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－ 50 － 
Chapter 4 Ship Design 
 
 
4.1 Lines and Key Plan 
 
4.1.1 Lines 
The word "lines" is a generic term for the hull form consisting of a set of sections cut in longitudinal, 
transverse and horizontal planes at given distances, including body plan, elevation or profile, half 
breadth plan, bow lines, and buttock lines (see Fig.4.1). 
Neumerical presentation of “lines” is called “table of offset” (see Fig.4.2) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(b) Profile (part) 
 
 
 
 
 
 
 
（a）Body Plan 
－ 51 － 
 
 
 
 
 
 
 
 
 
（c）Half-Breadth Plan（part） 
 
 
 
The lines are the basic drawings defining the form and performance of a ship. 
On the basis of the lines, various ship calculations provide basic hydrostatic curves to obtain the data 
necessary for ship operations. The wet surface area derived from the lines is used to determine the 
frictional resistance of the ship, leading to a determination of the required power of the engine in 
association with various hull form coefficients (see 3.6). 
 
4.1.2 Key Plans 
Once the hull form has been determined by use of “lines”, key plans determining the dimensions of 
main structural members and arrangement of cargo holds and tanks in the ship are developed. The key 
plans include general arrangement, midship section, construction profile, and shell expansion plan. 
Key plans are developed determining arrangement and scantlings of structural members by taking the 
results of strength calculations of hull structure in accordance with the Rules for the Survey and 
Construction of Steel Ships of this Society (hereafter they are called as the Rules) into account. The 
Rules address ships for unrestricted navigational areas and services and contains special requirements 
in scantlings and others for ships to have special class notations for hull structure and outfittings. 
Though very seldom there were ships with class notation CoC for reduced design scantlings subject to 
application of corrosion control system approved by the Society. 
Specifications of anchors, anchor chain cables and ropes for the intended ship are determied by a use 
of an equipment number to be obtained based on the displacement of the ship as well as the wind 
resistance on the hull. 
The principal informatin given above are to be contained in the ship’s key plans such as general 
arranagement, midship section, construction profile and shell expansion. Detailed drawings are to be 
developed based upon the information given in the key plans , which are further used for establishing 
yard plans containing instructions for fabrication of the structural components. 
Ship’s plans are drawn up with the ship’s bow on the right hand and its stern on the left hand. Ships 
are to be built synmetrically for port and starboard sides and it is sufficient to draw up a half side of 
the ship. Non-synmetrical parts, if any, are to be highlighted specifically. In sections the side is to be 
made identifiable by giving information whether the part is drawn up by “Looking Forward” or 
“Looking aft”. Port side is normally contained in plans. 
The scale of the plans are normally 1/25, 1/50, 1/100, 1/200 or 1/500.
Fig. 4.1 
－ 52 － 
 (HALF BREADTH) m (HEIGHT ABOVE BASE LINE) m 
Section 0.6W
L 
1.2W
L 
1.8W
L 
2.4W
L 
3.6W
L 
4.8W
L 
6.0W
L 
7.2W
L 
Main 
deck 
Super 
struct
0.6BL 1.2BL 2.4BL 3.6BL 4.8BL Main 
deck 
Super 
structure 
Section 
B (-1/4) - — - - - 1.058 1.848 2.760 6.604 7.408 9.706 - - 8.495 10.756 B (-1/4) 
A (-1/8) - - - - - - 0.525 1.697 2.553 3.558 6.062 6.629 8.187 10.80 - 8.437 10.687 A (-1/8) 
A.P.(0) - - - - - - 0.980 2.258 3.152 4.238 5.706 6.180 7.363 9.172 - 8.380 10.620 A.P.( 0)
1/4 0.170 0.117 - - 0.088 0.480 1.895 3.250 4.132 5-257 4.950 5.465 6.410 7.580 9.440 8.273 10.498 1/4
1/2 0.393 0.542 0.659 0. 758 1.027 1.562 2.813 4.120 4.903 5.950 1.500 4.120 5. 618 6.684 8.022 8.172 10.396 1/2 
3/4 0.757 1.074 1.310 1.508 1.930 2.590 3.717 4.890 5.527 6.435 0.368 1.510 4.526 5.886 7.094 8.080 10.310 3/4
1 1.174 1.650 1.986 2.270 2.820 3.570 4.560 5.538 6.030 6.790 0.118 0.627 2.689 4.840 6.278 7.995 10.230 1 
11/2 2.188 2.946 3.440 3.842 4.558 5-263 5.930 6.510 6.748 7.200 - 0.152 0-740 2.017 4.007 7.855 10.108 11/2 
2 3.458 4.350 4.902 5.343 5.998 6.462 6.807 7.064 7.150 7.315 - 0.030 0.217 0.675 1.666 7.744 10.019 2 
21/2 4.780 5.605 6.113 6.445 6.866 7.100 7.228 7.293 7.308 7.315 - 0.013 0.050 0.190 0.610 7.660 - 21/2
3 5.908 6.548 6.920 7.098 7.258 7.310 7.315 7.315 7.315 7.315 - 0.013 0.042 0.068 0.170 7.620 - 3 
4 6.875 7.236 7.315 7.315 7.315 7.315 7.315 7.315 7.315 - - 0.013 0.042 0.068 0.097 7.620 - 4 
5 6.875 7.236 7.315 7.315 7.315 7.315 7.315 7.315 7.315 - - 0.013 0.042 0.068 0.097 7.620 - 5 
6 6.875 7.236 7.315 7.315 7.315 7.315 7.315 7.315 7.315 - - 0.013 0.042 0.068 0.097 7.620 - 6 
7 6.620 7.08O 7.263 7.315 7.315 7.315 7.315 7.315 7.315 - - 0.013 0.042 0.068 0.097 7.827 - 7 
71/2 5.788 6.418 6.748 6.935 7.108 7.176 7.200 7.218 7.220 - - 0.013 0.042 0.068 0.182 8.00.8 - 71/2
8 4.520 5.264 5.600 5.983 6.336 6.533 6.670 6.773 6.849 - - 0.013 0.042 0.218 0.781 8.238 - 8 
8 1/2 2.958 3.680 4.160 4.480 4.958 5.312 5.600 5.858 6.118 - - 0.013 0.304 1.116 3.150 8.522 - 81/2
9 1.430 1.947 2.340 2.645 3.144 3.578 3.990 4.390 4.948 - 0.057 0.407 1.920 4.860 8.416 8.843 - 9 
91/4 0.807 1.200 1.500 1.757 2.210 2.620 3.030 3.456 4.168 5.222 0.345 1.200 4.140 7.590 10.41 9.025 11.260 91/4 
91/2 0.282 0.557 0.787 0.981 1.330 1.665 2.018 2.410 3.203 4.480 1.303 3.143 7.176 10.01 - 9.210 11.442 91/2
93/4 - 0.029 0.237 0.312 0.520 0.730 0.968 1.257 2.015 3.380 4.070 6.988 10.19 - - 9.408 11.652 93/4
F.P.(10) — — — — — — — ~ 0.560 1.728 9.728 11.14 — — — 9.620 11.890 F.P.(IO) 
 
Fig. 4.2 
4.2 Basic Calculations 
 
The basic calculations include determining displacement, trim, and other values necessary for the 
design of the ship, stability calculation, and launching calculation. 
Areas and centre of gravity of curved figures are to be calculated accurately to obtain those values. It 
is, however, practically difficult to obtain them analytically for the ship’s form in three-dimensional 
curvature and a well-established approximation procedures such as trapezoidal rule or Simpson’s First 
Rule (see Fig. 4.3) have been widely applied. Also specially designed tool such as planimeter had been 
widely utilised. Thanks to the development of the computer technology in recent years, those 
approximation procedures or the tools gave their position to numerical calculations by computer 
programs. There are various techniques in numerical calculations and the one given below is one of 
them. 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 4.3 
)4(
3 210
yyyhAreaAHID ++=
 I 
A E H 
D 
F 
h h
y0 
y1 
y2 
－ 53 － 
4.2.1 Calculation of Displacement 
 
4.2.1.1 Hydrostatic Curves 
Calculations using hydrostatic curves such as displacement, etc. are carried out to the form in mold, i.e. 
without shell plating, which is to be taken into account where necessary. 
The following coefficients are to be calculated to drafts and they are to be presented in Hydrostatic 
curves with drafts measured from the bottom of keel at midship in ordinate and the coefficients in 
abscissas. These calculations assume that the ship is afloat in sea water in even keel and adjustment is 
to be made for trimmed conditions. 
(1) Displacement ( )bCdBL ×××=∇=Δ γγ 
Where, 
 γ is the specific gravity of sea water, 1.025 
The volume displaced ∇ is obtained by integrating in the longitudinal direction the 
underwater sectional area given by the body plan. 
The additional elements to the displacement are shell plating, bilge keels, propeller boss, 
exposed shaft, propeller, rudder, etc. The cruisertype stern and bulbous bow are to be 
included in the mold displacement rather than inclusion in the additional elements. 
(2) Wetted Surface Area 
The wetted surface area, S, is obtained by integrating in the longitudinal direction the girth 
length of under water sections given in the body plan. Curves of waterlines in horizontal 
planes are to be taken into account and the girth length “g” is to be revised by θsec×g , 
the θbeing the angles of the water line to the centre line measured at the middle of the water 
line. 
(3) Water Plane Area 
Water plane area Aw is obtained by twice the integration in the longitudinal direction of the 
half width of sections given in the body plan. 
(4) Tons per Centimeter Immersion 
It is the increment of displacement with the draft increase uniformly by 1 cm. 
TPC=Aw/100 
(5) Fineness Coefficient 
Block Coefficient: ( )dBLCb ××∇= / 
Prismatic Coefficient: ( )LAC p ×∇= ⊗/ 
Midship Coefficient: )/( dBAC ×= ⊗⊗ 
Water Plane Coefficient: ( )BLAC ww ×= / 
Where, 
 ⊗A : under-water sectional area at midship 
 wA : water plane area at a given draft 
(6) Centre of Buoyancy 
The horizontal distance from the midship to the centre of buoyancy is given in B⊗ with 
the direction to aft a positive value. 
The position of the centre of buoyancy B⊗ is the centre of gravity of the area surrounded 
by the centre line of the ship and sectional area curve in the longitudinal direction.. 
(7) Centre of Floatation 
It is the centre of gravity of the water line area at a given draft. 
The horizontal distance from the midship to the centre of floatation is given in F⊗ with 
the direction to aft a positive value. 
－ 54 － 
The position of the centre of floatation F⊗ is the centre of gravity of the area of water 
plane obtained from the half breadth of the water line in each sectional area given in the 
body plan. 
(8) Centre of Buoyancy above Base Line (Top of Keel) 
The position of the centre of buoyancy above top of keel KB is the centre of gravity of the 
area surrounded by the centre line of the ship and the area of water plane curve in the 
direction of the ship depth. 
(9) Transverse Metacentric Radius 
It is the centre of curvature in up-right condition of the locus of centre of buoyancy for ship 
heeling. 
∇= /IBM 
where I is the moment of inertia of the area of water plane about the centerline of hull. I is 
given by the following formula with the longitudinal position of the sections in x axis and 
the half breadth of the area of water plane in y axis: 
 ∫= dxyI 332 
(10) Transverse Metacentric Height above Base Line 
BMKBKM += 
(11) Longitudinal Metacentric Radius 
It is the centre of curvature in up-right condition of the locus of centre of buoyancy for 
ship’s trim:. 
∇= /tILBM 
Where tI is the moment of inertia of the area of water plane about the horizontal axis 
through the centre of floatation. The following equation gives the tI with the moment of 
inertia about the transverse axis at the midship being ⊗I : 
 2FAII wt ⊗×−= ⊗ 
The moment of inertia about the transverse axis at the midship ⊗I is given by the 
following formula with the coordinate origin taken at the midship and x being the position 
of the sections and y being the half breadth of the water plane: 
 ∫=⊗ ydxxI 22 
(12) Metacentric Height above Base Line 
LBMKBLKM += 
(13) Moment to change Trim One Centimeter 
Inclining moment to change trim by one centimeter. 
 ( )LLGMMTC 100/×Δ= 
Where LMG is the longitudinal metacentric height. 
Normally LGM ≒ LBM and MTC is obtained by LLMTC t 100/025.1= 
 
4.2.1.2 Displacement of a ship afloat at any draft 
The displacement of a ship afloat with the aft trim da’, the forward draft df’ and the average draft at 
midship dm’ is obtained by the following procedures: 
(1) Correction of forward and aft trim (see Fig. 4.5) 
The following formulae give the forward draft df at the forward perpendicular and the aft draft 
da at the aft perpendicular: 
df = df’－(da’－df’)×lf／(L－la－lf) 
da = da’－(da’－df’)×la／(L－la－lf) 
where, 
－ 55 － 
fl is the horizontal distance between the position where the forward draft was 
measured and the forward perpendicular 
al is the horizontal distance between the position where the aft draft was measured 
and the aft perpendicular 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(2) Correction of hull bending deflection 
Bending deflection of the hull ( ) '2/ mfa ddd −+=δ 
 σ>0 for hogging condition 
 σ>0 for sagging condition 
Average draft ( ) 4/32/ δ±+= mam ddd 
 + is to be used for sagging condition 
 - is to be used for hogging condition 
 
 
AP FP 
da da’ df df’ 
la lf 
dm’ 
Fig. 4.5 
－ 56 － 
 
－ 57 － 
(3) Correction of trim (see Fig. 4.6) 
When the ship changes trim without changing the displacement the initial draft line and the 
draft line after the trim change cross at the centre of floatation F. The horizontal distance 
between the midship and the centre of floatation being F⊗ , the draft difference at the 
midship and the centre of floatation is to be obtained for calculating corresponding draft. 
 Draft difference ( ) LFddd fa /⊗×−=Δ 
 Equivalent draft ddd m Δ±= 
 The signs are to be: 
 + for trim by the stern with the centre of floatation is located aft of midship 
 - for trim by the head with the centre of floatation is located aft of midship 
 - for trim by the stern with the centre of floatation is located forward of midship 
 + for trim by the head with the centre of floatation is located forward of midship 
 
 
 
 
 
 
 
 
 
(4) Displacement 
Displacement corresponding to the equivalent draft 0Δ is to be obtained in the hydrostatic 
curves. 
(5) Correction of specific gravity 
The specific gravity of seawater is assumed to be 1.025 in the calculation and a correction is 
required when the measured specific gravity of seawater shows a different value. The measured 
specific gravity of seawater being γ, the displacement when measured is given by: 
 0025.1/ Δ×=Δ γ 
 
4.2.2 Calculation of Capacity 
 
4.2.2.1 Calculation of Cargo Capacity 
Grain capacity and bale capacity (see Section 3.4.7) are to be calculated for each cargo hold. Grain 
capacity only is to be calculated for a bulk carrier. 
 
4.2.2.2 Calculation of Tank Capacity 
Capacity curves and sounding tables are to be developed by determining the tank capacity to tank 
depths from the tank bottom for each tanks such as cargo tanks, bunker tanks, ballast tanks (see Fig. 
4.7). 
 
4.2.3 Trim Calculation 
Trim calculatins are to be carried out for light condition, full load conditions and empty hold (or 
ballast) conditions in departure and arrival conditions and further displacement, draft, metacentric 
height (GM) are calculated. 
The cetre of gravity in light condition is to be determined by means of the “inclining test” described in 
the latter part of this book. Centre of gravity in other loaded conditinos is to be calculated based upon 
the data at light condition taking the loading into account. 
 
