What is the angle of loll for this negative GM?

Formula, assumptions, and limits

A ship with KM 7.50 m, KB 3.00 m, KG 7.55 m and a free surface correction of 0.10 m has a fluid GM of −0.15 m and a BM of 4.50 m. On the wall-sided formula it lolls to 14.48° either side, and its effective GM at that angle is 0.310 m.

Formula

• $\theta_L$: angle of loll, in degrees, to port or to starboard.
• $GM$: fluid initial metacentric height, in meters. It must be negative for a loll; zero or positive gives none.
• $BM$: transverse metacentric radius at the upright waterline, in meters, always positive.
• $GM_L$: effective metacentric height at the angle of loll, the slope of the righting lever curve there, in meters.
• $KM$, $KB$, $KG$: heights of the metacenter, the center of buoyancy and the center of gravity above the keel, in meters.
• $\text{FSC}$: free surface correction, the virtual rise of G from slack tanks, in meters.

Derivation

When a ship whose sides are vertical through the waterline heels by $\theta$, the immersed and emerged wedges are equal triangles. Taking their moments about the upright center of buoyancy B puts the new center of buoyancy $B_1$ a distance $BM\tan\theta$ out to the low side and $\tfrac{1}{2}BM\tan^2\theta$ above B, measured on the ship’s upright axes. The righting lever is the horizontal distance from G to the vertical through $B_1$:

Since $BM - BG = GM$, this collapses to the wall-sided formula:

With GM negative, the bracket is negative at small angles, so the ship heels away from upright. GZ is zero at any angle other than upright where $GM + \tfrac{1}{2}BM\tan^2\theta = 0$, which gives $\tan^2\theta_L = -2GM/BM$ and the first formula above. The slope of the curve there is the derivative of GZ at $\theta_L$. The term carrying the bracket vanishes, and what is left is $BM\tan^2\theta_L/\cos\theta_L$, or $-2GM/\cos\theta_L$. The MCA’s Naval Architecture written examination syllabus, section 3.2, asks candidates to apply the wall-sided formula to a vessel with negative metacentric height and derive this expression for the angle of loll.

Assumptions

• The sides are vertical between the upright waterline and the waterline at the loll angle: no flare, no tumblehome, no deck edge under and no bilge out of the water.
• BM is the upright value at the present draft, and the ship does not trim as it heels.
• G lies on the centerline. Any off-center weight adds a list, which this calculator does not model.
• The free surface correction is constant over the heel. Real tank surfaces change shape with angle, and the stability booklet’s tables give the figure the loading computer uses.

Worked example

Take KM 7.50 m, KB 3.00 m, KG 7.55 m and a free surface correction of 0.10 m: an 18 m box section at 6 m draft has KB 3.00 m and BM $18^2/(12 \times 6)$ = 4.50 m, which is where these figures come from.

1. Fluid GM: 7.50 − 7.55 − 0.10 = −0.15 m.
2. BM: 7.50 − 3.00 = 4.50 m.
3. $\tan\theta_L$ = √(2 × 0.15 / 4.50) = √0.06667 = 0.2582.
4. Angle of loll: arctan 0.2582 = 14.48°.
5. Effective GM at loll: 2 × 0.15 / cos 14.48° = 0.30 / 0.9682 = 0.310 m.

The righting levers on the wall-sided formula confirm it: −0.0116 m at 5°, −0.0139 m at 10°, zero at 14.48°, +0.0506 m at 20° and +0.3000 m at 30°.

Edge cases and limits

A fluid GM of exactly zero gives no angle of loll: the only root of the wall-sided formula is upright, where the ship is neutral, and GZ = ½ BM sin θ tan²θ is positive at every small angle. A positive GM has no angle of loll at all, and a steady heel on such a ship is a list.

The formula stops holding once the deck edge goes under or the bilge comes out. The calculator compares the result with a box-section deck-edge angle, $\arctan(f/(B/2))$, when the breadth B and the freeboard f are entered. For the worked example’s 18 m breadth and 2.5 m freeboard that is 15.52°. Raising KG to 8.00 m on the same ship gives a fluid GM of −0.60 m and a wall-sided loll of 27.31°, well past the deck edge. The real angle then has to be read from the GZ curve built from the ship’s cross curves.

