695

1 INTRODUCTION

Dry bulk cargo constitutes the significant part of

maritime transport and includes any cargo carried in

bulk in solid form, e.g. coals and cokes, grains, bulk

minerals (e.g. sand, gravel), iron ore, cements,

chemical fertilizers or bauxite. Generally, ships

designed/built to carry such cargo are called bulk

carriers, although a part of them have independent

type name which is directly related to the type of

transported cargo e.g. ore carrier, ore-bulk-oil carrier

(OBO) or cement carrier. In Chapter IX of the SOLAS

Convention (the International Convention for the

Safety of Life at Sea) a bulk carrier is defined as “a

ship constructed with a single deck, top side tanks

and hopper side tanks in cargo spaces and intended to

carry dry cargo in bulk primarily; an ore carrier; or a

combination carrier” (IMO, 2013). It should be noted

here that some light dry bulk cargoes sometime are

carried also by general cargo ships. The EQUASIS

(Electronic Quality Shipping Information System)

provides information about the world’s merchant

fleet. Their database includes most of the world’s

merchant ships. Reports and statistics are published

every year and are generally available. According to

the EQUASIS 2021 database, the dry bulk carriers

includes 12 874 ships and represent 10.8% of the total

number of ships, however in terms of gross tonnage it

is already 34.4%.

Dry bulk cargo is transported unpackaged in large

quantities and the significant issue is determining the

mass of the cargo loaded on (unloaded from) the ship.

Practically in most cases, the basic method for

determining the mass of the transported dry bulk

cargo is the Draught Survey (DS) procedure.

The Draught Survey procedure is based on the

ship’s draughts read from the draught marks placed

The Second Trim Correction in Draught Survey

rocedure – Accuracy Analysis

W. Wawrzyński

Gdynia Maritime U

niversity, Gdynia, Poland

ABSTRACT: The basic method for determining the mass of dry bulk cargo, loaded on the ship or unloaded

from the ship, is the Draught Survey procedure. This procedure is quite simple, however is highly susceptible to

mistakes and accidental errors. Moreover, it is also affected by some systematic errors. The difference between

the mas of cargo at the loading and discharging port (cargo shortage) bigger than 0.5% then such cargo shortage

generally becomes the subject of a Cargo Claim. In this study the systematic error that concerns the formula of

second trim correction in Draught Survey procedure is discussed. The results of calculations performed with

the commonly used the second trim correction formula are compared with the results obtained during

calculations with the use of Bonjean Scale. The calculations carried out with the use of Bonjean Scale, have been

validated based on the ship’s documentation. For the purpose of this study the small bulk carrier was selected.

The results of performed calculations shows the si

gnificance of the systematic error of the second trim

correction for a ship with higher trim values.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of

Sea Transportation

Volume 18

Number 3

September 2024

DOI: 10.12716/1001.18.03.

696

on the ship’s hull, at the bow and stern and in the case

of large ships also at the midship. Using these

draughts, the current ship displacement is determined

where heel, trim, hull deformation and water density

are considered. The weight of loaded/unloaded cargo

usually is calculated as the difference between ship

displacement determined before and after

loading/discharging. Such an approach eliminates

from the calculations a number of masses that are

unknown or known with insufficient accuracy and

generally are called ship constant (silt and mud in

tanks, bilge water, minor changes in construction and

equipment made during stays at the shipyard,

auxiliary equipment and some supplies, etc.). A short

description of the standard Draught Survey procedure

is provided in the next Section, while a broad

description with additional notes about actions

performed under the DS procedure and possible

errors can be found in (UNECE Committee on Energy,

Working Party on Coal 1992; Isbester 1993; Dibble and

Mitchell 1994; The International Institute of Marine

Surveying 1998; Puchalski and Soliwoda 2008; Barras

and Derrett 2012 ).

Considering the number and total tonnage of bulk

carriers it is easy to notice the enormous importance

of the Draught Survey procedure accuracy. According

to Japan P&I Club (2016), if the quantity of loaded

cargo is shown by mass and the difference between

the quantity measured at the loading and discharging

port is bigger than 0.5% (Trade Allowance most often

used for dry bulk cargo) then such cargo shortage

generally becomes the subject of a Cargo Claim.

