Limit of Positive Stability (LPS)
I thought I might begin a thread to discuss this topic. I hope the knowledgeable designers amongst us will chime in with their thoughts and correct any errors I may introduce below:
A recent post inquired about the Limit of Positive Stabilty (LPS) of a particular boat model. LPS, sometimes referred to as the Angle of Vanishing Stability, is a measure of a boat's ultimate stability. The LPS figure, expressed in degrees, is supposed to approximate the point at which a particular boat will heel so much that it cannot right itself. At that point, the boat may continue to roll upside down. In theory the sailboat will eventually selfright, usually by completing the roll through 360 degrees (assuming no downflooding, etc). The ORC (Offshore Racing Council) recommends a minimum LPS of 120 degrees for offshore racing. An LPS of 120 degrees would mean that the boat could heel an additional 30 degrees past a perpendicular 90 degree knock down, and still right itself (again, in theory, since LPS figures are static calculations which don't reflect dynamic variables such as seastate, downflooding, loading, etc). In the other thread I mentioned, JeffH made the following observation: Quote:
Published stability tables for older production boats, as well as many newer ones, can be hard to come by. When I can find them (not all builders publish the LPS), it generally surprises me how low the LPS figures are. For instance, we are presently considering purchasing a larger sailboat for our family, and I was surprised to learn that the LPS of this popular 42 foot model was only in the range of 114 degrees. Our current 31 footer has an LPS of 139 degrees, which is among the lowest for all the models made by this manufacturer. Looking at the hull form of the Bayfield 29... http://farm3.static.flickr.com/2209/...bd06d060_o.jpg ...I would have thought the LPS to be higher than 105110  it just looks like a more stable hull form than many of the modern production boats. So, lacking the LPS tables from a builder, how do we guage the suitability of a design for coastal or offshore sailing? The figure of 120 degrees or better is considered desirable for offshore sailing. What is a minimum LPS figure for coastal sailing? 
JRP—
Hull form is just one factor in LPS. The amount of ballast, the weight of the rig the beam of the boat, all contribute... 
Quote:
Limit of positive stabiity is sometimes called the "point of no return." The boat heels over so much that righting arm shrinks to zero. At that point, once it heels further, which can happen by being tilted on a wave face at sea, the righting arm turns into an overturning arm and flips the boat upside down. A couple things to keep in mind: The calc of this vanishing point is usually based on the hull form, but the cabintop can provide reserve buoyancy (provided the companionways is shut) and increase its value. You'll routinely see pictures of capsized keelboats happily sitting upside down, their crew on the exposed hull bottom (or squirreled away inside). Because of hull form, a keelboat can be more stable upside down than it is right side up. That's partly because despite the weight of the keel, a keelboat's center of gravity typically is higher than the center of buoyancy when rightside up. What stops it from falling right over is that as it heels the center of buoyancy moves outboard, creating a righting arm. When a keelboat is upside down, the center of gravity is lower than the center of buoyancy, thus providing "good" stability. (Dinghies are typically the same, even moreso, as anyone who has turtled a dinghy knows.) Thus while a keelboat technically should be able to right itself, that isn't necessarily going to happen once it's upside down. Depending how much air has been captured in the hull (which affects overall buoyancy and the center of buoyancy as it rolls), it needs some external energy input to roll back rightside up. In storm conditions, waves can provide that. The bottom line is that if hull form is exploited through form stability to provide a high vanishing angle, it typically means that it takes a lot of force should the boat become inverted to turn it rightside up again. Steve Killing and I wrote about this in Yacht Design Explained, published by WW Norton in 1998. He could explain it all way better than I can, but that's a start. 
The angle of vanishing stability and capsize screening formals can be found at:
http://www.sailingusa.info/formula.htm They don't take into account nonstandard weight distributions of the hull or keel and they don't count the cabin house as part of the hull. Basically, narrow, heavy and deep boats get the best numbers. 120 degrees is usually considered the minimum AVS for offshore sailing. My 1960s keel/centerboarder is considerably over the minimum: it's narrow by present standards, and heavy, but not deep. I did own a boat that passed the formulas but I knew was too tender. I had an inclining test done (moving known heavy weights from rail to rail and measuring the change in list) the results of which alarmed the naval architect. Turns out the lead ballast on the plans (used for the formulas) was actually much lighter iron. A few hundred pounds of internal ballast helped but didn't solve the problem. Most quoted AVS are provided by the manufacturer and. looking at typical modern designs that are light and wide, one wonders at their accuracy. 
That is an interesting formula seemingly pretty accurate. It came back with a LPS of 128 for my 10,500 lb 38 ft boat, which is about right.
To comment on two points in Notoway's post, "Most quoted AVS are provided by the manufacturer and. looking at typical modern designs that are light and wide, one wonders at their accuracy." One of the great things about modern design software is that it is much easier to accurately model LPS and get a reasonably accurate LPS for a boat while it is being designed. Of course manufacturers of boats can tilt the numbers a bit by assuming full tanks and empty lockers and the like. For what it is worth when this technology has been applied to 1960's era keel/CB boats they generally have LPS angles well under 120 degrees, often down around 110 or so. Although narrow, they carried a larger percentage of their beam to the ends of the boat, and had lower freeboard both fators that results in a smaller limits of positive stability. As seen in the US Sailing estimate of stability, displacement itself has a comparatively small impact on angle of vanishing stability, and because heavy boats of the 1960's tend to have comparatively high vertical center of gravities, their weight works against them in dynamic resistance to capsize as well. Jeff 
