How to Draw Lines Plan From Table of Half Breadths

Design Waterline

Some tools

K.J. Rawson MSc, DEng, FEng RCNC, FRINA, WhSch , E.C. Tupper BSc, CEng RCNC, FRINA, WhSch , in Basic Ship Theory (Fifth Edition), 2001

Basic geometric concepts

The main parts of a typical ship together with the terms applied to the principal parts are illustrated in Fig. 2.1. Because, at first, they are of little interest or influence, superstructures and deckhouses are ignored and the hull of the ship is considered as a hollow body curved in all directions, surmounted by a watertight deck. Most ships have only one plane of symmetry, called the middle line plane which becomes the principal plane of reference. The shape of the ship cut by this plane is known as the sheer plan or profile. The design waterplane is a plane perpendicular to the middle line plane, chosen as a plane of reference at or near the horizontal: it may or may not be parallel to the keel. Planes perpendicular to both the middle line plane and the design waterplane are called transverse planes and a transverse section of the ship does, normally, exhibit symmetry about the middle line. Planes at right angles to the middle line plane, and parallel to the design waterplane are called waterplanes, whether they are in the water or not, and they are usually symmetrical about the middle line. Waterplanes are not necessarily parallel to the keel. Thus, the curved shape of a ship is best conveyed to our minds by its sections cut by orthogonal planes. Figure 2.2 illustrates these planes.

Fig. 2.1.

Fig. 2.2.

Transverse sections laid one on top of the other form a body plan which, by convention, when the sections are symmetrical, shows only half sections, the forward half sections on the right-hand side of the middle line and the after half sections on the left. Half waterplanes placed one on top of the other form a half breadth plan. Waterplanes looked at edge on in the sheer or body plan are called waterlines. The sheer, the body plan and the half breadth collectively are called the lines plan or sheer drawing and the three constituents are clearly related (see Fig. 2.3).

Fig. 2.3. Lines plan

It is convenient if the waterplanes and the transverse planes are equally spaced and datum points are needed to start from. That waterplane to which the ship is being designed is called the load waterplane (LWP) or design waterplane and additional waterplanes for examining the ship's shape are drawn above it and below it, equally spaced, usually leaving an uneven slice near the keel which is best examined separately.

A reference point at the fore end of the ship is provided by the intersection of the load waterline and the stem contour and the line perpendicular to the LWP through this point is called the fore perpendicular (FP). It does not matter where the perpendiculars are, provided that they are precise and fixed for the ship's life, that they embrace most of the underwater portion and that there are no serious discontinuities between them. The after perpendicular (AP) is frequently taken through the axis of the rudder stock or the intersection of the LWL and transom profile. If the point is sharp enough, it is sometimes better taken at the after cut up or at a place in the vicinity where there is a discontinuity in the ship's shape. The distance between these two convenient reference lines is called the length between perpendiculars (LBP or L _PP). Two other lengths which will be referred to and which need no further explanation are the length overall and the length on the waterline.

The distance between perpendiculars is divided into a convenient number of equal spaces, often twenty, to give, including the FP and the AP, twenty-one evenly spaced ordinales. These ordinates are, of course, the edges of transverse planes looked at in the sheer or half breadth and have the shapes half shown in the body plan. Ordinates can also define any set of evenly spaced reference lines drawn on an irregular shape. The distance from the middle line plane along an ordinate in the half breadth is called an offset and this distance appears again in the body plan where it is viewed from a different direction. All such distances for all waterplanes and all ordinates form a table of offsets which defines the shape of the hull and from which a lines plan can be drawn. A simple table of offsets is used in Fig. 3.30 to calculate the geometric particulars of the form.

A reference plane is needed about mid-length of the ship and, not unnaturally, the transverse plane midway between the perpendiculars is chosen. It is called amidships or midships and the section of the ship by this plane is the midship section. It may not be the largest section and it should have no significance other than its position halfway between the perpendiculars. Its position is usually defined by the symbol

.

The shape, lines, offsets and dimensions of primary interest to the theory of naval architecture are those which are wetted by the sea and are called displacement lines, ordinates, offsets, etc. Unless otherwise stated, this book refers normally to displacement dimensions. Those which are of interest to the shipbuilder are the lines of the frames which differ from the displacement lines by the thickness of hull plating or more, according to how the ship is built. These are called moulded dimensions. Definitions of displacement dimensions are similar to those which follow but will differ by plating thicknesses.

The moulded draught is the perpendicular distance in a transverse plane from the top of the flat keel to the design waterline. If unspecified, it refers to amidships. The draught amidships is the mean draught unless the mean draught is referred directly to draught mark readings.

Fig. 2.4. Moulded and displacement lines

Fig. 2.5.

The moulded depth is the perpendicular distance in a transverse plane from the top of the flat keel to the underside of deck plating at the ship's side. If unspecified, it refers to this dimension amidships.

Freeboard is the difference between the depth at side and the draught. It is the perpendicular distance in a transverse plane from the waterline to the upperside of the deck plating at side.

The moulded breadth extreme is the maximum horizontal breadth of any frame section. The terms breadth and beam are synonymous.

Certain other geometric concepts of varying precision will be found useful in defining the shape of the hull. Rise of floor is the distance above the keel that a tangent to the bottom at or near the keel cuts the line of maximum beam amidships (see Fig. 2.6).

Fig. 2.6.

Tumble home is the tendency of a section to fall in towards the middle line plane from the vertical as it approaches the deck edge. The opposite tendency is called flare (see Fig. 2.6).

Deck camber or round down is the curve applied to a deck transversely. It is normally concave downwards, a parabolic or circular curve, and measured as x centimetres in y metres.

Sheer is the tendency of a deck to rise above the horizontal in profile.

Rake is the departure from the vertical of any conspicuous line in profile such as a funnel, mast, stem contour, superstructure, etc. (Fig. 2.7).

Fig. 2.7.

There are special words applied to the angular movements of the whole ship from equilibrium conditions. Angular bodily movement from the vertical in a transverse plane is called heel. Angular bodily movement in the middle line plane is called trim. Angular disturbance from the mean course of a ship in the horizontal plane is called yaw or drift. Note that these are all angles and not rates, which are considered in later chapters.

There are two curves which can be derived from the offsets which define the shape of the hull by areas instead of distances which will later prove of great value. By erecting a height proportional to the area of each ordinate up to the LWP at each ordinate station on a horizontal axis, a curve is obtained known as the curve of areas. Figure 2.8 shows such a curve with number 4 ordinate, taken as an example. The height of the curve of areas at number 4 ordinate represents the area of number 4 ordinate section; the height at number 5 is proportional to the area of number 5 section and so on. A second type of area curve can be obtained by examining each ordinate section. Figure 2.8 again takes 4 ordinate section as an example. Plotting outwards from a vertical axis, distances corresponding to the areas of a section up to each waterline, a curve known as a Bonjean curve is obtained. Thus, the distance outwards at the LWL is proportional to the area of the section up to the LWL, the distance outwards at 1WL is proportional to the area of section up to 1WL and so on. Clearly, a Bonjean curve can be drawn for each section and a set produced.