Whereas the West of England P&I Club (2018) reports

that using Draught Survey procedure the accuracy of

calculated cargo mass usually varies between 0.5%

and 1%. However, Ivče et al. (2011) stated that the

accuracy of calculated cargo mass varies between

0.1% and 1%. This seems more reliable because the

range of 0.5-1% would mean that the Trade

Allowance limit is practically always exceeded.

Nevertheless, the error equal to 0.5-1% of total cargo

mass, at the maximum ship displacement,

corresponds to a value of 4 to 8 TPC (the mass in tons

required to change the ship mean draught by one

centimeter) where the accuracy of draught

measurement assumed in DS procedure is ±0.5 cm.

Obviously, in real conditions, such draught

measurement accuracy is most often not achievable.

The ship draught measurement accuracy was the

main issue of the study (Ivče et al. 2011). Although

Ivče et al. (2011) pointed that the ship draught

measurement error is one of the main errors causing

the DS procedure inaccuracy, the authors still claim

that measuring cargo mass by means of draught gives

a smaller error than measuring the mass by cargo

weighting. The reasons for the measurement error of

the ship’s current draughts may be: parallax

phenomenon, waves and related ship movements,

strong current, reduced visibility (e.g. night

conditions) or rain. Ivče et al. (2011), based on their

experimental discoveries, claims that error in draught

readings can be up to ±10 cm. This is likely but only in

extreme weather conditions. To reduce the error in

draught readings, authors suggest using the optical

fiber technology. They believe this may be a way to

eliminate errors that may occur during the visual

draught reading. Unfortunately, the study does not

contain any comparative data, based on the real

measurements.

No doubt, the draughts that have been measured

are a substantial factor since they are the only input

parameter for ship displacement calculations.

However, there are also other issues that may

significantly affect the accuracy of Draught Survey

procedure and finally the mass of cargo that is

determined. A considerable number of them have

been accurately indicated by the West of England P&I

Club (2018), however these are mainly operational

issues where potential errors may be classified as

mistakes (errors caused by inattention, inexperience,

carelessness, misjudgment, distraction) or accidental

errors. These issues concern: mass of ballast on board,

water density, unfactored masses (e.g.: bilge water,

water in swimming pool, anchor and anchor cable on

the seabed, silt and mud in the double bottom tanks),

squat, trim by the head (the tanks suctions and

sounding pipes are located at the aft end of the tanks),

the nature of cargo (for certain types of cargo, water

could migrate from the cargo to the hold bilges and be

subsequently pumped overboard) and others (West of

England P&I Club 2018).

The Draught Survey procedure not only is

susceptible to mistakes and accidental errors but it is

also affected by systematic errors. Systematic errors

may be caused by inaccuracies and even considerable

errors in the ship’s documentation - the hydrostatic

data and tank sounding tables may not be accurate

(e.g. because of changes to the ships structure made in

shipyards). Another source of the systematic error

may be one of the assumptions of the DS procedure,

that the hull deformation (deflection) is symmetrical,

that the deformation maximum is placed exactly at

the midship cross section. Whereas the location of the

hull deformation maximum very often is placed

outside the midship cross section. This issue most

often affects smaller ships because of large engine

room in relation to the hull size (West of England P&I

Club 2018) but it can also be caused by the cargo

distribution on large ships. The issue of correction for

hull deformation in the DS procedure was discussed

in (Soliwoda 2016; Wawrzynski 2011).

