25-Nav-A1 Fundamentals of Naval Architecture · May-98-Nav-A1 2016

Question 7 of 7: Inclining Experiment & Heel After Removing a Slack Tank's Free Surface

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 7: Inclining Experiment & Heel After Removing a Slack Tank's Free Surface

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Precautions during an inclining experiment

An inclining experiment must isolate the single relationship it is designed to measure — heel angle versus a known, precisely-located weight shift — from every other influence on trim and heel, so the standard precautions are: conduct it in calm water, sheltered from wind and current, with all mooring lines slack; use only 2–3 known test weights, moved a precisely measured transverse distance, with their exact position recorded at each stage; ensure all loose/movable weights (cranes, lifeboats, cargo-handling gear) are secured in a known position, and that no personnel move about during a reading; use two or more pendulums (or a manometer/inclinometer) at different locations to cross-check readings and average out local effects; press up (completely fill) or completely empty all slack tanks to eliminate free-surface effects, and record the actual conditions of any tank that cannot be pressed up; take readings only after the ship has settled and stopped oscillating following each weight shift; and record the actual draft/trim/water density at the time of the test so the measured $GM$ can be corrected to the required loading condition.

(b) Heel angle after removing the oil tank's free surface

Given. The inclining-experiment pendulum reading gives an apparent $GM$ that is reduced by the on-board oil tank's free-surface effect (the tank is only half full — 4.00 m of 8.00 m depth — hence slack); once that free surface is removed, the ship's actual (solid) $GM$ governs a second, different weight shift.

Find. The heel angle for the second weight shift, once the tank's oil has been completely removed.

Approach. Back out the apparent $GM$ from the pendulum reading, add the tank's free-surface correction to recover the ship's actual solid $GM$, then apply the small-angle heeling formula to the second (post-removal) weight shift, for which no tank free-surface term remains.

1. Apparent $GM$ from the inclining test. With $\tan\theta_1=w_1d_1/(\Delta\,GM_{app})$ and $\theta_1=1.3^\circ$ ($\tan1.3^\circ\approx0.022693$): $$GM_{app}=\frac{w_1d_1}{\Delta\tan\theta_1}=\frac{6.00(10.00)}{3500.00(0.022693)}\approx0.7554\text{ m}$$
2. Free-surface correction of the slack tank. The tank's own free-surface inertia is $i=\dfrac{l\,b^3}{12}=\dfrac{7.50(15.00)^3}{12}=2109.4\text{ m}^4$ (independent of the 4 m oil depth), and $\nabla=\Delta/\rho_{sw}=3500.00/1.025\approx3414.6\text{ m}^3$: $$FSC=\frac{0.8}{1.025}\cdot\frac{2109.4}{3414.6}\approx0.4821\text{ m}$$
3. Ship's actual (solid) $GM$. The pendulum reading during test 1 already reflects the reduced, fluid $GM$, so the ship's real $GM$ is recovered by adding back the correction: $$GM_{actual}=GM_{app}+FSC=0.7554+0.4821\approx\boxed{1.238\text{ m}}$$
4. Heel after emptying the tank and shifting the second weight. With the oil completely removed, the tank has no liquid and hence no free surface, so the ship's stability is governed directly by $GM_{actual}$ (wall-sided, small-to-moderate angle): $$\tan\theta_2=\frac{w_2d_2}{\Delta\,GM_{actual}}=\frac{25.00(10.00)}{3500.00(1.238)}\approx0.05772$$ $$\boxed{\theta_2\approx3.30^\circ}$$