CHAPTER 4HYDROSTATIC CURVES

Ships may experience many loading conditions resulting in different draught and trim values. It is important both for the designer and the operator that the underwater hull form characteristics for different loading conditions are known for the safety and efficiency of the ship. These characteristics are calculated for a number of waterlines covering all possible lodaing conditions for the particular ship. These waterlines are taken parallel to the baseline but hull form characteristics for different trim conditions can be calculed by some adjustments. The ship is assumed to be stationary in calm water, hence the calculations are known as the hydrostatic calculations.

The results of the hydrostatic calculations are plotted with suitable scales and fair curves are drawn through the plotted points. These curves are called the hydrostatic curves. The main hydrostatic curves which should be included in any hydrostatic calculations are:

1. Sectional area curves (Bon Jean sectional area curves)2. Sectional moment curves (Bon Jean sectional moment curves)3. Volume and displacement ( and ) curves4. Block coefficient (CB) curve5. Midship section area coefficient (CM) curve6. Prismatic coefficient (CP) curve7. Waterplane area coefficient (CWP) curve8. Waterplane areas (AWP) curve9. Longitudinal centre of buoyancy (LCB) curve10. Longitudinal centre of flotation (LCF) curve11. Vertical centre of buoyancy (KB) curve12. Transverse metacentric radius (BM) curve13. Longitudinal metacentric radius (BML) curve14. Tons per centimeter immersion (T1) curve15. Moment to trim one centimeter (MT1) curve16. Wetted surface area (S) curve

Additional hydrostatic curves may include the change in displacement for one centimeter trim curve and the moments of inertia in transverse and longitudinal directions.

The hydrostatic properties are calculated for a series of waterline levels upto the depth and presented both in tabular and graphical forms. The depth scale is usually selected larger than the length scale for a clear presentation.

The hydrodtatic curves represent an important document indispensable for all calculations concerning the loading conditions and stability of the vessel. All curves are drawn up to a certain scale. As different units appear, such as m (position of centres of buoyancy and flotation, radius of metacentre,…), m2 (waterplane area, sectional area, ..), m4 (moments of inertia), tm/cm (moment to chang trim 1 cm) these units must be presented clearly. Usually the scale is shown alongside the corresponding curve, in some cases the scales for the different quantities are shown underneath the base of the graph.

4.1

The range of draughts to be covered by the hydrostatic curves depends on the difference of draughts corresponding to the light ship displacement and the maximum draught in any practical loading condition. To obtain a smooth and complete run of the curves, they are often calculated for each waterline from baseline upto the highest waterline close to the deck level. For warships, fishing vessels, and other non transport ship types, however, with relatively small difference of light ship draught and loaded draught, the hydrostatic curves are calculated for the range of draughts occuring during practical operation.

The use of hydrostatic cırves is limited to parallel immersions or emersions of the vessel (parallel to the design waterline). For small angles of trim we can assume that the volume of the immersed wedge is equal to the volume of the emerged wedge. The displacement of the trimmed vessel is then equal to that corresponding to the mean draught. For large angles of trim, however, the error will be too large. In that case, we have to use another set of curves called Bonjean’s curves which will be explained in the following section.

4.1. BonJean Sectional Area Curves

A sectional area curve is the integral curve of a section, i.e. it shows the the area of the section upto any draught level as shown in Figure 4.1. Thus a sectional area is given by

where y(z) are the half breadths.

Figure 4.1. Derivation of sectional area curve

The principal use of sectional area curves is determining displacement volume at any level or trimmed waterline. In the case of a trimmed waterline, the trim line may be drawn on the profile of the ship and draughts read at which the Bonjean sectional area curves are to be entered. By

4.2

Y Y

XX

WL5 WL5

WL4 WL4

WL2

WL1

WL3

WL2

WL1

WL3

A2

A2

integrating the sectional areas along the length the displacement at any trimmed condition can be calculated.

Bonjean sectional area curves are usually plotted from association station lines in the profile plan against a common scale of draught as shown in Figure 4.2. A common horizontal scale is selected for the areas which will provide a clear presentation. This arrangement is convenient for placing and locating trim lines on the profile.