Fig. 2.8.

The volume of displacement, ∇, is the total volume of fluid displaced by the ship. It is best conceived by imagining the fluid to be wax and the ship removed from it; it is then the volume of the impression left by the hull. For convenience of calculation, it is the addition of the volumes of the main body and appendages such as the slices at the keel, abaft the AP, rudder, bilge keels, propellers, etc., with subtractions for condensor inlets and other holes.

Finally, in the definition of hull geometry there are certain coefficients which will later prove of value as guides to the fatness or slimness of the hull.

The coefficient of fineness of waterplane, C _WP, is the ratio of the area of the waterplane to the area of its circumscribing rectangle. It varies from about 0.70 for ships with unusually fine ends to about 0.90 for ships with much parallel middle body.

$C_{\text{WP}}=\frac{A_{\text{W}}}{L_{\text{WL}}B}$

The midship section coefficient, C _M, is the ratio of the midship section area to the area of a rectangle whose sides are equal to the draught and the breadth extreme amidships. Its value usually exceeds 0.85 for ships other than yachts.

$C_{\text{M}}=\frac{A_{\text{M}}}{BT}$

The block coefficient, C _B, is the ratio of the volume of displacement to the volume of a rectangular block whose sides are equal to the breadth extreme, the mean draught and the length between perpendiculars.

$C_{\text{B}}=\frac{\nabla }{BT{L}_{\text{PP}}}$

Mean values of block coefficient might be 0.88 for a large oil tanker, 0.60 for an aircraft carrier and 0.50 for a yacht form.

Fig. 2.9. Waterplane coefficient

Fig. 2.10. Midship coefficient

The longitudinal prismatic coefficient, C _P, or simply prismatic coefficient is the ratio of the volume of displacement to the volume of a prism having a length equal to the length between perpendiculars and a cross-sectional area equal to the midship sectional area. Expected values generally exceed 0.55.

$C_{\text{P}}=\frac{\nabla }{A_{\text{M}}{L}_{\text{PP}}}$

Fig. 2.11. Block coefficient

Fig. 2.12. Longitudinal prismatic coefficient

The vertical prismatic coefficient, C _VP is the ratio of the volume of displacement to the volume of a prism having a length equal to the draught and a cross-sectional area equal to the waterplane area.

$C_{\text{VP}}=\frac{\nabla }{A_{\text{W}}T}$

Before leaving these coefficients for the time being, it should be observed that the definitions above have used displacement and not moulded dimensions because it is generally in the very early stages of design that these are of interest. Practice in this respect varies a good deal. Where the difference is significant, as for example in the structural design of tankers by Lloyd's Rules, care should be taken to check the definition in use. It should also be noted that the values of the various coefficients depend on the positions adopted for the perpendiculars.

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Definition and Regulation

Eric C. Tupper BSc, CEng, RCNC, FRINA, WhSch , in Introduction to Naval Architecture (Fifth Edition), 2013

Defining the Hull Form

A ship's hull form determines many of its main attributes, stability characteristics and resistance, and therefore the power needed for a given speed, seaworthiness, manoeuvrability and load-carrying capacity. It is important, therefore, that hull shape be defined accurately. The basic descriptors such as length and beam used must be defined. Not all authorities use the same definitions, and it is important that the reader of a document, or user of a computer program, checks the definitions adopted. Those used in this chapter are those generally used in the United Kingdom and most are internationally accepted.

The Geometry of the Hull

A ship's hull is three dimensional and is usually symmetrical about a fore and aft plane. Throughout this book, a symmetrical hull form is assumed unless otherwise stated. The hull envelope shape is defined by its intersection with three sets of mutually orthogonal planes. The horizontal planes are known as waterplanes and the lines of intersection are known as waterlines. The planes parallel to the middle line plane cut the hull in bow and buttock lines, the middle line plane itself defining the profile. The intersections of the transverse, that is the athwartships, planes define the transverse sections.

Two sets of main hull dimensions are used. Moulded dimensions are those between the inner surfaces of the hull envelope. Dimensions measured to the outside of the plating are meant if there is no qualifying adjective. Moulded dimensions are used to find the internal volumes of the hull – a rough indicator of earning capacity.

Five different lengths used in naval architecture are defined below. The first three lengths are those most commonly used to define the ship form and are as follows (Figure 2.1).

Figure 2.1. Principal dimensions.

•

The length between perpendiculars (LBP or L _PP) is the distance measured along the summer load waterplane (the design waterplane for warships) from the after to the fore perpendicular. The after perpendicular is commonly taken as the line passing through the rudder stock. The fore perpendicular is the vertical line through the intersection of the forward side of the stem with the summer load waterline.

•

The length overall (LOA or L _OA) is the distance between the extreme points forward and aft measured parallel to the summer (or design) waterline. Forward the point may be on the raked stem or on a bulbous bow.

•

The length on the waterline (LWL or L _WL) is the length on the waterline, at which the ship happens to be floating between the intersections of the bow and after end with the waterline. If not otherwise stated the summer load (or design) waterline is to be understood.

•

The scantling length is used in classification society rules to determine the required scantlings. The scantling length is the LBP but is not to be less than 96% of LWL and need not be more than 97% of LWL

•

Subdivision length. A length used in damage stability calculations carried out in accordance with International Maritime Organisation (IMO) standards. It is basically the ship length embracing the buoyant hull and the reserve of buoyancy.

The mid-point between the perpendiculars is called amidships or midships. The section of the ship at this point by a plane normal to both the summer waterplane and the centreline plane of the ship is called the midship section. It may not be the largest section of the ship – in general this will be a section somewhat aft of amidships. The breadth of the ship at any point is called the beam. Unless otherwise qualified, the beam quoted is usually that amidships at the summer waterplane. Referring to Figure 2.2, the moulded beam is the greatest distance between the inside of plating on the two sides of the ship at the greatest width at the section chosen. The breadth extreme is measured to the outside of plating but will also take account of any overhangs or flare.

Figure 2.2. Breadth measurements.

The ship depth (Figure 2.2) varies along the length but is usually quoted for amidships. The moulded depth is measured from the underside of the deck plating at the ship's side to the top of the inner flat keel plate. Unless otherwise specified, the depth is to the uppermost continuous deck. Where a rounded deck edge, gunwhale, is fitted the convention used is indicated in Figure 2.2.

Sheer (Figure 2.1) is a measure of how much a deck rises towards the stem and stern. For any position along the length of the ship, it is defined as the height of the deck at side at that position above the deck at side amidships.

