Hydrostatic and Stability

Sailboat “Wayra “ Hydrostatic and Stability Calculations “

By boat owner: Gilberto Castro C. Calculations done in base at concepts from the book “ Principles of Yacht Design “ by Lars Larsson & Rolf E Eliasson.

CONTENT: A- Calculations done at the Design Water Line (DWL):

1 – Brief Introduction and definitions. 2 – Calculation methods 3 - Simpson’s Rule 4 – Calculation of displacement.

4.1 – Obtaining data with AutoCAD. 4.2 – Using Simpson’s rule.

4.3 – Calculation of the Prismatic Coefficient. 4.4 – Calculation of the Block Coefficient.

5 – Calculation of Water Plane Area 5.1 – Calculating TPC (Tons per centimetre of immersion) 6 – Longitudinal Centre of Buoyancy of canoe body (LCB) 7 – Longitudinal Centre of Flotation (LCF) 8 – Longitudinal Moment of Inertia (Il). 9 – Transverse Moment of Inertia (It). 10- Transverse stability at small angles 11- Longitudinal Stability at small angles

A- Calculations done at each draft line: Next web page update: soon.

1- Brief Introduction and definitions: Hydrostatics and stability represents perhaps the most important aspect for design and for safe of the boat since the properties of a yacht in these respects reflect its ability to carry the required weight and to withstand the heeling moment from the sails, but the exact knowledge of stability is not restricted to the static case with no waves on the water surface, therefore I will make also some dynamic calculations for “Wayra” that I will show here. Just to note that all calculations for my sailboat “Wayra” has been made for myself, them do not come together with the original plans from the yacht designer. (Bruce Roberts). I got the dimensions from the plans and from the real frames. First of all I write some quick and basic definitions and information to show how to make the calculations. Basic definitions used in yachts: Length Overall (LOA) The maximum length of the hull from the forwardmost point on the stem to the extreme after end. According to common practice, spars or fittings, like bowsprits, pulpits etc are not included and neither is the rudder. Length of water line (LWL) The length of the designed waterline (often referred to as the DWL). Length between perpendiculars (LPP) This length is not much used in yachting but is quite important for ships. For ships (cargo ships) it is very important because the Draft line marks are painted at certain distance from the Perpendiculars and the trim calculations will be done on this. The forward (FP) is the forward end of the designed waterline, while the aft perpendicular (AP) is the centre of the rudder stock. Beam (B or Bmax) The maximum beam of the hull excluding fittings, like rubbing strakes. Bean of waterline (BWL). The maximum beam at designed waterline. Draft (T) The maximum draft or the yacht when floating on the designed waterline, TC is the draft on the hull without the keel (The canoe body). Depth (D) The vertical distance from the deepest point of the keel to the sheer line (see below). Dc is without the keel.

Displacement Could be either mass displacement (m). The mass of the yacht, or volume displacement (V or D)The volume of the immersed part of the yacht. Mc, Vc, and Dc are the corresponding notation without keel. Midship section For ships, this section is located midway between the fore and aft perpendiculars. For yachts it is more common to put it midways before and aft ends of the waterline. The area of the amidships section (Submerged part ) is denoted Am, with and index ‘c’ indicating that the keel is not included. Maximum area section For yachts the maximum area section is usually located behind the amidships section. Its area is denoted Ax (Axc). Prismatic coefficient (Cp) This is the ratio of the volume displacement and the maximum section area multiplied by the waterline length. Cp = D/(Ax . Lwl). This value is very much influenced by the keel and in most yacht applications only canoe body is considered. The prismatic coefficient is representative of the fullness of the yacht. The ends, the larger the Cp. Its optimum value depends on the speed. Centre of buoyancy (B) The centre of gravity of the displaced volume of water. Its longitudinal position denoted as LCB and vertical position denoted by VCB. Centre of gravity (G) The centre of gravity of the yacht must be on the same vertical line as the centre of buoyancy. In drawing G is often marked with a special symbol created by a circle and a cross. This is used also for marking geometric centres of gravity. Sheer line The intersection between the deck and the topside. Traditionally, the projection of this line on the symmetry plane is concave, the ‘sheer’ is positive. Zero and negative sheer may be found on some extreme racing yachts and powerboats. Freeboard The vertical distance between the sheer line and the waterline. This is the reserve of buoyancy. 2-Calculation methods: Due to the shape curved of the frames, there is not too easy to calculate the frame areas. There are different ways to obtain the areas on closed curves: - One way might be to draw the closed curve on a square grid and just count the number of

squares. In most cases this method is accurate enough, but it is tedious.

