Quick Barge Trim, List and
GM (Initial Stability) Calculations
By: Brian Trenhaile, P. E., Naval Architect & Marine Engineer,
Hawaii Marine Company, last update, May 7, 2014.
This article is focused on quickly determining trim, list and initial stability of simplified box barges. There are several benefits this approach. First, because it is quick and simple, it is efficient. Second it helps a person to easily and firmly grasp complex hydrostatics concepts involving trim, list and stability. With this method one can approximate how barge deck cargo distribution affects operating trim, list and GM (metacentric height). Another use would be to estimate how much a barge trims and lists with a crane and a hook load present.
This article is primarily focused on initial stability. This type of stability involves utilizing a vessel’s metacentric heights, GMs, in order to get righting arms. Some names applied to this approach are “static stability” and “initial stability.” In some respects it could also be called the “Curves of Form” approach. These types of calculations normally apply from zero up to about ten degrees of heel. As a vessel rolls into larger angles of heel another calculative method is required. This second approach is not within the scope of this article. This second method often involves the use of a computer model for the hull. This later approach requires considerable labor. The second method is called the “dynamic stability” or the “secondary stability” approach. This method can also be called the “Cross of Curves” and “Curves of Statical Stability” approach. From a safety point of view, this second calculative is normally more important than the first method, especially with regard to sea going vessels and sailing crafts of all types. Note that a computer model can also be conveniently used with the first method, but it is usually necessary with the second method. A computer model is also more flexible in the variety of hull orientations that it can handle. For formal calculations, which are prepared for submittal to regulatory agencies and other maritime entities, a computer model is often required. The approach here is aimed at assisting with operational decision making, and it is focused on processing data quickly and simply without the use of a computer model.
Conditions for Applicability
There are important restrictions that apply to trim and list with this simplified method. Also some operating guidelines and their rationale are discussed in this part.
For box barges, no portion of the bottom of the hull should be allowed to be lifted out of the water. If any portion of the barge bottom lifts out of the water these simplified calculations do not compensate for this effect. It is important to remember that the trim is based on the longitudinal weight distribution. However, if weights are unbalanced in the transverse direction, say due to a crane lifting a load over the side, then the vessel will heel. So in cases like this, both trim and list effects require consideration. For calculative accuracy, no corner of the box barge’s bottom can be lifted out of the water.
Similarly, no portion of the barge deck maybe submerged. If the barge deck submerges the accuracy of the results diminish with this method. Water on deck is also unsafe for personal and equipment.
It is also good marine practice to keep trim and heel to a minimum. So, for safety reasons a vessel operator should keep vessel trim and heel to a minimum. Further operational margins should be established, so that the bottom never gets exposed to the air, and the deck never gets submerged.
Free surface effects of any liquids on board are not within the scope of this article. If large free surfaces are present, the operator should compensate for these effects.
Effects of Rakes and Sharpened Ends
The effects of rakes are ignored with this method. If forward and after rakes are present, and the length of these rakes are equal, this method will give conservative results. It is conservative, because this simplified method will predict deeper sinkage than what the vessel will actually experience, and greater trim and list than the actual responses experienced by the vessel. This is the result of ignoring the buoyancy provided by the rakes and the water plane surfaces provided by the rakes. This simple method gives reasonable estimates provided that the rakes are not a large portion of the barge’s overall length. Put another way, if the barge meets your requirements without accounting for the benefits of the rakes, it will more than likely meet your requirements with the rake effects included.
If there is only a forward rake or forward and aft rakes of unequal length results with this method will not be as accurate, due to the unbalanced hull envelope shape.
Special consideration is required when doing simplified calculations for barges that have a rake at only one end. That is the amidships in these simplified calculations (for weight centers and for curves of form or hydrostatics lengths) is normally different from the amidships on the vessels drawing. This is because on the drawings amidships is often based on half of the vessels entire length. In the simplified calculations amidships is based on the half length without the rakes present. For a forward rake, the amidships in the simplified calculations could well be half the rake length aft of the normal amidships location noted on the drawings. The longitudinal centers of gravity, LCG, for each weight input must be based on the amidships for the simplified calculations. Also the hydrostatic (or Curves of Form) length based inputs must also be based on the amidships for the simplified calculations.
