Ship Stability Formula

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Evaluate the stability of a ship in terms of: a. Range of Stability. Dynamic Stability. Maximum Righting Arm. Maximum Righting Moment. Angle at which Maximum Righting Moment Occurs. Create a Curve of Intact Statical Stability for a ship at a given displacement and assumed vertical center of gravity, using the Cross Curves.

By: Brian Trenhaile, P. E., Naval Architect & Marine Engineer, Hawaii Marine Company, 2004

  1. MERCHANT SHIP STABILITY FORMULAE. (WALL SIDED FORMULA): GZ = SINθ ( GM + 1.BM.TAN 2 θ ) 2 WIND HEELING MOMENT: Total Wind heeling moment = F.A.d 1000 GZ (at.
  2. For normal shaped vessels KB can be estimated with Formula A as follows: Formula A) KB = 0.55T The transverse moment of inertia of the vessels water plane needs to be computed next. The symbol for moment of inertia is I.
  3. Stability Formulae Sheet Page 3 of 4 www.officerofthewatch.co.uk BILGED H 2 PARALLEL AXIS CENTROIDAL AXIS 2 T L 3 L L L L L GM BB Tanθ I I As Intact Water Plane Area Volume of Bilged Compartment Permeability(μy Sinkage Effective length l μ 100 SF of Cargo SF of Cargo-Solid Factor Permeability ( μ ) RD 1 Solid Factor 100 Volume available for.
  4. We as a human being have a same intelligent, because we was born in the same condition, as a little baby (human: red). So we have same chance and potential to understanding something exist above the surface of the earth. Here the one of uncountable.

There are basically only two types of trimming calculations. Other trimming calculations are just variations of these two fundamental types. In Case I you know the vessel's weight and center of gravity (CG) location and you seek to find the forward and after drafts. In Case II you know the forward and after drafts and desire to know the vessel's weight and center of gravity (CG) location.The simple approach that is presented here should be readily understandable by anyone with a basic understanding of algebra and geometry.

This article should benefit anyone involved with designing or operating any type of floating vessel. Regulatory agents and classification society employees also need to understand these methods in order to check various designs for compliance. This article is also designed to give a good theoretical understanding of the calculative methodology utilized in the trim and stability sheets sold on this site.

The vessel's 'Curves of Form' are needed to perform these calculations. No attempt is made to describe the theory behind or how to construct the 'Curves of Form.' This article is only concerned with application of 'Curves of Form' data.

Vessel Geometry and Sign Conventions Adopted

Vessel geometry is defined in figure below. All items are drawn to show the positive sign conventions adopted for this method.
Some important characteristics of this picture are:

  • B is the Buoyancy Force provided by the vessel with level trim. Usually obtained from a 'Curves of Form' plot. In this procedure B is numerically equal to the vessel's displacement (i. e. B = D).

  • W is the total weight applied to the vessel. Usually obtained from a'Weights and Moments' analysis.

  • L is the length between the forward and after draft marks

  • LCG is the Longitudinal Center of Gravity location, normally obtained from a 'Weights and Moments' analysis.

  • LCB is the Longitudinal Center of Buoyancy location, usually obtained from a 'Curves of Form' plot.

  • LCF is the Longitudinal Center of Floatation, normally obtained from a 'Curves of Form' plot.

  • TA is the draft and the Aft Draft marks

  • TF is the draft at the Forward Draft marks

  • TM is the Draft Amidships, located midway between forward and aft draft marks

  • TLCF is the draft located at the Longitudinal Center of Floatation.

The sign conventions adopted for this presentation are:

  • Distances Aft of Amidships are Positive, applies to LCG and LCB

  • Trim by the stern is defined as Positive

  • Trim Moment causing trim by the stern is Positive

  • Trim Lever that causes trim by the stern is Positive

Now that the geometry is defined and the sign conventions are stated, we can proceed with the two basic trimming cases.

Case I - Displacement & CG Location Known Find Forward & Aft Drafts

This option is utilized over and over again in design and operational stages to determine a vessel's responses to various loading conditions.

