Recently was refreshing my limited knowledge of NA and after going to my go to reference Principles of Naval Architecture (classic SNAME reference, Comstock editor) and sorting through the issue I decided to check the new-fangled sources.
I think you could do worse than this series as a quick review.
https://www.youtube.com/user/NEECvideos/playlists
And this was pretty clear on the GZ curve
Good stuff.
At 4:04 the term counterflexure is used for what is usually (at least in my experience) is called the âpoint of inflectionâ. The term counterflexure is a term used for beams, Iâve not seen it used for graphs.
. A point of inflection in a beam is a point where the moment changes sign, and where the center of curvature changes from one side of the beam to the other. A point of inflection is also called a point of counterflexure.
I was surprised to learn that this point, the point of counterflexure, is the point of deck edge immersion. From LaDage and Gemert, which is what I studied, they seem to be saying that the point of maximum righting arm is the point of deck-edge immersion.
But reading it now it says that
"it must be understood that the angle of maximum stability is intimately associated with the angle of deck-edge immersion.
Seems all tied up with waterplane area which changes somewhat drastically when deck edge immersion happens, no?. This may be an erroneous assumption on my part due to long absence from the subject and the simple level I am re-starting out on.
Yes, I think thatâs right, Iâm at point where this stuff starts to get a little fuzzy.
LeDage says the âwhen the deck edge is immersed a wedge of buoyancy has been gained on the immersed side. The center of buoyancy consequently has moved out, creating righting arms. But after the deck edge is immersed, no additional buoyancy can be gained.â
Reading on a bit I see it says the max righting arm âoccurs near the angle of deck edge immersion.â
LaDage uses âimmersed formâ rather than âwaterplane areaâ
From LaDage again:
Angle of Maximum List
The angle of maximum stability is of interest from another vital point of view. It indicates in most cases the angle of maximum list. (bold in the original). A ship listed to any particular angle is in equilibrium; that is, the force inclining the ship is equal to the moment. If the inclining moment exceeds the righting moment which the ship possesses, the ship will obviously list over to the angle at which the two moments are again equal. But suppose the inclining moment is greater than the maximum righting arm moment which the ship possesses? Then the ship will capsize. But the maximum righting power of a ship occurs near the angle of deck-edge immersion. Therefore if a ship lists her deck edge under, she will probably capsize immediately.
This is from Marine Insight:
The area moment of inertia of this waterplane area about its centroidal axis is the transverse moment of inertia of waterplane at the corresponding draft.
So not just the area of the waterplane but the shape of the waterplane. Area further from the axis has more righting power. (top drawing)
The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis.
Itâs the thing with the skater spinning. The âareaâ of the skater is the same with arms out or in. but with the arms âinâ the "moment of inertia is reduced.
When a figure skater draws her arms and a leg inward, she reduces the distance between the axis of rotation and some of her mass, reducing her moment of inertia. Since angular momentum is conserved, her rotational velocity must increase to compensate.
But the maximum righting power of a ship occurs near the angle of deck-edge immersion. Therefore if a ship lists her deck edge under, she will probably capsize immediately.
I donot know who LaDage is but he sure makes two funny statements, both wrong.
The angle of deck immersion or the point of inflection is identified on the curve as the point where the curve trend changes from increasing steepness to decreasing steepness or from convex to concave. This can clearly be noted at point âAâ in the above curve. The ship will capsize in C on the curve and not in A as LaDage suggests.
La Dage is: Stability and Trim for the Shipâs Officer by John La Dage and Lee Van Gemert. Itâs the standard text book.
Looks like the curve you posted is from Marine Insight? I donât use that site as I donât think itâs reliable. Is that curve an actual ship? In the video the "shipâ looks like a rectangle.
Iâll stick with the text book.
Also La Dage and Gemert are are discussing a list, not a roll.
Iâm going to assume heâs talking specifically about modern commercial vessel designs as there are numerous large vessels I know of that have a maximum righting arm equal to or greater than 90°.
According to the La Dage and Gemert if the GZ is positive at 90 degrees the vessel will not capsize near deck edge immersion.
Maybe itâs an NAS designed OSV? 
Iâll stick with Ladage too, though considering the misery I go through with NA/stability questions from Cadets, Iâm going to be checking out all these videos to see if I can teach it any clearer when they come on my boat with their projects.
According to Applied Naval Architecture by R. Monro-Smith:
For large angles of inclination:
a) the upright and inclined walterlines do not intersect on the middle line,
b) the metacentre does not remain in a constant position.
The determination of the magnitude of GZ involves a considerable amount of calculation and there are several methods which can be adopted.
The methods that are outlined are Barnesâ, Benjaminâs, Blomâs, direct-wedge, Lelandâs, and Reechâs. Seems like the simplest way is to consult the cross curves.