F Equivalent Draft 
Mean Draft 
Δd 
Fig. 4.6 
－ 58 － 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The procedures for trim calculations are as follows: 
(1) The position of the ship’s centre of gravity is given by the horizontal distance G⊗ from the 
midship, which is otained by devision of the summension of moment of light weight, cargo weight, 
bunker oil weight, etc around the midship by their total weight. The position of the centre of gravity 
of each element is given in negative sign when located forward of the midship and positive sign 
when located aft of the midship. 
(2) dcorr (draft in trim 0) for the displacement is to be calculated and the positions of the centre of 
buoyancy B⊗ , the centre of floation F⊗ and MTC (moment to change trim onecentimeter) at 
this draft. 
(3) Horizontal distance between the centre of buoyancy B⊗ and the centre of graviey G⊗ is 
given by BGHBG ⊗−⊗= 
Trim is to be obtained by the following formula with Δ for the displacement and HBG×Δ for 
the trim moment: 
 Trim= ( )100/ ××Δ MTCHBG (m) 
Positive sign of trim denotes trim by the stern and negative sign denotes trim by the bow. 
(4) Forward and aft drafts is given by the following formulae: 
Forward draft trim
L
FLdd corrf ×⊗+−= 2/ 
Aft draft trim
L
FLdda corr ×⊗−−= 2/ 
(5) Light weight, cargo weight, bunker weight, etc. are to be multiplied by the height of their centre of 
gravity individually to obtain moments and the total of the moments devided by the toal weight 
gives the height of gravity KG of the ship. 
(6) Transverse Metacentric Radius KM above the base line for drafts corresponding to the loading 
conditions is to be calculated and metecentric height GM is given by KGKMGM −= 
(7) Virtual centre of gravity rises with free surface in bunker tanks or fresh water tanks. The amount of 
rise is given by 
Δ
×= γiGG0 
Fig. 4.7
U
llage 
C
apacity 
Sounding 
U
llage 
Sounding
KG Curve 
Capacity 
Capacity Curve 
KG 
－ 59 － 
 Where, 
 i : moment of inertia of free surface of liquid in a tank 
 γ : specific gravity of liquid in a tank 
 Δ : displacement of a ship 
Metacentric height is to be corrected for free surface by 00 GGGMMG −= with the 0GG for 
the total of tanks having free surface. (see Section 4.3.1.2) 
 
4.2.4 Longitudinal Position of Centre of Gravity 
When a ship is afloat in even keel the longitudinal position of the centre of gravity locates in the 
vertical plane penetrating the centre of floation. In case of the ship is afloat with the aft trim da and 
the forward trim df the longitudinal position of the centre of gravity is given by the following 
procedures: 
Calculate the equivalent draft by ( )fafa ddLFddd −⊗++= 21 and the displacemnt W, the 
centre of buoyancy 1B⊗ and the moment to change trim one centimeter MTC for the draft d1 are 
obtained by reading them in Hydrostatic curves. Giving Z to the crossing point of the vertical line 
penetrating the centre of buoyancy B1and the horizontal line penetrating the centre of gravity G, the 
trim moment causing the trim fa ddt −= is described as tMTCGZW ×=× , which gives: 
 
W
tMTCGZ ⋅= 
The longitudinal position of the centre of gravity is given by the following formula: 
 
( )
W
ddMTC
BGZBG fa
−⋅−⊗=−⊗=⊗ 1 
 
4.2.5 Deadweight Scale and Capacity Plan 
The master of a ship must know beforehand what the values of draft, trim, and GM will be during 
navigation. Therefore, a deadweight scale should show displacements, deadweights, TPC and MTC 
at any given draft and the capacity plan should show the capacities of each cargo hold tanks such as 
cargo tanks and bunker tank as the data necessary for making calculations. 
 
4.2.6 Bonjean Curves 
The Bonjean curves gives sectional area of a ship at each square section. They were conceived by 
Bon-Jean, a French and so named. The curves facilitate obtaining displacement in a large trim 
condition. 
 
 
－ 60 － 
 
 
Fig. 4.8 
 
 
 
 
 
 
 
 
 
 
Fig. 4.9
 
－ 61 － 
4.3 Stability 
 
Stability or righting force of a ship is the resistance against further inclination or the condition to 
restores a ship from an inclined position when the cause of inclination has been removed. The 
inclination is in both transverse and longitudinal directions. The ship might also be damaged or intact. 
Though stability in longitudinal direction can be assessed similarly with stability in transverse 
direction, the position of longitudinal metacenter is significantly higher than the position of transverse 
metacenter (LKM≒L) and the stability in longitudinal direction does not cause a problem. Trim in 
longitudinal direction is to be evaluated, in particular, in damage condition and further in the case of 
loss of buoyancy leading the ship to sink. Stability in longitudinal direction, therefore, is dealt with as 
subdivision of spaces. 
In general, however, stability may be regarded as intact stability in the transverse direction, except for 
a special case to assess damage stability. 
Stability is one of the most important technical elements of a ship. Propulsion, structural strength, fire 
protection, accommodation are among various matters that require close technical assessment. 
Stability, however, attracted special attention in the long maritime history due to its contribution to 
serious casualties involving capsizing in case of a lack of appropriate attention. A number of standards 
therefore have been developed by IMO. 
Classification societies generally had stood on side line with regard to stability requirements for 
classification. The meeting of the Council of IACS held in June 1988 agreed that a new ship of 24 
meters or more in length may be classified only when it demonstrates that it has an appropriate intact 
stability and each classification society decided to deal with intact stability based on the IMO 
Resolution as a classification requirement from 1 January, 1991. Consequently, the Society prescribed 
newly "Part U Intact Stability" in the Rules and the Guidance, and the "Guidance for stability 
information for master" was also prescribed. Before these Rules and Guidance on stability were 
prescribed, although the stability of any Japanese flag ship was to be governed by the Japanese 
Government on the basis of the Ship's Safety Law, the Society was deeply concerned about stability 
and mental provisions on stability performance were included in the Rules (C 1.1.6). 
Accordingly, even in that time, an inclining test was carried out in the classification survey to establish 
the values for stability performance and a set of stability information in the form approved by the 
Society was required to be supplied to the master as guidance on the stability of his ship under various 
conditions. With the addition of Part U to the Rules, the provisions on stability were also revised and 
stability information have been required to be kept on board subject to the approval of the Society. 
SOLAS 1974 and ILLC 1966 also require all ships to have stability information. 
 
4.3.1 Initial Stability 
 
4.3.1.1 Statical Stability 
As shown in Fig. 4.10, when the ship heels by an angle θ only from the upright position, the 
intersection of the line of buoyancy through the center of buoyancy 'B' and the ship's center line is 
assumed to coincide with a fixed point M for small angles of θ. This fixed point M is the center of 
curvature when the ship with the locus of buoyancy center is upright, and is called the transverse 
metacenter. The radius of curvature BM is called the metacentric radius, and is determined as given 
below. 
At small angles of heel, the volumes of the exposed part and the immersed part are equal. If this 
volume is taken as v, and the center of gravity of the former as g, and the center of gravity of the latter 
as g', then the center of gravity g of a part of the displaced volume v, moves from g to g' because of the 
small angle of heel. Thus the shift of the center of gravity of the entire displacement V, that is, the shift 
－ 62 － 
of center of buoyancy becomes:BB'／／gg' and 
V
ggvBB '' ×= 
At small angles of heel, BB' can be taken as equal to BM tan θ, therefore, we get: 
θθ tan
'
tan
'
V
ggvBBBM ×== 
 
 
 
 
 
 
 
 
 
 
 
 
 
Next, the shift moment of a part of the displaced volume v x gg' is determined. Considering an 
infinitesimal length of the waterplane in the length direction as dx, and taking half its width as y, the 
shift moment of this infinitesimal part becomes: 
⎟⎠
⎞⎜⎝
⎛ ××⎟⎠
⎞⎜⎝
⎛ ⋅⋅⋅=×= 2
3
2tan
2
1' ydxyyggvMomentShiftingθ 
If this shift moment is integrated along the entire length of the ship, then v x gg' can be determined. 
∫ ∫⋅=⋅=× dxydxyggv 33 32tantan32' θθ 
However, since ∫ dxy332 is the second moment of inertia I of the water plane about the hull center 
line, 
θtan' ⋅=× Iggv 
Accordingly, 
V
IBM = 
If a vertical line GZ is dropped from the center of gravity of the ship G to the line of action of 
buoyancy, the moment of the couple that restores the ship to the upright position becomes W×GZ, 
where GZ is called the righting arm or the righting lever. The righting moment at small angles of heel 
when point M does not move far away from the metacenter is called the initial restoring force and can 
be obtained from the following equation: 
Initial restoring force = W×GZ = W･GM･sinθ 
GM is the distance between the center of gravity and the metacenter, and is called the metacentric 
height. 
This righting moment is called the statical moment or statical stability, and the quantity of work 
required to heel the ship from the equilibrium position to a specific angle is called the dynamic 
restoring force or dynamical stability. 
 
4.3.1.2 Free Surface Effect and Free Suspended Weight Effect 
Water having a free surface (surface in contact with air) is called free water. Let us consider the case 
when a ship of displacement W (volumetric displacement V, specific gravity γ, center of gravity G) 
M 
θ
γV 
G 
Z 
B' B 
W 
L' 
W' 
W 
g 
g’ L
Fig. 4.10 
－ 63 － 
and having free water of weight w (volume v, specific gravity γ', center of gravity g0) in the ship heels 
by a small angle θ. At this stage, the intersection of the line of action through the center of gravity of 
free water acting on the center of gravity g1 after inclination of the free water, and the line of action of 
weight before the heeling passing through g0, is taken as m. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Taking the moment to restore the ship to the upright position as g0g1 = g0m･sinθ, we get 
θsin' 010 ⎟⎠
⎞⎜⎝
⎛ −=×−×= mg
V
vGMVggwGZWMomentRighting
γ
γ
γ 
That is, the GM reduces only by mg
V
v
0
' ⋅
γ
γ
 because of the effect of free water (the CG of the ship 
has risen apparently). Since g0m is the same as the BM of section 4.3.1.1, 
v
img =0 i is the second moment of inertia around the longitudinal axis of the free surface. 
Accordingly, the reduction in GM due to free surface of water can be found by the equation below. 
V
img
V
v ⋅=⋅
γ
γ
γ
γ ''
0 
If there are several tanks with free water, then the effects of the free water of each tank will be added 
up, and if the virtual center of gravity is taken as G0 considering the effect of free water, then: 
∑−= ViGMMGVirtual γγ'0 
Similar to free water, the GM reduces (virtual CG increases) when the ship heels and a swinging 
suspended object exists. Similar to free water, if the weight of the suspended object is w, its center of 
gravity g0, and if the upper end of the suspension wire is taken as m, then the virtual rise in the center 
of gravity GG0 can be expressed by the equation below. 
W
wmgGG ×= 00 
This is the same as lifting the weight w to the upper end of the suspension wire. Even if an object at 
the bottom of the ship is lifted slightly by a derrick, it is equivalent to lifting the object up to the upper 
θ 
W' 
W 
B 
L' 
L 
m
g 1 g 0 
B’ 
Z 
G 
M 
γV 
W 
Fig. 4.11 
－ 64 － 
end of the suspension wire, and such an action has a very negative effect on the stability of the ship. 
 
4.3.1.3 Inclining Experiment 
The Inclining Experiment is conducted to determine the height of center of gravity of a ship in the 
light condition. The Inclining Experiment is conducted by shifting solid weights or ballast, measuring 
the angle of heel of the ship, determining the transverse metacentric height from the angle of heel and 
then the center of gravity of the ship. 
This experiment is to be performed with the ship in the completed condition as far as possible, and at 
calm locations where the ship will not be subjected to external forces. Various measurements are to be 
performed before conducting the Inclining Experiment. Firstly, the draught is to be measured at the 
forward and aft parts of the ship and amidship on the port and starboard sides, and the specific gravity 
of sea water is to be measured at the time of the experiment to determine the displacement, KB, and 
BM of the ship. Next, to determine the light condition from the experimental condition, the weight and 
center of gravity of water, oil, and bilges in tanks, water and oil in the machinery spaces, 
miscellaneous objects and not-yet loaded weights are calculated and studied. Generally, to improve the 
accuracy of the Inclining Experiment, tanks should be either empty or full so as to eliminate the free 
surface effect, and mooring lines should be slack so that tension in the lines does not affect the 
experiment. 
As shown in Fig. 4.12, a solid weight w is shifted through a distance y in the transverse direction and 
the heel of the ship θ is measured. The angle of heel is measured from the distance of shift of 
the "plumb bob" of length l installed in a hold, or by measurement using a U-tube or a clinometer. The 
height of center of gravity KG is determined from the equation below. When the experiment is 
performed by shifting ballast water, changes in the center of gravity may occur in the vertical direction, 
and these effects should be considered. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
θtan0 W
ywMG ⋅= 
l
S=θtan 
Where, W is the displacement 
 w and y are the weight of the solid object to be shifted and the distance shifted in the breadth 
direction of the ship respectively 
 θ is the change in the angle of heel due to the heeling moment w･y 
 G0M is the GM inclusive of the free surface effects of fluids 
GM is determined by deducting the free surface effect during the experiment from G0M obtained. 
Fig. 4.12 
S 
W
y 
θ l 
－ 65 － 
KG is determined from this GM by the following equation: 
KG = KB + BM – GM 
Here, KB is the height of the center of buoyancy of the ship. 
 BM is the metacentric radius. 
W, KB and BM are determined from the hydrostatic curves prepared beforehand. 
The position in the longitudinal direction of the CG of the ship in the light condition is determined by 
finding the position of the CG during the Inclining Experiment in the method described in Sec. 4.2.4, 
and by performing corrections for items other than lightweight such as water in tanks by calculation. 
 
4.3.2 Stability at Large Inclination Angle 
When the angle of heel is less than 10°, the line of action of buoyancy may be assumed to pass 
through the metacenter M, so the restoring force can be judged with the metacentric height GM, as 
described earlier. However, when the angle of heel becomes large, the line of action of buoyancy no 
longer passes through the metacenter M, and the righting level GZ can no longer be assigned by the 
value GM sinθ. 
 