Step 1 - Obtain Equilibrium

  • For equilibrium the vessel weight must equal the vessel's displacement, W = D.

  • With this displacement enter the 'Curves of Form' and obtain a draft. This draft (TLCF) obtained is the draft present at the LCF location.

Step 2 - In the 'Curves of Form' at this LCF draft obtain the following:

  • Moment to Trim

    • MTI Moment to Trim one Inch, for English Units of long tonsfeet/inch or

    • MTC Moment to change Trim one Centimeter, for Metric Units of
      metric tonsmeters/cm.

  • Longitudinal Center of Buoyancy, LCB, feet or meters, with aft of amidships defined as positive

  • Longitudinal Center of Floatation, LCF, feet or meters, with aft of amidships defined as positive

Step 4 - Find the Trim.

  • First the trim lever is defined mathematically as Trim Lever = LCG - LCB, in either feet or meters. If this value is positive trim by the stern should be produced. If it is negative the vessel should trim by the bow. Sign convention consistency is extremely important. For instance if the LCG is 5 feet aft of amidships and the LCB is 2 foot forward, the trimming lever would be equal to 5 - (-2) = positive 7 feet. Since the numerical value is positive this scenario will cause trim by the stern.

  • The applied trimming moment is defined mathematically as TM = D(LCG - LCB)

  • The hydrostatic response trimming moment is defined mathematically as:

    • TM = TRIMMTI for English units

    • TM=TRIMMTC for metric units.

  • For equilibrium to occur, the applied trimming moment must equal the response trimming moment. The previous defined equations are combined, algebraically rearranged with the following expressions for trim obtained:

    • TRIM = D(LCG-LCB)/MTI for English units of inches, the value obtained must be converted to feet, by dividing by 12, prior to applying it in the formulas which follow.

    • TRIM = D(LCG-LCB)/MTC for metric units of centimeters, the value obtained must be converted to meters, by dividing by 100, prior to applying it in the formulas which follow.

    • When the above expressions are satisfied, there is corresponding subtle hydrostatic physical reality for the trimmed vessel condition. This reality is that the LCB has moved to a new location that is either directly above or below the LCG location. However, the initial LCB that must be applied in these trim calculations correspond to the vessel in a level condition (i. e. obtained from 'Curves of Form' values).

Step 5 - Find the Forward and After Draft Via Geometry

  • This method involves the use of similar triangles and the position of the LCF.

  • For the forward draft the similar triangles present yield the following expression TRIM/L = dTF/(LCF+L/2), solve this for dTF to obtain dTF = (TRIM/L)(L/2+LCF) = TRIM(1/2+LCF/L), then apply the following formula from geometry to obtain the forward draftTF = TLCF - dTF = TLCF - TRIM(1/2+LCF/L).

  • For the aftdraft the similar triangles present yield the following expression TRIM/L = dTA/(L/2-LCF), solve this for dTA to obtain dTA = (TRIM/L)(L/2-LCF) = TRIM(1/2-LCF/L), then apply the following formula from geometry to obtain the forward draft TA = TLCF+dTA = TLCF+ TRIM(1/2-LCF/L).

  • Alternatively, based on geometry, the after draft may be more simply computed as follows:
    TA = TF+ TRIM.

  • With the forward and aft drafts known the mean draft can be quickly computed as follows:
    TM = (TF+ TA)/2    .

Step 6 - Important Points to Remember

  • If the LCB is aft of the LCG the vessel will trim by the bow. If the LCB is forward of the LCG then the vessel will trim by the stern. These principles apply regardless of the position of the LCF.

  • Sign convention consistency is extremely important. If they are not followed the formulas presented here will not work properly.

Step 7 - Improvements Made to this site's Trim and Stability Sheets

  • The MTI or MTC values that are presented in the 'Curves of Form' are based on the assumption that metacentric radius in the longitudinal direction is equal to the metacentric height in the longitudinal direction (i. e. BML = GML). This assumption yields approximations for moment to trim values. These approximations are normally adequate since in most cases there is not much difference between the BML and GML values. Furthermore the approximations must be made because the VCG values are not known at the time that the 'Curves of Form' are made.