Camber or round of beam is defined as the rise of the deck in going from the side to the centre as shown in Figure 2.3. Decks are cambered to enable water to run off them more easily. For ease of construction, and reduce cost, camber is applied usually only to weather decks, and straight line camber often replaces the older parabolic curve.

Figure 2.3. Section measurements.

The bottom of a ship, in the midships region, is usually flat but not necessarily horizontal. If the line of bottom is extended out to intersect the moulded breadth line (Figure 2.3), the height of this intersection above the keel is called the rise of floor or deadrise. Most ships have a flat keel and the extent to which this extends athwartships is termed the flat of keel or flat of bottom.

In some ships, the sides are not vertical at amidships. If the upper deck beam is less than that at the waterline, it is said to have tumble home, the value being half the difference in beams. If the upper deck has a greater beam the ship is said to have flare. Most ships have flare at a distance from amidships particularly towards the bow.

The draught of the ship at any point along its length is the distance from the keel to the waterline. If a moulded draught is quoted, it is measured from the inside of the keel plating. For navigation purposes, it is important to know the maximum draught. This will be taken to the bottom of any projection below keel such as a bulbous bow, propeller or sonar dome. If a waterline is not quoted, the design waterline is usually intended. To aid the captain, draught marks are placed near the bow and stern and remote reading devices for draught are often provided. The difference between the draughts forward and aft is referred to as the trim. Trim is said to be by the bow or by the stern depending upon whether the draught is greater forward or aft, respectively. Often draughts are quoted for the two perpendiculars. Being a flexible structure, a ship will usually bend slightly fore and aft, the curvature varying with the loading. The ship is said to hog or sag when the curvature is concave down or up respectively. The amount of hog or sag is the difference between the actual draught amidships and the mean of the draughts at the fore and after perpendiculars. In calculations, the curvature of the ship due to hog or sag is usually taken to be parabolic in shape.

Air draught is the vertical distance from the summer waterline to the highest point in the ship, usually the top of a mast. This dimension is important for ships that need to go under bridges in navigating rivers or canals or when entering port. In some cases, the topmost section of the mast can be struck (lowered) to enable the ship to pass under an obstruction.

Freeboard is the difference between the depth at side and the draught. That is, it is the height of the deck at side above the waterline. The freeboard is usually greater at the bow and stern than at amidships due to sheer. This helps create a drier ship in waves. Freeboard is important in determining safety and stability at large angles.

In defining the relative position of items, the following terms are used:

•

Ahead and astern (aft) are used for an item either forward of or behind another.

•

In way of to denote proximity to. (Note: It does not denote an obstruction.)

•

Abreast to denote to the side of.

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Flotation

Eric C. Tupper BSc, CEng, RCNC, FRINA, WhSch , in Introduction to Naval Architecture (Fifth Edition), 2013

Underwater Volume

Once a ship form is defined the underwater volume can be calculated. If the immersed areas of a number of sections throughout the length of a ship are calculated, a sectional area curve can be drawn as in Figure 4.2. The underwater volume, or volume of displacement, is given by the area under the curve and is represented by:

Figure 4.2. Cross-sectional area curve.

$\nabla =\int A\mathrm{d}x$

If immersed cross-sectional areas are calculated to a number of waterlines parallel to the design waterline, then the volume up to each can be determined and plotted against draught as in Figure 4.3. The volume corresponding to any given draught T can be read off provided the waterline at T is parallel to those used in deriving the curve.

Figure 4.3. Volume curve.

One method of finding the underwater volume is to use Bonjean curves. These are curves of immersed cross-sectional areas plotted against draught for each transverse section. They are usually drawn on the ship profile as in Figure 4.4. Suppose the ship is floating at waterline WL. The immersed areas for this waterline are obtained by drawing horizontal lines from the intercept of the waterline with the middle line of a section to the Bonjean curve for that section. Having the areas for all the sections, the underwater volume and its longitudinal centre of buoyancy can be calculated.

Figure 4.4. Bonjean curves.

When displacement was calculated manually, it was customary to use a displacement sheet. The displacement up to the design waterline was determined by using Simpson's rules applied to half ordinates measured at waterlines and at sections. The calculations were done in two ways. First the areas of sections were calculated and integrated in the fore and aft direction to give volume. Then areas of waterplanes were calculated and integrated vertically to give volume. The two volume values had to be the same if the arithmetic was correct. The displacement sheet was also used to calculate the vertical and longitudinal positions of the centre of buoyancy. This text has concentrated on calculating the characteristics of a floating body. It is helpful to have these concepts developed in more detail using numerical examples and this is done in Appendix B. The calculation lends itself very well to the use of Excel spreadsheets.

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Cubic Splines

Adrian Biran , in Geometry for Naval Architects, 2019

7.9 Exercises

Exercise 7.1 Drawing a ship section, 1

The following points belong to Station 19 of the previously cited INSEAN model 1189. The dimensions are in mm.

Point	Draught	Half-breadth	Point	Draught	Half-breadth
1	0.0	0.0	7	242.2	55.7
2	12.1	24.2	8	302.7	79.9
3	24.2	29.1	9	363.3	111.4
4	60.5	33.9	10	423.8	145.3
5	121.1	38.8	11	484.4	188.9
6	181.6	43.6	12	523.1	218.0

Your tasks are:

1.

Plot the points.

2.

Fit the polynomial defined by the given points and plot it over the previous plot.

3.

Add the corresponding MATLAB spline.

Exercise 7.2 A forward station of a fishing vessel

The following points belong to station 20 of the INSEAN fishing-vessel model 647 (INSEAN, 1962). The dimensions are in metres.

z	0.192	0.192	0.262	0.421	0.525
y	0.000	0.015	0.015	0.086	0.141

1.

Try to model this station using a cubic spline.

2.

Calculate the half breadth at $z=0.35$ m.

Exercise 7.3 The design waterline of INSEAN model 1189

The following points approximate the design waterline of the INSEAN model 1189. The coordinates are measured in metres.

Station	Aft	0	5	10	15	20
x	−0.232	0.000	1.161	2.322	3.483	4.644
y	0.000	0.141	0.363	0.388	0.300	0.000

In this exercise we are going to check the influence of parametrization.

1.

Use the spline $y=f(x)$ to plot the waterline and calculate the half breadths at stations 3, 7, 17. Do not forget the command axis equal.

2.

Plot the waterline using parametric splines, $x(t)$ , $y(t)$ , with uniform parametrization. The resulting plot is inacceptable.

3.

Plot the waterline, over the plot obtained at 1, using parametric splines with chord-length parametrization. Calculate the half breadths at stations 3, 7, 17. Identify the slight differences relative to the results obtained with the $y=f(x)$ spline.

Exercise 7.4 The design waterline of the INSEAN fishing-vessel model 467

The following points approximate the design waterline of the INSEAN model 1189. The coordinates are measured in metres.