4.3.2.1 Stability Curve 
As shown in Fig. 4.13, the curve expressing the restoring force W x GZ versus the angle of heel of the 
ship is called the stability curve or curve of stability. Since the displacement remains constant during 
the inclination, a similar curve is obtained by using the righting lever GZ instead of the restoring force. 
Generally, the curve of stability refers to the curve of righting levers. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The angle of heel at which the restoring force reaches the maximum restoring force after increasing 
along with the angle of heel from the upright position is called the angle of maximum stability. When 
the angle of heel increases further, the restoring force decreases and intersectsthe horizontal axis 
where the restoring force becomes zero. The angle of heel at this stage is called the angle of vanishing 
stability. If the angle of heel increases further, the restoring force becomes negative, and a moment 
tending to incline the ship further occurs and capsizes the ship. The range in which the angle of heel 
has positive values of restoring force is called the range of stability. 
At very small angles of heel, the initial stability or initial restoring force is given by GZ = GM sinθ = 
GM･θ (θ: radians), as mentioned earlier. The tangent of the angle of heel of the curve at the origin is 
given by: 
Range of Stability 
1rad 
GM
Vanishing A
ngle Heel Angle 
Stability 
Righting Lever M
axim
um
 Stability 
Fig. 4.13 
－ 66 － 
GM
d
dGZ =⎟⎠
⎞⎜⎝
⎛=
=0
tan
θθα 
As shown in Fig. 4.13, the height of the intersection of the tangent at the origin and the vertical line at 
θ = 1 radian (57.3 degrees) becomes the metacentric height. 
Generally, when GM or GZ is large, the ship may be considered to be stable, but if the angle of 
vanishing stability is small, then even a small heel may capsize the ship. If the ship heels by a large 
angle, cargo and objects within the ship may shift, sea water may enter through various topside 
openings, and a restoring force different from the one calculated, may occur. 
When the ship is navigating through waves, the shape of the waterplane differs from the corresponding 
shape when the ship is in calm water. Particularly, if the ship travels at approximately the same speed 
as the following waves in such waves, the restoring force reduces when the crest of the wave is 
amidship. If this dangerous condition continues for a long period, the ship is likely to capsize. 
Furthermore, the steerability is lost in comparatively small ships traveling through following waves or 
oblique following waves, broaching occurs and the ship cannot be controlled. In such a condition, the 
ship may ride on the waves and turn around abruptly, heel to a large angle and may capsize. 
 
4.3.2.2 Cross Curves of Stability 
When preparing the curves of stability showing the restoring force at various angles of heel at a 
constant displacement, it is difficult to establish the position of the inclined waterline retaining a 
constant displacement. Moreover, preparing another set of curves again for a different displacement is 
fairly troublesome. 
For this reason, the righting lever at a constant angle of heel is determined at various waterlines, and at 
the same time, the displacement at these waterlines are also determined for every angle of heel to 
obtain the curves shown in Fig. 4.14, which are called the cross curves of stability or simply cross 
curves. Cross curves are determined for a certain assumed height of center of gravity (CG). This 
assumed height of CG is always indicated on the figure. The assumed height of CG should be kept as 
low as possible so that the cross curves do not extend below the base line and a positive righting lever 
is obtained even at an angle of heel of 90°. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Righting Lever GZ
60° 
45° 
30° 
75° 
90° 
15° 
W1 W２ W3 W4 
Displacement 
Fig. 4.14 
Assumed Centre of Gravity = m 
Fig. 4.15 
Amount of Correction 
Za 
Z 
M 
G 
Ga 
K
θ 
－ 67 － 
 
 
 
To determine the stability curves for any arbitrary displacement, an upright line may be erected at the 
required displacement on the horizontal axis of the cross curves, and the values of GZ at each angle of 
heel may be read off immediately. 
The GZ correction for actual height of center of gravity is given by the equation below from Fig. 4.15. ( ) θsinaaa KGKGZGGZ −−= 
Here, G is the actual position of center of gravity 
 Ga is the assumed position of center of gravity. 
 
4.3.2.3 Dynamic Stability 
Dynamic stability is the work required to heel the ship from the position of equilibrium to a specific 
angle of heel. Dynamic stability is considered to be the product of the moment of the restoring couple 
W x GZ and the angle in radians over which the ship has been heeled. Accordingly, the work required 
to heel the ship from the upright to the angle θ in radians, or the dynamic stability, is given by: 
θθ θθ dGZWdGZWDst ⋅=⋅⋅= ∫∫ 00 
This is the area of the statical stability curve (W·GZ curve) until θ. (See Fig. 4.16.) 
Even if the angle of vanishing stability is large, if the dynamic stability is small, the ship may capsize 
when subjected to a small external force. 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.3.3 Stability criteria 
The criteria prescribed in Part U of the Rule are based on IMO Assembly Resolutions A167 including 
its amendment by A.206 and A562, and are outlined below 
 
4.3.3.1 General Stability criteria (IMO Assembly Resolution A.167) 
The following requirements are to be met by the stability curves shown in Figure 4.17 for a ship not 
carrying lumber cargoes on deck: 
 
(1) A1≧0.055m･rad 
(2) A2≧0.03m･rad 
(3) A1+A2≧0.09m･rad 
(4) GZ≧0.20m at the heel angle θ≧30° 
(5) θmax≧25° 
 
Fig. 4.16 
Dynamical Stability (Dst) 
Statical Stability Curve Dynamical Stability Curve 
Stability 
Heel Angle 
－ 68 － 
(6) G0M≧0.15m 
Where, 
A1 is the area under the GZ curve for the heel angel between 0o and 30 o 
A2 is the area under the GZ curve for the heel angel between 30°and θu 
θu is 40°or the downflooding angle, whichever is the smaller 
GZmax is the maximum righting lever GZ 
θmax is the heel angel where GZ is the maximum 
G0M is the initial metacentric height corrected by free surface effect. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
*If lumber cargo is carried on deck, the requirements of A206 should be met. These requirements are 
relaxed compared to A167. 
 
4.3.3.2 Stability Requirement in Severe Wind and Rolling (IMO Assembly Resolution 
A.562(14)) 
Stability curves and heel moment curves of all ships are to satisfy the following requirements in Fig. 
4.18: 
(1) θ0≦16°or 80% of the angle of deck immersion, whichever the smaller 
(2) Area b≧Area a 
where， 
θ0: angle of heel under action of steady wind 
θ1: angle of roll to windward due to wave action (Calculation formula is available) 
θ2: downflooding angle, θc or 50°, whichever is the smaller 
θc: angle of second intercept heeling lever lw2 and stability curve 
θr: rolling angle 
lw1: heeling lever of wind corresponding to θ0 (Calculation formula is available) 
lw2: heeling lever of a gust of wind (= 1.5 lw1) 
a: Area surrounded by the stability curve, lw2 and θr 
b: Area surrounded by the stability curve, lw2 and θ2 
 
 
 
 
 
 
G0M 
GZmax 
GZ 
θma θu 30° 
θ 
Fig. 4.17 
－ 69 － 
 
 
 
 
 
 
 
 
 
 
4.3.4 Damage Stability 
 
4.3.4.1 Deterministic Approach 
Ships subject to the provisions of B-60 and B-100 freeboards of Regulation 27 of the International 
Convention on Load Lines, 1966, oil tankers subject to the provisions of Chapter 3 Part 3 of the Rules 
for the Marine Pollution Prevention Systems, bulk carriers subject to the provisions of Chapter 31A 
Additional Requirements for New Bulk Carriers or Chapter 31B Additional Requirements for Existing 
Bulk Carriers of Part C of the Rules for the Survey and Construction of Steel Ships, ships carrying 
liquefied gas in bulk subject to the provisions of Part N of the Rules for the Survey and Construction 
of Steel Ships, ships carrying dangerous chemicals in bulk subject to the provisions of Part S of the 
Rules for the Survey and Construction of Steel Ships, and passenger ships subject to the provisions of 
SOLAS, are required to comply with the respective provisions for damage stability and flooding 
calculations are required to be performed. 
The method of calculation of damage stability in these Rulestakes the name of a deterministic method. 
Calculations for one-compartment and multiple-compartment flooding are performed according to the 
Rules by this method, to judge whether the stability and equilibrium condition prescribed in the rules 
exist or not in all the flooding calculations. 
Flooding calculations are calculations that help to determine whether a ship in which a compartment is 
holed and flooding by sea water occurs, will capsize or sink, or will float in the equilibrium condition 
after the flooding. 
There are two methods to determine the state of equilibrium of a ship by flooding calculations: the 
Added Weight Method and the Lost Buoyancy Method. In the Added Weight Method, the water 
flooding the ship is considered to be a part of the ship and the displacement is considered to increase 
only by the weight of this flood water. On the other hand, in the Lost Buoyancy Method, the damaged 
compartment is considered a part outside the ship and the buoyancy of the part of the compartment 
under water is considered to be lost. Thus, when no cargo is loaded in the damaged compartment, the 
center of gravity position and the displacement are the same as before the damage to the ship. The 
final state of equilibrium will be the same whatever be the method used for calculation, but the 
condition in any intermediate stage of flooding can be calculated only by the Added Weight Method. 
Various rules prescribe the extent of assumed damage of the bottom and side shell, the permeability of 
each compartment by application, the progressive stages of flooding due to damage, and the survival 
requirements in the final stage. 
For reference, an overview of the requirements of Part N "Ships Carrying Liquefied Gas in Bulk" of 
the Rules for the Survey and Construction of Steel Ships and the criteria prescribed in these are given 
below. 
(1) The extent of assumed damage is as prescribed in Table 4.1 and Table 4.2. 
(2) The permeability is as prescribed in Table 4.3. 
(3) The criteria for damage are prescribed according to the type of ship; for instance, for Type 1G 
Fig. 4.18 
θ 
θc θ2 
θ1 
b
lw2 
θ0 θr 
lw1 
a 
G 
－ 70 － 
and Type 2G ships, the following are the requirements: 
(a) Type 1G ship It is assumed that damage occurs to any kind of part in the 
longitudinal direction. 
(b) Type 2G ships of L>150 m It is assumed that damage occurs to any kind of part in the 
longitudinal direction. 
(c) Type 2G ships of L≦150 m It is assumed that damage occurs to any kind of part in the 
longitudinal direction excluding any of the bulkheads at 
either end of an machinery space in a ship. 
(4) Survival requirements are as given below. 
(a) Survival should be possible in a condition of stable equilibrium 
(b) The conditions at any stage of flooding 
• The waterline should be below the lower edge of openings likely to be flooded. 
• The maximum angle of heel due to asymmetric flooding should be below 30 degrees. 
• The residual stability at intermediate stages of flooding should not fall below the 
requirements of (c) given below. 
(c) In the final state of equilibrium after flooding; 
• The curve of righting levers should have a range of stability of at least 20 degrees from the 
equilibrium state, and should have the maximum residual righting lever of 0.1 m within the 
range of 20 degrees. The area under the stability curve in this range should be at least 0.0175 
m·rad. Moreover, unprotected openings that may cause flooding should not be immersed in 
this range. 
(d) Emergency power should be capable of being activated in the final stage of equilibrium of 
flooding. 
 
Table 4.1 Extent of side damage 
Extent Extent of damage 
Longitudinal extent l/3Lf2/3 or 14.5m, whichever is less 
Transverse extent B/5 or 11.5m, whichever is less (measured inboard from the ship's side 
at right angle to the centreline at the level of the summer load line) 
Vertical extent Unlimited upwards (measured from the upper surface of the bottom 
shell at the ship's centreline) 
 
Table 4.2 Extent of bottom damage 
Extent of damage Extent 
Within 0.3L/ from the forward 
perpendicular of the ship 
Any other part of the ship 
Longitudinal extent l/3Lf2/3 or 14.5m, whichever is less l/3Lf2/3 or 5m, whichever is less 
Transverse extent B/6 or 10m, whichever is less B/6 or 5m, whichever is less 
Vertical extent B/15 or 2m, whichever is less 
(measured from the upper surface 
of the bottom shell at the ship 
centreline) 
B/15 or 2m, whichever is less 
(measured from the upper surface 
of the bottom shell at the ship 
centreline) 
－ 71 － 
Table 4.3 Permeability 
 
 
 
 
 
 
 
 
 
 
Note * The permeability of partially filled compartments is to be consistent with the amount of liquid 
carried in the compartment 
 
4.3.4.2 Probabilistic Approach 
Requirements for compartments of cargo ships and damage stability were newly established in Part 
B-1 of the amendments to SOLAS 1974, which were adopted in May 1990 and which entered into 
force on 1 February 1992. These provisions were applicable initially to cargo ships of compartment 
length Ls greater than 100 m engaged in international voyages other than ships mentioned below 
(hereafter referred to as "dry cargo ships"), but with the amendments to SOLAS, they also became 
applicable to dry cargo ships of gross tonnage greater than 500 tons and Ls greater than 80 m 
constructed after 1 July 1998, engaged in international voyages. 
(i) Oil tankers subject to the provisions of Part 3, Chapter 3 of the Rules for the Marine 
Pollution Prevention Systems 
(ii) Ships carrying liquefied gases in bulk subject to the provisions of Part N of the Rules 
for the Survey and Construction of Steel Ships 
(iii) Ships carrying dangerous chemicals in bulk subject to the provisions of Part S of the 
Rules for the Survey and Construction of Steel Ships 
(iv) Bulk carriers subject to the provisions of B-60 and B-100 freeboards of Regulation 27 
of the International Convention on Load Lines, 1966 
(v) Offshore supply ships subject to the provisions of IMO Resolution A.469 (VII) 
(vi) Special purpose ships subject to the provisions of the IMO Resolution A.534 (13) 
Provisions for "subdivision" were newly established in Part C, Chapter 4 of the Rules for the Survey 
and Construction of Steel Ships in 1991 by the Society in response to the establishment of the new 
SOLAS requirements. 
The method of performing flooding calculations according to these Rules is called the probabilistic 
method, and an overview of the method of evaluation and flooding calculations is given below. 
(1) Based on the statistics of marine casualties of 296 ships, and the results of statistical analysis 
of the length and position of holed openings, the probability Pi (probability of flooding a 
certain compartment) of occurrence of a holed opening of a certain size and at a certain 
position in the hull is determined. 
(2) For the summer load line condition (deepest subdivision load line) and 60% partially loaded 
draught condition, flooding calculations are performed for a compartment considered to be 
damaged, and the survival probability Si is determined. 
Survival probability is the probability of survival when a ship in waves neither capsizes nor 
sinks because of flooding of certain compartments, and as shown in the equation below, it is 
evaluated using 0≦Si≦1. 
))((5.0 max RangeGZCSi = 
Spaces Permeability 
Store 0.60 
Accommodation 0.95 
machinery 0.85 
Voids 0.95 
Consumable liquids 0 ~ 0. 95* 
Other liquids 0 ~ 0. 95* 
－ 72 － 
1=C θe≦25º 
0=C θe＞30º 
5/)30( eC θ−= 25º＜θe≦30º 
Here, θe is the final equilibrium angle of heel after flooding. Range is the range of positive 
stability from θe to the angle of flooding. If the angle exceeds 20°, it is taken as 20°. GZmaxis 
the maximum righting lever in this range. If its value exceeds 0.1 m, it is taken as 0.1 m. The 
flooding angle is the angle at which an opening that cannot maintain weathertightness becomes 
submerged. 
(3) The permeability of each compartment during flooding calculations is given in Table 4.4 
according to the application of the compartment. 
 
Table 4.4 Permeability 
Compartment Permeability 
Store 0.60 
Accommodation space 0.95 
machinery space 0.85 
Cofferdam 0.95 
Cargo space 0.70 
Liquid Loading space 0 or 0.95 
 
 
(4) By integrating the product of the flooding probability Pi and survival probability Si of a 
compartment over the entire length of the ship, the survival probability of the ship can be 
obtained. The survival probability is calculated for the summer load line condition and the 
60% partially loaded draught, and the average for these two conditions is called the Attained 
Subdivision Index. 
(5) The Attained Subdivision Index should be greater than the Required Subdivision Index, as 
required by the Rules. The Required Subdivision Index R, in case Ls is 100 m or above, is 
given by the equation below. 
R = (0.002 + 0.0009Ls)1/3 
This Required Subdivision Index is based on the calculation examples of Attained Subdivision 
Index submitted to the Maritime Safety Committee (MSC) of the IMO by various countries. 
Taking it as the function of length for subdivision of the ship, the requirements prescribe the 
central value of this Index as the target. 
(6) If complying with the Rules, the limiting G0M can be prepared by joining G0M (= TKM - 
KG0) for two loading conditions (summer load line draught dF and 60% partial loaded draught 
dp) by a straight line on the allowable G0M curves according to the intact stability requirements, 
as shown in Fig. 4.19. 
The G0M satisfying the damage stability requirements will be displayed to the right side of the hatched 
part, and while the ship is in service, it should be loaded such that G0M falls to the right side of the 
hatched part. 
 
