR and no o use he wa e pa ame e .
Figu e 3. Accu acy o he new load o mula ions (New ules) compa ed o linea di ec compu a ions (Rec.34 Re .2)
3.3. Hull gi de loads
The hull gi de loads a e de ined by one o mo e cha ac e is ic loads using he empla e o equa ion (7) and by
one o mo e dis ibu ion unc ions. The longi udinal dis ibu ions a e desc ibed by sinusoidal unc ions, o i as
much as possible o he calcula ions. The pa ame e s 𝑥𝑎 and 𝑥𝑓 a e i ed o minimize he e o in all sec ions. The
o mula ions p esen ed in equa ions (8) o (12) a e i ed o he linea calcula ions. In a second s ep, CFD
compu a ions a e used o co ec o non-linea e ec s o e ical bending momen and shea o ce (see [8]). Non-
linea e ec s will a ec he ampli ude o he loads using he non-linea ac o 𝑓𝑛𝑙 and he longi udinal dis ibu ion
by adjus ing he pa ame e s 𝑥𝑎 and 𝑥𝑓.
3.3.1. Ve ical wa e bending momen
The o mula ion o e ical bending momen (VBM) is gi en in equa ion (8). I is buil om he maximum
alue along he ship, 𝑀𝑤𝑣−𝑚𝑎𝑥, and a dis ibu ion unc ion 𝑓𝑚(𝑥′).
𝑀𝑤𝑣=𝑀𝑤𝑣−𝑚𝑎𝑥 𝑓𝑚(𝑥′)
𝑀𝑤𝑣−𝑚𝑎𝑥=0.118 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿2.3𝐵0.7𝐶𝑊𝑃
1.4 𝑓𝑚(𝑥′)=𝑠𝑖𝑛2(𝜋𝑥′)
𝐿𝑟𝑒𝑓=358𝐶𝑊𝑃
−1.3 𝑓𝑝=1.56 𝐿−0.4𝐶𝑤𝑝
−1.25
𝑥𝑎=0.34+0.3𝐶𝐵−0.6𝐶𝑊𝑃−𝑇/𝐿 𝑥𝑓=0.95+0.5𝐶𝐵−0.4𝐶𝑊𝑃+0.1𝑇/𝐿
(8)
8
This can be compa ed o he o iginal CSR [3] o mula ion gi en in equa ion (9):
𝑀𝑤𝑣=𝑀𝑤𝑣−𝑚𝑎𝑥 𝑓𝑚(𝑥)
𝑀𝑤𝑣−𝑚𝑎𝑥=0.19𝐶𝑤−𝐶𝑆𝑅𝑓𝑝𝑓𝑛𝑙𝐿2𝐵𝐶𝐵 𝑓𝑚(𝑥)=
{
𝑥
0.4𝐿 𝑓𝑜𝑟 0≤𝑥<0.4𝐿
1 𝑓𝑜𝑟 0.4𝐿≤𝑥<0.65𝐿
1−𝑥/𝐿
0.35 𝑓𝑜𝑟 𝑥≤0.65𝐿<𝐿
(9)
Figu e 4 shows he di e en s eps o he ule op imiza ion p ocess. Figu e 4 (a) shows he compa ison o he
compu ed e ical bending momen and he linea CSR bending momen (wi hou 𝑓𝑛𝑙), and he co esponding e o
(exp essed as a pe cen age o 𝑀𝑤𝑣−𝑚𝑎𝑥). The e a e ob iously a lo o e o s, no only a ound he midship a ea, bu
also in he o e and a pa . Figu e 4 (b) shows he same compa ison whe e he new o mula ion is used o
𝑀𝑤𝑣−𝑚𝑎𝑥 bu he CSR dis ibu ion unc ion is kep . While he e o is la gely educed a ound midship, his ule
o mula ion is la gely o e -conse a i e in he o e and a a ea. The compu ed bending momen dis ibu ions a e
smoo h and do no ma ch a all he CSR dis ibu ion. This is why we ha e ied o use a sinusoidal dis ibu ion.
Figu e 4 (c) shows his compa ison whe e he dis ibu ion unc ion is 𝑓𝑚(𝑥)=𝑠𝑖𝑛2(𝜋𝑥/𝐿). The i is much be e .
Howe e , i is now qui e clea ha , in he a pa , he ule o mula ion unde es ima es he bending momen o he
ull loading condi ions, while i o e -es ima es he bending momen o ballas loading condi ions. In he o e pa
he ule o mula ion unde es ima es he bending momen o CSR ships, while i o e es ima es he bending momen
o non-CSR ships. To co ec o he e ec o loading condi ion and ship ype, he inal dis ibu ion unc ion is
now de ined as unc ion o 𝑥′. The e ec o 𝑥′ is o s e ch he sinusoidal unc ion, depending on he ship
pa ame e s (𝐶𝐵, 𝐶𝑊𝑃 o 𝑇/𝐿). The inal compa ison is shown in Figu e 4 (d): The accu acy o he ule o mula ion
is much be e . The e o is now simila a all longi udinal sec ions, wi h no signi ican e ec o ship ype o loading
condi ion.