Apart from papers mentioned earlier, a few more

studies that address the accuracy of the Draught

Survey procedure can be found in (Li et al. 2014;

Elnoury and Gaber 2018; Xu et al. 2018; Canımoğlu

and Yıldırım 2021), but generally the number of

advanced studies is quite small. In (Li et al. 2014) the

fuzzy comprehensive evaluation method was adopted

to analyze the DS procedure error. Authors of this

study claim that this method can be used to calculate

the error risk in different condition, ensuring the DS

procedure error will be below 0.5%. Canımoğlu and

Yıldırım (2021) used the extended fuzzy analytic

hierarchy process (FAHP) method to develop the

hierarchical structure establishing the

recommendations for reducing errors in the DS

procedure. In this study, like in (Ivče et al. 2011; Xu et

al. 2018), it is stated that errors occurring during the

draught reading stage are the main source of the DS

procedure errors. The errors priority weights have

been defined, where for draught readings it is 0.40, for

ballast measurement 0.29 and only 0.12 for the error

made during displacement calculations. However,

these values are questionable since it seems that

697

Canımoğlu and Yıldırım (2021) assume that the

mathematical formulas used in the calculation part of

DS procedure are fully valid (free from systematic

errors).

Returning to the issue of systematic errors in the

Draught Survey procedure, the inaccuracies in the

ship’s documentation or deformation maximum

outside the midship cross section may or may not be

present (this depends on the ship documentation

or/and the current loading condition). However, there

is another systematic error that directly concern the

mathematical formula used in the calculation part and

more specifically the second trim correction. This

analysis shows the significance of this error based on

calculations performed for the selected bulk carrier.

Finally, it should be highlighting that this study

omitted alternative procedures for determining the

ship's displacement on the base ship’s draughts. For

example, these procedures may use data of a trimmed

ship (Firsov diagram) or ready-made the

displacement correction tables which are sometimes

included in the ship stability booklet. Although some

reference to these procedures will be made in the

conclusions section

2 STANDARD DRAUGHT SURVEY PROCEDURE

Regardless of the Draught Survey procedure is

susceptible to mistakes and accidental errors the

calculation part of it is very simple. Generally, there

are two versions of the DS calculation part that differ

in the way the hull deformation is considered.

However, in this research the trim correction is the

issue, therefore it was assumed that the hull is free of

deformation. If so then both DS calculation versions

come down to one where only the draughts at aft and

fore marks are used.

To consider the possible ship list, the arithmetic

mean of the mark draught on the port and starboard

sides are calculated, independently for the stern and

bow. Next, both arithmetic means are converted into

draughts on the perpendiculars (T

aft and Tfore) and then

the mean draught T

M (draught at midship) is

calculated:

aft fore

(1)

The mean draught is the parameter the

displacement is taken from the Hydrostatic Table of

even-keel ship (HyTa). For trimmed ship this

displacement is incorrect due to the trimming axis is

not at midship but at the cross section of the center of

flotation (geometric center of the waterplane area).

The longitudinal coordinate of the center of flotation

(LCF) is given in HyTa.

As stated above the displacement taken from HyTa

is incorrect, so the two trim corrections are applied,

called 1

and 2

trim correction (ΔD1 and ΔD2). First

trim correction ΔD

1 (2) considers the difference in

draught between the midship and LCF cross section.

The idea of this correction is logical and obvious.

However, in the formula (2) the LCF taken from the

Hydrostatic Table of even-keel ship is used. For

trimmed ship the waterplane area changes its shape

and size. Consequently, the center of flotation changes

its position too. To take this into account the second

trim correction (3) is used. It should be noted here that

formula (3) was developed using some simplifications

and is no longer so obvious as formula (2).

100

D TPC LCF

∆= ⋅

(2)

( )

2 0.5 0.5

Tm Tm

D MTC MTC

+−

∆=⋅ −

(3)

In the above formulas t is the trim (negative for aft

trim and positive for forward trim), LCF is the

longitudinal center of flotation measured in relation to

midship (negative when towards the stern and

positive when towards the bow), L

PP is the length

between perpendiculars, TPC is the mass in tons

required to change the ship mean draught by one

centimeter and MTC the moment to change the trim

one centimeter. The remarks on the trim sign given in

brackets are very important. Unfortunately, for most

ships, the arrangement of trim sign and LCF sign that

is used gives the wrong sign of 1

trim correction. To

avoid this error a simple rule can be used: if the trim

and the position of center of flotation relative to

midship are in the same “direction” than the sign of

trim correction is positive, otherwise it is negative.