- One other way is to make use of the planimeter, this method is fast and accurate but few

amateur designers have access to this handy instrument. - By using CAD (Computer Aided Design) programs: CAD techniques are always used

nowadays by professional designers. The CAD programs without much work fair hull can be produced rapidly and different requirements my be satisfied such as a given prismatic coefficient or longitudinal centre of buoyancy, displacement, stability at small and large heel angles.

- The best choice for many designers is to compute the area using a simple numerical method This method is called Simpson's rule

Figure 2.1 3-Simpon’s Rule Simpson’s rule is quite popular in naval architecture. Since the sequence of operations is always the same when applying Simpson’s rule as special scheme, shown in Fig. 1. The distance between the end points of the interval, in this case Xo and X10, is divided into an even number of equidistant steps, in this case 10. The step size is denoted S. Values of the function Y are computed for all X-values and may be inserted into the table in the column “Ordinate value”. By multiplying each value by its Simpson multiplier, 1 for the end values and 4 and 2 alternating for the others, and adding all the products the ‘sum of products ‘ is obtained. The area A under the curve Y is the simply obtained as this sum multiplied by the step size divide by 3.

A = S / 3. (Sum of products) 4- Calculation Displacement at DWL (canoe body displacement) 4-1 obtaining frame areas with AutoCAD. According to Archimedes’ principle: When an object floats in a liquid, the weight of the volume of liquid displaced by the object is equal to the weight of the object. Thus, if you know the volume of the displaced liquid, the weight of the body is found multiplying the volume by the density of the liquid, in this case fresh water or salt water. To obtain the total Displacement of all yacht when the hull is submerged until the DWL, the area of each section until DWL has to be obtained first. With the “ Wayra” lines drawing and Using AutoCAD program I got the area of each section (Figure 4.2), then Using Simpson’s rule I obtain the Displacement as shown here:

S.M.14242424241

Sum of products

S.M. = Simpson´s multipliers

4Y72Y84Y9Y10

YoProduct

4Y12Y24Y32Y44Y52Y6

9 Y910 Y10

7 Y78 Y8

5 Y56 Y6

3 Y34 Y4

1 Y12 Y2

Ordinate No. Ordinate Value0 Yo

Note that the frame 0 is minimum submerged and frame No.10 is completely out, and the keel is not taken in account for calculations. The Keel volume I will calculate later.

Figure 4.2

4 – Calculating displacement. 4.2 – Using Simpson’s rule. Here the area of each section I obtained dividing each frame with 10 lines, spaced S distance. Once obtained the areas, I use again Simpson's rule to obtain the Displacement. Figure 4.3

Section N.0 Section N.1

Ordinate No.

Ordinate value A

S.M (B) Product (AXB)

Ordinate No.

Ordinate value A S.M (B) Product (AXB)

Y0 0 1 0,00 Y0 0 1 0,00 Y1 0,0028 4 0,01 Y1 0,128 4 0,51 Y2 0,0055 2 0,01 Y2 0,2172 2 0,43 Y3 0,0082 4 0,03 Y3 0,2823 4 1,13 Y4 0,0109 2 0,02 Y4 0,3327 2 0,67 Y5 0,0136 4 0,05 Y5 0,3746 4 1,50 Y6 0,0162 2 0,03 Y6 0,4115 2 0,82 Y7 0,0188 4 0,08 Y7 0,4436 4 1,77 Y8 0,0212 2 0,04 Y8 0,4711 2 0,94 Y9 0,0236 4 0,09 Y9 0,4939 4 1,98 Y10 0,026 1 0,03 Y10 0,5119 1 0,51

Suma of products = 0,40 Suma of products = 10,27 A = S/3 x (Sum of Products) A = S/3 x (Sum of Products)

S= 0,0074 S= 0,0539 S/3 0,002466667 S/3 0,017966667 S/3 x (sum of products) = 0,0010 S/3 x (sum of products) = 0,1845 As = A x 2 (both sides)= 0,0020 As = A x 2 (both sides)= 0,3689

Section N.2 Section N.3 Ordinate No.

Ordinate value A


Ordinate No.




S= 0,0696 S= 0,078 S/3 0,0232 S/3 0,026 S/3 x (sum of products) = 0,4524 S/3 x (sum of products) = 0,7250

As = A x 2 (both sides)= 0,9049 As = A x 2 (both sides)= 1,4501


Ordinate value A


Ordinate No.




S= 0,0828 S= 0,0836 S/3 0,0276 S/3 0,027866667 S/3 x (sum of products) = 0,9538 S/3 x (sum of products) = 1,0959



Ordinate value A


Ordinate No.