(a) (b) (c) (d)
Figu e 4. Ex eme e ical bending momen en elope compa ed o di e en ule o mula ions: (a) is o iginal CSR, (b) is he
new o mula ion o 𝑀𝑤𝑣−𝑚𝑎𝑥 combined wi h he CSR dis ibu ion unc ion, (c) is he new o mula ion o 𝑀𝑤𝑣−𝑚𝑎𝑥 combined
wi h he sinusoidal dis ibu ion unc ion 𝑓𝑚(𝑥)=𝑠𝑖𝑛2(𝜋𝑥/𝐿), (d) is he new o mula ion o 𝑀𝑤𝑣−𝑚𝑎𝑥 and he new dis ibu ion
unc ion 𝑓𝑚(𝑥′).
3.3.2. Ve ical wa e shea o ce
The o mula ion o e ical shea o ce (VSF) is gi en in equa ion (10). I is buil om h ee cha ac e is ic
alues (maximum VSF in he a pa 𝑄𝑤𝑣−𝑎𝑓𝑡, minimum VSF in he midship egion 𝑄𝑤𝑣−𝑚𝑖𝑑, and maximum VSF
in he o e pa 𝑄𝑤𝑣−𝑓𝑜𝑟𝑒) and h ee dis ibu ion unc ions 𝑓𝑎(𝑥′), 𝑓𝑚(𝑥′) and 𝑓𝑓(𝑥′).
𝑀𝑤ℎ=𝑄𝑤𝑣−𝑎𝑓𝑡 𝑓𝑎(𝑥′)+𝑄𝑤𝑣−𝑚𝑖𝑑 𝑓𝑚(𝑥′)+𝑄𝑤𝑣−𝑓𝑜𝑟𝑒 𝑓𝑓(𝑥′)
𝑄𝑤𝑣−𝑎𝑓𝑡=0.395 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿1.3𝐵0.7𝐶𝑊𝑃
1.1 𝑓𝑎(𝑥′)={𝑠𝑖𝑛2(2𝜋𝑥′) 𝑓𝑜𝑟 0<𝑥′<0.5
0 𝑓𝑜𝑟 𝑥′≤0 𝑜𝑟 𝑥′≥0.5
(10)
9
𝑄𝑤𝑣−𝑚𝑖𝑑=0.09 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿1.7𝐵0.3𝐶𝑊𝑃
1.2 𝑓𝑚(𝑥′)={𝑠𝑖𝑛2(2𝜋(𝑥′−0.25)) 𝑓𝑜𝑟 0.25<𝑥′<0.75
0 𝑓𝑜𝑟 𝑥′≤0.25 𝑜𝑟 𝑥′≥0.75
𝑄𝑤𝑣−𝑓𝑜𝑟𝑒=0.718 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿𝐵𝐶𝑊𝑃
0.9 𝑓𝑓(𝑥′)={𝑠𝑖𝑛2(2𝜋𝑥′) 𝑓𝑜𝑟 0.5<𝑥′<1.0
0 𝑓𝑜𝑟 𝑥′≤0.5 𝑜𝑟 𝑥′≥1.0
𝐿𝑟𝑒𝑓=375𝐶𝑊𝑃
−1.3 𝑓𝑝=1.37 𝐿−0.35𝐶𝑊𝑃
−0.6
𝑥𝑎=0.39+0.4𝐶𝐵−0.8𝐶𝑊𝑃−1.0𝑇/𝐿 𝑥𝑓=0.89+0.4𝐶𝐵−0.2𝐶𝑊𝑃+0.3𝑇/𝐿
3.3.3. Ho izon al wa e bending momen
The o mula ion o ho izon al bending momen (HBM) is gi en in equa ion (11). I is buil om he maximum
alue along he ship, 𝑀𝑤ℎ−𝑚𝑎𝑥, and a dis ibu ion unc ion 𝑓𝑚(𝑥′).