 
 
 
 
－ 73 － 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.3.5 Grain Stability 
When grain is loaded in the cargo holds, the surface of the grain inclines because of the motion of the 
ship. For this reason, the position of center of gravity of the grain moves away from the center line of 
the ship, and creates a moment to incline the ship. If this inclining moment becomes large, it may lead 
to the risk of capsizing the ship. To prevent this risk, the International Grain Code referred to in PART 
C, Chapter VI of SOLAS, has prescribed a method for calculating the heeling moment due to grain 
and also prescribed stability standards. For ships carrying grain in bulk, the heeling moment due to 
grain for each cargo hold should be calculated according to the Rules. These moments should be 
summed up to determine the heeling moment of the entire ship, stability calculations performed, and 
the results of the calculations confirmed to ensure that they satisfy the standards. 
Here, grains refer to wheat, corn, oats, rye, barley, rice, beans, seeds and their processed products, with 
their physical characteristics resembling the physical characteristics of the grains before processing. 
Before loading the grain, it should be trimmed such that the spaces below the decks and hatch covers 
are filled as much as possible. Even if the space is not fully loaded with grain, the surface of the grain 
must be made level. However, according to the 1991 amendments (May) of SOLAS (entered into 
force on 1 January 1994), if the grain is untrimmed and the heeling moment is calculated with the 
grain loaded in the cargo holds, and if the stability requirements are satisfied, then the ship can be 
exempted from trimming. In the untrimmed condition, the void up to the hull structure above the 
surface of the grain becomes larger compared to the void in the trimmed condition, and the heeling 
moment also increases. However, this untrimmed condition is not a rule requirement; if approval is 
obtained, grain can be loaded without trimming. 
In the calculations of heeling moment in the trimmed condition, the surface of the grain is assumed to 
incline at an angle of 15° when grain is fully loaded in the compartment, and assumed to incline at an 
angle of 25° when it is not fully loaded in the compartment. In the calculations of heeling moment 
when grain is not trimmed, the surface of the grain is assumed to incline by 15° in case of the hatched 
part generally, and to incline by 25° for all other parts. 
The intact stability requirements of grain in bulk are as below. 
(1) The angle of heel of a ship due to shift of grain in bulk should not exceed 12°. 
(2) In the statical stability curve, the area of the part enclosed by the curve of heeling levers and 
the curve of righting levers until the difference in the ordinates of these two curves becomes 
the maximum angle, that is 40°, or the angle of flooding, whichever angle is smaller, should be 
at least 0.075 m-rad for all loading conditions. 
dF 
dP 
Limit G0MCurve（Damage） 
Allowable G0MCurve（Intact）
Fig. 4.19 
G0M 
d 
－ 74 － 
(3) The initial metacentric height (G0M) after considering the free surface effects of liquids in the 
tanks should be at least 0.30 m. 
λ0，λ40，θ' in Fig. 4.20 are as follows: 
）（）（
）（
tntDisplacemetmFactorStorage
mngMomentGrainHeeli
×= /3
4
0λ 
 λ40 = 0.8×λ0 
θ' is the angle at which the difference in the ordinate of the two curves, namely the curve of 
heeling levers and the curve of righting levers becomes maximum, that is 40° or the flooding 
angle, whichever is smaller. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.4 Power Requirements for Propelling Machinery 
 
Although estimations of the necessary power of a main engine to propel a ship have long been studied 
as one of the main field of naval architecture, no simple, reliable method has been published. 
Methods of estimating required power of propelling machinery are outlined below. 
 
4.4.1 Hull resistance and effective horsepower 
 
4.4.1.1 Kinds of resistance 
When a ship moves through water, it encounters resistance from both water and air. 
To maintain the required sea speed, the ship should be provided with main eng-ine(s) of sufficient 
power to overcome such resistance. 
The horsepower of the main engine is determined in such a way that the horsepower necessary to 
overcome resistance the ship encounters when it runs straight ahead at a given speed in still water with 
no wind and waves is calculated first and then some margin of the power is added, so-called sea 
margin, so that the sea speed can be maintained even when resistance increases due to the effects of 
wind and waves, bottom fouling, etc. 
Because air resistance is very small compared with that of water, it is usually included in the sea 
margin to save calculation time for ships other than those with a high speed having a large 
superstructure. 
The resistance a ship encounters when it goes straight ahead in still water can be divided into 
λ0 
40°
λ40λθ'
R
ighting Lever 
Righting Lever Curve 
Residual Dynamical Stability 
Heeling lever curve due to 
grain cargo shift in 
Heel Angle 
Fig. 4.20 
Heel due to grain shift 
θ' 
－ 75 － 
wave-making resistance and viscous resistance. 
Owing to efforts to develop ship forms with less wave- making resistance in recent years, ships built 
recently have very small wave-making resistance at about 5% of total resistance for low-speed full 
ships and about 20% for high-speed ships, and most of the resistance a ship encounters is viscous 
resistance. 
 
4.4.1.2 Viscous Resistance 
 (1) Component of viscous resistance 
In calculating the viscous resistance of a ship advancing straight ahead, the frictional resistance 
of a ship was, formerly,regarded to be equal to that of a flat plate having the same length and 
surface as those of the submerged part of the ship. At present, as a ship is a three-dimensional 
body, its frictional resistance is thought to be larger than that of a flat plate and the increment to 
be taken into consideration is represented by a ratio of the resistance of a ship to that of a flat 
plate, K, which is referred to as a form factor. 
In addition, the surface of the hull cannot be free of unevenness, which increases frictional 
resistance to some extent compared to a completely flat plate. This increment of resistance is 
expressed as a roughness allowance. Then, viscous resistance is given by the following formula: 
Rv = (1+K)RF + ΔRF 
where: 
RF = frictional resistance of a flat plate 
ΔRF = roughness allowance 
(2) Frictional resistance coefficient 
Frictional resistance is considered a function of the density of water, the wet surface of the hull 
and the speed of the ship, and is given by the following formula: 
SVCR FF
2
2
1 ρ= 
where: 
CF = frictional resistance coefficient 
ρ = density of water 
V = speed of ship 
S = wetted surface 
The frictional resistance coefficient is nondimensional and a function of Reynolds 
number "Rn ". Rn is defined as follows: 
vVLRn /= 
where: 
V= speed of ship (m/sec) 
L = length of ship (m) 
v = coefficient of kinematic viscosity (m /sec) 
Kinematic viscosity is dependent upon temperature and normally one at 150C is used for 
calculation. Kinematic viscosity of sea water is 1.1883×10-6m2/sec. 
Frictional resistance coefficient CF can be obtained by various formulae available. The one of 
Schönherr that is widely used and another of ITTC (International Test Tank Conference) 1957 are 
shown below: 
Schönherr’s formula：Approximately CF = 0.4635(log10Rn)-2.6 
ITTC1957：CF = 0.075(log10Rn-2)-2 
As given in the formulae above greater the Rn the smaller is the frictional resistance coefficient 
CF. It means that for a given speed the greater is the ship length the smaller is the CF and for a 
given ship length the greater is the ship speed the smaller is the CF.. 
－ 76 － 
(3) Form factor 
A number of methods are available to obtain the value of the form factor of a ship through tank 
tests. One of them is to measure the resistance, R, of the model ship in a low-speed area with little 
or no wave-making resistance and consider it equal to viscous resistance. 
Viscous resistance coefficient C is calculared from the measured resistance R by following 
formula: 
 
 
 
As the wave-making resistance is zero, the following formula holds: ( ) FCKC += 1 
In this case roughness allowance FRΔ is assumed to be zero. 
When the frictional resistance coefficient of a flat plate, CF , is obtained, K can be given by the 
following- formula: 
1−=
FC
CK 
For low-speed full ships, the value of K is 0.25 ~ 0.5. 
(4) Roughness allowance coefficient 
The increment of frictional resistance due to surface roughness, FRΔ , is given by the following 
formula in a similar manner to frictional resistance. 
SVCR FF
2
2
1 ρΔ=Δ 
where FCΔ , is the roughness allowance coefficient. 
FCΔ , cannot be determined by tank tests or calculations. In practice, the total resistance 
obtained by analyzing the results of a ship trial are compared with the total resistance o f its 
model ship and FCΔ is determined from the difference. Accordingly, FCΔ includes, by nature, 
various measuring errors made at the time of sea trial, in addition to the increased resistance due 
to the roughness of the hull. 
 
4.4.1.3 Wave-making resistance 
If the wave-making resistance is Rw, then Rw, is given by the following formula: 
SVCR ww
2
2
1 ρ= 
where 
Cw, is the wave-making resistance coefficient. 
ρ is density of water (kg·sec2/m4) and 104.6 for sea water 
V is speed of ship (m/sec) 
S is wetted surface (m2) 
Wave-making resistance is a function of the Froude number, Fn, and the ship form, and cannot be 
expressed by such a simple formula as that for the frictional resistance coefficient or the form factor. 
Fn, is defined as follows: 
Lg
VFn = 
where, 
V is speed of ship (m/sec) 
L is length of ship (m) 
" g " is gravitational acceleration (m/sec2) 
2
2
1 SV
RC w
ρ
=
－ 77 － 
 
Conveniently, ships with similar hull forms have the same wave-making resistance when Fn is the 
same. This feature is utilized to determine Cw, of an actual ship on the basis of the results of tank tests. 
A model ship with a similar hull form to the actual ship is towed in a towing tank and the total 
resistance is measured. Because Rw can be obtained by subtracting the calculated value of viscous 
resistance from the measured value, Cw, is given by the following formula: 
SV
RC ww
2
2
1 ρ
= 
An example of relationship between Cw, and Fn is shown in Fig. 4.21. As can be seen from the figure, 
when the value of Fn increases, the value of Cw increases gradually with peaks and troughs, and 
reaches the maximum value at about 5.0=nF . Beyond that, the value of Cw, decreases gradually with 
an increase of the value of Fn. The ship is designed to operate at points in the troughs of Cw,. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
When a ship advances straight ahead at a given speed in still water, divergent waves and transverse 
waves are generated around the ship as shown in Fig. 4.22. Any difference in ship speed causes a 
change in the wave length of the transverse waves. Regarding interference between two transverse 
waves generated at bow and stern, it can be said that when both waves interact synergistically, the 
curve Cw, produces a peak and when the opposite occurs, the curve Cw, produces trough. 
Fig. 4.21 
－ 78 － 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
When the underwater geometry of the bow is rounded, the pressure of the water flow decreases, 
resulting in a lowering of the bow transverse wave. The bulbous bow is an idea to reduce the 
wave-making resistance by utilizing this phenomenon. 
 
4.4.1.4 Total resistance 
As described above, the total resistance is the sum of Rv and Rw, as given by the following formula: 
wFFwv RRRKRRR +Δ++=+= )1( 
Because the calculation formula described above takes the effects of viscous resistance due to the 
three-dimensional plane of hull surface into consideration when converting the results of tank tests 
into the values of actual ships, it is called a three-dimensional extrapolation. 
In contrast, the method of calculating the total resistance by classifying it into the frictional resistance 
of an equivalent flat plate and the other resistance is called a two-dimensional extrapolation. 
In the two-dimensional extrapolation, the resistance obtained by subtracting frictional resistance from 
total resistance is called residual resistance, and the total resistance is expressed as follows: 
RFF RRRR +Δ+= 
Residual resistance coefficient CR is considered to be equal for an actual ship and a model ship when 
the Froude number is equal. 
With this method, some viscous resistance due to the effects of hull form is included in the residual 
resistance, in addition to the wave-making resistance. A disadvantage of this method is that, if the 
residual resistance coefficient CR of the model ship is applied to the actual ship without correction, the 
value of resistance obtained can be larger than the actual value. This tendency is significant, especially 
in ships with a large coefficient of fineness. In high-speed ships with a small coefficient of fineness, 
however, the two-dimensional extrapolation is appreciably adopted. 
Fig. 4.23 shows the inter-relationship in resistance coefficients of model and full scale ships. 
 
 
 
Fig. 4.22
－ 79 －4.4.1.5 Effective horsepower ; (EHP) 
If resistance is expressed as R (kg) and the speed of the ship is V (m/sec), then the work-load 
necessary to advance the ship at this speed is R • V. Because 1PS = 75 kg • m, this is expressed as 
75
VR×
 in horsepower (PS). This horsepower is referred to as effective horsepower (EHP) 
 
4.4.2 Propulsive performance 
 
4.4.2.1 Delivered horsepower ; (DHP) 
The output of the main engine is delivered to the propeller to generate thrust, causing the ship to 
advance. The horsepower generated by the main engine, however, suffers a loss due to friction at 
bearings etc. The net horsepower obtained by subtracting these losses from the engine output and 
actually delivered to the propeller is referred to as delivered horsepower. The ratio of the delivered 
horsepower to horsepower of the main engine is called transmission efficiency, Tη 
 
4.4.2.2 Propulsive efficiency 
(1) Definition of propulsive efficiency 
If the ratio of effective horsepower to delivered horsepower is expressed by η, then 
DHP
EHP=η 
"η" is propulsive efficiency and a ship with a larger η has a higher propulsive performance. 
In some cases 
)(SHPBHP
EHP
is called propulsive coefficient and 
DHP
EHP
 is called quasi-propulsive 
coefficient. 
The horsepower transmitted to the shaft coupling is referred to as break horsepower, BHP and that 
transmitted to the intermediate shaft is referred to as shaft horsepower, SHP. 
In calculating" horsepower, procedures usually used are that the effective horsepower at each ship 
Fig. 4.23 
Model ship total 
resistance 
coefficient curve 
Total resistance of 
a ship 
Frictional resistance 
of a plate 
LV
－ 80 － 
speed is determined first and then the propulsive coefficient, η, is obtained by various methods. The 
delivered horsepower is calculated by η
EHP and finally, the required horsepower of the main engine 
for each speed can be obtained from
T
DHP
η . 
(2) Component of effective coefficient 
Because a ship travels forward due to the thrust produced by a propeller, propulsive efficiency is apt to 
be simply deemed the ratio of horsepower generated by the propeller independently of the ship to the 
horsepower delivered to the propeller, i.e. propeller efficiency in open water. In practice, however, 
when the propeller is installed at the stern of the ship, a complicated interaction occurs between the 
propeller and ship hull, and, as a result, the propulsive efficiency η is given as : 
η＝η0×α 
where: 
η0 = propeller efficiency in open water 
α= coefficient depending on interaction between ship hull and propeller 
 
(3) Coefficients depending on propeller and ship hull 
Among a series of coefficients showing the interaction between hull and propeller explained above, 
thrust deduction fraction, t, wake fraction, w, and relative rotative efficiency,ηH, are called 
self-propulsion factors. 
 