Moreover, it is worth noting here that the formula

(3) is incomplete. This is easy to see when looking at

the units. Performing unit’s calculation of (3) we will

get (t∙m) while it should be (t). The complete formula

of 2

trim correction has the form (4) however, due to

ΔT is one meter, it is commonly reduced to (3). The

ΔT equal to 1 m implies that 2

trim correction is

dedicated for the small values of trim.

( )

0.5 0.5

Tm Tm

MTC MTC

L Tm

+−

−

∆=⋅ ⋅

∆=

(4)

Both trim corrections are added to the

displacement taken from the Hydrostatic Table of

even-keel ship (5):

( )

D DT D D= +∆ +∆

(5)

The last step is to consider the water density that is

measured at the draught reading stage (ρ

m). If ρm is

equal to water density used in HyTa (ρ

HyTa) then the

final value of displacement D=D’ and if not then:

HyTa

(6)

This study is intended to show the accuracy of the

trim correction (3), (4).

698

3 SHIP SELECTED FOR ANALYSIS

For the purposes of trim correction analysis in the

Draught Survey procedure the small bulk carrier Armia

Krajowa was chosen. Ship length between

perpendiculars L

pp=176.65 m, breadth B=30.00 m,

summer maximum draught T

max(ρ=1.025 t/m

)=10.50

m and maximum displacement D=49 212.7 t. As basic,

it was assumed that the data provided in the ship's

documentation are correct and accurate.

The volume of immersed part of the hull

calculated using Bonjean Scale (BS) is the moulded

volume of hull. It does not include volumes of shell

plates and appendages (bilge keel, rudder, propeller,

propeller shafts and struts, roll fins). On the other

hand, the hull volume calculated using Bonjean Scale

should be reduced by the volumes of tunnels that

accommodate bow thrusters. Volumes of the hull

given in the ship Hydrostatic Table the most often are

the moulded volumes.

To determine the total ship displacement, the

moulded volume calculated using the Bonjean Scale or

taken from HyTa, should be multiplying not only by

the water density but also by an additional coefficient

sa (shell and appendages coefficient). For operational

ships draughts, the volume of immersed appendages

is almost constant while the volume of immersed part

of the hull is highly depend on the current draught.

Therefore, c

sa is dependent on the draught too. It is

worth noting that for many small ships, especially

general cargo ships, most often a constant value of c

is assumed.

For ship used in the presented research the c

dependence on draught is shown in Fig. 1. The graph

was developed based on the ship HyTa (even-keel),

using a simple formula:

( )

HyTa

⋅

(7)

where D(T) and V(T) are the displacement and

moulded volume taken from HyTa (even-keel) for the

draught T and ρ

HyTa is the water density used in HyTa.

Figure 1. Shell and appendages coefficient csa for the

bulkcarier Armia Krajowa

Unfortunately, the data in HyTa of Armia Krajowa

are given only for draughts every 0.5 m. To increase

the accuracy of the displacement calculations for

intermediate draught values, c

sa coefficient has been

approximated by the following formula:

( )

2345

0.0187742 0.0258258 0.0675878 0.0763044 0.0308097

1.00084

TTTT

=+−+−+

(8)

In the range of Armia Krajowa operational draught,

the formula (8) gives the c

sa values with a deviation

below 0.003%, in points used for approximation.

4 CALCULATION METHOD AND ITS

VALIDATION

The moulded volume of hull underwater part, for the

ship without heel, was calculated using the Bonjean

Scale. In Armia Krajowa documentation, the Bonjean

Scale includes curves of cross section area (presented

as a function of draught) for up to 44 cross-sections

and the volume calculation formula is as follows:

( ) ( )

nn n n

AT A T

−

∑

(9)

where A

n(Tn) is the transverse area of n cross section

taken from BS for the draught at this section T

n, dn is

the distance between cross sections (n and n+1) and m

is the total number of cross sections.