Determining Barge LCG with Known Forward and After Drafts
Great accuracy in results will occur if an accurate longitudinal center of gravity, LCG, for the barge is used as an input. The default value for LCG in these simplified calculations is usually zero. This assumption is based on the box barge being perfectly level in the forward and aft direction. In reality empty barges, in the light condition, often have some trim present. Especially barges with rake at only on one end. In these cases the barge profile can be drawn to scale along with the waterline present. This water line is based on the forward and aft drafts that are known. A CAD program can be used to find the center of area of the hull profile that is below the waterline. Then the distance between amidships of the barge and the center of gravity can be determined and input into the calculations as the empty barge’s LCG. If a CAD computer program is not available, the same results can be obtained by drawing a scaled barge profile along with the waterline and amidships location on a piece of card board. Then cutting out the portion of the profile below the water line and balancing the cutout on a knife edge to find the center of area and then scaling off the distance from the area center and the amidships location to obtain the input LCG.
Weights and Moments Calculations
Each weight, wi along with the location of its
center in space, defined by three coordinates, xi, yi and
zi, needs to be entered into a weights and moments table. When all
the weights are entered the total weight effect can be determined. The total of
all the weights and the combined center of all the weights are then utilized by
the trim, list and initial stability calculations. This analysis carried out
using the following formulas.
The total weight W = Swi where the total weight equals the summation of, S, each of the individual weights. Similarly the centers of gravity are found below.
The vertical center of gravity of all weights VCG = Swi x yi / W.
The longitudinal center of gravity of all weights LCG = Swi x xi / W.
The transverse center of gravity of all weights TCG = Swi x zi / W.
For meaningful results consistency is required. The coordinate system selected for the weights and moments analysis must be exactly the same as the coordinate system selected for the curves of form (or hydrostatics) calculations, which are discussed next.
Simplified Curves of Form Calculations (Hydrostatics)
Calculations for curves of form, or hydrostatics, characteristics are very simple for a box barge. These calculations are described in this part.
The longitudinal center of buoyancy, LCB, and the longitudinal center of floatation, LCF, for a box barge are both equal to zero when amidships is selected as the origin of the axis system. While amidships is normally defined as the axis system center, some select the forward end or the aft end. For whatever coordinate system is selected, consistency is required with the hydrostatic and weight calculations.
By definition, the water plane coefficient, CW, for a box barge is equal to one.
By definition, the block coefficient, CB, for a box barge is equal to one.
L is defined as the length, B is defined as the beam, D is defined as the depth and T is defined as the draft of the box barge.
Displacement is based on the formula D = CB x L x B x T.
For equilibrium to occur the displacement must equal the weight, D = W.
Combining and reworking the previous two formulas the draft can be found through the following formula: T = W / (CB x L x B).
The longitudinal moment of inertia of the water plane can be calculated by the following formula: IL = CW2 x B x L3/12. This formula is exact for a box barge, for other types of vessels it is often a fair approximation.
The transverse moment of inertia of the water plane can be calculated by the following formula: IT = CW2 x L x B3/12. This formula is exact for a box barge, for other types of vessels it is usually a good approximation.
The vertical center of buoyancy for a box barge is half the draft. This is expressed mathematically as: VCB = T / 2. Sometimes KB is used interchangeably with VCB. Therefore VCB = KB = T / 2.
The volume displaced is dependent on the density of the water in which the vessel is floating. The density for salt water is gSW = 35 cubic feet / long ton. The density for fresh water is gFW = 36 cubic feet / long ton. The formula for displaced volume is expressed mathematically as V = gi x D, where i depends on water type the vessel is floating in.
The longitudinal metacentric radius is defined mathematically as BML = IL / V.
The transverse metacentric radius is defined mathematically as BMT = IT / V.
Refer to a fluid dynamics or a naval architecture text books for derivations of the metacentric radius formulas above. We are only concerned with application here.
The distance from the metacenters to the keel are then
calculated and follows:
For the longitudinal direction: KML = VCB + BML.
For the transverse direction: KMT = VCB + BMT.
The curves of form inputs are T, LCB, LCF, KMT and KML.
Trim, Heel and Initial Stability Calculations.