Station	1/2	1	5	10	15	20
x	0.0781	0.1563	0.7813	1.5625	2.3438	3.1250
y	0	0.0580	0.3330	0.4188	0.3784	0.0151

1.

Use the spline $y=f(x)$ to plot the waterline and calculate the half breadths at stations 3, 7, 17. Do not forget the command axis equal.

2.

Plot the waterline, over the plot obtained at 1, using parametric splines with chord-length parametrization. Use the MATLAB spline to calculate the half breadths at stations 3, 7, 17. Mind the slight differences relative to the answers to the first question.

Exercise 7.5 Trawler sheer line

Below are the coordinates of six points belonging to the deck line of a trawler model (SNAME, 1966, Model 68). We omitted the aft extremity to simplify the exercise. Plot the deck line using the MATLAB function spline. If you can use a surface modelling program, such as MultiSurf, try to model the line in this software too and compare the results.

Station	0	5	10	15	20	Stem
x	0.000	0.619	1.238	1.860	2.470	2.510
y	0.130	0.249	0.260	0.200	0.020	0.000
z	0.320	0.308	0.308	0.340	0.420	0.430

Exercise 7.6 Curve of statical stability

In this chapter we have dealt with cubic splines as a tool for drawing ship lines. Cubic splines, however, are a general tool for interpolating points for plotting. An important application in Naval Architecture is in the drawing of the curve of statical stability. For this concept see, for example, Biran and López-Pulido (2014), Chapter 5.

1) Use the MATLAB function spline to plot the curve of the righting arms given below; they belong to a real vessel. Hint: intervals of 2.5 degrees are usually suitable for a smooth appearance.

Heel angle	Righting, arm	Heel angle	Righting, arm
ϕ, degrees	$\overline{GZ}$ , m	ϕ, degrees	$\overline{GZ}$ , m
0.000	0.000	50.000	1.102
5.000	0.212	60.000	1.083
10.000	0.403	70.000	0.911
20.000	0.697	80.000	0.671
30.000	0.904	90.000	0.389
40.000	1.037

2) Calculate the area in m⋅rad between 0 and 30 degrees, and the area between 0 and 40 degrees. These data are required by Rahola's criterion of stability and the IMO code of intact stability (see, for example, Biran and López-Pulido, 2014, Chapter 8).

Exercise 7.7 Curve of resistance, INSEAN model 1189

The data shown below are derived from the basin tests of the INSEAN model 1189 (INSEAN, 1965). $F_n$ is the Froude number and $R_D$ the non-dimensional resistance defined by

$F_n=\frac{V}{\sqrt{gL}},R_D=\frac{R_t}{\mathrm{\Delta }}$

where V is the model speed, g the gravity acceleration, $R_t$ the total resistance, and Δ the displacement of the model. Use the MATLAB spline to plot $R_D$ against $F_n$ .

F n	0.305	0.327	0.347	0.365	0.383	0.390	0.400
R D	10.129	11.857	14.276	18.011	22.504	24.883	28.499

Exercise 7.8 Curve of resistance, Trawler model

The data below result from the basin tests of a trawler model (SNAME, 1966, Model 68). Plot the curve of non-dimensional resistance versus Froude number using the MATLAB spline for interpolation.

$F_n$	$R/\mathrm{\Delta }$	$F_n$	$R/\mathrm{\Delta }$
0.10	1.17	0.24	7.07
0.12	1.67	0.26	8.53
0.14	2.26	0.28	10.17
0.16	2.96	0.30	12.35
0.18	3.78	0.32	15.04
0.20	4.74	0.34	18.35
0.22	5.83	0.36	28.59

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Flotation and stability

In The Maritime Engineering Reference Book, 2008

3.4.1 Determination of Displacement from Observed Draughts

Suppose draughts at the perpendiculars are T _a and T _f as in Figure 3.13. The mean draught will be T=(T _a+T _f)/2 and a first approximation to the displacement could be obtained by reading off the corresponding displacement, Δ, from the hydrostatic curves. In general, W₀L₀ will not be parallel to the waterlines for which the hydrostatics were computed. If waterline W₁L₁, intersecting W₀L₀ at amidships, is parallel to the design waterline then the displacement read from the hydrostatics for draught T is in fact the displacement to W₁L₁. It has been seen that because ships are not symmetrical fore and aft they trim about F. As shown in Figure 3.13, the displacement to W₀L₀ is less than that to W₁L₁, the difference being the layer W₁L₁L₂W₂, where W₂L₂ is the waterline parallel to W₁L₁ through F on W₀L₀. If λ is the distance of F forward of amidships then the thickness of layer=λ×t/L where t=T_a −T_f .

Figure 3.13.

If i is the increase in displacement per unit increase in draught:

(3.18) $\begin{array}{c}\text{Displacement of layer}=\lambda \times ti/L\text{and the actual}\\ \text{displacement}\\ =\Delta -\lambda \times ti/L\end{array}$

Whether the correction to the displacement read off from the hydrostatics initially is positive or negative depends upon whether the ship is trimming by the bow or stern and the position of F relative to amidships. It can be determined by making a simple sketch.

If the ship is floating in water of a different density to that for which the hydrostatics were calculated a further correction is needed in proportion to the two density values, increasing the displacement if the water in which ship is floating is greater than the standard.

This calculation for displacement has assumed that the keel is straight. It is likely to be curved, even in still water, so that a draught taken at amidships may not equal (d _a+d _f)/2 but have some value d _m giving a deflection of the hull, δ. If the ship sags the above calculation would underestimate the volume of displacement. If it hogs it would overestimate the volume. It is reasonable to assume the deflected profile of the ship is parabolic, so that the deflection at any point distant x from amidships is δ[l−(2x/L)²], and hence:

(3.19) $\text{Volume correction}=\int b\delta [1-{(2x/L)}^2]\text{dx}$

where b is the waterline breadth.

Unless an expression is available for b in terms of x this cannot be integrated mathematically. It can be evaluated by approximate integration using the ordinates for the waterline.

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ENERGY-EFFICIENT SHIP DESIGN

E.K. Pentimonti , in Energy and Sea Power, 1981

Hydrodynamics Performance

The compromises related to optimizing hull form, speed length ratio, propeller, etc., must be carefully analyzed in today's changing economics. In the past, simple and more producible designs were always chosen over the sophisticated hull form details requiring more labor and materials in construction. However, with increasing fuel prices, steel and labor have become relatively cheap. Sleek hull form characteristics and hydrodynamic drag-saving details are now being sought out by ship owners, and today the hydrodynamicist is enjoying newfound importance. This ship designer must pay close attention to the fundamentals of the hull form and appendages, and must evaluate the construction and operating costs as he proceeds through the various design decisions. In this area of hull form, savings of up to 15% to 20% in power (fuel costs) can be realized for an optimally designed vessel.