It needs to be highlighted that the volume

calculations accuracy significantly depends on the

number of cross sections in the stern and bow part of

hull, where the greatest variability of the hull cross

sections shape can be observed. To check if the

number of sections given in BS of Armia Krajowa is

sufficient, the calculations of the hull moulded

volumes were carried out for even-keel ship at

draughts same as given in HyTa. Moulded volumes

calculated with the use of BS and taken from HyTa

turned out to be very close, so to show the divergence,

the graph of moulded volumes differences (ΔV=V

HyTa -

BS) will be better (Fig. 2). In Fig. 2 the solid line

shows ΔV for calculations performed according to

formula (9) where the interpolation between A

n(Tn)

and A

n+1(Tn+1) is linear and the dashed line shows ΔV

for calculations performed using spline interpolation

(a curve between every two points were

approximated by a second degree polynomial based

on 3 consecutive points):

( ) ( )

n nm

V f x dx f x dx

= −

−





= +





∑

∫∫

(10)

Figure 2. Differences of moulded volumes ΔV, calculated

using Bonjean Scale and given in Hydrostatic Table of the

even-keel ship (ΔV =V

HyTa-VBS). Solid line – linear

interpolation; dashed line – spline interpolation

In Fig. 2 its clear that the linear interpolation

provides results closer to HyTa, so this method has

been chosen. Additionally, Fig. 2 shows that the

moulded volumes difference increases with draught

699

(volume of immersed part of hull) increase. However,

if the difference of moulded volumes showed in Fig. 2

will be replaced by the coefficient of relative

difference (11) than it can be seen that the

convergence of the V

HyTa and VBS increases with

draught increase (Fig. 3, solid line). This is fully

justified since the shape of the hull shows greater

variability in the range of small draughts.

( )

HyTa

∆

= +

(11)

Figure 3. Coefficient of the moulded volume relative

difference c

mv. Solid line – actual; dashed line –

approximated by formula (12)

Because the exact values of coefficient cvm can be

calculated only for draught selected in HyTa table, for

convenience, a function (12) approximating its value

was developed. The curve of the approximated values

of c

mv is shown in Fig. 3 (dashed line). Despite minor

differences between actual and approximated value of

cmv, the use of (12) caused that ΔV in any case was

not greater than 4.6 m

(Fig. 4). This corresponds to

approximately 0.1 TPC.

( )

2345

0.018593 0.112466 0.370059 0.496174 0.223627

1.0017

xxxx

=+−+−+

(12)

Figure 4 Difference between the moulded volume given in

Hydrostatic Table (even-keel) and calculated with the use of

Bonjean Scale, after using the coefficient of relative difference

mv (12)

In the Armia Krajowa stability booklet, apart from

even-keel HyTa, the Hydrostatic Table for trimmed ship

(HyTa_t) are also given, for trim -1.00, -2.00 and -3.00

m. Fig. 5 shows difference between the moulded

volume given in the trimmed ship Hydrostatic Table

and calculated using Bonjean Scale and the c

coefficient (12), for trim -2.00 and -3.00 m. This time

the differences are slightly greater than for even keel

ship and are clearly related to the trim value.

Nonetheless, it can still be concluded that the applied

calculation procedure gives satisfactory results since

the ship average TPC is close to 50 t/cm (hull volume

change is close to 50 m

for 1cm draught change).

Figure 5. Difference between the moulded volume given in

trimmed ship Hydrostatic Table and calculated using Bonjean

Scale and the coefficient of relative difference c

mv (12).

Volume 10 m

corresponds to approximately 0.2 TPC

5 CALCULATIONS

The displacement of trimmed ship was calculated

using two methods, with the use of Bonjean Scale and

Draught Survey procedure. Calculations were carried

out for the mean draught (draught at midship) from

5.00 to 11.80 m with the spacing 0.2 m and trim from -

6.00 m (aft trim) to 2.00 m (forward trim) with the

same 0.2 m spacing. The differences between ship

displacement calculated by both methods are shown

in Fig. 6. However, because the difference in

displacement expressed directly in tones is strongly

related to the ship size, it was considered better to

present those differences in TPC units. It should also

be noted, due to the difference sign, that the

displacement difference Δ

disp was calculated as

follows:

disp Bonjean Scale Draught Survey

DD∆= −

(13)