(a) Thrust deduction fraction 
To advance at a given speed, overcoming the resistance of water, a ship must be given 
propelling power in forward direction. This power is produced by the revolutions of propeller 
and is called thrust (kg). If the advance speed of propeller through water is Va (m/sec), then the 
work done by thrust T of the propeller is 
T • Va , i.e. 
75
aVT ⋅ expressed in horsepower. 
This horsepower is called horsepower developed by stern propeller thrust THP. The advance 
speed of the propeller does not agree with that of the ship. 
If the thrust necessary to produce a given speed is T (kg) and the resistance is R (kg), then the 
necessary thrust is larger than R to some extent. The reason is that when the propeller rotates at 
the stern of the ship, the surrounding water is pressed aftward resulting in an increase of water 
speed which produces a low-pressure portion at the stern and, as a result, a force is exerted on 
the hull to drag it aftward, which gives the apparent resistance has increased. This increase in 
resistance is T—R (kg"), which can be taken, from the opposite viewpoint, as a reduction in 
effective thrust by T-R from T. Therefore, the ratio of decrement to thrust, T, is called the thrust 
deduction fraction and is given as: 
ｔ＝
T
RT −
 
"T" can be calculated by the following formula, when ”t" is determined. 
 T ＝
t
R
−1 
After measuring the resistance of the model ship and the thrust produced by the model 
propeller, "t" can be obtained from the above formula. This fraction is affected little by scale and 
can be applied to the actual ship without correction. Several approximate expressions are 
available for this fraction. 
－ 81 － 
(b) Wake fraction 
When a ship advances, water adjacent to the hull surface flows forward. This flow is called the 
wake. 
An important factor in the propulsive performance of a ship is the amount of wake at the 
location of the propeller. Assuming a wake at the propeller of a ship Vw advancing at a speed of 
V, the wake fraction, w, is defined as follows: 
V
Vw W= 
Forward speed through the water of the propeller can be expressed by the following formula: 
VA = V－VW 
Therefore, the wake fraction is given as: 
V
VV
V
Vw AW −== 
From this, the following formula can be derived: 
VA =（1－w）V 
Therefore, the advance speed of the propeller can be estimated when the wake fraction is 
obtained. 
Several approximate expressions for the wake fraction are available. This fraction is largely 
affected by scale and is smaller with larger ships. 
(c) Relative rotative efficiency 
Because a propeller installed at the stern of a ship rotates in a complex wake, its efficiency is 
different from that of a propeller operating- solely in a uniform flow. The efficiency of the 
propeller installed at the stern is called propeller efficiency behind ship,ηB, and is expressed as 
follows: 
DHP
THP
B =η 
The ratio ofηB toη0 is called relative rotative efficiency,ηR. 
0η
ηη BR = 
(d) Hull efficiency 
When a ship is driven forward by a propeller, the thrust required to drive it becomes larger than 
the resistance, which is a disadvantage. On the other hand, the advance speed through the water 
of the propeller, Va , is smaller than the advance speed of the ship due to the wake, which is an 
advantage. The advantage and the disadvantage for propulsion caused by the interaction 
between hull and propeller can be predicted using the following formula: 
A
H VT
VR
THP
EHP
⋅
⋅==η 
whereηH is hull efficiency and is derived from formulae as follows with R=(1-t)T, VA=(1-W)V: 
w
t
H −
−=
1
1η 
 
(4) Relationship between propulsive efficiency, propeller efficiency in open water and 
self-propulsion factor 
Propulsive efficiency can be transformed as follows: 
DHP
THP
THP
EHP
DHP
EHP ×==η 
－ 82 － 
THP
EHP
 is hull efficiency, which is expressed by 
w
t
H −
−=
1
1η . 
On the other hand; 
DHP
THP
 is propeller efficiency behind ship, which is expressed by ηB=η0×ηR 
Then propulsive efficiency is expressed by the following formula 
RHRw
t
DHP
THP
THP
EHP ηηηηηη ⋅⋅=⋅−
−=×= 001
1
 
In propulsive efficiency,ηH×ηR; shows the value of the interaction between hull and propeller. 
As can be seen from formula above, for the improvement of propulsive efficiency, it is 
important, in addition to improve propeller efficiency, to improve various coefficients 
indicating the interaction between hull and propeller, which are mainly dependent on hull 
form and location of the propeller. 
 
4.4.2.3 Method of calculating horsepower 
(1) Method using admiralty coefficient 
Admiralty Coefficient, Cadm, is defined by the following formula: 
PS
VCadm
33/2 ⋅Δ= 
where: 
Δ= displacement (t) 
V = ship speed (knot) 
PS = required horsepower at speed V 
The results of trials and tanktests for ships with similar hull forms arranged by plotting them on a 
drawing, taking speed-length ratio, V／ Ｌ on the abscissa and Cadm on the ordinate, can be 
used to develop the required horsepower using the following formula with a given displacement,Δ, 
and ship speed, V, of the ship under consideration on the assumption that Cadm is identical to ships 
with compatible speed-length ratios. 
admC
PS
33/2 V⋅Δ= 
(2) Method using tank test 
The most accurate method of predicting- the horsepower of a ship is to utilize the results of tank 
tests on a model of the ship. This method, however, has a disadvantage in that it can only confirm 
the initial design, because there is usually no time to amend the design of a hull form on the basis of 
the results. Tank tests comprise the following three: 
(a) Resistance test 
Tow a model of the ship to determine the relationship between speed and resistance. The 
wave-making resistance and the form factor can be obtained by analyzing the results of the 
tests. 
(b) Self-propulsion test 
Install a propeller in the model ship and measure thrust and revolutions of the propeller and 
torque of propeller shaft. 
Because the frictional resistance coefficient of a ship is smaller than that of the model ship, a 
force corresponding to the difference in frictional resistance between the ship and the model 
must be added to the model from outside during the self-propulsion test. 
The thrust deduction fraction can be determined from the thrust measured during the 
self—propulsion test and the resistance of the model measured during the resistance test. Wake 
－ 83 － 
fraction and relative rotative efficiency can be obtained from a comparison of propeller 
efficiency in open water with that behind the ship measured during the self-propulsion test. 
(c) Propeller test in open water 
A scale propeller of the ship under design without hull model is run at a constant rotational 
speed and variable advance speed or with a constant advance speed and variable revolutions, 
and the propeller efficiency in open water is obtained from the measured values of propeller 
thrust and torque. 
The power curves of the ship can be developed directly from the results of the self-propulsion 
test. In Japan, however, it is common practice to determine wave-making resistance coefficient, 
form factor, self-propulsion factor, and propeller efficiency in open water of the model ship by 
the tank test. These data are converted into those for the actual ship to develop the predicted 
power curves. In this case, however, the roughness correction and the scale effect of the wake 
cannot be obtained by the tank test. To obtain these factors, therefore, it is required comparative 
data with the ship, because power curves show different results depending on the data, even if 
the same data of tank test is used. 
(3) Method using nomogram 
A method to determine required horsepower is available using a nomogram by estimating 
effective horsepower and propulsive efficiency of the ship designed. The nomogram published 
now is a diagram developed from the results of tank tests conducted by varying the ratios of 
principal dimensions, Cb etc., systematically, most of which are determined by 
two-dimensional extrapolation. 
 
4.4.2.4 Factors affecting" hull resistance 
(1) Hull dimensions 
With an increase of L, total resistance increases and frictional resistance coefficient and 
wave-making resistance coefficient decrease. With an increase in B, total resistance increases. 
(2) Hull form 
Where Cb is small, total resistance decreases. As for the lines of the frame, total resistance is 
smaller for V-shape than U-shape. 
(3) Appendage 
Stern frame, rudder, and bilge keel are causes of increased resistance. 
(4) Wave 
Navigation in waves causes increased resistance. 
(5) Depth of water 
Shallow water causes increased resistance, and is referred to as a shallow water effect. 
(6) Bottom fouling" 
Bottom fouling causes increased resistance. 
 
 
4.5 Hull Strength 
 
4.5.1 Waves 
Forces generating sea waves include wind, earthquake, solar and lunar gravitation, etc., of which 
wind generates waves having the closest relation with the strength of a ship. Waves built up by wind 
force are called wind waves and waves advancing from the generated area are called swell. 
 
4.5.1.1 Basic characteristic of regular waves 
The representation of waves mentioned above is for regular waves and conserved waves on the 
－ 84 － 
basis of classic hydrodynamics, as one cannot change the magnitudes of waves with place or have 
them build up over time. 
(1) Sine wave 
A sine wave is the most fundamental regular wave. Its length, height, amplitude, and period 
are shown in Fig. 4.24. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
They have the following relationship for deep sea waves, i.e. the water depth is greater than the wave 
length: 
cf
T λω
π === 21 
T
f ππω 22 == 
2
2
Tgπλ = 
π
λ
2
gc = 
g
T πλ2= 
where: 
T = wave period (sec) 
f = wave frequency 
ω= circular frequency 
λ = wave length (m) 
c = wave velocity (m/sec) 
g = gravitational acceleration (m/sec2) 
This wave has a smaller height compared with its length, and the particles of water in the wave have 
an orbital motion and do not advance with the wave. 
(2) Stokes wave 
If disturbances of the water surface become large, it is difficult to explain a wave as a sine wave from 
the viewpoint of hydrodynamics. Stokes rigorously analyzed this problem under a given boundary 
condition, and the result is called the Stokes wave. 
The wave form has a slightly sharper crest and flatter trough than that of the sine wave as shown in 
Fig. 4.25. Although this wave is fundamentally the same as the sine wave, the particles of water do not 
Wave 
Amplitude 
Wave 
Height
Wave Length λ
Wave Velocity c
Distance 
Time 
Wave Period T 
Fig. 4.24 
－ 85 － 
have a perfect circular movement, but shift gradually toward the advance direction. This results in the 
real transportation of the wave, which is significant in the generation of waves due to wind. 
 
 
 
 
 
 
 
(3) Trochoidal wave 
Stokes wave is difficult to use in practice due to its complicated formula. Gerstner also tried to 
represent trochoidal waves hydrodynamically. This representation completely satisfies the boundary 
condition from a theoretical point of view, except that it is not an irrotational motion, which is a 
prerequisite for an ideal fluid. The stokes wave will be a trochoidal wave if the restriction of irrotation 
is removed. However, the trochoidal wave is usually used in practice, because actual fluid can create a 
rotational movement and the trochoidal wave is also very convenient for developing form, structure, 
pressure distribution, etc. of the wave. 
A trochoid is an orbit generated by a point on the radius, r, of a circle with a radius R, as the circle rolls 
along a horizontal straight line (Fig. 4.26) and expressed by the following formulae 
θθ sinrRx += . 
θcosrRz += 
 
 
 
 
 
 
 
 
 
 
 
 
4.5.1.2 Wind and waves 
Waves are generated when wind blows over the water surface. The higher the wind velocity, the longer 
is the distance over which the wind blows (fetch), the larger is the height of the wave generated, and 
the longer is the wave period. For constant wind velocity and constant fetch, the time the wind 
continues to blow becomes prolonged, the wave height increases, waves develop, and a steady state is 
finally reached. Waves develop because their energy is replenished by wind. 
The wind transfers energy to waves in two ways. Firstly, the transfer is due to normal forces. If wind at 
a speed of about 3 m/sec blows on a still water surface, it cause the surface to develop a 
concave-convex shape. When more wind blows onthis kind of surface, the weather side of the wave 
crest is depressed, while low pressure occurs on the leeward side; thus, the wind transfers energy to 
waves through its horizontal component. The other method of transfer is by the tangential force due to 
friction on the wave surface. 
If the particles of the wave move in a perfect circle and their positions are in variable, as in a sine 
wave, then the frictional force does not perform work in one cycle. If this is a Stokes wave, the 
particles of water are transported, and thus, the energy is transferred by friction. 
 
Fig. 4.25 
Sine Wave
Stokes Wave 
Fig. 4.26 
－ 86 － 
Together with the development of waves, energy is dissipated because of the internal viscosity and by 
the breaking of the waves. When the energy replenished by the wind and the energy dissipated balance 
each other, the wave development stops and a steady state is reached. 
 
4.5.1.3 Ocean waves 
It was difficult to find a rule covering the sea surface, because ocean waves are very irregular. During 
World War II, however, studies on wave prediction were actively promoted, resulting in successfully 
analyzing the irregularity of waves using statistical procedures. 
 
(1) Significant wave height 
Because ocean waves are very irregular, it should be represented by a mean value in some way. 
Sverdrup—Munk defined the significant wave height as the mean value. This is the mean height of 
waves, observed for a given period, arranged in order of height, and taken one-third from the highest. 
The reason that this mean value is considered in representing the wave is that, when waves are 
observed, higher waves are likely to be attractive and, consequently, those having the heights near the 
significant wave height are deemed average. The feature of the significant wave height that is almost 
constant irrespective of the observation period is another reason. Longuet-Higgins observed irregular 
waves and discovered that the following relationship holds between the results of observations and the 
significant wave height: 
Significant wave height : 1.00 
Arithmetic mean : 0.64 
Mean for the range of one tenth from highest : 1.29 
The significant wave is different from the classic wave. As mentioned before, the classic wave is 
stationary, has no energy change, and the wave height does not change with place and time. On the 
other hand, the length of the significant wave changes with location, that is, the wave length is shorter 
on windward and longer on leeward sides, while it becomes stable and wave length does not change 
with time when a light wind blows for a long time over a lake with a limited area. When a strong wind 
blows for a short time, the wave length is uniform but increases with time except the extreme 
windward region. 
 
(2) Spectrum of irregular waves 
Observation of the cross section of waves that are called "wind waves" excepting swells, at one fixed 
point and in a single direction on the sea surface shows that there is no regularity in the wavelength 
and the wave height. Looking at a plane section, there are few crest widths that are perpendicular to 
the direction of the wave and that continue for a long period; generally, the crests are small. Such an 
irregular sea surface can be expressed by superposing an extremely large number of regular 
component waves. Each of these component waves have their own amplitude, direction of travel, 
frequency and phase lag, and they obey the Small Amplitude Wave Theory. Energy spectra assigned to 
energy distributions (proportional to the square of the wave amplitude) of components in small spaces 
of wave frequencies (ω = 2π/T) can be considered. Short-crested irregular waves can be expressed by 
superposing each spectrum considering that the energy spectra mentioned above come from various 
directions in the range of about 180 degrees on the weather side. 
That is, if the spectrum of long-crested irregular waves is expressed by S(ω) then the spectrum for 
short-crested irregular waves is: 
 θωθω nkSS cos)(),( = 
22
πθπ ≤≤− 0 at all other angles 
 k and n are constants. 
The spectrum of long-crested irregular waves is generally given by the equations below, which modify 
－ 87 － 
the Pierson Moskowitz spectrum proposed by Pierson and Moskowitz based on a large number of 
wave data observed at fixed points in the Pacific Ocean, and by using the ISSC spectrum shown in Fig. 
4.27. It is extremely common to use n=2 and k=2/π in the equations mentioned above, to obtain the 
spectrum for short-crested irregular waves. 
 ⎭⎬
⎫
⎩⎨
⎧−= −− 452 )
2
(44.0exp)
2
(
2
11.0)( VVvS TTTHS π
ω
π
ω
πω （m
2･sec） 
 Hs is the significant wave height (m) 
 Tv is the mean wave period (sec) 
 ω is the frequency (1/sec). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(3) Characteristic of the spectrum 
Apart from the above, Longuet-Higgins verified the following relationship between spectral 
distribution and irregularity of waves. When waves are observed for some period and are arranged in 
order of wave heights, a curve is drawn as shown in Fig. 4.28. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 4.27 
Fig. 4.28
－ 88 － 
 
According to Longuet-Higgins, when the frequency range is not large, calculation 
ωω d2
0
)][S(=E ∫∞ 
shows that the following relationship holds irrespective of the shape of a spectral distribution. E is 
cumulative energy density. 
(a) Relationship between mean values 
Amplitude of most frequent wave 0.707 E 
Amplitude of mean wave h 0.886 E 
Amplitude of significant most frequent wave h 1/3 1.416 E 
Amplitude of significant most frequent wave h 1/10 1.800 E 
(b) Distribution of wave amplitude 
10% for all the following ranges: 
0 E ～ 0.32 E 
0.32 E ～ 0.47 E 
0.47 E ～ 0.60 E 
0.60 E ～ 0.71 E 
0.71 E ～ 0.83 E 
0.83 E ～ 0.96 E 
0.96 E ～ 1.10 E 
1.10 E ～ 1.27 E 
1.27 E ～ 1.52 E 
1.52 E or above 
(c) Number of waves passed and amplitude of maximum wave 
Amplitude of maximum wave among 20 waves 1.87 E 
Amplitude of maximum wave among- 50 waves 2.12 E 
Amplitude of maximum wave among 100 waves2.28 E 
Amplitude of maximum wave among- 200 waves 2.43 E 
Amplitude of maximum wave among 500 waves2.61 E 
Amplitude of maximum wave among 1,000 waves 2.74 E 
Amplitude of maximum wave among 10,000 waves 3.13 E 
 
4.5.2 Response of hull to irregular waves 
Generally, the short-term response in irregular waves can be reproduced by linear superposition of 
responses in regular waves. Therefore, the hull response in irregular waves during short-term sea 
conditions can be predicted using short-term predictions. Moreover, the maximum value or the 
long-term distribution of hull responses can be predicted using the long-term occurrence frequency of 
short-term sea conditions. 
Short-term predictions here refer to predictions of responses in comparatively short-term sea 
conditions and can be obtained assuming that the wave heights and wave periods of irregular waves 
are statistically the same. Moreover, long-term predictions are predictions of responses in various 
statistically different sea conditions that appear over a long period over the service period of a ship. 
 