The box-section deck-edge angle is this site’s screening check, not a figure from the stability booklet. Flare, sheer and camber all change the real angle, and the booklet’s GZ curves include them.

Regulatory basis

No IMO instrument states the angle of loll formula or sets a limit on the angle. The 2008 IS Code, adopted by resolution MSC.267(85) on 4 December 2008, closes off the condition from the other side: Part A, paragraph 2.2.4, requires the initial metacentric height GM₀ to be not less than 0.15 m, and paragraph 2.1.2 requires free surface effects to be accounted for in all conditions of loading. A ship that meets 2.2.4 in its fluid GM cannot loll on the wall-sided model. The MCA’s MSIS 43 (Rev 05.23, 2023) reproduces the code text with its later amendments, among them resolutions MSC.413(97) and MSC.415(97). The exam syllabuses are the documents that name the loll: the MCA’s Chief Mate/Master syllabus, section 12, covers the dangers of an angle of loll, the difference from a list and the correct procedure for correcting it.

Common errors

• Entering KM where BM is asked for. On the worked example that gives 11.31° in place of 14.48°.
• Dropping the square root: $\tan\theta = -2GM/BM$ gives 3.81° for the same ship.
• Using solid GM, KM − KG, when tanks are slack. Here solid GM is −0.05 m and gives 8.48°, about six degrees short.
• Writing the effective GM with the signed GM, which makes it negative. It is $2\lvert GM \rvert/\cos\theta_L$, a positive figure.
• Correcting the heel as a list, by moving weight to the high side.

Angle of loll and angle of list compared

The MCA’s Officer of the Watch syllabus, section 8(g), asks candidates to understand the difference between the two and the corrective measures for each. The figures that separate them:

The tell on board is the flop. The US Naval Academy’s EN400 text, section 4.9.4.1, describes a lolling ship oscillating between its two loll angles, and notes that an empty tanker can carry its center of gravity high enough to loll.

Free surface as the cause of loll

Slack tanks are the usual route to a negative fluid GM in service, because the free surface correction raises G virtually without any weight moving. The worked example’s ship shows the effect in three steps:

A 0.10 m correction takes a ship with a positive GM of 0.10 m to neutral. It also turns an 8.48° loll into a 14.48° one, because the loll angle grows with the square root of the GM deficit, not in proportion to it.

Loll angle against GM

For a BM of 4.50 m, the wall-sided loll angle at each fluid GM:

The first 0.05 m of negative GM produces the largest step, 8.48°. Each further 0.05 m adds less: 3.42°, then 2.58°, 2.12°, 1.83° and 1.63°. On the worked example’s 18 m by 2.5 m freeboard section, every GM from −0.20 m down puts the computed loll past the 15.52° deck-edge angle, where the table stops being a reliable answer.

Correcting an angle of loll

The correction lowers G. Moving weight across the ship does not help, because G is already on the centerline. The order follows from the formulas above:

1. Confirm the heel is a loll: the ship lies to either side, no off-center weight explains it, and the loading condition shows a negative fluid GM.
2. Remove free surface where it lowers G. A slack tank that sits low and on the centerline, such as a double-bottom tank, can be pressed up or emptied; either removes its free surface correction. A slack tank that sits high, or on the high side, is a different case: filling it adds weight above G or a moment toward the side the ship is not lying to.
3. Lower weights or take ballast low, into a subdivided double-bottom tank on the low side. The added weight then sits on the side the ship already lies to, so as GM passes zero the ship comes back toward upright from that side rather than across. It then lies listed to that side, and filling the matching tank on the other side brings it upright.
4. Expect the heel to grow while an empty tank fills: the new free surface counts against GM under the 2008 IS Code, Part A 2.1.2, until the tank is pressed up.

Ballast into a high-side tank adds a moment toward the side the ship is not lying to. With GM still negative, that moment can send the ship across to loll on the other side, through upright and with the roll that carries it there.

Where to find each input

KM and KB come from the hydrostatic particulars in the approved stability booklet, read at the present mean draft or displacement; BM is their difference. KG comes from the loading condition, worked by the loading computer or by hand from the light ship KG and the weights on board. The free surface correction comes from the tank tables in the same booklet, as a free surface moment for each slack tank divided by the displacement. Breadth is the molded breadth amidships, and the freeboard is the distance from the waterline to the deck edge amidships at the present draft.