It can be seen in Fig. 6 that apart from small and

medium values of trim (

< 2.00 m) where Δdisp is

small and in the most cases negligible (Δ

disp < 0.25

TPC) the displacement difference quite increases for

bigger trim values. Also, it’s easy to see that for small

draughts (T

M < 7.40 m) the displacement difference is

positive and for higher ones it is negative. Although a

certain tendency of changes can be identified, the

problem is to observe the exact relation between Δ

disp

and the draught. For example, for mean draughts 7.60

and 11.80 m the displacement differences are quite

similar while for T

M = 9.00 m Δdisp is clearly bigger.

Figure 6. Difference between ship displacement calculated

using Bonjean Scale and Draught Survey procedure for the

mean draught from 5.00 to 11.80 m with 0.2 m spacing, as a

function of trim, in TPC units

The calculation results showed in Fig. 6, but in a

different arrangement, are presented in Fig. 7 and 8.

700

These figures show the displacement difference for

different trim values as a function of the ship mean

draught. In Fig. 7 it is clearly visible that the changes

may be difficult to describe with a mathematical

formula. Moreover, an unfavorable phenomenon is

the Δ

disp sign change. For TM < 7.20 m the sign of Δdisp is

positive while above is negative. This mean that

displacement calculation inaccuracy for ship before

and after loading will add up.

Fig. 8 shows the same graphs as Fig. 7 but for the

trim absolute value limited to 3.00 m. Visible, the

small local deviations of graphs are probably caused

by the discrete data format used in the ship

Hydrostatic Table and Bonjean Scale as well as the

precision of this data. Moreover, some of the visible

differences may also be caused directly by the

calculation method used. Apart from the issue of

deviations, Fig. 8 strengthens the conviction that

describing the Δ

disp using a mathematical formula may

be impossible.

Finally, it is worth adding that since the 1

trim

correction is logical and obvious, Δ

disp presented in

Fig. 6, 7 and 8 is directly related to the not entirely

correct 2

trim correction and may be considered as

an error of 2

trim correction.

Figure 7. Difference between ship displacement calculated

using Bonjean Scale and standard Draught Survey procedure,

for trim from -6 m (aft trim) to 2 m (forward trim) with 0.2

m spacing, as a function of draught, in TPC units

Figure 8. Difference between ship displacement calculated

using Bonjean Scale and standard Draught Survey procedure,

for trim from -3 m (aft trim) to 2 m (forward trim) with 0.2

m spacing, presented as a function of draught, in TPC units

6 SECOND TRIM CORRECTION

In the previous section it was stated that the

difference between ship displacement calculated

using Bonjean Scale and Draught Survey procedure

(Δ

disp) may be difficult or even impossible to describe

using a mathematical formula. Moreover, this

approach would require to develop an additional

correction formula to the Draught Survey procedure

trim correction). Focusing on the differences in the

results obtained using two different calculation

methods (Fig. 6, 7 and 8) is not always conducive to

finding a solution.

Fig. 9 shows the 2

trim correction calculated

according to the formula (3) while Fig. 10 shows the

difference between the ship displacement calculated

with the use of Bonjean Scale and the displacement

taken from HyTa increased by 1

trim correction (2):

( )

21BS HyTa M

DD D T D

′

∆ = − +∆

(14)

Figure 9. Second trim correction in the Draught Survey

procedure calculated according to formula (3)

Figure 10. Difference between the ship displacement

calculated with the use of Bonjean Scale and the

displacement taken from HyTa increased by 1

trim

correction (2) → 2

trim correction calculated on the base

of Bonjean Scale.

Considering what was written earlier, it can be

assumed that Fig. 10 shows the correct values of 2

trim correction

( )

′

∆

. Therefore, it is worth

considering whether the formula (3) can be modified

to obtain the calculation results as close as possible to

those presented in Fig. 10.