S= 0,0806 S= 0,0728 S/3 0,026866667 S/3 0,024266667 S/3 x (sum of products) = 1,1479 S/3 x (sum of products) = 1,0273 As = A x 2 (both sides)= 2,2958 As = A x 2 (both sides)= 2,0545


Ordinate value A


Ordinate No. Ordinate value A



Suma of products = 40,53 Suma of products

= 37,46 A = S/3 x (Sum of Products) A = S/3 x (Sum of Products)

S= 0,05393 S= 0,0362 S/3 0,017976667 S/3 0,012066667 S/3 x (sum of products) =

0,7285

S/3 x (sum of products) = 0,4520


Calculation of the Volume Displacement (Dc=Canoe body displacement)

Ordinate No. Area section (A) S.M (B) Product (A X B) 0 0,002 1 0,002 1 0,3689 4 1,4756 2 0,9049 2 1,8098 3 1,4501 4 5,8004 4 1,9076 2 3,8152 5 2,1918 4 8,7672 6 2,2958 2 4,5916 7 2,0545 4 8,218 8 1,4571 2 2,9142 9 0,9040 4 3,616

10 0 1 0 Sum of Products = 41,01

S = distance between sections Dc = S/3 X (sum of products) S = 1,29 S/3 = 0,43 Dsw = Displ. In Salt water = Dc x 1.025

DC = 17,59 m3 Dsw

= 18,03 Metric Tons

Note: There is one small difference between Displacement calculated using AutoCAD to obtain sectional areas with the displacement obtained using Simpson's rule also to obtain the sectional areas. Displacement obtained with AutoCad program = 18.15 m3 Displacement obtained using Simpson’s rule = 17.59 m3

4.3 – Calculation of Prismatic Coefficient (Cp). This is the ratio of the volume displacement and the maximum section area multiplied by the waterline length. This value is very much influenced by the keel and in most yacht applications only the canoe body is considered. The prismatic coefficient is representative of thefullness of the yacht. The fuller the end, larger the Cp. Its optimum value depends on the speed. Here I consider only the Canoe body for Sialboat Wayra.

Ax = 2,3804 m ² = 25,622 foot ²

Dc = 18,15 m3 = 640,96 cubic feet

LWL = 12,87 mts = 42,224 Feet

Ax = Maximum sectional area

[Section No. 6]

Cp = 0,59 4.4 – Calculation of Block Coefficient (Cb).

In general ship hydrodynamics this coefficient is quite important, but it is not commonly used in yachts design. The

volume displacement is now divided by the volume of a circumscribed block (only canoe body value is relevance). T = 0.852 mts. (Draft without keek. Only canoe body) LWL = 12.87 mts. BWL = 3.99 mts Cb = 0.41

5 – Calculation of the Water Plane Area:

The area inside the designed waterline (DWL) is important in several respects:

1- Its size determines the weight per mm immersion. It is the weight required to sink the hull a certain

distance. 2- Its centre of gravity is located on the axis around which the hull is trimmed, when moving a weight

longitudinally on board. 3- The so-called moment of inertia (called also second moment of area) around a longitudinal axis

determines the stability at small angles of heel. 4- The moment of inertia around a transverse axis through the centre of gravity (of the area) yields the

longitudinal stability, it is the moment required to trim the hull a certain angle. The calculation of the area is done using Simpson's rule as shown here:

Water Plane area

0,0000

0,5000

1,0000

1,5000

2,0000

2,5000

1 2 3 4 5 6 7 8 9 10 11

Sections

Are

as

" Wayra " Water Line Plane Area (ADWL). Calculation using Simpson´s rule. Calculo del area del plano en la linea del calado de diseño. Usando " El metodo de Simpson"

Ordinate No. Half beam of each Section at DWL SM Product

0 0,0028 1 0,0028 1 0,5200 4 2,0800 2 0,9680 2 1,9360 3 1,3560 4 5,4240 4 1,6490 2 3,2980 5 1,8490 4 7,3960 6 1,9960 2 3,9920 7 1,9160 4 7,6640 8 1,6850 2 3,3700 9 1,1290 4 4,5160

10 0,0000 1 0,0000 Sum of Products = 39,6788

A total = 2 x S/3 (Sum of products)

S = 1,287 Distance between sections. (Distancia entre cuadernas) S/3 0,43 Adwl = 34,04 M²