These trim, heel and initial stability calculations meaningfully combine the results obtained from the weights and moments and curves of form calculations. For additional background on this approach, see Reference A.
The metacentric heights can now be calculated.
For the transverse direction: GMT = KMT – VCG.
For the longitudinal direction: GML = KML – VCG.
The trimming lever is calculated next. The trimming lever or arm is difference between the LCG and the LCB. The sign system has a positive value if it causes the vessel to trim by the stern and a negative value if it causes it to trim by the bow. This is mathematically defined as Trim Lever = LCG – LCB.
The listing lever or arm is difference between the TCG and the TCB. For the sign system selected this has a positive value if it causes the vessel to list to starboard and a negative value if it causes it to list to port. Where List Lever = TCG – TCB.
Note that the LCB and TCB applied in the levers above are those that are present prior to any trim or list that is they are those that apply to the vessel in the level condition. Once the trim occurs the LCB has moved to have the same value as the LCG. Once list has occurred the TCB has moved to have the same value as the TCG. This subtlety is often not mentioned, but these movements are required for the vessel to remain at equilibrium.
Trim is caused by a longitudinal moment. This longitudinal trimming moment = W x Trim Lever. The longitudinal restoring moment = D x GML x tanq, where the trim angle is expressed by tanq = TRIM / L. Equilibrium requires that the longitudinal trimming moment equal the longitudinal restoring moment. Reworking these three equations yields TRIM = L x (Trim Lever) / GML.
For the longitudinal direction, trim may alternatively be calculated using MTI (moment to trim one inch), MTF (moment to trim one foot), MTcm (moment to trim one centimeter), or MTM (moment to trim one meter) but the results will be the same as with the simpler formula that is given above.
List is caused by a transverse moment. This transverse list moment = W x List Lever. The transverse restoring moment = D x GMT x tanf, where the list angle expressed by tanf = LIST / B. Equilibrium requires that the transverse list moment equal the transverse restoring moment. Reworking these three equations yields LIST = B x (List Lever) / GMT.
For the transverse direction, a shortcut is often taken. This involves calculating the list moment based on the weight, w, causing the list, and its transverse heeling arm, y. This method has a list moment = w x y. This heeling moment is then set equal to the transverse restoring moment (the formula containing GM) and calculations to determine list are then made. With this method it is implied that the heel weight was initially on board, but located along the centerline when the vessel’s weights and moments calculation were carried out. In the regular method, the first part of the previous paragraph, the effects of this weight being off centerline are already incorporated in the weights and moments analysis made for the entire vessel. In the regular method the heeling moment arm is for the entire vessel and the heel weight is that of the entire vessel. But both methods should agree in the magnitude of the heeling moment, provided that there are no other unbalanced weights on board. The first method has a smaller arm and a larger weight and the second method has a smaller weight and a larger arm, but when each respective weight and arm is multiplied together the moment values obtained should be exactly the same.
For the longitudinal direction, the trimmed waterline goes through the LCF. The forward and aft drafts are resolved through the use of similar triangles (for details see Reference B). Barges are a simplified case because the LCF occurs at amidships and the similar triangle approach is not necessary, though it can be used. Likewise for the transverse direction, similar triangles are not necessary because the heeled waterline goes right through the intersection of the centerline and the initial waterline. This occurs with wall sided vessels like a barge. This would not be the case if the vessel had flare or tumblehome. If the vessel had flare, the heeled waterline would normally run just a little below the intersection of the initial waterline and the centerline.
This is a very methodological approach and there are all kinds of shortcuts available, but this method can be applied to all possible cases. So it may take longer in some cases, but is a comprehensive approach. If this approach is understood, it will be easier to understand the shortcut methods that are available.
To make it easier, these concepts are all brought together in pre-made spreadsheets that are available on this website. Their web addresses are at www.hawaii-marine.com/templates/Products/QBrg-e/description.htm and www.hawaii-marine.com/templates/Products/QBrg-m/description.htm. These spreadsheets also conveniently show waterline profiles so users can quickly determine if the deck is submerged or the bottom has emerged above the water surface. If either of these cases occur, weights should be changed (and/or redistributed) or a new barge be selected such that these conditions do not occur. In fact it is proper and expedient that there be margins present so that the barge deck never submerges and the barge bottom never emerges from the water.