A look at the past designs of the LASH vessels shows why basic compromises must be made to optimize hull form. Penalties are being paid in fuel efficiency on these vessels for their blunt transom sterns, which were selected for a cargo-handling tradeoff. The barge-handling requirements of this single-screw design had led to a very wide waterplane at the transom at the original design waterline of 28 feet. The original stern design also required an excessively full U-shape underwater section which rises slowly as it leads aft and becomes essentially flat over and behind the propeller. Since these LASH vessels in their barge configuration had a 35–foot full load draft, the original designer compromised fuel efficiency at deeper drafts for these cargo handling requirements.

American President Lines has decided that improvements to the hull form, even in the mid-life of these vessels, are both necessary and cost-effective. Therefore, model tests have been performed for a modified stern section which minimizes drag at deeper drafts. The model test results of these proposed modifications predicted full-scale power savings of more than 20% at the operating drafts, and this redesigned hull can steam up to 1-1/2 knots faster at full power. From these results, it appears that the design compromise that led to the original lines may not have included sharp focus on all the hydrodynamics issues. This example clearly illustrates the effect of hydrodynamics on fuel-efficient ship design.

To find a universal solution to the problem of optimizing hydrodynamic form, the engineer and designer should allocate considerable time and money to the design and testing requirements of the hydrodynamic form for a first design. A one-year $1 million program could be typical for optimizing the relationships between hull characteristics and power requirements.

This time and money spent on the front end of a ship design and construction program can likely provide the highest return on investment of any investments available to a ship owner today, providing for drastic savings of operating costs. Typically, three or four design iterations are necessary to assure the maximum tradeoff between the capacity and optimum resistance characteristics considering the hull alone. Even today in the computer age, these iterations require slow but accurate physical model tests to assure meaningful results. At each iteration, an analysis needs to be made to test why the results change with input changes.

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The Hull Surface — Graphic Definition

Adrian Biran , in Geometry for Naval Architects, 2019

2.3 Main Dimensions and Coefficients of Form

The terms defined in this chapter belong to the basic terminology of Naval Architecture. Therefore, let us start with the beginning. In English, for a ship we use the pronoun she. Forward we have the bow, aft, the stern. Looking from the stern toward the bow, the right-hand side of the vessel is the starboard side, the left-hand side is the port side. The concepts of main dimensions and coefficients of form are related to the lines drawing. In this chapter we give the definitions of the most important terms. For more figures, comments, and translations into other languages we refer the reader to Chapter 1 in Biran and López-Pulido (2014). The lines are designed in the first iterations of the ship-design process when structural details are not yet known. In later iterations the hull surface defined by these lines is the internal surface of the plating. This surface is smooth, a qualifier that should be taken here in the most general sense. For reasons of strength, usually the plates have different thicknesses. Then, the external surface of the shell is not smooth and defining the main dimensions on it would be awkward. The internal surface of the shell is the moulded surface of the ship and dimensions defined on it are known as moulded dimensions.

In Section 2.2.1 we defined the length between perpendiculars, $L_{pp}$ . The midpoint of this length is called midships, and the station that contains it is the midship section. The symbol used in English literature for this term is shown in Fig. 2.9 in the appropriate place. In German literature we find a simplified symbol consisting of a circle and an inscribed X. French Naval Architects place a perpendicular in the midship section and call it perpendiculaire milieu (Rondeleux, 1911; Hervieu, 1985). Italian Naval Architects call this 'perpendicolare al mezzo' and note it by MP (Miranda, 2012). The following main dimensions are defined in the midship section.

Breadth,

B — This is the breadth of the DWL at draught T. Alternative terms are moulded breadth and moulded beam. More than often B is different from the maximum breadth of the waterline that can occur in another section. Also, the deck breadth may be larger than the moulded breadth and this is the case of vessels with side flare.

Draught,

T — This is the distance between the lowest point of the moulded hull and the DWL . One common definition is 'the distance between the upper surface of the keel and the design waterline'.

Depth,

D — This is the distance between the lowest point of the moulded hull and the deck-at-side line.

Freeboard,

f — This is the distance between the DWL and the deck-at-side line. In algebraic notation we can write $f=D-T$ .

Part of the bottom of many ships is flat and horizontal. Other ships display a rise of floor measured in Fig. 2.8 as $r_f$ . An alternative term is deadrise. The corresponding German word is 'Aufkimmung' (Rupp, 1981), and the Italian term is 'stellatura' (Miranda, 2012). The deck of many ships display a curvature in the transverse sections. This property is called camber and it is measured in Fig. 2.8 as c. The standard value for c is 1/50 of the local deck breadth. The transverse section of a deck with camber is a parabola. A complete view in the sheer plan would show two deck lines:

Figure 2.8. Dimensions defined in the midship section

•

the deck at centre, which is the intersection of the deck surface with the centreplane;

•

the deck at side, which is the intersection of the deck and side surfaces.

The above lines meet at their ends. Usually, only the deck-at-side line is shown. Certain ships do not have camber because this feature would interfere with their function. Examples of such vessels are containerships and aircraft carriers.

In Section 2.2.1 we defined the aft perpendicular, AP, as the vertical line passing through the intersection of the centreplane contour of the moulded-hull surface and the DWL. This definition is mainly employed for warships. For merchant ships it is usual to assume the axis of the rudder as aft perpendicular, as shown in Fig. 2.9. As to the forward perpendicular, we can now be more specific and explain that it is usual to place it at the intersection of the external surface of the stem and the DWL. This definition is an exception as it uses a point that is not on the moulded surface. Other lengths defined in Fig. 2.9 are

Figure 2.9. Definitions of lengths and sheer

•

the length overall, $L_{OA}$ , measured between the ship extremities above water;

•

the waterline length, $L_{WL}$ , a property important in hydrodynamical calculations.

For ships with submerged appendages, such as a bulbous bow or a sonar, it may be necessary to define a length overall submerged and a draught under the appendage. Many times the projection of the deck line in the sheer plan is curved. This curvature, called sheer, helps in preventing the water ingress on deck. The curved line is composed of two arcs of parabola that have a common, horizontal tangent in the midship section. Referring to Fig. 2.9 the standard values of the sheer at AP and FP are

(2.1) $S_a=25(\frac{L_{pp}}{3}+10),S_f=50(\frac{L_{pp}}{3}+10)$

where $S_a$ and $S_f$ are measured in mm, while $L_{pp}$ is measured in m. Again, there is no sheer in ships where this feature would interfere with their function.