5.1 – Calculation of TPC (Tons per centimetre of immersion) at ADWL The additional displacement when sinking the hull 1 cm is 0.01 X Adwl m2 X The mass of this volume, corresponding to the applied mass on the hull, is: TPC = d x 0.01 x Adwl where d is the water density. ( 1.025 for salt water). TPC = 1.025 x O.O1 m x Af (m2) TPC = 1.025 tons/m3 x 0.01m x 34.04 m2 TPC = 0,348 tons [ 348 kg por centimeter] This means that at the DWL to sink the hull one centimeter, will be necessary to add 348 kg. Weight to the “ Sailboat Wayra”. 6 – Longitudinal Centre of Buoyancy of canoe body ( LCB ) at DWL. The Centre of gravity of the displaced volume of water is called Centre of buoyancy. The longitudinal position is denoted by LCB. The vertical position is denoted by VCB. The moment created by a force with respect to a perpendicular axis is the product of the force and the distance to the axis (the lever arm). This concept can be used for finding the centre of gravity of a body. By definition the centre of gravity is the point where the mass of the body may be assumed concentrated. The gravitational force bay be assumed acting at this point. To calculate the distance to the centre of gravity from one arbitrary axis, is to add the moments of the different parts of the body with respect to the axis. This gives the resulting moment, which must be equal to that of the concentrated mass at the centre of gravity. The axis chosen is located athwart ships at the FP. I will use this method to find the Sailboat Wayra Centre of Gravity (CG) later on, before I launch to the water the boat, I will have to calculate all the installed equipments weights. To calculate the real Sailboat Wayra CG will take a some time from now. (Dec. 2007) A corresponding computation can be done for the centre of gravity of the displaced volume of water: Centre of buoyancy. Therefore I calculate first the longitudinal position of it, LCB using the FP as reference axis. Each section of the hull may be considered as contributing to the moment by an amount proportional to its area multiplied by its distance from the FP. Thus a ‘curve of sectional moments’ can be constructed in a similar way to the curve of sectional areas. The area on the new curve represent the total moment, from which the position of the centre of buoyancy can be obtained as explained in the figures and calculation below.

Calculating Longitudinal center of Buoyancy of Canoe body at DWL

Ordinate No.

Area of each section

(A)

dist.from Fp to each

section ( S)

Moment (A x S)

S.M Product

(AxS) x SM

0 0,0016 0,0000 0 1 0,0000

1 0,3920 1,2870 0,504504 2 1,0090

2 0,9626 2,5740 2,4777324 4 9,9109

3 1,5227 3,8610 5,8791447 2 11,7583

4 2,0083 5,1480 10,3387284 4 41,3549

5 2,3176 6,4350 14,913756 2 29,8275

6 2,3804 7,7220 18,3814488 4 73,5258

7 2,1456 9,0090 19,3297104 2 38,6594

8 1,5828 10,2960 16,2965088 4 65,1860

9 0,7304 11,5830 8,4602232 2 16,9204

10 0,0000 12,8700 0 1 0,0000

Suma of products = 288,15

Am = S/3.(Sum of products)

S = 1,287

S/3 = 0,429

X lcb = Am/Dc

Am = 123,62

Dc = 18,60 Dc=Displacement

X lcb = 6,65 Mtrs. from Forward Perpendicular.

X lcb = Distance from FP to the Centre of buoyancy (El centro de carena se encuentra a 6,65 metros desde la Perpendicular de Proa )

7 – Longitudinal Centre of Flotation (LCF ) The sum of moments obtained from multiply distance from each Section to the Forward Perpendicular axis and divided between the sum of the distances will give the geometric centre of gravity of water plane Area, usually called, Centre of flotation. For now I made calculation at the DWL. Later I will make the calculations for all draft lines.

Longitudinal Center of Flotation (LCF)

Section Beam

(A)

dist.from Fp to each section

( B) Product (AXB)

0 0,0560 0,0000 0 1 1,0400 1,2870 1,33848 2 1,9360 2,5740 4,983264 3 2,7120 3,8610 10,471032 4 3,2980 5,1480 16,978104 5 3,6980 6,4350 23,79663 6 3,9920 7,7220 30,826224 7 3,8320 9,0090 34,522488 8 3,3700 10,2960 34,69752 9 2,2580 11,5830 26,154414

10 0,0000 12,8700 0

26,1920 183,768156

LCF= S moments / S beams

LCF = 7,02 mts. From FP.

