By Mazlan Muslim, MEng

Principal Specialist

UniKL MIMET

Weight and Location of Center of Gravity It is important that the weight and the location of the

center of gravity be estimated at an early stage in the design of a ship. The weight and height of the center of gravity are major factors in determining the adequacy of the ship’s stability. The weight and longitudinal position of the center of gravity determine the drafts at which the ship will float.

The distance of the center of gravity from the ship’s centerline plane determines whether the ship will have an unacceptable list. It will be clear that this calculation of weight and center of gravity, although laborious and tedious, is one of the most important steps in the successful design of ships.

Detailed Estimates During the early stages of design, the weight and the height

of center of gravity for the ship in light condition are estimated by comparison with ships of similar type or from coefficients derived from existing ships. At later stages of design detailed estimates of weights and centers of gravity are required.

It is often necessary to modify ship dimensions or the distribution of weights to achieve the desired optimum combination of a ship’s drafts, trim and stability, as well as to meet other design requirements such as motions in waves and powering. A summary of only the three major components of light ship weights for two typical ships is given in Table 1.

Light Ship Weights & Centres Dry Cargo Ship Containership Weight VCG LCG Weight VCG LCG t m m t m m Steel 4968 8 82.2 4557 9 94.6 Outfit 2099 13.8 81.5 1739 13.9 94.3 Machinery 1004 6 96.4 827 9 151.5 8071 9.3 83.8 7123 Weight Margin 511 76 KG Margin 0.5 1 0.1 Total w/o Ballast 8582 9.8 84.8 7199 Fixed Ballast — — 3329 2.8 80.9 Total 8582 9.8 84.8 10,528 7.9 94.6 Notes: 1. Commercial ships are assumed to be transversely symmetrical, and

therefore transverse centers and moments are not included. LCG measured from F.P. Margins adjusted to agree with inclining experiment results.

Detailed Estimates of Weights Detailed Estimates of Position of Center of Gravity. Ordinarily in design the horizontal plane of reference is taken

through the molded baseline of the ship. The height of the center of gravity above this base is referred to as KG and its position as VCG (vertical center of gravity). Sometimes, after a ship’s completion, the reference plane is taken through the bottom of the keel.

The plane of reference for the longitudinal position of the center of gravity may be the transverse plane at the midship section, which is midway between the forward and after perpendiculars. In this case the longitudinal position of the center of gravity, called the LCG (longitudinal center of gravity), is measured forward or abaft the midship section. This practice involves the possibility of inadvertently applying the measurements aft instead of forward, or vice versa, and a more desirable plane of reference is one through the forward perpendicular.

Systematic classification The plane of reference for the transverse position of the

center of gravity is the vertical centerline plane of the ship, the transverse position of the center of gravity being measured to port or starboard of this plane.

In weight estimates it is essential that an orderly and systematic classification of weights be followed. Two such classifications are in general use, that of the U.S. Maritime Administration (MarAd, 1962), and that of the U.S. Navy (NAVSEA, 1978). Both of these use the three broad classifications of hull steel, outfit, and propelling machinery. A detailed description of either classification may be obtained from the respective organizations. Some design offices may use systems differing in detail from either of these, but the general classification will be similar.

Weight and Center of Gravity Margins The weight estimate will of necessity contain many

approximations and, it may be presumed, some errors. The errors will generally be errors of omission. The steel as received from the mills is usually heavier, within the mill tolerance, than the ordered nominal weight. It is impossible, in the design stages, to calculate in accurate detail the weight of many groups, such as piping, wiring, auxiliary machinery, and many others.

For these and similar reasons, it is essential that margins for error be included in the weight estimate. The amount of these margins is derived from the experience of the estimator, and varies with the accuracy and extent of the available information.

Table 2: Margins Margin of weight (in percent of light-ship weight) Cargo ships 1.5 to 2.5 Tankers 1.5 to 2.5 Cargo-passenger ships 2.0 to 3.0 Large passenger liners 2.5 to 3.5 Small naval vessels 6.0 to 7.0 Large naval vessels 3.5 to 7.0

Margin in VCG Meters Cargo ships 0.15–0.23 Cargo-passenger ships 0.15 Tankers 0.15–0.23 Large passenger liners 0.23–0.30 Small naval vessels 0.15–0.23 Large naval vessels 0.15–0.23

Amount of Margin The amount of margin will also depend on the

seriousness of mis-estimating weight or center of gravity. For example, until the advent of the double bottom for tankers there was no real need for any margin at all in the VCG of a conventional tanker, because such ships generally have considerably more stability than is needed.

On the other hand, if there were a substantial penalty in the contract for overweight or for a high VCG, a correspondingly substantial margin in the estimate would be indicated.

Monitoring of weights The above margins apply to estimates made in the

contract-design stage, where the calculations are based primarily on a midship section, arrangement drawings, and the specifications. In a final, detailed finished-weightcalculation, made mostly from working drawings, a much smaller margin, of 1 or 2 percent, or even, if extremely detailed information is available, no margin at all may be appropriate.