The moulded volume of displacement at a given draught, $T_0$ , is the volume enclosed by the moulded hull surface up to the waterplane corresponding to $T_0$ and that waterplane. We use for it the notation ∇. To classify ship lines and to relate them to certain hydrostatic and hydrodynamic properties Naval Architects use a set of non-dimensional numbers called coefficients of form. The most important coefficient is the block coefficient defined as

(2.2) $C_B=\frac{\mathrm{\nabla }}{LBT}$

The waterplane coefficient is the number

(2.3) $C_{WL}=\frac{A_W}{LB}$

where $A_W$ is the waterplane area. The definition of the midship coefficient is

(2.4) $C_M=\frac{A_M}{BT}$

where $A_M$ is the area of the midship section. The prismatic coefficient is given by

(2.5) $C_P=\frac{\mathrm{\nabla }}{A_{M}L}$

and the vertical prismatic coefficient by

(2.6) $C_{VP}=\frac{\mathrm{\nabla }}{A_{W}T}$

Researchers define sometimes separate coefficients for the fore- and afterbody. An advantage of using non-dimensional numbers is that their values do not depend on the system of coordinates used in the design, as long as it is consistent. More advantages appear in the research and presentation of relationships between various ship parameters.

The volume of displacement is related to the displacement mass by Archimedes' principle

A body immersed in a fluid is subjected to an upwards force equal to the weight of the fluid displaced

The force predicted by this principle is the buoyancy. If the body floats freely, for equilibrium of forces the weight of the body must equal the force exercised by the fluid. Then, noting by Δ the mass of the body, and by ρ the density of the fluid, we can write

(2.7) $\mathrm{\Delta }=\rho C_{B}LBT$

Eq. (2.7) sub-estimates the mass of real ships because it is based on moulded dimensions. The thickness of the shell and various appendages, such as the rudder, propeller and propeller struts, contribute additional volumes, i.e. an additional buoyancy force. This problem is discussed in Biran and López-Pulido (2014), Section 4.3.

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Special Problems in Sea Petroleum Engineering for Beaches and Shallow Sea Areas

Huacan Fang , Menglan Duan , in Offshore Operation Facilities, 2014

5.4.4.1 Shallow Draft Dragging Supply Vessels

Take the shallow draft 2400 kW dragging supply vessels as an example here for description; they are designed and built by Dagang oil field.

1.

The main purposes are to:

a.

Drag the offshore oil drilling platform in place;

b.

Deliver fuel, fresh water, cement, barite equipment and other equipment and materials for the platform

c.

Undertake the mission of guarding and rescuing the shallow drilling platform

d.

Weigh anchor and drop anchor for the drilling platform of beach/shallow sea areas

e.

Supply an external fire capacity of 1200 m³/h of water, foam

f.

Tow barges

2.

Size parameters

The total length of the ship is 59.4 m; the length of the design waterline is 57.0 m; the length between perpendiculars is 53.7 m; the width of the beam is 12.0 m; and the depth is 4.0 m.

3.

Technical performance

a.

Draft and displacement

Draft of a light load is 9 m; displacement is 940 t; heavy draft is 2.6 m; and displacement is 1388 t.

b.

Navigation

In deepwater navigation area the draught is 1.9 m, and the maximum speed is 13.5 kn; in the light-load condition in a deep-water navigation area, wind is 13.8 m/s, and the speed of dragging the platform is 5.0 kn. In the condition of overloaded navigation service speed, endurance is 1000 n mile; the self-sustaining force is not less than 20 d.

c.

Bollard pulls

The bollard pull of the sham air district is 400 kN; and the bollard pull of beach/shallow sea trade area is 250 kN.

d.

Cargo capacity

Cargo capacity of overloaded deck is 250 t, and amount of bulk cement (canned) is 100 t.

e.

Dynamic power

3 MWMTBD620V8 diesel engines made in Germany are the host engines, with a rated speed of 1500 r/min. Rated power is 829 kW. Generator sets are 275M × DFCC made in United States, with NTA855G3 and HCM534C engines, and a total of three units. Their speed is 1500 r/min and capacity is 275 kW. The emergent generator is CCFJ50Y-1500, whose engine is the 4135AD; there is only one unit generator, the TFHX with a speed of 1500 r/min and capacity of 50 kW.

4.

Weight

The total weight of the entire ship is 902.1 t; and the net weight is 270.6 t.

5.

Structural composition

The structure of the shallow draft 2400 kW dragging supply vessel is shown in Figure 5-28. The ship is a beach/shallow sea supply vessel which has a long side, single continuous deck, single bottom, closed cabin; forward side column, Transom stern, and three paddle machines and twin rudders.

a.

The hull

The hull has a welded steel structure, single bottom, single deck, and transverse framing. The carrying capacity of the cargo deck is 50 kN/m².

b.

The turbine system

The turbine system includes three main propulsion units, three propulsion shafts, and a set of bow thrusters. The main propulsion unit consists of three diesel engines. Propulsion shafts are arranged in parallel with the ship's three machines propeller, ducted propeller, propeller rotation direction of internal rotation and external rotation. Thus, the middle propulsion shaft is arranged in the midline plane. But the port and starboard propulsion shafting is arranged 3.5 m away from the midline surface, three promote axis are parallel, 1.2 m away from the baseline. The bow thruster is in its own compartment, and is furnished with a lateral conduit propeller by catheter, gearbox, motor, control cabinet, and so on. Other equipment, such as generator sets, deck machinery, life-saving equipment, cabin oily wastewater treatment systems, sewage treatment equipment, ballast water, fresh water and oil fuel transfer systems, powder pneumatic conveying systems, fire systems, and electrical and communications equipment, are the same as for the ordinary ship, so it will not be repeated here.

FIGURE 5-28. 2400 kW shallow draft ship structure.

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Ship dimensions, form, size, or category

D.J. Eyres M.Sc., F.R.I.N.A. , G.J. Bruce M.B.A, F.R.I.N.A., MSNAME. , in Ship Construction (Seventh Edition), 2012

The hull form of a ship may be defined by a number of dimensions and terms that are often referred to during and after building the vessel. An explanation of the principal terms is given below:

After Perpendicular (AP): A perpendicular drawn to the waterline at the point where the after side of the rudder post meets the summer load line. Where no rudder post is fitted it is taken as the center line of the rudder stock.

Forward Perpendicular (FP): A perpendicular drawn to the waterline at the point where the fore-side of the stem meets the summer load line.

Length Between Perpendiculars (LBP): The length between the forward and aft perpendiculars measured along the summer load line.

Amidships: A point midway between the after and forward perpendiculars.

Length Overall (LOA): Length of vessel taken over all extremities.

Lloyd's Length: Used for obtaining scantlings if the vessel is classed with Lloyd's Register. It is the same as length between perpendiculars except that it must not be less than 96% and need not be more than 97% of the extreme length on the summer load line. If the ship has an unusual stem or stern arrangement the length is given special consideration.