A useful design curve is the curve of sectional areas which is calculated by integrating the sectional area of each section upto the waterline in question along the ship length, as shown in Figure 4.2. The shape of the curve of sectional areas represent the distribution of buoyancy along the length. The area under the curve of sectional areas is equal to the displacement volume upto the waterline in question.

Figure 4.2. Curve of sectional areas

4.3

Ax

Ax

x

4.2. BonJean Sectional Moment Curves

Bonjean sectional moment values are determined by calculating the moment of sectional areas about the baseline at each waterline level. The total vertical moment about the baseline may be found by integrating the sectional moments along the length of the ship. The vertical height of cetre of buoyancy, KB, can be found by dividing the total vertical moment by the displacement volume. The vertical sectional moments are also useful in flooding calculations.

Figure 4.3. Sectional area and moment curves

4.3. Volume and Displacement Curves

The immersed volume for each waterline level is a measure of the volume of fluid displaced by the ship while floating at a draught upto that waterline. The displacement volume can be calculated in two different manners:

by integrating the sectional areas along the ship length

by integrating the waterplane areas along the depth direction

The displacement is equal to the displacement volume times the density of water in which the ship floats.

4.4

Sectional area curve

Sectional moment curve

The displacement volume calculated directly from the offsets is called the moulded displacement volume. This value must be corrected by adding the volumes of the shell and appendages like bilge keels, raudder, shafts, etc. to define the total displacement. This correction depends on the type of the ship but is usually around 0.006 - 0.020

The volume and displacement curves are plotted from the aft perpendicular on the left hand side of the graph.

4.4. Block Coefficent (CB ) Curve

The block coefficient is defined as the ratio of the displacement volume upto any waterline to the volume of rectangular prism with length, breadth and mean draught of the ship at that waterline. Thus.

These curve indicates the change in fullness at different draughts. The curve is drawn from the AP with a suitable scale which is usually the same for all other form coefficients.

4.5. Midship Section Coefficent (CM ) Curve

The midship section coefficient at any draught is the ratio of the immersed area of the midship section to that of a rectangle with breadth and draught of the ship at that waterline. Thus

The curve is drawn from the AP with the same scale of other form coefficients.

4.6. Prismatic Coefficent (CP ) Curve

The prismatic coefficient gives the ratio between the displacement volume () and a prism whose length equals the waterline length of the ship and whose cross section equals the midship section area. Thus

The prsimatic coefficient is a measure of the longitudinal distribution of a ship’s buoyancy. If two ships with equal length and displacement have different prismatic coefficients, the one with the smaller value of CP will have the larger midship sectional area and hence a larger concentration of the volume of displacement amidships.

The curve is drawn from the AP with the same scale of other form coefficients.

4.7. Waterplane Area Coefficient (CWP ) Curve

4.5

The waterplane area coefficient is defined as the ratio between the area of the waterplane and the area of a circumscribing rectangle. Thus

The curve is drawn from the AP with the same scale of other form coefficients.

4.8. Waterplane Area (AWP ) Curve The waterplane area is required to determine the change in draught when small weights are loaded

and discharged. The waterplane area at a given height is

where y is the sectional offset values.The waterplane areas must be calculated at a sufficiently large number of waterlines to allow a

well-defined curve to be drawn over the range of draughts covering the operational profile of the ship.

4.9. Longitudinal Centre of Buoyancy (LCB) Curve The position of buoyancy affects the stability and trim of a ship. LCB is the distance of the centre

of buoyancy from a specified transverse reference plane, usually the midship section, as shown in Figure 4.4. In some cases LCB may be measured from the FP or AP, so the reference axis should be specified in the drawing.

4.10. Longitudinal Centre of Flotation (LCF) Curve The centroid of each waterplane is also the centre of flotation for that waterplane. A weight added

to a vessel at the centre of flotation would produce paralle sinkage, with no change of trim or heel. The longitudinal position of flotation is required to calculate changes in draughts at bow and stern as a results of changes in trim due to loading, discharging or shifting weights aboard the ship. It is usually plotted with respect to midships along with the longitudinal centre of buoyancy as shown in Figure 4.4.