4.5.2.1 Response of hull to regular waves 
The assumption here is that regular waves can be expressed by small amplitude waves. It is also 
－ 89 － 
assumed that the amplitude of hull response is small. The hull response to small amplitude in small 
amplitude waves can be estimated by the linear theory. 
During hull response calculations in regular waves, ship motions are estimated first. In recent years, 
ship motions are generally estimated by the strip method. A slender hull is generally cut into circular 
slices (strip (belt) form), and two-dimensional hydrodynamic forces actingon each strip are integrated 
along the length of the ship to estimate the three-dimensional hydrodynamic force acting on the hull in 
this method, hence it is called the strip method.. 
The strip method used may be the Ordinary Strip Method (OSM), the New Strip Method (NSM) or the 
Salvensen Tuck Faltinsen' Method (STFM). The estimation of two-dimensional hydrodynamic force 
may be performed by either the Ursell-Tasai method in which the hull section is expressed by Lewis 
form or the singular point distribution method. 
If ship motions are determined, the acceleration and relative water levels at arbitrary positions can be 
estimated. Wave-induced loads such as wave-induced dynamic pressure, shear force, and bending 
moment can also be determined. Furthermore, the hull structural response due to these wave-induced 
loads can also be calculated. 
Fig. 4.29 shows a calculation example of longitudinal bending moment by varying the wave length λ 
and the angle of encounter of the ship and the wave. The response in regular waves is expressed as the 
response per unit wave height obtained by dividing the response value by the wave height. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.5.2.2 Short term forecast 
The variance of hull response σ2 or the cumulative energy density E is used as parameter to express 
the short-term distribution of hull response. The relationship between E and σ is: σ2=E 
If the frequency response function of hull response in long-crested irregular waves (RAO: Response 
Amplitude Operator) and the wave spectrum of short-crested irregular waves S(ω,θ) are given, the 
energy density function of hull response can be determined by the energy spectrum method using the 
linear superposition theory. 
The response amplitude operator of hull response RAO(ω,χ) is calculated by the method given in Sec. 
4.5.2.1. The wave spectrum is according to Sec. 4.5.1.3(2). 
Here, ω is the circular frequency of regular waves, χ is the angle of encounter between the 
predominant wave and the ship's course, and θ is the angle between the predominant wave and the 
component of wave. 
0.0E+00
2.0E+04
4.0E+04
6.0E+04
8.0E+04
1.0E+05
1.2E+05
1.4E+05
1.6E+05
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
( L /λ )1/2
M
y/
ζ
Dir. = 0 Deg.
Dir. = 30 Deg.
Dir. = 60 Deg.
Dir. = 90 Deg.
Dir. = 120 Deg.
Dir. = 150 Deg.
Dir. = 180 Deg.
Fig. 4.29 
－ 90 － 
The energy density function of hull response is given by: 
 [ ] [ ] [ ] θωθωθχωθωθχω ddSRAOddS 222 ),(),(),,( += 
The variance of hull response is given by: 
 [ ] [ ] θωθχωχσ ππ ddS
2
0
2 ),,()( ∫ ∫− ∞= 
The spectrum of irregular waves is treated as a narrow-band spectrum, therefore, naturally the hull 
response in such irregular waves will have a narrow band spectrum, and the characteristics as given in 
Sec. 4.5.1.3(3). 
4.5.2.3 Long term forecast 
If the significant wave height H1/3 and the mean wave period T0 are assigned as functions of the 
long-term occurrence probability density function p(H1/3,T0), then the long-term cumulative 
probability Qx1 at which the hull response of a ship plying on a fixed course in such waters exceeds x1 
is given by the equation below, as a function of the angle of encounter. 
 ( )∫ ∫∞ ∞ ⎟⎟⎠
⎞
⎜⎜⎝
⎛ −=
0 0 03/103/12
2
1
1 ,2
exp)( dTdHTHpxQx σχ 
If the long-term occurrence probability of the angle of encounter is distributed uniformly between 0 to 
2π (calling this as "All Heading), the equation becomes: 
 ( )∫= π χχπ
2
0 11 2
1 dQQ xx 
Fig. 4.30 shows examples of the long-term wave occurrence probability table for the North Atlantic 
Ocean; while Fig. 4.31 shows the examples of results of long-term prediction of longitudinal bending 
moment. 
 
 
Mean wave period (sec) 
 0-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-
0 - 1 0.9 102.6 999.1 2085.5 1540.8 556.9 122.6 19.1 2.3 0.2 0.0 5430
1 - 2 0.2 74.7 1584.0 6332.1 8225.3 4871.2 1655.5 378.7 65.2 9.2 1.1 0.1 0.0 23197
2 - 3 0.0 11.5 503.1 3618.4 7751.9 7078.1 3505.5 1114.6 256.4 46.5 7.1 1.0 0.1 0.0 23894
3 - 4 1.8 138.2 1550.9 4825.4 6069.0 3964.9 1603.8 455.4 99.4 17.8 2.8 0.4 0.1 0.0 18730
4 - 5 0.3 36.2 584.2 2457.8 4000.7 3265.4 1602.5 538.6 136.3 27.8 4.8 0.7 0.1 0.0 12655
5 - 6 0.1 9.2 202.8 1101.5 2229.5 2196.9 1269.9 492.5 141.3 32.3 6.2 1.0 0.2 0.0 7683
6 - 7 0.0 2.3 66.3 449.5 1099.8 1277.1 851.9 374.5 120.0 30.2 6.3 1.1 0.2 0.0 4279
7 - 8 0.6 20.6 170.3 492.9 662.3 502.0 247.0 87.4 24.0 5.4 1.1 0.2 0.0 2214
8 - 9 0.1 6.2 60.6 204.1 312.8 266.1 145.0 56.2 16.8 4.1 0.9 0.2 0.0 1073
9 - 10 0.0 1.8 20.5 79.0 136.4 129.0 77.2 32.5 10.5 2.7 0.6 0.1 0.0 490
10 - 11 0.0 0.5 6.6 28.8 55.5 57.8 37.7 17.2 5.9 1.7 0.4 0.1 0.0 212
11 - 12 0.1 2.0 9.9 21.2 24.2 17.1 8.4 3.1 0.9 0.2 0.1 0.0 87
12 - 13 0.0 0.6 3.3 7.7 9.5 7.3 3.8 1.5 0.5 0.1 0.0 0.0 34
13 - 14 0.0 0.2 1.0 2.6 3.5 2.9 1.6 0.7 0.2 0.1 0.0 13
14 - 15 0.1 0.3 0.9 1.2 1.1 0.7 0.3 0.1 0.0 0.0 5
15 - 16 0.0 0.1 0.3 0.4 0.4 0.2 0.1 0.0 0.0 2
Si
gn
ifi
ca
nt
 w
av
e 
he
ig
ht
 (
m
) 
16 - 0.1 0.2 0.2 0.1 0.1 0.0 0.0 1
 1 191 3273 14469 26613 26724 17188 7835 2721 761 179 37 7 1 0 100000
 
 
 
 
Fig. 4.30 
－ 91 － 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.5.3 Longitudinal Strength 
Longitudinal strength refers to the strength of the hull when considered as a single beam to resist 
bending, shear, and torsion. Bending and shear may be in the vertical and horizontal directions, but 
generally longitudinal strength refers to longitudinal bending and longitudinal shear strength in the 
vertical direction. 
 
4.5.3.1 Still-water longitudinal bending moment and shearing force 
When the hull is floating on still water, the weight and displacement of the ship coincide, and on the 
whole, the ship is in equilibrium in the vertical direction. However, the distribution of weight and 
displacement is not the same; therefore, these two items need not always be in equilibrium at each 
section in the longitudinal direction, and thus, shear forces and bending moments act at each section. 
This can be explained by an actual example. In Fig. 4.32(a), the comb-type distribution in solid line is 
the weight distribution, while the dotted smooth-line distribution is the buoyancy distribution. If the 
difference between these two forces is determined at each section, and integrated from the aft end to a 
specific position, the result will the shear force Fs at position. Integrating this way until the forward 
end results in the shear force distribution, which is a sharp distribution in solid line shown in Fig. 4.32 
(b). If the shear force distribution is again integrated from the aft end to a specific position, then the 
bending moment Ms at that position is obtained. If integrated to the forward end, then the longitudinal 
bending moment distribution, shown by the smooth dotted line in the figure can be obtained. 
Fig. 4.31 
－ 92 － 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.5.3.2 Wave-induced longitudinal bending moment and shearing force 
A ship traveling through waves is subjected to longitudinal bending moment and shear force in waves, 
in addition to the longitudinal bending moment and shear force in still water. The condition of the ship 
wherein a wave crest occurs amidship causing tensile stress in the deck is called Hogging, while the 
condition wherein the wave crest occurs at the fore and aft ends of the ship with the trough amidship 
Fig. 4.32(a） 
Fig. 4.32(b） 
－ 93 － 
causing compressive stress in the deck is called Sagging. (Fig. 4.33.) 
 
 
 
 
 
 
 
 
 
The longitudinal bending momentin waves MW is given by the equation below, established as part of 
the Unified Requerments of IACS related to the midship part of a ship. As mentioned in the method 
given in Sec. 4.5.2, the calculation equations have been decided from two aspects: after studying the 
maximum values based on the results of long-term predictions over about 20 years (occurrence 
exceedance probability 10-8) of longitudinal bending moment determined by the strip method for a 
large number of ships and based on the actual results of ships in service. 
 ( ) ( )mkNBCLCCM bW −+=+ '212119.0 Hogging Moment 
 ( ) ( ) ( )mkNCBLCCM cW −+−=− 7.011.0 '2121 Sagging Moment 
The distribution in the longitudinal direction of the ship, as given by the coefficient C2, is shown in Fig. 
4.34. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The shear force in waves, given by the equations below and determined according to the same method 
as used for the longitudinal bending moment in waves, is established in the Unified Rules. 
 ( ) ( ) ( )kNCBLCCF bW 7.03.0 '131 ++=+ 
 ( ) ( ) ( )kNCBLCCF bW 7.03.0 '141 +−=− 
The distribution in the longitudinal direction of the ship, as given by the coefficients C3 and C4, is 
shown in Fig. 4.35. 
 
（a）Hogging （b）Sagging 
Fig. 4.33 
Midship Part 
Fore end of L Aft end of L 0.3L 0.4L 0.7L 0.65L 
Fig. 4.34 
C2 
0 
1.0 
－ 94 － 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( )7.0110190'
'
+b
b
C
C
 
 
 
 
 
 
 
 
 
 
 
 
 
4.5.3.3 Bending Strength 
From the beam theory, the bending stress can be obtained by dividing the bending moment by the 
section modulus. At the upper deck and at the bottom shell plating, the tensile and compressive 
stresses obtained, become maximum. Thus, these stresses should be smaller than the stress level that 
does not cause any damage to the hull, that is, smaller than the permissible stress. 
The IACS Unified Requirements prescribes the required section modulus taking the permissible stress 
as 175 N/mm2 in case mild steel is used. That is, the section modulus of each hull section considered 
at the midship part of the hull (0.3 L to 0.7 L) should be greater than the value Zσ obtained from the 
two equations below, in all designed loading and ballast conditions of the ship. 
Zσ = 5.72|Ms＋Mw(+)|（cm3） 
Zσ = 5.72|Ms＋Mw(–)|（cm3） 
In the equations above, Ms is the still water bending moment at the considered position, while Mw(+)，
Mw(–) are the longitudinal bending moments in waves. 
1.0 
1.0 
( )7.0110 19092.0 '
'
+
×
b
b
C
C
 
0.7 
C3 
0.7 
Aft end of L 
Aft end of L 
0.2L 0.4L 0.85L 0.6L Fore end of L 
Fore end of L 0.85L 0.7L 0.6L 0.3L 0.2L 
0.7L 
0
0 
0.3L 
0.4L 
Fig. 4.35 
C4 
－ 95 － 
Furthermore, the IACS Unified Requirements requires that the section modulus at a hull section in the 
middle of L should not be less than the value of Wmin given below, which is the minimum section 
modulus. 
 ( ) ( )3'211min 7.0 cmCBLCW b += 
The requirement for this minimum section modulus leads to the equation below, from the previous 
equation for longitudinal bending moment in waves. 
 
( )( )2min /110 mmNMW w −= 
That is, the requirement can be interpreted as follows: "Even if the loading condition of a ship is such 
that the still water bending moment can be ignored, the section modulus should be such that that the 
longitudinal bending stress that occurs in the hull due to the longitudinal bending moment in waves is 
always less than 110 N/mm2." 
Moreover, since the hull needs not only strength but also rigidity, the moment of inertia of the hull 
section in the middle of L should be at least the value given below. 
 3WminL1 （cm4） 
The shape of the hull section varies slightly at each section even amidship, but in a conventional hull 
form, section modulus is calculated at the hull sections at the 0.3L, 0.4L, 0.65L, and 0.7L positions 
from the aft end of L, and the section modulus at these sections should satisfy the required section 
modulus. 
The longitudinal strength at the forward and aft peaks need not be calculated; it is adequate if the 
effective section areas of the strength deck decrease gradually. 
 
4.5.3.4 Shear Strength 
When studying longitudinal strength, the shear strength needs to be studied in addition to bending 
strength. Particularly in bulk carriers where holds are alternatedly loaded, the strength of shell plating 
near transverse bulkheads where distribution of cargo weight changes abruptly often becomes an issue. 
The shear stress of thin section beams as in the hull structure can be precisely calculated by the shear 
flow theory, but it can be calculated approximately by the equation given below. 
 