For more detailed information on U.S. Navy practice with regard to margin, the reader is referred to Weight Control of Naval Ships (NAVSEA, 1978). Monitoring of weights (including load items) and the coordinates of the center of gravity is continued during ship construction, in order to ensure that all applicable weight and stability requirements will be satisfied in the completed ship.

Loading of Ship Variation in Displacement and Position of Center

of Gravity With Loading of Ship.

The total weight (displacement) and position of the center of gravity of any ship in service will depend greatly on the amount and location of the deadweight items —cargo, fuel, fresh water, stores, etc. Hence, the position of the center of gravity is determined for various operating conditions of the ship, the conditions depending upon the class of ship.

Metacentric Height

The Transverse Metacenter and Transverse Metacentric Height.

Consider a symmetric ship heeled to a very small angle, δ,shown, with the angle exaggerated, in Fig. 13. The center of buoyancy has moved off the ship’s centerline as the result of the inclination, and the lines along which the resultants of weight and buoyancy act are separated by a distance, GZ,the righting arm.

A vertical line through the center of buoyancy will intersect the original vertical through the center of buoyancy, which is in the ship’s centerline plane, at a point M, called the transverse metacenter, when δ→0. The location of this point will vary with the ship’s displacement and trim, but, for any given drafts, it will always be in the same place.

Fig. 13: Metacenter and righting arm

Righting arm Unless there is an abrupt change in the shape of the

ship in the vicinity of the waterline, point M will remain practically stationary with respect to the ship as the ship is inclined to small angles, up to about 7 or, sometimes, 10 deg.

As can be seen from Fig. 13, if the locations of G and Mare known, the righting arm for small angles of heel can be calculated readily, with sufficient accuracy for all practical purposes, by the formula

Fig. 14: Locating the transverse metacenter

Adequate GM The distance GM is therefore important as an index of

transverse stability at small angles of heel, and is called the transverse metacentric height. Since GZ is considered positive when the moment of weight and buoyancy tends to rotate the ship toward the upright position, GM is positive when M is above G, and negative when M is below G.

Metacentric Height (GM) is often used as an index of stability when preparation of stability curves for large angles has not been made. Its use is based on the assumption that adequate GM, in conjunction with adequate freeboard, will assure that adequate righting moments will exist at both small and large angles of heel.

Location of the Transverse Metacenter When a ship is inclined to a small angle, as in Fig. 14, the

new waterline will intersect the original waterline at the ship’s centerline plane if the ship is wall-sided in the vicinity of the waterline, since the volumes of the two wedges between the two waterlines will then be equal, and there will be no change in displacement.

If υ is the volume of each wedge, ∇ the volume of displacement, and the centers of gravity of the wedges are at g1 and g2, the ship’s center of buoyancy will move:

In a direction parallel to a line connecting g1 and g2.

A distance, BB1, equal to (υ·)/∇.

Determination of BM As the angle of heel approaches zero, the line , and

therefore BB1, become perpendicular to the ship’s centerline. Also, any variation from wall-sidedness becomes negligible, and we may say

If y is the half-breadth of the waterline at any point of the ship’s length at a distance x from one end, and if the ship’s length is designated as L, then, since the area of a section through the wedge is 1/2(y) (y tan δ) and its centroid is at a distance of 2×2/3y from the centroid of the corresponding section on the other side

Moment of Inertia

Moment of Inertia The expression

is the moment of inertia of a figure bounded by a curve and a straight line with the straight line as the axis. If we consider the straight line to be the ship’s centerline, then the moment of inertia of the entire waterplaneabout the ship’s centerline (both sides) designated as IT, is

Calculation of KM Therefore, when δ→0,

This theorem was derived by the French hydrographerPierre Bouguer while on an expedition to Peru to measure a degree of the meridian near the equator. It appeared in his Traité du Navire published in Paris in 1746.

The calculation of the height of the transverse metacenterabove the keel, usually called KM. This distance is the sum of BM, or IT/∇, and KB, the height of the center of buoyancy above the keel. The height of the center of gravity above the keel, KG, is found from the weight estimate or inclining experiment

Longitudinal GM Then,

The Longitudinal Metacenter and Longitudinal Metacentric Height.The longitudinal metacenter is similar to the transverse metacenter except that it involves longitudinal inclinations. Since ships are usually not symmetrical forward and aft, the center of buoyancy at various even-keel waterlines does not always lie in a fixed transverse plane, but may move forward and aft with changes in draft. For a given even-keel waterline, the longitudinal metacenter is defined as the intersection of a vertical line through the center of buoyancy in the even-keel attitude with a vertical line through the new position of the center of buoyancy after the ship has been inclined longitudinally through a small angle.

GML The longitudinal metacenter, like the transverse

metacenter, is substantially fixed with respect to the ship for moderate angles of inclination if there is no abrupt change in the shape of the ship in the vicinity of the waterline, and its distance above the ship’s center of gravity, or the longitudinal metacentricheight, is an index of the ship’s resistance to changes in trim.

For a normal surface ship, the longitudinal metacenteris always far above the center of gravity, and the longitudinal metacentric height is always positive.