I actually sailed on this lube oil tanker which had a very small freeboard.
The hull resembles the rectangle shape of the video model rather well. I donot know about the reliability of Marine Insight. Here is a similar GZ curve from another web site.
The point is that the maximum righting power of a ship normally does not occur near the angle of deck-edge immersion as LaDage says.
Furthermore the statement âTherefore if a ship lists her deck edge under, she will probably capsize immediatelyâ is not correct. The point of vanishing stability in the above curve takes place at 73° and not at 35° as LaDage says. Well that is how I read it.
This is precisely what I was taught, too. I wish my camera was working. I have a diagram that shows how it changes with the geometry from fore to aft on a ship with a fine bow and a boxy mid-section. If people are interested, I will scan it when I go to the library.
To my eye the DEI point of contrfleture shown on this graph looks like the rate of change of GZ is increasing? By La Dage the inflection should be the other way at DEI.
La Dage is not saying max list is at the point of vanishing stability. Itâs at the point of max GZ. Thatâs because a ship listing is at equilibrium, Itâs a list, not a roll.
La Dage:
âA ship listed to any particular angle is in equilibrium; that is, the force inclining the ship is equal to the moment which the ship possesses, the ship will obviously list over to the angle at which the two moments are again equal. But suppose the inclining moments is greater than the maximum righting moment which the ship possesses? Then the ship will capsize.â
I think that this is the definition for the angle of loll.
It similar in that that the ship with a list is at equilibrium. But the angle is loll is caused by negative GM, the initial righting arm is less than zero, so the ship heels until righting arm is sufficient to achieve equilibrium.
Hereâs a random righting arm diagram:
If the ship was loaded off-center a moment would be created and the ship would list. As the ship lists the righting arm increases. When righting arm moment = moment caused by off-center weights the ship will be at equilibrium, say at the dot at 10 degrees.
Say more weights are added = increased list = increased righting moment= new equilibrium point, say 20 degrees.
Once the top of the curve is reached, say in this case about 42 degrees, when weight is added = increase list = righting arm decreases. = no equilibrium.
yes, I see that, now. Thanks. But isnât it true that the curve also changes as you increase the displacement? I think the unladen curve is different from the curve when the first weight is added, again another curve after the next weight, etc? If the weights are shifted, rather than added I reckon you can stay on the same curve, which is the inclining experiment.
Yes, thatâs right. The shifting / adding weights is covered in La Dage, I presume all stability books cover this subject.
In the section quoted from La Dage, the point is, from a stability standpoint, what is a dangerous list and what is a dangerous roll? According to the book the general rule of thumb used is a dangerous roll is about 1/2 the angle of vanishing stability or roughly about the angle of maximum righting arm.
The rule of thumb for a dangerous list is 1/2 the angle of maximum righting arm.
Iâd expect trouble before then but with regards to abandon ship decisions, maybe helpful.
This is where deck edge immersion comes in. According to La Dage, DEI occurs, as a rule of thumb, roughly at max righting arm.
So as a guide, according to La Dage, dangerous roll is when the deck edge rolls under, dangerous list is half that.
Presumably this rule of thumb was meant as a guide for a mariner at sea, in trouble. YMMV, check your shipâs stability curves.
====================
This is from the IMO
2.3.1 The ability ofa ship to withstand the combined effects of beam wind and rolling shall be
demonstrated, with reference to figure 2.3.1 as follows:
.1 the ship is subjected to a steady wind pressure acting perpendicular to the shipâs
centreline which results in a steady wind heeling lever (lUâi);
.2 from the resultant angle of equilibrium (<Po), the ship is assumed to roll owing to
wave action to an angle of roll (<pd to windward. The angle of heel under action
of steady wind (<Po) should not exceed 16° or 80% of the angle of deck edge
immersion, whichever is less;
.3 the ship is then subjected to a gust wind pressure which results in a gust wind
heeling lever âŚ
EDIT: Stability aside at lists greater than DEI there is risk from hull openings that may not be sufficiently weathertight.
Iâm happy, as long as they remember to give the basement creatures enough time to emerge. Last ship our clinometer wasnât even remotely on the centreline. If they had given the offset, we could have done the trig⌠but⌠now thinking about climbing the escape trunk if the angle means the ladder is overhanging. Escape trunks should have two ladders at 90 degree angles, I reckon. How do I amend the SOLAS?
The trick is to use sufficient margins. I use 7 meter seas and 50 kt winds, or less depending on (mostly) the load. For TCs itâs 35 kt winds.
There are a lot of things that can go wrong in heavy weather well short of survivability limits.
Iâve been (one time only) in 12-14 meter seas and 70 kts of wind but I had some difficulties (discovered the ship rolls heavily in short steep head seas).