It
Fm=τ 
 τ is the shear stress 
 F is the shear force acting at the section 
 m is the moment of area of the section about the horizontal neutral axis for longitudinal members 
on the opposite side of the neutral axis from the horizontal line passing through the position 
at which stress is calculated 
 I is the moment of inertia of the section 
 t is the plating thickness 
The Rules for the Survey and Construction of Steel Ships prescribe the thickness of side shell plating 
by the equations below such that the permissible shear stress becomes 110 N/mm2 for total shear 
forces including shear force in still water and shear force in waves for all designed loading and ballast 
conditions at each section over the entire length of the ship. Where members that share shear loads 
such as longitudinal bulkheads exist, the load sharing factor for shear force of the side shell plating 
and longitudinal bulkheads is determined, and the thickness of these members are then determined. 
 
I
mFFt ws )(455.0 ++= 
 
I
mFFt ws )(455.0 −+= 
－ 96 － 
 
4.5.3.5 Torsional Strength 
In tankers or general cargo ships where the hatch opening width is small, torsional strength of the hull 
is not an issue, but in ships with cargo hatch openings of large width such as container ships, torsional 
strength needs to be studied. If the torsional strength is inadequate, damage is likely to occur at the 
corners of hatch openings. This kind of damage is found not only in container ships but also in large 
bulk carriers with comparatively large hatch openings. 
The Rules for the Survey and Construction of Steel Ships prescribe combined longitudinal stresss in 
which warping stress due to torison of hull is considered in addition to longitudinal bending stress and 
horizontal bending stress for each section from the forepeak bulkhead to the machinery space forward 
bulkhead in container ships. 
 
4.5.4 Transverse Strength 
 
4.5.4.1 Transverse Strength 
The overall strength of the ship must be discussed if the ship is considered a single solid structure, but 
it is difficult to perform structural analysis considering the entire ship as a single body. Therefore, for 
convenience, the strength of the ship has been treated by dividing it into longitudinal strength and 
transverse strength. 
Transverse strength is the strength of transverse bulkheads, frames, transverse girders, and so on, to 
resist loads such as water pressure and cargo loads. These transverse strength members have the role 
of receiving loads directly through plate panels and small stiffeners, and at the same time, to ensure 
that the initial hull form does not collapse. Transverse strength members form indeterminate structures 
mutually connected to and affecting each other, therefore, it is not as easy to determine their response 
to loads unlike longitudinal strength members. 
The Rules for the Survey and Construction of Steel Ships prescribe loads and permissible stresses for 
establishing scantlings by direct strength calculations using Finite Element Method (FEM) for oil 
tankers, ore carriers, and bulk carriers.4.5.4.2 Finite Element Method 
The Finite Element Method is an effective tool that can be used for structural analysis at a high 
accuracy, wherein the applicable structure can be modeled as-is and analyzed. Furthermore, the user 
can perform structural analysis using the computer program as a Black Box even if the user does not 
know the theory of the Finite Element Method. However, the results of the analysis are largely 
influenced by the range of the model, the division into meshes, and the method of assigning the 
boundary conditions, hence the user must become familiar with these parameters. 
The Finite Element Method is a method of dividing a structure into elements of finite size, and 
analyzing the assembly of elements. Thus, the three conditions of equilibrium of forces at the nodes of 
each element (equilibrium condition), the continuous deformation of each node (conformance 
condition), and a specific relationship between the stress and strain within an element should be 
satisfied if the finite element problem is to be solved. 
An overview of the finite element method taking an example of the most basic triangular plate element 
is given here. 
The coordinates of each node, the nodal displacements, and the nodal forces are expressed as shown in 
Fig. 4.36. 
The nodal forces and displacements are expressed as shown below. 
－ 97 － 
Nodal force { }
⎪⎪
⎪⎪
⎭
⎪⎪
⎪⎪
⎬
⎫
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
=
3
3
2
2
1
1
y
x
y
x
y
x
f
f
f
f
f
f
f Displacement { }
⎪⎪
⎪⎪
⎭
⎪⎪
⎪⎪
⎬
⎫
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
=
3
3
2
2
1
1
v
u
v
u
v
u
u 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The displacement u in the x direction and the displacement v in the y direction of the element are 
expressed by the following equations: 
 ycxccu 321 ++= 
 ycxccv 654 ++= 
 
If the above are expressed using coordinates and displacements of each node are expressed, then the 
following matrix is obtained: 
{ } [ ]{ }cF
c
c
c
c
c
c
yx
yx
yx
yx
yx
yx
u ≡
⎪⎪
⎪⎪
⎭
⎪⎪
⎪⎪
⎬
⎫
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
=
6
5
4
3
2
1
33
33
22
22
11
11
1000
0001
1000
0001
1000
0001
 
The relationships between normal strains εx,εy, the shear strain γxy, and the displacements are as shown 
below. 
 
x
v
y
u
y
v
x
u
xyyx ∂
∂+∂
∂=∂
∂=∂
∂= γεε 
 
Accordingly, the strain vector ｛ε｝, can be expressed as shown below. 
u1(fx1) 
u3(fx3) 
v3(fy3) 
y 
(x1,y1) 
(x3,y3) 
v2(fy2) 
v1(fy1) 
(x2,y2) 
x 
u2(fx2) 
Fig. 4.36
－ 98 － 
 [ ]{ }cS
c
c
c
c
c
c
xy
y
x
≡
⎪⎪
⎪⎪
⎭
⎪⎪
⎪⎪
⎬
⎫
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
6
5
4
3
2
1
010100
100000
000010
γ
ε
ε
 
This can be expressed by the following equation: 
 { } [ ]{ } [ ][ ]{ } [ ]{ }uBuFScS ≡== −1ε 
The strain coefficient matrix [B] can be calculated by the following equation: 
 
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
−−−−−−
−−−
−−−
211213313223
123123
211332
000
000
2
1
yyxxyyxxyyxx
xxxxxx
yyyyyy
A
 
 A is the area of the triangular element. 
Next, the relationships between the normal stresses σx, σy, the shear stress τxy, and the strains can be 
expressed by the following matrix: 
 
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎣
⎡
−−=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x E
γ
ε
ε
νν
ν
ντ
σ
σ
2
100
01
01
1 2
 
This can be expressed by the following equation: 
 { } [ ]{ }εσ D= 
 [D] is the stress-strain matrix. 
 E is the elastic modulus. 
 ν is Poisson's ratio 
Next, the stiffness matrix is determined based on the principle of virtual work, by which the external 
work constituted by external forces is equal to the internal work constituted by stresses in an element. 
Firstly, since external work is the sum of the product of the nodal forces and displacements at each 
node, it can be expressed as follows: 
 { } { }fuU Te = 
On the other hand, since internal work is given by the product of stresses and strains, it can be 
expressed as follows: 
 { } { } { } { } [ ]{ }( ) [ ][ ]{ }∫ ∫∫ ∫∫=== tdxdyuBDuBtdxdydVU TTTi σεσε 
 V is the volume of the element 
 t is the plate thickness of the element. 
Since external work is equal to internal work, therefore Ue = Ui 
 { } { } [ ]{ }( ) [ ][ ]{ }∫∫= tdxdyuBDuBfu TT 
From the above, the stiffness matrix [K] can be determined as below. 
 { } [ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ]{ }∫∫ ≡== uKuBDBAtdxdyuBDBtf TT 
 [K] is the stiffness matrix given by [ ] [ ][ ]BDBAt T 
The procedure to determine the displacement of a node and the stress in the element by finite element 
method is as follows: 
(1) Divide the structure into elements. 
－ 99 － 
(2) Determine the [B] matrix and the [D] matrix of an element. 
(3) Calculate the [K] matrix of an element. 
Repeat the procedures (2) and (3) for all the elements. 
(4) Superpose the stiffness matrix of each element, and create the stiffness matrix of the entire 
structure. 
(5) Distribute the given load to the different nodes, assign boundary conditions, and create the 
equation of equilibrium. 
(6) Solve the equation and determine the displacement of each node. 
(7) If the displacement at the node is known, the stress in the element can be determined from { } [ ]{ } [ ][ ]{ }uBDD == εσ 
The element described above is a very simple element. In an general-purpose structural analysis 
program, many other different kinds of elements can be processed. 
 
4.5.5 Local Strength 
Local strength refers to the strength that resists loads acting on elements such as small stiffeners 
including longitudinals or stiffeners attached to shell plating or bulkheads, or elements such as panels 
of plating members surrounded by frames. Local strength may also refer to the local strength of 
members subjected to stress concentration because of the abrupt change in form or ends of members. 
 
4.5.5.1 Strength of Small Stiffeners 
The strength of small stiffeners can be evaluated by bending strength according to the beam theory. 
Where beam ends are fixed in place by brackets or by lugs (including clip fixing by chamfering the 
face plate), they may be sniped, and the position and magnitude at which bending moment becomes 
maximum may vary depending on the method by which the beams are fitted. 
Strength evaluation is performed not by evaluating small stiffeners alone, but by considering the 
included plating attached to the stiffeners also. The width of the included steel plate is called the 
effective width. The Rules for the Survey and Construction of Steel Ships prescribes "0.1l of the width 
on either side of the stiffener" as the width of the included plating. However, 0.1l of the width shall not 
exceed half the distance up to the adjacent stiffener. Here, l is the length of the stiffener. 
 
4.5.5.2 Bending Strength of Panel 
Strength evaluation is performed for the case in which a plate panel surrounded by stiffeners and 
girders is removed and a distributed load is applied normally on the face of the panel. The periphery of 
the panel on which the distributed load is applied is close to the fixed condition, therefore, the panel is 
treated as one in which the four sides are fixed. Since the maximum stress occurs at the center of the 
longer side, this stress is used in the strength evaluation. 
To evaluate loads that are encountered only once in the lifetime of a ship as in panels, such as in 
watertight bulkheads, strength is evaluated based on the plastic collapse of panel and not by elastic 
analysis.4.5.5.3 Buckling Strength of Panel 
If an excessively large in-plane compressive load is applied on the panel, it causes compressive 
buckling of the panel. If an excessively large shear load is applied on the periphery, shear buckling of 
the panel occurs. In the actual panel of the hull structure, both compressive load and shear force act 
simultaneously, so evaluation using buckling interaction equations is necessary. 
The buckling strength of panel is proportional to the square of the plating thickness. If high tensile 
steel has been used, the panel thickness becomes small, therefore, adequate precautions for buckling 
strength need to be taken. 
－ 100 － 
4.5.5.4 Fatigue Strength 
If a fluctuating load that varies with time is received, even if the maximum stress is much smaller than 
the strength of the material, the member is likely to fail after a specific number of repetitive cycles of 
the load. Such a failure is called fatigue failure. Most of the cracks that occur in the hull structure are 
cracks due to fatigue. 
The evaluation of fatigue strength is generally performed according to Miner's rule (linear damage 
rule) by determining the frequency of occurrence of stress of a certain magnitude from among 
arbitrary, irregular, fluctuating stresses, calculating the cumulative fatigue damage using S-N curves 
assuming that fatigue life does not change even if the acting stress sequence changes, and concluding 
that failure occurs when the cumulative fatigue damage becomes 1. 
The S-N curves are logarithmic curves of the fluctuating stress range plotted on the vertical axis and 
the fatigue life (cycles) on the horizontal axis, based on the results of a large number of fatigue tests 
carried out in the laboratory. Thus, it is important to select the appropriate S-N curves to suit the 
members to be evaluated for strength. 
To calculate the cumulative fatigue damage, the long-term stress frequency distribution for the 
relevant member is determined either by the method described in Sec. 4.5.2 or by other simple 
methods, and the distribution is superposed with the S-N curves as shown in Fig. 4.37. 
When a fluctuating stress σi is applied, a crack is likely to occur at the location whether the number of 
cycles Ni intersects the S-N curve. However, in practice, if the number of cycles of fluctuating stress σi 
 
in the member to be evaluated is ni, the fatigue damage received by the said member is:
i
i
N
n
 
 
This is calculated for all the fluctuating stresses, and the sum is expressed by D below, which is the 
called the cumulative fatigue damage. 
∑=
i i
i
N
nD 
The permissible cumulative fatigue damage need not always be 1.0. Depending on the importance of 
the member, this value may vary over a wide range from 0.5 to 1.5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.5.6 Corrosion of Steel 
The wastage due to corrosion of structural hull members varies depending on the type of member and 
its maintenance level. Generally speaking, the corrosion in tankers is excessive compared to that in 
cargo ships; moreover, it is more frequently found on the deck than on the bottom shell plating. 
S-N Curve 
ni 
Variable Stress 
Frequency Distribution of 
Stresses in Structure 
Cycles 
Fig. 4.37
σi 
Ni 
－ 101 － 
The table below shows the results of study when a hole was made in an actual ship. In the table, ⊗ 
indicates the mean value. The arrows indicate a range that includes 50% of all the data. As is evident 
from these results, the annual wastage of most of the ships is in the range of 0.1 mm to 0.20 mm. 
However, the extent of corrosion depends largely on the environment of the location where the steel is 
used, and varies from no corrosion at all to an annual wastage of 0.3 to 0.5 mm. In the extreme case, 
an annual wastage of 1.4 mm has also been recorded. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
All hull structural members except members within the oil tanks are required to be painted according 
to the Rules. However, at locations where bilge is likely to accumulate, where paint is likely to peel off 
due to contact with cargo, or in unpainted cargo oil tanks where sea water ballast is filled, special 
precautions against corrosion are necessary. Generally, the progress of wastage is noticeably fast in 
members exposed to high-temperature and high-humidity environments especially in parts with high 
stress. Next, the general trends of members that suffer excessive corrosion are described below. 
(1) (Note: This almost never occurs in recent years) The progress of corrosion is fast because of the 
following reasons: the underside of the deck in way of cargo oil tanks of tankers is exposed to 
the heat of the midday sun, creating a high temperature and high humidity environment inside 
the tank. Sea water at high temperature and high pressure is used to clean the tanks. Oil film 
sticking to steel is removed. The longitudinal bending stress in the hull girder is high in the 
upper deck. Crude oil washing became mandatory when the MARPOL Convention entered into 
force, and corrosion has decreased, but the filling of sea water as ballast in cargo oil tanks has 
not disappeared completely, therefore care needs to be taken against corrosion. 
(2) (Note: Extremely high) In the past, anti-corrosive measures by heavy coats of paint applied to at 
least the upper parts of ballast tanks including the underside of the tank top in association with 
cathodic protection were frequently adopted, but in recent years, anti-corrosive paint is required 
on all the surfaces of ballast tanks. The anti-corrosive effect degrades with the passage of time. 
Since ballasting with sea water is performed repeatedly, generally, the progress of corrosion is 
noticeable. 
(3) Corrosion and wastage is also rapid in cargo hold members of timber carriers. This is because 
the loading and unloading of wood logs is generally performed roughly, and wood logs are 
floated on the surface of the sea water and then loaded, therefore, the content of sea water in the 
wood is large. As a result, the paint in the hold is lost within a short time, or it becomes wet 
because immersion in the fluid given off from the wood logs. Thus, an environment that 
deteriorates and leads to corrosion is created. Special care is necessary to inspect the area near 
the lower ends of frames and the lower parts of bulkheads. 
－ 102 － 
(4) The side shell plating in way of forepeak tank and cargo holds is subject to repeated wave 
impact loads. As a result, excessive corrosion and wastage may be observed near side shell 
panels with high stresses, that is, along the lines whether the frames are secured to the shell 
plating. 
(5) The progress of corrosion of upper deck at parts where water drainage is poor, such as at the 
corners of the deck house and lower parts of deck machinery and equipment on exposed decks 
may be abnormally high. 
(6) In addition to the above, local pitting corrosion of bottom shell plating or internal members in 
the area near the bell - mouth at the cargo oil intake port of tankers is frequently found because 
of cavitation and the complex flow of cargo oil. Thus, precautions are necessary against pitting 
corrosion, which is likely to develop at the rate of more than 1 mm per year. 
 