Formula (3) is a direct derivative of formula (4)

where it was assumed that ΔT is one meter. This

assumption implies that 2

trim correction (3) is

dedicated for small values of trim, and it must be

admitted that for the such trim values it gives

satisfying results (Fig. 6 and 8). Nevertheless, it can

seem that formula (4) with small modification can be

better for larger trim values, maintaining good results

for small and medium values of trims. To test this,

formula (4) was modified to:

( )

0.5 0.5

2.1

TT TT

MTC MTC

+ ⋅∆ − ⋅∆

−

∆=⋅⋅

∆

(15)

The idea of formula (15) was to consider changes

in the size and shape of the waterplane area over the

bigger range of current ship draughts, not only over

1 m range. This can be important since the aft part of

stern can begin more than 0.5 m above the horizontal

701

plane of mean draught T

M. In the first attempt it was

assumed that

Tt∆=

(Fig. 11). Fig. 11 clearly shows

that for such assumption formula (15) did not give

good results. The graphs are smoothed and the

characteristic changes, visible in Fig. 10 in the draught

range from 7 to 11 m, did not show up.

Figure 11. Second trim correction in the Draught Survey

procedure calculated according to formula (13)

To test the formula (15) more thoroughly, the ΔT

value was varied from 0.1 to 1 of the trim absolute

value. The summary results of calculations performed

for trim -6, -4 and -2 m are presented in Fig. 12. It can

be seen that ΔT value bigger than 1 m only slightly

affect the maximum difference between 2

trim

correction calculated using Bonjean Scale and formula

(15):

2 2.1

trim corr

D DD

′

∆ =∆ −∆

(16)

For

∆≥

m and t =-6 m, the total value of

trim corr

D∆

varies with a deviation smaller than 0.2

TPC and for t =-2 m smaller than 0.02 TPC. Also,

worth noting is that reducing ΔT value below 1 m

causes a clear and fast increase in value of

trim corr

D∆

The different values of ΔT, connected with vertical

shift, were also tested, but without satisfying results.

The results of all calculations which were performed

suggest that the LCF changes caused by the ship trim

can’t be accurately determined using MTC changes.

Figure 12. Maximum difference between 2

trim correction

determined using Bonjean Scale and calculated using

formula (15)

Of course, when analyzing the 2

trim correction

for a specific ship, as in the case of this work, formula

(3) can be modified based directly on the results of

calculations which were carried out. In Fig. 7 and 8 it

is enough to consider the draught at which the graphs

change the sign. The modified formula (3) has now

the form:

( )

2.2 0.5 0.5

Tm Tm

PP M

D MTC MTC

+−

∆=⋅⋅ ⋅ −

(17)

where T

+/− is the draught at which the graphs change

the sign.

Although the modification placed in formula (17)

is based on the observation of changes in the graphs

in Fig. 7 and 8, it seems reasonable. The ship trim has

a greater impact on the waterplane area (size, shape,

geometric center) for smaller draughts than for bigger

ones. This is related to greater variability of the hull

shape in the range of small and medium draughts and

can be confirmed by the LCF graph (Fig. 13).

Figure 13. Longitudinal center of flotation (LCF) of the bulk

carrier Armia Krajowa, relative to midship

However, the value of T+/− in formula (17) is not

obvious. This is because the individual graphs change

the sign at slightly different draught (Fig. 7 and 8). For

the ship which is used in this research the calculations

were performed for several values of T

+/− and the best

results were obtained for T

+/−=6.60 m (Fig. 14 and 15).

Comparing the differences in the ship

displacement obtained after using the standard 2nd

trim corr. formula (3) and the proposed one (17), it is

clearly visible that formula (17) gives better results.

Although for

3tm≤

, the maximum differences

practically don’t change (Fig. 8 and 15), at bigger

values of the trim the improvement if quite significant

(Fig. 7 and 14). For trim -6 m, the maximum spread of

differences in Fig. 7 is almost 5 TPC while in Fig. 14 it

is less than 3 TPC.