  • However in the 'Trim and Stability Sheets,' that are available on this website, the VCG values are known for the conditions at hand, so the moment to trim values are computed accurately. Three basic formulas are applied. First, by definition, the restoring moment = GMLDTanq. Second geometry present requires that Tanq = Opposite/Adjacent = TRIM/L. Three, by definition GML = KML - VCG, where KML is obtained from the 'Curves of Form' instead of MTI or MTC. All three of these equations are combined and rearranged yielding: MTF = (KML - VCG)D/L. MTF in this case is moment to trim one foot, where TRIM equals one foot. Note that ML can be obtained from the following formula: KML = BML + VCB. The 'Curves of Form' may just give BML and VCB, but this is alright since these can be summed to obtain the KML value. Another article in this website, 'Understanding Stability' explains the theory discussed in this paragraph. However, a little adaptation is required by the reader because that article applies to stability in the transverse direction and this article applies to stability in the longitudinal direction.


Case II - Forward & Aft Drafts Known, Find Displacement &LCG Location

This option is used by naval architects, yacht and boat designers, marine surveyors, marine inspectors and others for deadweight surveys and for stability tests. It is also used by dock masters, by captains, mates, fisherman and others who may want to determine a vessel's weight and center of gravity location.

The first goal of this analysis is to find the LCF draft. This draft is needed because the 'Curves of Form' are based on the LCF draft and not the mean draft. After this draft is determined, the primary goals of obtaining a displacement and the LCG location are easily determined through the use of the 'Curves of Form' data.

Step 1 - Calculate the Mean Draft & Trim Present

  • Compute the mean draft present, where TM = 1/2(TF + TA). Remember the 'Curves of Form' are not based on this mean draft but on the LCF draft. However this mean draft serves its purpose as a close estimate for the LCF draft and is initially used to retrieve preliminary data from the 'Curves of Form.'

  • Compute the trim present, with this formula TRIM = TA - TF.

  • These values of draft and trim are now used to help determine the LCF draft (TLCF).

Step 2 - Obtain the LCF Draft Through Iteration

  • At TM go into the Curves of Form and obtain a initial value for LCF.

  • An expression for the LCF draft needs to be derived. Fortuitously the waterline slope (or Tanq = TRIM / L) and the ship length (L between forward and aft draft marks) are known. From similar triangles we have dTLCF / TRIM = LCF / L. From geometry we have TLCF = TM + dTLCF. Combining the preceding two equations we have: TLCF = TM + (TRIM)LCF / L

  • Compute the initial guess for LCF draft through application of TLCF = TM + (TRIM)LCF / L

  • Go back to the 'Curves of Form' with initial TLCFjust computed and obtain a new value for LCF.

  • Recompute the LCF draft, by using the LCF value just obtained into the following formula: TLCF = TM + (TRIM)LCF / L.

  • The LCF just obtained should be close to the one previously calculated. If not, repeat this process using the most recent LCF draft value to enter the 'Curves of Form' to get a new LCF value. Recompute another LCF draft using the formula TLCF = TM + (TRIM)LCF / L and compare it with the preceding LCF draft computed, they now should be very close. Usually only need to iterate once. The last value for LCF draft is the considered the actual LCF draft and it is applied in the rest of this analysis.

Step 3 - Obtain 'Curves of Form' Data Based on the LCF Draft

  • With the last TLCF value enter the 'Curves of Form' and obtain the following:

  • Displacement, D

  • Longitudinal Center of Buoyancy, LCB

  • Moment to Trim, MTI or MTC. Which term depends on applicable units system.

    • MTI for English units of inches, the value obtained from 'Curves of Form' must be converted to feet, by dividing by 12, prior to applying it in the formulas which follow.

    • MTC for metric units of centimeters, the value obtained from 'Curves of Form' must be converted to meters, by dividing by 100, prior to applying it in the formulas which follow.