4.5.7 Scantlings prescribed in the Rules for the Survey and Construction of Steel Ships 
The loads for establishing the scantlings required for hull structural members according to the Rules 
for the Survey and Construction of Steel Ships are assumed to be the severest loads anticipated during 
the operation of the ship for each structural member such as shell plating, deck plating, double bottom, 
bulkhead and tank structural members. Accordingly, the loads need not always be loads at special 
conditions for all hull structural members. 
General cargoesloaded in the cargo holds are generally considered to have a specific gravity of 0.9 
and distributed uniformly, unless specified otherwise. Consequently, corrections are to be made to 
ships loaded with ore or other heavy cargoes, or ships loaded with cargoes that give rise to 
concentrated loads. Special loading conditions such as the above are entered normally in the Midship 
Section Drawing and also in the Loading Manual. 
The prescribed scantlings are determined based on the loading conditions assumed as mentioned 
above by the most rational analysis method when the Rules were established, which includes the 
experience of NK, by deciding permissible stresses and safety factors, etc., for each member, and also 
anticipating the margin for corrosion. The margin for corrosion is generally 2.5 mm for plating 
(general structural members) to 3.5 mm (cargo oil tanks of tankers), and for members used in section 
modulus calculations, it is about 20%. 
The scantlings of members such as the plate of shell plating, deck, bulkheads and so on are decided by 
prescribing the thickness and for frames, beams, stiffeners, girders and so on, by prescribing the 
section modulus and the moment of inertia. Since frames and other stiffeners are attached to plating, 
the required strength is prescribed by the section modulus of not only the frames alone but also 
including that of the attached steel plating, as described in "Section 4.5.5.1 Strength of small 
stiffeners." The effective width of attached steel plating includes a width of steel plating equivalent to 
1/10 the span on either side of the stiffener. However, this width shall not exceed half the distance to 
the adjacent member. 
Since structures need to have not only strength but also rigidity, the moment of inertia is prescribed in 
addition to section modulus for some of the members, or in case of girders and built-up steel sections, 
the depth to span ratio is prescribed instead of the moment of inertia. 
For instance, the Rules require that if the depth to length ratio of transverse frames in the holds is less 
than 1/24 or if the same ratio for frames in the forepeak is less than 1/22, then the scantlings of the 
frames need to be appropriately increased. The Rules also require that the height of the centerline 
girder plate in the double bottom area be greater than B/16, except when specially approved by the 
Society. The longitudinal strength of the hull girder is required to be such that the moment of inertia of 
the hull section is greater than 3WminL (Wmin is the minimum required section modulus). 
When high tensile steel is used, the section modulus can be reduced, but care is necessary when 
deciding the moment of inertia of the section. 
－ 103 － 
 
4.6 Shipbuilding steel material 
 
A great variety of materials are used in the shipbuilding industry and those used for parts requiring 
strength are specified in the Rules to be manufactured by a manufacturing process approved by the 
Society and to be tested and inspected in the presence of the Society's Surveyor. Steel plates and 
angles used for hull structure and casting or forged steel used for rudders or components are as 
follows; 
 
4.6.1 Rolled steel material 
Rolled steel includes the following: 
Rolled steel for boilers For pressure vessels used at high temperatures and boilers 
Rolled steel for pressure vessels For pressure vessels used at atmospheric temperature 
Rolled steel for low temperature service For hull structures around tanks and tanks of ships carrying 
liquefied gases in bulk, and for hull structures at locations 
exposed to low temperatures such as refrigerated cargo 
ships 
Rolled stainless steel For tanks in low temperature service or corrosion-resisting 
service 
Round bars for chain For manufacturing chains 
Rolled steel bars for machine structure For machine structure such as shafts and bolts 
High strength quenched and tempered For offshore structures, tanks in ships carrying liquefied 
rolled steel for structures gases in bulk and for pressure vessels for processes 
Stainless clad steel plate For tanks of ships carrying dangerous chemicals in bulk and 
hull structures around tanks, and for corrosion-resisting 
tanks 
Here, in this section, the following is described: 
Rolled steel for hull Steel material of thickness below 50 mm for use in hull 
structures. 
For the hull, rolled steel only up to a thickness of 50 mm is prescribed, but in ships such as large 
container ships, steel plates of thickness greater than 50 mm is used. Therefore, the Rules for the 
Survey and Construction of Steel Ships include "Special Regulations for Rolled Steel Plate for Hull of 
Thickness Exceeding 50 mm but Less Than 100 mm." 
Requirements for rolled steel for hull are established in the IACS Unified Rules. Test methods, 
chemical composition, and material standards are also prescribed. The Unified Rules also prescribe 
thickness tolerance of the steel. 
 
－ 104 － 
4.6.1.1 Mild Steel 
Mild steel has mechanical properties as shown below, and it can be classified into steel of Grade A, 
Grade B, Grade D and Grade E. 
 
Tensile test Impact test 
Minimum mean absorbed energy 
(J) 
Material 
Notation 
Yield Point 
（N/mm2） 
Tensile 
strength 
（N/mm2） 
Elongation 
（%） 
Test 
temper. 
（℃） L T 
KA ‐ ‐ ‐ 
KB 0 
KD -20 
KE 
235 and 
above 400 to 520 
22 and 
above 
-40 
27 20 
 
Grade A steel is the most widely used type of steel. Impact tests are not required for this grade of 
steel. 
Grade B steel has better notch toughness than Grade A steel. Impact tests at 0°C are to be 
conducted. 
Grade D steel has better notch toughness than Grade B steel. Impact tests at -20°C are to be 
conducted. 
Grade E steel has superior notch toughness compared to Grade D steel. Impact tests at -40°C are to 
be conducted. Grade E steel is also called "crack arrestor" and it has the role of preventing cracks in 
the hull from propagating. 
Impact test is also called the Charpy test. A test specimen with an artificially created notch is broken 
by hitting the specimen using a hammer, and the energy required for breaking the specimen is 
measured to evaluate the notch toughness. 
The required value of energy absorbed in the impact test is the same for all grades of steel. The grades 
are classified by changing the test temperature making use of the property of steel that the lower the 
temperature, the more brittle it gets. Two cases are prescribed - the longitudinal direction of the 
specimen parallel to the rolling direction of the material (L direction) and normal to it (T direction). 
The following carbon equivalent is used as index of the weldability of steel. 
 (%)
1556
CuNiVMoCrMnCCeq ++++++= 
Considering the chemical composition for ensuring weldability of mild steel, the value of 
6
MnC + is 
not to exceed 0.40% according to the Rules. 
The Rules prescribe at which part of the hull structure each of these grades of steel is to be used. The 
applicable usage of steel are prescribed according to the importance of the member in the Rules for the 
Survey and Construction of Steel Ships. 
 
－ 105 － 
4.6.1.2 High Tensile Steel 
High tensile steel is classified into groups by yield point as shown in the table below. Similar to mild 
steel, it is categorized by notch toughness into Grade A steel, Grade D steel and Grade E steel (no 
Grade B steel), with a further addition of Grade F steel. The numerals in the material notation are 
values of the yield point in gravimetric units. 
 
Tensile test Impact test 
Minimum mean absorbed energy 
(J) 
Material 
Notation 
Yield Point 
（N/mm2） 
Tensile 
strength 
（N/mm2） 
Elongation 
（%） 
Test 
temper. 
（℃） L T 
KA32 0 
KD32 -20 
KE32 -40 
KF32 
315 and 
above 440 to 590 
22 and 
above 
-60 
31 22 
KA36 0 
KD36-20 
KE36 -40 
KF36 
355 and 
above 490 to 620 
21 and 
above 
-60 
34 24 
KA40 0 
KD40 -20 
KE40 -40 
KF40 
390 and 
above 510 to 650 
20 and 
above 
-60 
41 27 
 
Conventional methods for manufacturing high tensile steel required high carbon equivalents to ensure 
the strength, pre-heating, restrictions on heat input, adequate management of welding materials, 
restrictions on small beads, and many other factors that hindered the welding work. The use of high 
tensile steel was limited to some large ships, but in 1980 in Japan, Thermo-Mechanical Controlled 
Processing (TMCP) was developed, which combines controlled rolling in which the temperature range 
through which rolling is performed is controlled minutely with accelerated cooling after rolling, which 
enabled low carbon equivalent values to be realized. As a result, the hindrance factors mentioned 
above were eliminated to a large extent, welding similar to mild steel was enabled, and in addition, 
high heat input welding was enabled, and the usage rate of high tensile steel increased by leaps and 
bounds. TCMP is now also used in the manufacture of mild steel. 
 
4.6.1.3 Steel Sections 
Steel sections and built-up members are used in side frames, deck beams, bulkhead stiffeners and 
longitudinals, and these members account for about 10 to 15% of the total steel materials used. Similar 
to steel plates, steel sections may be made of mild steel or high tensile steel. 
Steel sections may of the following kinds: 
(1) Flat bars and slab longitudinals 
Flat plate with no face bar, hence, if the section area is the same, the strength of stiffeners is the 
least among steel sections. Used in deck longitudinals in which axial force is large but bending 
load is small. The ratio of depth to thickness is maintained below 15 to prevent compressive 
buckling. 
(2) Bulb Plate 
Making use of the advantage of the small flange part in the shape, bulb plate sections are used, 
for instance, in cross deck beams of bulk carriers loaded with grain which is likely accumulate 
in these beams, beams of holds in cement ships, bilge keels of small ships, and so on. 
－ 106 － 
(3) Equal-angle steel sections, unequal angle steel sections, unequal angle unequal thickness steel 
sections 
The section performance is large for a given section area, thus these sections form the 
mainstream of various kinds of stiffeners in small and medium-sized ships. Used in 
longitudinals of double bottom and upper part of the hull in large ships. 
(4) Built-up members 
Used mainly in parts of large ships at locations where the section performance cannot be 
covered by unequal angle unequal thickness steel sections. Steel plates are cut and built up to L 
and T sections. L sections are divided into L2 and L3 sections according to the difference in the 
direction of assembly of face flat, but L2 is generally used. Built-up members have no restriction 
on size, and this freedom in selection of size is their biggest advantage. Conversely, however, 
the disadvantages are the excessive welding and cutting costs compared to rolled materials. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.6.2 Castings and Forgings 
The stern frame and rudder parts are made of steel castings assembled in the hull structure. The 
anchor is also a steel castings. Large stern posts are assembled with steel castings and steel plates. The 
rudder is also assembled using steel castings and steel plates. 
The rudder stock and rudder pintle may be made of steel forgings. 
The carbon content of steel castings and steel forgings that are to be welded should be less than 0.23%. 
 
 
4.7 Welding 
 
4.7.1 Welding Methods 
An overview of the main welding methods used in shipbuilding is given here. 
 
Flat Bar Bulb Plate Angle Bar 
Built-up Members 
Fig. 4.38 
L２ L3 T 
－ 107 － 
4.7.1.1 Covered Arc Welding 
This is a welding method in which an arc is struck between a coated electrode with flux coated on the 
core wire and dried, and the parent metal. In most cases, the worker holds the holder in hand and 
welds the parts, so this method is generally referred to as hand welding method. Fig. 4.39 shows a 
sketch of the covered arc welding method. 
Covered arc welding is widely used because the method is simple, but since the efficiency is low, most 
industries have switched over to metal active gas (MAG) welding, described below. However, gravity 
welding in which a simple jig is used, is highly efficient, so it is still used widely in fillet welds even 
today. In gravity welding, welding is performed making use of gravity such that the electrode material 
drops naturally by gravity as it travels along the fillet corner, and one worker can operate jigs of 4 to 8 
welding machines. 
－ 108 － 
4.7.1.2 Metal Active Gas (MAG) Welding 
Metal Active Gas is abbreviated to MAG. In MAG welding, a wire wound around a spool normally 
(solid wire and flux-containing wire) is automatically fed while simultaneously gas is supplied (carbon 
dioxide gas or 80% Ar + 20% CO2 gas) through a nozzle to generate an arc between the wire tip and 
the parent metal, so that welding is performed while the wire and parent metal melt. Fig. 4.40 shows a 
sketch illustrating MAG welding. 
Since the shield gas includes a large percentage of oxygen, it is called active (activated) gas welding, 
but when inactive gases such as Ar or He is included, then in most cases it is called inert gas arc 
welding. These welding methods are sometimes called semi-automatic welding methods since in many 
cases, the welder normally takes the welding torch in hand and performs the work. A major share of 
MAG welding in the shipbuilding industry is carbon dioxide gas MAG welding. 
Wire may be either a solid wire or a flux-containing wire, but the latter is mostly used in the 
shipbuilding industry. During MAG welding, wire can be continuously fed, and it can be mounted on 
the equipment with comparative ease, therefore, it is being increasingly used after assembly with 
automatic welding machines and welding robots. 
If the object to be welded is too large and cannot be inverted, one-side MAG welding using a backing 
plate may be performed. Thus, MAG welding can be used for small assemblies, large assemblies, ship 
on cradle or other processes, and also for fillet welding or butt welding, and it is thus widely used. 
 
－ 109 － 
4.7.1.3 Submerged Arc Welding 
Welding flux and welding wire are fed separately and automatically, arc is generated between wire and 
parent metal, and wire, flux, and parent metal are melted and welded together in this method. Since the 
arc is covered by flux, it is called the submerged arc welding method. It is sometimes simply called 
automatic welding or Union Melt. Fig. 4.41 shows a sketch illustrating submerged arc welding. 
This welding method generally uses a high current and has high efficiency, but considerable time is 
required for preparations because the welding equipment is large, therefore this method is unsuitable 
for welding short joints. It is used in processes where medium/long steel plate joints are to be welded. 
For instance, reversing panels of 10 to 20 m in length is difficult, therefore, one-sided submerged arc 
welding with a backing plate is used; for plates shorter than mentioned above and which can be 
reversed, two-sided submerged arc welding of the panel joints is used. 
 
4.7.1.4 Other Welding Methods 
Electrogas arc welding is used mainly for butt-welding comparatively thick plating such as side shell 
plating in the vertical position. 
Consumable guide electroslag welding (CES welding) for joints with short welding lengths is used for 
welding thick plates for short lengths in the vertical position, such as slab longitudinals of deck. 
－ 110 － 
4.7.2 Welding Notations 
Although thereare many kinds of weld joints, welding notations for typical butt welds and fillet 
welds used in shipbuilding are shown in Fig. 4.42. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
In the figure, plate joints are expressed by the notations indicated in Fig. 4.43
(d)Continuous fillet weld 
(d)Staggered intermittent fillet 
weld 
Fig. 4.42 
(a)I-butt weld 
(b)V-butt weld 
Fig. 4.43 
(c)X-butt weld 
－ 111 － 
－ 112 － 
 
	4.1nullLines and Key Plan
	4.2nullBasic Calculations
	4.3nullStability
	4.4nullPower Requirements for Propelling Machinery
	4.5 Hull Strength
	4.6nullShipbuilding steel material
	4.7nullWelding