It might be noted that the point of maximum righting moment, where the couple GZ is largest, is at deck edge immersion. Further heeling results in a decrease in the righting moment, which is the displacement times GZ. While the vessel is still stable and possesses righting moment, that tendency to right is diminished with every degree past deck edge immersion. Obviously the size and shape of deck structures as well as the flooding of cockpits, etc...only come in to play at these large angles of inclination.
Initial stability, as measured with an inclining experiment, is a measure of the vessel's stability at small angles of heel where hull form and freeboard are much less a factor in ultimate stability. Initial stability will give us the sense of whether the vessel is tender or stiff as well as the resultant motion in a seaway. High freeboard, whatever it's other detractions, does impart a large angle of heel prior to deck edge immersion and the resultant decrease in stability. Significant deadrise to the underwater hull form raises the center of buoyancy as well which will provide a greater righting moment at large angles of heel. Flared bows will also increase stability at large angles of inclination. To the extent that beam plays a role in stability, it is perhaps best understood to be one of more rapidly vanishing stability after the point of deck edge immersion. A beamy, high initial stability boat that feels quite stiff may fail to offer significant stability much past the angle of deck edge immersion or, perhaps better stated, after deck edge immersion the beamy vessel may experience rapid decreases in positive righting moment and start to feel very tender rather quickly. As a result, I am somewhat puzzled with the fascination with beamy, plumb bowed boats, although open to enlightenment. 
Without getting into a lot of detail but, there is a number of mistaken assumptions in your posting. First of all, on most ballasted keel boats, the maximum stability is around 90 degees of heel, at which point there is the maximum spread between the vertical center of gravity and the instantanteous center of bouyancy. Depending on the specifics of the design, the deck hits the water somewhere between 4555 degrees of heel.
Deepening deadrise lowers the vertical center of buoyancy at low heel angles reducing inititial stability, but within normal, second half of the twentieth century, designs has little impact on stability at high angles of heel. At high angles of heel, the boat pretty much floats on its topsides and so the portion of the hull where the deadrise occurs is located is typically out of the water. Adding buoayancy in the form of beam increases the amount of force required to get the boat to its limit of positive stability, but it also increases the amount of force required to bring the boat back up again once its passes its its limit of positive stability. When a boat carries its beam towards its ends there is more buoyancy outboard and so it has more form stability and as a result it takes greater force to right than a boat with a identical beam which occurs only at a single point. (edit shown in Italics) In hindsight, as I thought about yesterday's post, I thought that this matter of the increase in inverted form stability that results from carrying beam towards the end needs more explanation. If we think about the plan form of a 1960's era boat such as the keel/ centerboarder mentioned earlier in this thread, they carried their beam very far towards their ends compared to more modern IMS/IRC derived designs. If you looked at these boats from above, the 1960's era boats are closer in form to a rectangle and IMS/IRC derived modern boats are more triangular in form. So while the more modern design may have a greater beam, it rarely has as much deck area as the same length 1960's era design. If you think about calculating form stability, (assuming similar amounts of flare in the topsides which is reasonable since both 1960's era and IMS/IRC derived designs have very little flare) in its simpliest form, the force to over turn is proportionate to the deck area times the lever arm. So for the sake of simplifying our example, we can assume that equal length boats of both eras have similar deck areas (which is not really a fair assumption since modern designs of equal length typically have smaller deck areas) and the modern boat has 20% more beam which is pretty typically the case, and by way of simplifying things the 1960's boat is a rectangle and the modern boat is a triangle. The center of the area of the rectangle form would be at a point that is half its width, while the center of area of the triangle would be at a point one third of its width. In effect, in this simple model, the rectanglar form of the 1960's boat would have 20% more form stability than the triangular form of the modern hull form [1.2= (1 divided by 2)/ (1.2 divided by3)] Of course this is a bit of an over simplification, since 1960's boats are not literally rectangles and modern boats are not triangles, but what it does show is that modern IMS/IRC derived boats, while somewhat beamier than 1960's era boats do not necessarily have greater inverted form stability,and it fact, they often are carefully modeled to have poor inverted form stability inorder to achieve CE open ocean classifications. The fascination with current crop of plumb bow, moderate beam, ultra low vertical center of gravity, carefully modelled hull forms is substantially better seakeeping, higher stability forces required to achieve any given heel angle, often higher angles of positive stability, lower resistance through the water permitting smaller sail plans, much more comfortable motions than similar length older style boats and of course greater speed. Feeling enlightened......Gotta go 
Beware that you're also assuming that the vessel design will be selfrighting which is not always the case, especially with socalled 'modern' design cruising hulls which are actually perfectly stable when upsidedown.

Especially boats like mine... takes a lot to get it to heel past 15˚, but things get really hairy after that... :)

SD, I imagine that your boat would be perfectly stable upsidedown! :p

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