𝑀𝑤ℎ=𝑀𝑤ℎ−𝑚𝑎𝑥 𝑓𝑚(𝑥′)
𝑀𝑤𝑣−𝑚𝑎𝑥=0.24 𝜌𝑔𝐶𝑤𝑓𝑝𝐿2.2𝐵−0.1𝑇0.9 𝑓𝑚(𝑥′)=𝑠𝑖𝑛2(𝜋𝑥′)
𝐿𝑟𝑒𝑓=358(𝐿/𝑇)0.3 𝑓𝑝=2.44 𝐿−0.55𝐵0.3𝐶𝐵1.6
𝑥𝑎=−0.31+0.4𝐶𝐵 𝑥𝑓=1.0
(11)
3.3.4. To sion momen
The o mula ion o o sion momen is gi en in equa ion (12). I is buil om he maximum alue along he
ship, 𝑀𝑤𝑡−𝑚𝑎𝑥, and h ee dis ibu ion unc ions 𝑓𝑎(𝑥′), 𝑓𝑚(𝑥′) and 𝑓𝑓(𝑥′).
𝑀𝑤𝑡=𝑀𝑤𝑡−𝑚𝑎𝑥(𝑓𝑎(𝑥′)+0.25𝑓𝑚(𝑥′)+0.76𝑓𝑓(𝑥′))
𝑀𝑤𝑡−𝑚𝑎𝑥=1.43 𝜌𝑔𝐶𝑤𝑓𝑝𝐿0.6𝐵1.4𝑇𝐶𝑊𝑃
𝑓𝑎(𝑥′)={𝑠𝑖𝑛2(𝜋𝑥′/0.6) 𝑓𝑜𝑟 0<𝑥′<0.6
0 𝑓𝑜𝑟 𝑥′≤0 𝑜𝑟 𝑥′≥0.6
𝑓𝑚(𝑥′)={𝑠𝑖𝑛2(𝜋(𝑥′−0.3)/0.4) 𝑓𝑜𝑟 0.3<𝑥′<0.7
0 𝑓𝑜𝑟 𝑥′≤0.3 𝑜𝑟 𝑥′≥0.7
𝑓𝑓(𝑥′)={𝑠𝑖𝑛2(𝜋(𝑥′−0.4)/0.6) 𝑓𝑜𝑟 0.4<𝑥′<1.0
0 𝑓𝑜𝑟 𝑥′≤0.4 𝑜𝑟 𝑥′≥1.0
𝐿𝑟𝑒𝑓=375(𝐿/𝑇)0.3 𝑓𝑝=0.303 𝐿0.2𝑇−0.45
𝑥𝑎=−0.05 𝑥𝑓=1.07
(12)
3.3.5. Compa isons wi h compu ed alues
Figu e 5 shows he compa ison o hese linea ule o mula ions wi h he linea compu a ions. The e o s a e
also included in he i les o each g aph. We can see he e y good ag eemen o VSF and VBM o all ypes o
essels, wi h RMSE in he o de o 10%. Compa isons o he non-linea coe icien s a e epo ed in [8].
Fo o sion and HBM i was ound di icul o i he o mula o all ship ypes and loading condi ions. This is
again because o oll mo ion ha could inc ease signi ican ly hese loads, especially o ballas cases. The ule
upda e is he e o e based on CSR essels and ull load condi ion as he momen was ound o inc ease wi h d a :
he ule o mula ions a e app oxima ely p opo ional o he d a 𝑇, i has been checked ha hese loads a ull d a
a e always highe han he alue a ballas d a , despi e he e ec o oll. When conside ing only he CSR essels
in ull load, he RMSE o HBM is 8.6% o ex eme loads and 6.0% o a igue loads, while he RMSE o o sion
is 9.9% o ex eme loads and 11.8% o a igue loads.
16
6. Fa igue pe iod
6.1. Me hodology
In he cu en CSR, he a igue damage is compu ed om he 10-2 s ess ange and he s ess pe iod 𝑇=
4log10𝐿. This s ess pe iod only depends on he ship leng h and is he same o all he s uc u al de ails onboa d
he ship.
When a igue damage is compu ed using a spec al app oach, he mean cycle pe iod migh be di e en o each
de ail onboa d he ship. I is compu ed om he s ess RAO o his pa icula de ail. In o de o de i e a ule
o mula ion, we should compu e s ess RAOs on mul iple de ails onboa d many di e en ships. Howe e , he 10-
2 s ess ange being compu ed om he s ess anges compu ed on he 5 a igue Equi alen Design Wa es, we can
assume ha he mean s ess pe iod is equal o he mean pe iod o he go e ning load ( he go e ning pa ame e o
he design wa e maximizing he s ess ange). Hence i makes sense o compa e he mean cycle pe iod o he 4
go e ning load pa ame e s (Ve ical bending momen , o sion momen , oll mo ion and wa e line p essu e a
midship). Figu e 10 shows his compa ison. The mean cycle pe iods o he go e ning loads a e e y di e en om
he ule pe iod 4log10𝐿. The mean oll pe iod is usually highe han he ule pe iod, bu he mean pe iods o he
o he loads a e always smalle , which mean ha he ules a e unconse a i e.