8 – Longitudinal Moment of Inertia (Il ). Moment of Inertia of the water plane is a measure of a vessel's resistance to rolling motion about the vessel's longitudinal axis. For example, the moment of inertia of a log would be very small, because there is little resistance to rolling motion about its longitudinal axis. A broad-beamed barge, on the other hand, would offer a great deal of resistance to rolling motion about its longitudinal axis. It would have a large moment of inertia. This moment that resists motion is actually made up of an infinite number of moments, each of which are composed of the product of the elementary area and the square of the distance from the axis. The FP is used as transverse axis to construct the curve of sectional moments of inertia, which each ordinate is the product of the waterline half- width and the square of distance from the FP as shown here:

Ordinate No.

bo = Half-width of each section

(A)

X ² (distan.FP to each Section)

(B) A X B S.M

Product (AxB)xSM

0 0,0280 0 0 1 0,0000

1 0,5200 1,6563 0,861276 4 3,4451

2 0,9680 6,6254 6,4133872 2 12,8268

3 1,3560 14,9073 20,214299 4 80,8572

4 1,6490 26,5019 43,701633 2 87,4033

5 1,8490 41,4092 76,565611 4 306,2624

6 1,9960 59,6292 119,01988 2 238,0398

7 1,9160 81,1620 155,50639 4 622,0256

8 1,6850 106,0076 178,62281 2 357,2456

9 1,1290 134,1658 151,47319 4 605,8928

10 0,0000 1 0,0000

Sum of Products = 2313,9985

S = 1,287 ILFP = 2 * S/3 * (Sum of products)

ILFP = 1985,41 M 4

Moment of inertia around center of fluotation

bo = Half beam

Awp = 34,04 m ²

IL = ILFP - (Aw. Xf ²) Aw = Waterplane area

Xf = 7,02 m

Xf ² = 49,23 Xf = Distance from FP to center of flotation

Awp.Xf ² = 1675,90

IL = 309,51 M4

9 – Transverse Moment of Inertia ( It ). The transverse moment of inertia It around the longitudinal axis, needed for the transverse stability, could be computed in similar way than the Inertia longitudinal but to do this, the water plane area would have to be divided into a set of longitudinal strips, which could be treated like the transverse ones as above. This division is impractical, however, since it is not used in any other calculation, for reasons of symmetry the longitudinal axis has to pass through the centre of flotation, so no correction need be applied.

Calculation Of the transverse moment of inertia (It)

Ordinate No. bo = Half beam

each section (A) bo ³ (B) S.M Product (A X B)

0 0,028 0 1 0,0000 1 0,520 0,1406 4 0,5624 2 0,968 0,9070 2 1,8141 3 1,356 2,4933 4 9,9733 4 1,649 4,4840 2 8,9679 5 1,849 6,3214 4 25,2855 6 1,996 7,9521 2 15,9042 7 1,916 7,0337 4 28,1350 8 1,685 4,7841 2 9,5682 9 1,129 1,4391 4 5,7563

10 0,000 0,0000 1 0,0000 Sum of Products = 105,9668

S = 1,287 bo half beam in watel line A = S/3 * (Sum of products) A = 45,46 IT = 2/3 x A

IT = 30,31 M4

transverse moment of inertia (It) in the water line

01234

56789

1 2 3 4 5 6 7 8 9 10 11

Sections

10- Transverse stability at small angles The transverse stability of a yacht may be explained with reference at figure 10.1. When the yacht is heeled the centre of buoyancy moves to leeward from B to B `. The buoyancy force, upwards, then creates a couple with the equally large gravity force acting downwards at G. The lever arm is usually GZ and the righting moment is m.g. GZ, since the gravity force is the mass, m, times the acceleration of gravity, g (9,81 m/s 2). There is another important point marked in the figure is called transverse metcentre, M. This is the intersection between the vertical line through B` and the symmetry plane of the yacht. For small angles of heel this point may be assumed fixed, which simplifies the calculations considerably. The distance between G and M, GM, is called the metacentric height and BM is the metacentric radius. A fundamental stability formula says that the metacentric radius is equal to the ratio of the transverse moment inertial It and volume

displacement Dc. Using this formula and some simples geometric relation the fighting moment may be obtained as explained in the figure 4.9 Since the stability of the yacht is proportional to GM there are two principal ways to increasing it. Either G may be lowered or M may be raised. Figure 10.1

11- Longitudinal stability at small angles. The method of calculating the longitudinal stability corresponds exactly to that of the transverse stability. Thus, the restoring moment when the hull gets a trim angle, may be computed from the formula of the figure 11.1, which correspond to those of the transverse figure. There is also a formula for computing the trim angle obtained when moving a weight longitudinally on board the yacht.

A- Calculations done at each draft line and More: Next web page update: soon………….

Hydrostatic and Stability

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