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HYDROSTATIC CURVES

From day to day a ship may be loaded to different drafts and different trims. Therefore, underwater hull form characteristics over a range of loading conditions need to be calculated. This is done by calculating each characteristic at each loading condition (different waterlines). The results of these calculations are plotted on closely spaced grid paper. These curves are called hydrostatic curves or curves of form. The following figure shows such a set. Vertical scale shows the ship’s draft.

Displacement (salt water and fresh water) VCB : vertical center of buoyancy) LCB : longitudinal center of buoyancy LCF : longitudinal center of floatation CB : block coefficient CP : prismatic coeficient CM : midship section coeficient WS : wetted surface KM : location of transverse metacentre above the baseline MT1 : moment to trim one inch (or one centimetre) Tons per inch (or one centimetre) immersion

Sample hydrostatic curves

For the convenience of the deck officers , much of the numerical information shown on the hydrostatic curves is repeated in the form of tables., which most people find easier to use. In cargo ships this information is incorporated in the capacity plan, which also shows the volume of each hold and tank and its centre of gravity. With that information at hand, the officers can predict the ship’s drafts, fore and aft and stability characteristics for any proposed condition of loading.

Example –Numerical area calculation for a waterline Lets illustrate finding the area of the waterplane whose shape is shown in Fig.

Since the form is symmetric we need analyze only one side, being careful to multiply by two before we finish . The first task is to divide the baseline-which in this case is the ship's centerline-into some number of equal spaces. In real life these would typically number twenty, but we shall use only six here in order to simplify the explanation. As you may recall from our discussion of the lines drawing , these fore-and-aft dividers are called stations. In the figure we have identified them with numbers 0 to 6. We are now ready to put Mr. Simpson to work. Using an appropriate scale, we measure the full-size distances from the centerline to the curve at each station. In general terms these distances are called offsets. In this particular case they are called halfbreadths.These distances are shown in the second column of Table.

The third column in the table, identified as "SM," shows Simpson's multipliers (the 1, 4, 2, 4 , etc., numbers explained below).

S is the station spacing

The final column shows the product of the half-breadth measurements and Simpson's multipliers. The sum of all those products, when multiplied by two-thirds the station spacing, will yield a close approximation to the waterplane area-which is what we set out to find.

We have just explained how to apply the principles of numerical analysis to approximate the area of a waterplane. Naval architects use exactly the same procedure to find the area of any station below the design waterline. That is, instead of analyzing a horizontal area they analyze a vertical area. They do this for several stations along the vessel's length. These cross-sectional areas are then plotted to some convenient vertical scale against their fore-and-aft location, as shown in Fig. 5.13. The smooth line that you see drawn through those data points is what naval architects call the sectional area curve, an important tool in ship design.

In this case we have derived a sectional area curve from a set of lines. If we now apply Simpson's rule to that curve's offsets, I can derive the ship's volume of displacement and its longitudinal center of buoyancy. In actual practice, naval architects often work in the opposite direction.That is, they start by drawing what they know to be a good sectional area curve and use that to develop the individual stations, and then fair up the complete lines drawing.This brings up the question of what is meant by "a good sectional area curve"? It is one that will provide the required displacement with a longitudinal center of buoyancy that will lead to minimum wavemaking resistance. It will also result in acceptable trim fore and aft when the ship is in full-load condition. In this lesson we have given you an introduction to numerical analysis in naval architecture.

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From day to day a ship may be loaded to different drafts and different trims. Therefore, underwater hull form characteristics over a range of loading conditions need to be calculated. This is done by calculating each characteristic at each loading condition (different waterlines). The results of these calculations are plotted on closely spaced grid paper. These curves are called hydrostatic curves or curves of form. The following figure shows such a set. Vertical scale shows the ship’s draft.

Displacement (salt water and fresh water) VCB : vertical center of buoyancy) LCB : longitudinal center of buoyancy LCF : longitudinal center of floatation CB : block coefficient CP : prismatic coeficient CM : midship section coeficient WS : wetted surface KM : location of transverse metacentre above the baseline MT1 : moment to trim one inch (or one centimetre) Tons per inch (or one centimetre) immersion

Sample hydrostatic curves

For the convenience of the deck officers , much of the numerical information shown on the hydrostatic curves is repeated in the form of tables., which most people find easier to use. In cargo ships this information is incorporated in the capacity plan, which also shows the volume of each hold and tank and its centre of gravity. With that information at hand, the officers can predict the ship’s drafts, fore and aft and stability characteristics for any proposed condition of loading.

Example –Numerical area calculation for a waterline Lets illustrate finding the area of the waterplane whose shape is shown in Fig.

Since the form is symmetric we need analyze only one side, being careful to multiply by two before we finish . The first task is to divide the baseline-which in this case is the ship's centerline-into some number of equal spaces. In real life these would typically number twenty, but we shall use only six here in order to simplify the explanation. As you may recall from our discussion of the lines drawing , these fore-and-aft dividers are called stations. In the figure we have identified them with numbers 0 to 6. We are now ready to put Mr. Simpson to work. Using an appropriate scale, we measure the full-size distances from the centerline to the curve at each station. In general terms these distances are called offsets. In this particular case they are called halfbreadths.These distances are shown in the second column of Table.

The third column in the table, identified as "SM," shows Simpson's multipliers (the 1, 4, 2, 4 , etc., numbers explained below).

S is the station spacing

The final column shows the product of the half-breadth measurements and Simpson's multipliers. The sum of all those products, when multiplied by two-thirds the station spacing, will yield a close approximation to the waterplane area-which is what we set out to find.

We have just explained how to apply the principles of numerical analysis to approximate the area of a waterplane. Naval architects use exactly the same procedure to find the area of any station below the design waterline. That is, instead of analyzing a horizontal area they analyze a vertical area. They do this for several stations along the vessel's length. These cross-sectional areas are then plotted to some convenient vertical scale against their fore-and-aft location, as shown in Fig. 5.13. The smooth line that you see drawn through those data points is what naval architects call the sectional area curve, an important tool in ship design.

In this case we have derived a sectional area curve from a set of lines. If we now apply Simpson's rule to that curve's offsets, I can derive the ship's volume of displacement and its longitudinal center of buoyancy. In actual practice, naval architects often work in the opposite direction.That is, they start by drawing what they know to be a good sectional area curve and use that to develop the individual stations, and then fair up the complete lines drawing.This brings up the question of what is meant by "a good sectional area curve"? It is one that will provide the required displacement with a longitudinal center of buoyancy that will lead to minimum wavemaking resistance. It will also result in acceptable trim fore and aft when the ship is in full-load condition. In this lesson we have given you an introduction to numerical analysis in naval architecture.

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