Figure 4.4. LCB and LCF curves

4.11. Vertical Centre of Buoyancy (KB) Curve

4.6

LCB

LCF

KB is the height of the centre of buoyancy (B) from the baseline or keel (K). The height of buoyancy affects the initial stability of the ship, hence it is one of the most important hydrostatic characteristics. KB can be calculated in two different manners

For each waterline the total moment of sections about the baseline is divided by the volume, or The total moment of waterplanes upto the level of the waterline in question

4.12. Transverse Metacentric Radius (BM) Curve

The transverse metacentric radius is the vertical distance between the centre of buoyancy (B) and the transverse metacentre (M). It can be calculated as

where I is the transverse moment of inertia of the waterplane area about the longitudinal centreline and is the displacement volume.The height of the transverse metacentre above the baseline is called KM, which is found by adding the height of the centre of buoyancy KB to the metacentric radius BM. Thus

KM=KB+BM

The curves of KB and BM are drawn with the same scale, thus the KM values can be read directly.

Figure 4.5. KB and BM curves

4.13. Longitudinal Metacentric Radius (BML) Curve

4.7

KM

KB BM

AP FP

KB 1 cm= x mBM 1cm= x m

The longitudianl metacentric radius is the vertical distance between the centre of buoyancy (B) and the longitudinal metacentre (ML). It may be calculated as

where IL is the longitudinal moment of inertia of the waterplane area about a transverse axis through the longitudinal centre of flotation (LCF).

For conventional ship forms hence the transverse and longitudinal moments of inertia are

the smallest and the greatest, respectively. This can be shown in the following figure.

Figure 4.6. Transverse and longitudinal moments of inertia

4.14. Tons per Centimeter Immersion (T1) Curve

When a weight is loaded on a ship, the displacement must increase by exactly the same amount in order that the law of flotation is maintained. This fact is used to define a useful hydrostatic property, the tons per centimeter immersion which is used to determine small changes in draught of a ship that result from loading or discharging relatively small weights.

Let us assume that a small weight is loaded at the centre of flotation so that a parallel sinkage will occur, i.e. the increase in draught is the same throught the length of the ship. The volume of the immersed layer is

4.8

y

c

dAx

Y

X

v

F

where AWP is the waterplane area and t is the parallel sinkage. The buoyant force created by parallel sinkage must be equal to the added weight

This equation is valid only if the ship’s sides are vertical, i.e. wall-sided. However, if the parallel sinkage is relatively small, the approximation is acceptable.

The tons per centimeter immersion can be calculated by eqauting the added weight to the buoyant force of a parallel sinkage layer of 1 cm thickness

Figure 4.7. Waterplane area and tons per centimeter immersion curves

4.15. Moment to Change Trim 1cm (MT1) Curve

The moment necessary to change trim by a fixed quantity is an important characteristic of a vessel and one frequently used for loading studies. This can be found by

where is displacement in metric tons, GML is the longitudinal metacentric height in meters, and L is the length of waterline in meters.

4.16. Wetted Surface Area (S) Curve

4.9

Waterplane area 1cm = x m2

tons per centimeter immersion1cm = x t/cm

For a ship floating at a given waterline, the total area of the outer surface in contact with the water is known as the wetted surface. The wetted surface is required in resistance calculations where the frictional resistance of the ship is directly proportional to the wetted surface area. The wetted surface may also be used in estimating the amount of paint required to coat the ship’s bottom upto a specifid waterline. The wetted surface area is also useful for estimating the weight of the shell.

Several empirical formulae have been proposed to determine the wetted surface area of ships upto the design waterline. Some of these expressions are given in the following table

Reference Expression Unit systemFroude metric

Mumford metric

DennyT

LT7.1S

metric

Admiralty)

metric

Wageningen metric

The calculation of wetted surface area requires an expansion of the moulded surface upto the desired waterline. This will require to measure the distance along the contour of each section from the centerline at the bottom upto any given waterline. This distance is known as the half girth of the section upto that waterline. On a scale drawing of body plan, the half girths may be measured by rolling a map measurer, bending a flexible thin batten, or a straight measuring scale or strip of paper may be placed in contact with the curve of the section at the starting point and thereafter kept in contact with and tangent to the curve at successive points, by rotating the strip of paper slightly with the paper held in place at the point of contact by the point of a pencil.