Register Length: The length of ship measured from the fore-side of the head of the stem to the aft side of the head of the stern post or, in the case of a ship not having a stern post, to the fore-side of the rudder stock. If the ship does not have a stern post or a rudder stock, the after terminal is taken to the aftermost part of the transom or stern of the ship. This length is the official length in the register of ships maintained by the flag state and appears on official documents relating to ownership and other matters concerning the business of the ship. Another important length measurement is what might be referred to as the IMO Length. This length is found in various international conventions such as the Load Line, Tonnage, SOLAS and MARPOL conventions, and determines the application of requirements of those conventions to a ship. It is defined as 96% of the total length on a waterline at 85% of the least molded depth measured from the top of keel, or the length from the fore-side of stem to the axis of rudder stock on that waterline, if that is greater. In ships designed with a rake of keel the waterline on which this length is measured is taken parallel to the design waterline.

Molded dimensions are often referred to; these are taken to the inside of plating on a metal ship.

Base Line: A horizontal line drawn at the top of the keel plate. All vertical molded dimensions are measured relative to this line.

Molded Beam: Measured at the midship section, this is the maximum molded breadth of the ship.

Molded Draft: Measured from the base line to the summer load line at the midship section.

Molded Depth: Measured from the base line to the heel of the upper deck beam at the ship's side amidships.

Extreme Beam: The maximum beam taken over all extremities.

Extreme Draft: Taken from the lowest point of keel to the summer load line. Draft marks represent extreme drafts.

Extreme Depth: Depth of vessel at ship's side from upper deck to lowest point of keel.

Half Breadth: Since a ship's hull is symmetrical about the longitudinal centre line, often only the half beam or half breadth at any section is given.

Freeboard: The vertical distance measured at the ship's side between the summer load line (or service draft) and the freeboard deck. The freeboard deck is normally the uppermost complete deck exposed to weather and sea that has permanent means of closing all openings, and below which all openings in the ship's side have watertight closings.

Sheer: A rise in the height of the deck (curvature or in a straight line) in the longitudinal direction. Measured as the height of deck at side at any point above the height of deck at side amidships.

Camber (or Round of Beam): Curvature of decks in the transverse direction. Measured as the height of deck at center above the height of deck at side. Straight line camber is used on many large ships to simplify construction.

Rise of Floor (or Deadrise): The rise of the bottom shell plating line above the base line. This rise is measured at the line of molded beam. Large cargo ships often have no rise of floor.

Half Siding of Keel: The horizontal flat portion of the bottom shell measured to port or starboard of the ship's longitudinal center line. This is a useful dimension to know when dry-docking.

Tumblehome: The inward curvature of the side shell above the summer load line. This is unusual on modern ships.

Flare: The outward curvature of the side shell above the waterline. It promotes dryness and is therefore associated with the fore end of ship.

Stem Rake: Inclination of the stem line from the vertical.

Keel Rake: Inclination of the keel line from the horizontal. Trawlers and tugs often have keels raked aft to give greater depth aft where the propeller diameter is proportionately larger in this type of vessel. Small craft occasionally have forward rake of keel to bring propellers above the line of keel.

Tween Deck Height: Vertical distance between adjacent decks measured from the tops of deck beams at ship's side.

Parallel Middle Body: The length over which the midship section remains constant in area and shape.

Entrance: The immersed body of the vessel forward of the parallel middle body.

Run: The immersed body of the vessel aft of the parallel middle body.

Tonnage: This is often referred to when the size of the vessel is discussed, and the gross tonnage is quoted from Lloyd's Register. Tonnage is a measure of the enclosed internal volume of the vessel (originally computed as 100 cubic feet per ton). This is dealt with in detail in Chapter 30.

Deadweight: This is defined in Chapter 1. It should be noted that for tankers deadweight is often quoted in 'long tons' rather than 'metric tons (tonnes)'; however, MARPOL regulations for oil tankers are in metric tons.

The principal dimensions of the ship are illustrated in Figure 2.1.

Figure 2.1. Principal ship dimensions.

TEU and FEU: Indicate the cargo-carrying capacity of container ships. TEU (twenty-foot equivalent unit) indicates the number of standard shipping containers that may be carried on some shipping routes; container ships may carry standard containers that are 40 feet in length. FEU is forty-foot equivalent unit.

An indication of the size by capacity of oil tankers, bulk carriers, and container ships is often given by the following types:

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Floating Offshore Platform Design

John Halkyard , in Handbook of Offshore Engineering, 2005

7.6.6 TLP Hull Structure

General

While the structural design is more generally covered in Section 7.8, several aspects of structural design as they specifically relate to TLPs should be discussed in this section. Unlike semi-submersibles, the API code, API RP2T is specifically developed for TLP design and is more prevalently used than Classification Rules. However, apart from the more global aspects and specific TLP issues, the structural response is not altogether different from that of semi-submersibles. The classification rules are applicable, particularly as discussed in Section 7.8, and are being more increasingly recognised thus. While many a major oil company owners have very specific additional requirements, as noted in the semi-submersible discussion, the regulating authorities are increasingly reliant on the classification societies for certification review.

In the same way as semi-submersibles, a TLP hull structure design is taken at two levels: Local Strength and the Global Strength. There are design issue differences from semi-submersibles at both levels and those are discussed here. Perhaps more important for a TLP than for a semi-submersible is considerations of loading from construction and installation. Ocean transit of the hull from a remote builder to the installation site can pose structural design challenges (see fig. 7.33).

Due to permanent siting, the longevity, inspectability and repair issues noted in the semi-submersible discussion are highly significant factors for TLPs. Typically TLPs are designed and built to highly considered design bases developed for a specific application and do not suffer from the less than specific, but versatile requirements needed for a mobile unit. The design standards for mobile units reflect the anticipation of highly variable use for well in excess of 20 years.

While more prevalently built by fabricators in the past, there is a persistent shift to shipbuilders for TLP hull construction. Therefore, many of the comments made for semi-submersible fabricating practice are relevant to TLPs, particularly those regarding cross-stiffened panel construction and block fabrication and erection practices. Also, the discussion of column and pontoon configuration is relevant.

Local Strength of TLPs

For a TLP, local strength follows largely the same issues as discussed for semi-submersibles but there are some notable differences. For TLPs also, 80–85% of all hull steel is a consequence of local loading. This may be even more true of TLPs in that they tend to have significantly deeper draft, requiring higher pontoon and lower column design pressures. Also, since TLPs require relatively little ballast, the internal spaces tend to be empty. Most internal subdivision therefore need not be designed for tank service; i.e. they are "void spaces." Generally the watertight flats and bulkheads are designed for watertight integrity only, a less demanding criteria. Also, for the shell plating, external pressure is usually the controlling designed pressure where as for a semi-submersible the internal pressure of a tank with a particularly high overflow vent may govern.