Step 4 - Derive Relationships Between Trim and LCG

  • Two relationships for trimming are presented and then equated to each other, the combined result is then manipulated to give an expression for computing LCG.

  • First the applied trimming moment is defined as TM = D(LCG - LCB).

  • Second the hydrostatic response moment is defined as TM = MTITRIM.

  • These equations are equated to each other and solved for LCG to obtain the following result:
    LCG = LCB + MTI TRIM /     D

Step 5 - Calculate the LCG Value

  • With the LCB, TRIM, MTI (converted to per foot or meter) and displacement compute the LCG using the formula just derived in Step 4.

Step 6 - Important Points to Remember

  • It trim value is positive, the vessel is has trim by the stern (the stern is submerged deeper than the bow) then the LCG must be located aft of the LCB.

  • If trim value is negative, the vessel has trim by the bow (the bow is submerged deeper than the stern), then the LCG must be located forward of the LCB.

  • Sign convention consistency remains extremely important! If they are not followed exactly the formulas presented here will not work properly.

Update: Another article on this website, 'Understanding Ship and Boat Stability (Stability & Trim - Part 1)' is meant to be a prelude to this article. This article apples to stability in the longitudinal direction, whereas the prelude article applies to stability that is in the transverse direction.

Simple box barges also present an interesting and quick way to learn about stability, trim, list, weights and moments. There is an article entitled 'Barge Trim, List and Initial Stability (GM - metacentric heights)' that should be helpful.

Understanding the parallel axis theorem is also very useful for both stability and structural analysis. This subject is comprehensively discussed in an article entitled 'Parallel Axis Theorem.'

Application: The concepts described in this article are utilized in the following templates:

Naval architecture
  • Weight and buoyancy
  • Metacentric stability
  • General arrangement
  • Resistance and propulsion
  • Maneuverability
  • Ships in waves
  • Strength of ships
  • Laws and regulations for safety
  • Ship design procedure

Concept of the metacentre

One would think, at first sight, that the average surface ship, with its weight concentrated above its point of support (considered as the centre of buoyancy), would fall over like a top that has stopped spinning. If properly designed it does not do so, because as the ship is inclined transversely, say by a strong wind, the centre of buoyancy of the immersed portion of the hull shifts sideways, in the same direction as the inclination. This is because the volume of the wedge of emersion shifts to the low side. At the new inclined waterline, the centre of buoyancy moves to the low side as well. The total buoyancy force continues to act upward along the true vertical. Provided no weights shift as the ship inclines, the centre of gravity remains at G, in the keel-to-deck centreline. The buoyancy force acting upward and the weight force acting downward through G produce a righting moment which acts to return the ship to its upright position.

Submarines, Ships, and other Watercraft: Fact or Fiction?
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For small angles of inclination, say less than 10°, the verticals through the shifted centres of buoyancy intersect the ship centreline at or close to a point M, called the metacentre. The stability provided by the action described is therefore called metacentric stability. For the situation most often encountered in practice, it is transverse metacentric stability. For a similar situation in which the ship is inclined or trimmed in its fore-and-aft centre plane, it is longitudinal metacentric stability.

If the centre of gravity of the ship is too high, the righting moment for any inclination is negative; that is, it acts to incline the ship still further. The ship then has transverse metacentric instability. Whether it will capsize or not depends upon whether the revised position G is overtaken by the vertical through the revised B as the inclination increases. If so, the ship remains in that inclined position, with a righting moment that is practically zero.

Ship's Stability Calculation (excel File)

Indexes of metacentric stability

The distance GM for positive stability, known as the metacentric height, is taken as one index of the degree of metacentric stability. The other is the range of stability, or the angle of inclination at which the metacentric height diminishes to zero. For example, when a ship is heeled transversely until the depressed deck edge goes underwater, the centre of buoyancy cannot move to the inclined side far enough to keep the metacentre well above the centre of gravity on the ship centre plane. At a critical inclination the metacentre lies at the centre of gravity and the righting moment disappears. For inclinations beyond this the metacentric height becomes negative, the righting moment becomes a capsizing moment and the ship rolls over.