The half girths at selected sections may be plotted as ordinates along the ship length. A fair curve passing through these points will enclose an area equal to the half of the total wetted surface upto the given waterline. This area may be calculated by numerical integration. The half girths may also be calculated by using numerical integration techniques.

The distance along the contour of a curve defined by y=f(x) is given by

Consider the typical ship section given in Figure 4.8. We assume that this section may be represented by a second order polynomial between z=0 and z=2h, as follows

4.10

Figure 4.8. Typical ship section.

The vertical spacing between waterlines is assumed to be equal. Then the unknown coefficients a0, a1 and a2 can be determined from the boundary conditions

Then the unknown coefficients are

where m=y1-y2 and n=y1-y3.

In order to apply the Simpson’s first rule we define . Then the half girth for the first

three ordinates is

where

4.11

y

z

H

0

h

y0

y1

y2

This can be achieved in a tabular format as follows

Section Ordinate m,n SM Product0 y0 m=y0-y1

n=y0-y2

1 Z0

1 y1 4 4Z1

2 y2 1 Z2

=Z0+4Z1+Z2

Total girth :

4.17. Draught Marks

Draught marks are used for determining displacement and other properties of the ship for hydrostatics, stability, and damage stability. The draught marks indicate the depth of the baseline below the waterline. Usually Roman numerals 3” in height or Arabic numerals 6” in heaight are used for draught marks.

4.18. Draught Diagram

The draught diagram is a nomogram used for determining the following hydrostatic properties

ship’s displacement moment to trim one cm (MT1) tons per cm immersion (T1) height of metacentre (KM) longitudinal centre of flotation (LCF) longitudinal centre of buoyancy (LCB)

A typical draught diagram is shown in Figure 4.9.

4.12

Forward draught marks

(m)8

7

6

5

Displacement (t) KM(m)

4000

38004000

3600

3400

3200

9.0

8.0

7.0

6.0

3 2 1

LCF

Tons per cm

immersion (t/cm)

12

11

10

9

8

7

6

LCB (m)

2.0

1.0

0.00.5

1.5

0.5

1.0

1.5

2.0

After draught marks

(m)8

7

6

5

Moment to change

trim 1 cm (tm)

200

180

160

140

120

100

LBP

Figure 4.9. Typical draught diagram

In order to use the draught diagram aft and forward draught marks are read and a straight line connecting these marks is drawn on the diagram. The displacement, KM, T 1, MT1, LCB and LCF values are read offf rom the intersection of this straight line with corresponding scales.

4.19 Sample tables for hydrostatic curves calculations

1) The sample calculations of the sectional areas and moments are given in the following table:

SECTION 1 SECTION 2WL yi SM Prod. M.A Prod. WL yi SM Prod. M.A Prod.

0 ½ 0 0 ½ 0½ 2 ½ ½ 2 ½1 3/2 1 1 3/2 1

4.13

2 4 2 2 4 23 2 3 3 2 34 4 4 4 4 45 1 5 5 1 5

SECTION 3 SECTION 4WL yi SM Prod. M.A Prod. WL yi SM Prod. M.A Prod.

0 ½ 0 0 ½ 0½ 2 ½ ½ 2 ½1 3/2 1 1 3/2 12 4 2 2 4 23 2 3 3 2 34 4 4 4 4 45 1 5 5 1 5

SECTION 5 SECTION 6WL yi SM Prod. M.A Prod. WL yi SM Prod. M.A Prod.

0 ½ 0 0 ½ 0½ 2 ½ ½ 2 ½1 3/2 1 1 3/2 12 4 2 2 4 23 2 3 3 2 34 4 4 4 4 45 1 5 5 1 5

Sectional Areas:

Sectional Moments:

Where ( i ) refers to the number of the sections and (h) is the spacing between waterlines.

2) The sample calculations of the displacement and displacement volume are given in the following table:

SECTION SECT.AREA

SM PROD. M.A PROD. SECT.MOMENT

SM PROD.

