One limitation of the ods presented above is that the section spacing has to be the same for all sections. This method is suitable only when the number of ordinates is 4, 7, 10, 13 etc. If we take a unit with four equally spaced sections as the unit for calculation of the volume, then the resulting formula is called Simpson’s Second Rule which is given by: Numerical integration of Section Areas along the length of the vessel gives Hull Volume.Numerical integration of offsets along the depth of a section gives Section Area.Once the section areas are obtained for equally spaced stations along the length of the vessel, then the numerical integration can be applied again to provide the volume of the hull as well. What if we want to calculate the individual section areas too using numerical methods? The same process applies – divide the section using equally spaced offsets along with the depth of the section (odd-numbered for Simpson’s first rule), and use the correct multipliers to get the section areas. If we can tabulate the section areas and their respective multipliers, then the calculation can be done in a simple template as shown below: We notice that every odd-numbered section has a multiplier of 4, while every even-numbered section has a multiplier of 2. The formula for several sections will be. ![]() The total number of sections has to be odd (3,5,7 etc.).Thus, we note the following about Simpson’s first rule: The volume of the second unit onwards will be given by This method is called Simpson’s first rule, and it considers the variation between the three sections to be approximated by a curve of the third order.įrom Simpson’s first rule, the volume of the first unit is given by What if we take a unit with three sections as the unit of calculation? This is demonstrated below: The Trapezoidal rule is based on a linear approximation by using a section with two ends as the unit of calculation. Simpson’s Rulesįor cases where the change in section area is more rapid (say, for the complicated fwd and aft ends of the vessel), we need a more accurate estimation. However, if there’s a rapid change is the section area between two ends of a section, then this method leads to greater inaccuracy, as it assumes the variation between the two ends to be linear which may not be the case always as the ship is a curved body. We can see that the formula will give accurate results if the number of sections is high. The above formula is called the Trapezoidal rule of integration to get the volume of the hull. The final result of the volume of the hull will be V3 = ½ x h x (A2 + A3), where A3 is the section area of the fwd end of section 3 V2 = ½ x h x (A1 + A2), where A2 is the section area of the fwd end of section 2 Similarly, the volume of the second section will be If the area of its aft end is A0 and the area of its fwd end is A1, with the length being h, then the volume of the section is given by: ![]() If we divide the length into 10 equally sized sections, then the length of each section is h = L/10, where L is the length of the ship. We can see that this approximation of a section as a trapezoid is feasible only when the section length is small, i.e., we have to divide the length into many sections (usually more than 10) to be able to use the trapezoid approximation. In this rule, we simply divide the length of the vessel into a number of equally spaced sections, with each section resembling a trapezoid. The simplest form of integration is the trapezoidal rule. If we can divide the entire body of the ship into several number of sections along its length, then each section becomes a 3D shape in itself, with each section approximately making a trapezoid having different section areas at its ends. However, the section of a hull keeps changing along its length. ![]() Mathematically speaking, the volume of a body with a fixed section shape is given by Volume = Section Area x Depth (see figure below). The approach for complex geometries is called ‘integration’, which means dividing the geometry into several numbers of smaller sized pieces, calculating the volume of each piece, and adding them up together (integration) to give the volume of the entire geometry. How do we find a property, say the volume of a complex shape like the hull? We’ll take a detailed look at this article. is not possible through simple formulae unlike standard shapes like cuboid or a cylinder. The hull of a ship is a complex 3D geometry, and finding out its simple properties like volume, centroid, etc.
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