6.2. P oposed o mula ions
4 o mula ions a e p oposed o he mean cycle pe iod: one o each go e ning load. The mean pe iod o be used
o each EDW is gi en in he able below, oge he wi h he quan i ica ion o he e o .
Table 8. Fa igue pe iod o mula ions
EDW
Rule o mula ion
RMSE
HSM and FSM
0.8 √𝐿−𝐿/50+1.84 𝐶𝐵
2.9%
OST
3.41𝑙𝑜𝑔10(𝐿)
5.9%
BSP
5.71 𝐵0.07𝐶𝐵0.15
4.2%
BSR
𝑚𝑖𝑛(0.7 𝑇𝑟𝑜𝑙𝑙+2.8 ;6 𝐿0.13)
5.4%
Figu e 10. Compa ison o he a igue ule pe iod and he go e ning loads mean pe iod
6.3. Selec ion o he a igue pe iod
In p inciple we could compu e he a igue damage o each o he 5 design wa es, using o each o hem he 10-2
s ess ange and he co esponding mean pe iod, and ake he highes a igue damage. I is howe e simple o
selec he go e ning design wa e as being he one maximizing he quan i y Δ𝜎𝑖3
𝑇𝑖.
17
7. Conclusion
Following he upda e o he No h A lan ic sca e diag am on he Recommenda ion n°34, his pape explains
how he CSR load o mula ions ha e been upda ed. The objec i e was o achie e a good consis ency be ween he
ule loads and he loads es ima ed using a di ec app oach om he sca e diag am.
The i s s ep was o build a huge hyd odynamic da abase o loads RAOs o mo e han 250 essels o majo
ship ypes (oil anke s, bulk ca ie s, con aine ships, gas ca ie s, o e ca ie s, c uise ships…), in bo h ull load and
ballas condi ions. By combining RAOs and he sca e diag am, ex eme and a igue loads we e hen compu ed.
The second s ep was o i he ule o mula ions o loads en elopes o mo ions accele a ions and hull gi de
loads, and he co esponding LCF and sea p essu e unde he di e en EDWs. Each ime he p ocedu e was simila :
ying o minimize he e o be ween he ule o mula ion and he compu ed alue, while keeping ela i ely simple
o mula ions. The accu acy o he ule o mula ion is, o each o mula ion, quan i ied by he RMSE.
The esul is a se o o mula ion which is consis en wi h he ecommenda ion n°34 assump ions on he
en i onmen al (encoun e ed sea s a es) and ope a ing (heading and speed) condi ions.
A he ime o he inaliza ion o his pape , he consequence assessmen o hese new loads on he ship
scan lings is s ill ongoing. The plan is o adop he Rule Change P oposal in July 2027, o a en a i e da e o en y
in o o ce in July 2029.
Acknowledgmen s
All wo k pe o med in his s udy was sponso ed by IACS h ough he Hull Panel PT PH49 p ojec g an o
upda e he p esc ip i e wa e loads in CSR and UR S11/S11A.
Re e ences
[1]
IACS, Common S uc u al Rules o Double Hull Oil Tanke s, 2006.
[2]
IACS, Common S uc u al Rules o Bulk Ca ie s, 2006.
[3]
IACS, Common S uc u al Rules o Bulk Ca ie s and Oil Tanke s, 2014.
[4]
IMO, "Goal-based ship cons uc ion s anda ds o bulk ca ie s and oil anke s," in Resolu ion MSC.287(87).
[5]
IACS, Recommenda ion 34 S anda d Wa e Da a - Re .1, 2000.
[6]
IACS, Recommenda ion 34 S anda d Wa e Da a - Re .2, 2022.
[7]
H. N. Aus e jo d, G. de Hau eclocque, M. C. Johnson and T. Y. Zhu, "Upda o wa e s a is ics s anda ds o
classi ica ion ules," in MARSTRUCT, Go henbu g, 2023.
[8]
T. Lande , G. de Hau eclocque, K. Sugimo o, Q. De banne and H. Aus e jo d, "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," in PRADS, Ann A bo , 2025.
[9]
IACS, UR S11A: Longi udinal S eng h S anda d o Con aine Ships, 2015.
[10]
G. de Hau eclocque and Q. De banne, "Gene alized wa e pa ame e o ules o mulae," in PRADS, Copenhagen,
2016.
[11]
Q. De banne, G. S o haug, V. Shiguno , G. Xie and G. Zheng, "Rule o mula ion o e ical hull gi de wa e loads
based on di ec compu a ion," in PRADS, Copenhagen, 2016.