For a TLP, dynamic wave pressure can be significant. Unlike semi-submersibles and floaters, TLPs cannot rise with the crest (heave) and thereby reduce pressures. A TLP will, in fact, go through set-down. The heave of floaters actually mitigates enough of the dynamic pressure from a passing wave crest that the 20 ft code-based minimum design head is sufficient for the upper columns of semi-submersibles. This is not sufficient for a TLP and dynamic pressures must be considered. The underwater hull also endures significantly higher pressures. Figure 7.50 shows the external pressures applicable to a particular TLP column. Superposed over the static design head curve is the dynamic capacity for that static design head. The constant upper part reflects that the 20 ft minimum static head has a dynamic capacity of 26.7 ft. However, also shown is a combined static and dynamic wave pressure curve for the indicated wave crest. From about 68 ft below the design waterline to about 14 ft above it, the combined static and dynamic pressures exceeds the capacity required for static criteria (elsewhere, the curve is shown with a heavy dash). It is notable that below about 70 ft (e.g. pontoons), wave pressures do not sufficiently exceed the static pressures capacity. (The code basis for this is discussed more specifically in Section 7.8) While this illustration is specific to TLPs, similar effects will occur for minimal heave structures (e.g. SPARs).

Figure 7.50. TLP column external pressure example

Although a consideration for transit and installation (criteria adjusted as appropriate), stability is not an in-place design issue for a TLP. Not only is there less internal subdivision, high damaged stability waterlines are not needed for the shell plating design. Rather than stability, the internal subdivision is dictated by a need to prevent slack tendons. This is less demanding than damage stability compartmentation.

TLP framing is subject to considerable compressing. Figure 7.51 shows two typical framing sections for a TLP. On each is shown the distributed hydrostatic load transferred to the frames from the intervening plate and stiffeners. Notably for TLPs and deep draft semi-submersibles, particularly high pressures are developed. Particularly for TLPs, as noted above, there generally is no compensating internal fluid pressures. In the case of the pontoon section, the frame is not only highly loaded in flexure, but each frame element is under considerable compression. In the case of the circular column, there is a circular, unsupported ring frame. The strength of this type of frame is governed by buckling and not flexure. However, if there are internal supports (e.g. bulkheads, struts, etc.), the frame is governed by flexure, as is the pontoon frame.

Figure 7.51. Typical hydrostatic frame loading for a TLP

Global Strength of TLPs

All of the TLPs use an unbraced arrangement. Although the superstructure of TLPs is in a truss form, the behaviour is flexure and similar to the column and pontoon behaviour. In all cases of this discussion, the subject is a 4-column, closed array pontoon TLP. The deck moment is coupled to the column tops.

Figure 7.52 shows a profile of a side of a typical TLP with an elastic frame superposed over it. Also shown is the still water gravity/buoyancy load system corresponding to conditions shown in fig. 7.47 given previously. The hull loading includes the column weight, the upward bottom pressures, the net of pontoon buoyancy and weight, and the downward tendon loads. The deck loading included a distributed load and concentrated load corresponding to the well system (risers and/or drilling equipment).

Figure 7.52. Global loading – gravity/buoyancy load

This load system is shown in fig. 7.53. The corresponding distortion pattern is shown in fig. 7.54. What should be evident is the hog in the pontoons, the sag in the deck and the constant moment in the columns. Shear is small in the columns and there is no torsion in the pontoons and columns. By symmetry, this is the same in all four faces. Although the variable load can change, it is important to note that this is the background stress level under any additional environmental loading.

Figure 7.53. Global loading – gravity/buoyancy load, shear and moment

Figure 7.54. Global loading – gravity/buoyancy load deformation pattern

Figure 7.55 schematically shows the load system corresponding to the node centred environmental loading shown in the lower part of fig. 7.48, given previously. As with the gravity/buoyancy loading, fig. 7.56 shows the moments and shears and fig. 7.57 shows the distortion pattern. As can be seen, a very large part of the wave surge force is resisted by the deck inertia reactions. In the extreme, lateral deck accelerations can exceed 0.2 g. The frame deformation clearly is a side-sway shear. The proportions on a TLP are such that this loading can be particularly severe for the pontoons, which tend to be on the small side (relative to semi-submersibles).

Figure 7.55. TLP global loading – node centred wave

Figure 7.56. Node centred wave – shear and moment

Figure 7.57. Node centred wave – deformation pattern

This load system is sometimes called the "horizontal shear load" and is dominated by large horizontal shear loads between the deck and column tops. This is, of course, created by lateral acceleration on the mass of the deck. It is notable of TLPs that there is a high concentration of mass high in the structure. This is a result of the fact the TLPs are comparatively tall, are essentially empty within the hull, and are not stability limited. This can be the controlling load case for some parts of the structure.

The most severe loading for a closed array pontoon structural system, for TLPs, and semi-submersibles, is the crest (or trough) centred, oblique seas racking conditions. This is actually a form of squeeze/pry loading and corresponds to the upper part of fig. 7.48, given previously. The essence of this load pattern is seen in figs. 7.58 and 7.59. The former shows the pontoon array sitting atop a wave crest that is essentially trying to push the up- and down-wave corners outward. Forces on the pontoons indicated in fig. 7.58, are more specifically shown in profile in fig. 7.59. The consequent deformation pattern is shown in fig. 7.60.

Figure 7.58. TLP Global loading – oblique crest centred wave ("Squeeze/Pry")

Figure 7.59. Oblique crest centred wave – column forces

Figure 7.60. Oblique crest centred wave – pontoon deformation pattern

The horizontal plane moments are shown in fig. 7.58. There are also shears and significant axial force (not shown). However, it is not quite this simple and vertical plane bending and shear are considerable, not to mention torque. One part of the vertical plane action is an imbalance of vertical force on the column bottoms. Figure 6.60 summarises this action. As would be expected, the column force shown in fig. 7.59 imposes end moments at the pontoon connection. Figures 7.61 and 7.62 summarise the action of these moments. The column end moments create reacting moments and torques in the pontoons as shown.

Figure 7.61. Pontoon vertical plane bending from crest centred oblique wave – pontoon component

Figure 7.62. Pontoon vertical plane bending from crest oblique wave – column bending component

The foregoing has been largely qualitative and is given as a guide to strength analyses as well as design. For preliminary sizing (prior to analysis), simple estimater can be made of various force magnitudes and some sizing estimates can be made. However, what is most important is that the analyses include the controlling load cases and that the results be correctly interpreted. As will be discussed later, various modelling techniques can be employed for strength and fatigue analysis. It should be noted in closing that the moments, shears, etc. (stress resultants) used in this discussion apply to the large-scale hull elements actually used for the hull. Figure 7.63 schematically shows the relationship. Given the end stress resultants, and load distribution between the ends, local global stress within the element can be determined, including shear flow from torsion as well as shear and axial stress from biaxial bending and axial load. While the hull girder theory, particularly shear lag considerations, will improve accuracy, the engineering theory of bending shear and torsion are generally adequate unless the segments are multi-cellular.

Figure 7.63. Hull elements as components of a TLP hull

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How to Draw Lines Plan From Table of Half Breadths

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