Ship Stability Calculation

A greater range of metacentric stability can be built into a ship by raising the uppermost watertight deck to a higher position above the calm-water plane of flotation. The ship can then heel to a greater angle before water comes over the lower deck edge. For a craft like a sailing yacht, with a deep, heavy, stabilizing keel, the deck edge can actually go underwater and the range of positive stability can extend to large angles, provided water is kept out of the hull as the water level climbs higher and higher over the inclined deck.

Ship

The desirable value of transverse GM varies with the type, size, and service of the ship, but within limits it is still subject to the experience and judgment of the naval architect. It averages from 0.04 to 0.06 of the beam but may be as high as 0.10 of the beam for combatant vessels subject to heavy damage. For fishing vessels it may have two values, one for the outgoing and one for the return or loaded part of the voyage. The range of positive transverse GM for a normal ship to run in the open sea is usually in excess of 40° and may run as high as 70° or more, provided the hull remains intact and the weights do not shift.

Stability

Vertical position of metacentre

The height of the metacentre above the keel, or other selected point, depends upon the shape and size of the underwater body and of the at-rest waterline. The total height KM is the sum of the height KB of the centre of buoyancy above the keel and the height BM of the metacentre above the centre of buoyancy. The latter is known as the metacentric radius. The distance KB can be estimated by an approximate formula, it can be calculated by procedures applying to irregular volumes, or it can be determined by a mechanical integrator. The metacentric radius BM depends entirely on the geometry of the underwater hull and can be calculated from the formula, BM = I/V where I is the transverse moment of inertia of the waterline plane about the centreline axis, and V is the immersed volume of the hull. This relationship was first derived by Pierre Bouguer in the 18th century. For a given ship length and underwater volume, both I and BM are proportional to the cube of the beam, so that the latter is an important factor in transverse metacentric stability.

Vertical position of centre of gravity

The vertical position of the centre of gravity is as important as that of the metacentre in determining metacentric stability and the behaviour of the ship at sea. For a cargo ship, or even for a warship, this position may change by many feet, depending upon the nature, amount, and vertical position of the loads, fuel, and stores. Ballast may be used to increase the total mass moving in a seaway or to place G in a more advantageous position. Ice accumulating on the upper works of fishing boats, trawlers, and other small craft may be so thick, so heavy, and so high above the normal centre of gravity, as well as so difficult to remove at sea, that G rises above M and the craft capsizes.

Inclining experiment

Fortunately, the naval architect is able to make a full-scale check of the predicted or calculated metacentric stability before the completed ship goes to sea. By shifting liquids or solid masses whose weights and offset positions are known accurately, the centre of gravity of the whole ship is shifted and the ship is heeled. This shift is sideways for a determination of transverse and lengthwise for a measurement of longitudinal. The angle of inclination of the ship for each such shift is measured accurately with special devices. Then the actual metacentric height is determined for that loading condition. The height of the centre of gravity above the keel is then calculated from the relation.

Miller Stability Formula

Stability of submarines

When a submarine is in surface condition, its metacentric stability situation is the same as that of any surface ship. When diving or submerging, its water-plane area diminishes progressively to a negligible amount. The transverse moment of inertia of this plane likewise diminishes and with it the value of BM. Moreover, as the main-ballast tanks are flooded and the pressure hull is taken under, buoyancy is transferred from a low to a high position, so B rises in relation to the hull. Since G remains in its original position, M drops to the vicinity of G. Indeed, for some moments during diving and surfacing, it may be below G, with a negative GM. The flooding, venting, and blowing of the main-ballast tanks, with separate port and starboard controls, enable the submarine crew to counteract any list that may develop during this brief transition period.

Ship Stability Formulas Pdf

When the submarine is under the surface, the water-plane area is zero, as is BM. This means that the centre of buoyancy of the pressure-proof hull (B1) and M coincide. In this condition B1 is higher than G, so that the action of the buoyancy force and the weight force produces a stable situation, and the craft is said to have pendulum stability.





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