Figure 14. Difference between ship displacement calculated

using Bonjean Scale and Draught Survey procedure with 2

trim corr. calculated using formula (17) where T+/−=6.60 m,

for trim from -6 m (aft trim) to 2 m (forward trim) with 0.2

m spacing, presented in TPC units

Figure 15. Difference between ship displacement calculated

using Bonjean Scale and Draught Survey procedure with 2nd

702

trim corr. calculated using formula (17) for T+/−=6.66 m and

trim from -3 m (aft trim) to 2 m (forward trim) with 0.2 m

spacing, presented in TPC units

7 CONCLUSIONS

At the beginning, it should be noting that the

conclusions given below apply to the ship used in the

research. To treat them as general conclusions, similar

calculations should be performed for a large group of

ships (of various types).

In Section 4, it was shown that the ship volume

(displacement) calculated with the use of quite simple

formula (9) based on the Bonjean Scale placed in the

ship documentation gives accurate results. For even

keel ship the displacement differences in relation to

Hydrostatic Table were smaller than 0.1 TPC. For

trimmed ship differences were bigger and increases

with increasing trim, however for 3.00 m stern trim,

the maximum difference is smaller than 0.25 TPC. It is

quite small error since the assumed accuracy of the

draughts measurement, during the Draught Survey

procedure, is ±0.5 cm. However. at this point it should

be emphasized that Armia Krajowa documentation

includes Bonjean Scale with data for up to 44 cross-

sections. This is a significant and rather the rare

number of cross sections applied.

The standard version of 2

trim correction (3) used

in the Draught Survey procedure provide good results

for the ship with trim not bigger than 3 m. For such

trim the value of error caused by 2

trim correction

(Δ

disp) should be smaller than 0.5 TPC (Fig. 8 and 15).

Moreover, for

2tm≤

the error should be smaller

than 0.25 TPC and for

tm≤

the error should be

smaller than 0.1 TPC. For ship trim bigger than 3 m

the error increasing rapidly in accordance with trim

(Fig. 6), for most values of the ship mean draught T

However, within a small range of T

M the error of 2

trim correction can be very small and almost

independent of the ship trim (Fig. 7, draught between

7.00 and 7.20 m)

The commonly used form of 2

trim correction (3)

uses the difference of MTC taken for draughts spread

equal to 1 m, independently from the current ship

trim value. Using formula (15), where the draughts

spread value is assumed equal to the absolute value of

current ship trim, does not reduce the calculation

error. Moreover, using formula (15) for ΔT <1 m

causes an increase in error.

Fig. 7 shows an additional issue regarding the 2

trim correction and consequently the calculation part

of DS procedure. For small draughts the sign of the

error (Δ

disp) is positive while for bigger draughts is

negative. This means that displacement calculation

errors for ship before and after loading will add up.

The ship used in this work is a bulk carrier. It is

known that the hull of bulk carriers has the quite full

form and the shape of the stern and bow shows less

variability with respect to draught than for e.g.

container or ro-ro ships. This means that differences

between displacement calculated for trimmed ship

using Bonjean Scale and standard Draught Survey

procedure can be significantly bigger then obtained in

the presented study. Obviously, during standard

operation of container or ro-ro ships the Draught

Survey procedure is not needed. Nonetheless, if there

was a need to determine the displacement of a

trimmed ship, a larger error should be expected when

using the standard Draught Survey procedure.

Regardless of the analysis presented in the paper,

one wonders. Why in the case of ships for which the

Draught Survey procedure may be necessary for

practical use, the tables with accurate values of 2

trim correction or summed both trim corrections are

not included in the documentation. The calculations

discussed in the paper can be performed by the

shipyard's design office without much trouble. In this

study 2

trim corr. is presented in the graph form

(Fig. 10) however it can have the table form as well.

Another solution could be, a bit forgotten, Firsov

Diagram. Actually, the ship documentations with

ready-made trim correction (summed 1

+ 2

) can be

found, but values of this correction are calculated

using the standard Draught Survey procedure,

formulas (2) and (3).

ACKNOWLEDGMENTS

This study was financed by the Gdynia Maritime

University, the research project: WN/2023/PZ/03.

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