0 ½ -5 ½½ 2 -4.5 21 3/2 -4 3/22 4 -3 43 2 -2 2

4.14

4 4 -1 45 2 0 26 4 1 47 2 2 28 4 3 49 3/2 4 3/2

91/2 2 4.5 210 ½ 5 ½

Displacement Volume:

Displacement: Longitudinal center

Of buoyancy :

Vertical center of

Buoyancy :

Where (s) is the spacing between sections.By using the above equations, the block coefficient, the midship area coefficient and the prismatic

coefficient may be calculated as follows:

3) The sample calculations of the AWP, BM, LCF and BML are given in the following table:

SECTION yi SM PROD M.A PROD. M.A PROD. yi3 SM PROD.

0 ½ -5 -5 ½½ 2 -4.5 -4.5 21 3/2 -4 -4 3/22 4 -3 -3 43 2 -2 -2 24 4 -1 -1 45 2 0 0 2

4.15

6 4 1 1 47 2 2 2 28 4 3 3 49 3/2 4 4 3/2

91/2 2 4.5 4.5 210 ½ 5 5 ½

Water plane area :

By using the water plane area, tons per centimeter immersion and water plane area coefficient can be calculated as follows:

Tons per centimeter immersion :

Water plane area coefficient :

Transverse moment of inertia :

Transverse radii of metacenter:

Longitudinal center of flotation:

Moment of inertia due to amidship:

Moment of inertia due to centerOf flotation :

Longitudinal radii of metacenter:

Moment to trim one centimeter:

4) The sample calculations of the wetted surface area are given in the following table:

SECTION 1WL y m&n Z SM PROD WL y m&n Z SM PROD

0 1 0 1½ 4 1 41 1 2 1

4.16

1 1 2 12 4 3 43 1 4 1

3 1 4 14 4 5 45 1 6 1

5 1 6 16 4 7 47 1 8 1

This calculation should be repeated for every section.

WL1 WL2 WL3 WL4 WL5 WL6 WL7 WL8SEC SM G PR G PR G PR G PR G PR G PR G PR G PR0 ½½ 21 3/22 43 24 45 26 47 2

4.17

8 49 3/2

91/2 210 1/2

WS2= WS3= WS4= WS5= WS6= WS7= WS8=

FINAL TABLE FOR HYDROSTATIC CURVES

DEFINITIONS WL1 WL2 WL3 WL4 WL5 WL6

4.18

I.T.U.FACULTY OF NAVAL ARCHITECTURE AND OCEAN ENGINEERING, DEPARTMENT

OF OCEAN ENGINEERING

SHIP THEORY

PROJECT 1. HYDROSTATIC CURVES

4.19

Dr. Hakan AKYILDIZ

SHIP’s LENGTH : BREADTH : DEPTH : DRAUGHT :

STUDENT’s NAME : FACULTY NO :

The hydrostatics curves will be calculated and plotted on a paper(min. size A3) with suitable scale. The presentation of the project is as follows:Page 1: Cover pagePage 2: Offset tablePage 3: Area and moment calculations for station 0Page 4: Area and moment calculations for station ½Page 5: Area and moment calculations for station 1Page 6: Area and moment calculations for station 2Page 7: Area and moment calculations for station 3Page 8: Area and moment calculations for station 4Page 9: Area and moment calculations for station 5Page 10: Area and moment calculations for station 6Page 11: Area and moment calculations for station 7Page 12: Area and moment calculations for station 8Page 13: Area and moment calculations for station 9Page 14: Area and moment calculations for station 91/2Page 15: Area and moment calculations for station 10Page 16: Calculations for waterline 1Page 17: Calculations for waterline 2Page 18: Calculations for waterline 3Page 19: Calculations for waterline 4

4.20

Page 20: Calculations for waterline 5Page 21: Calculations for waterline 6Page 22: Wetted surface calculationsPage 23: Final tablePage 24: Hydrostatic curves

Hydrostatic Curves to be included:

Sectional area curves (Bonjean curves) Sectional moment curves Waterplane area curve (AWP) Volume and displacement curves Vertical center of buoyancy curve (KB) Longitudinal center of buoyancy curve (LCB) Midships section area coefficient curve (CM) Block coefficient curve (CB) Prismatic coefficient curve (CP) Waterplane area coefficient curve (CWP) Longitudinal center of flotation curve (LCF) Tons per centimeter immersion curve (T1) Transverse metacentric radius curve (BM) Longitudinal metacentric radius curve (BML) Moment to trim one centimeter curve (MT1) Wetted surface area curve (WS)

4.21