Angle of Vanishing Stability and Capsize Screening Formulae

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    Would any Member be able to say if the following formulae can be applied to eighteenth and nineteenth century sailing ships, and what DV1/3 stands for? The formulae are found on the Sailing US website:

    Angle of Vanishing Stability:
    This is the resistance to capsize and heel. One of the best predictors of ultimate stability is the “angle of vanishing stability” or the angle to which the boat can heel and still right itself. A dinghy will have a stability range of about 80 degrees, an inland water boat should have a stability range of 100 degrees, and an offshore boat of at least 120 degrees. Boats which have a stability angle of less than 140 degrees may be left floating upside down once capsized. Boats with a higher angle will usually right themselves.
    The following is the formula used to calculate The Angle of Vanishing Stability:
    Screening Stability Value ( SSV ) = ( Beam 2 ) / ( BR * HD * DV1/3)

    BR: Ballast Ratio ( Keel Weight / Total Weight )
    HD: Hull Draft

    DV: The Displacement Volume in cubic metres. DV is entered as pounds of displacement on the webpage and converted to cubic metres by the formula:

    ( Weight in Pounds / 64 )*0.0283168 = Displacement Volume in Cubic Metres
    The Beam and Hull Draft in this formula are in metres. These values are entered in feet on the webpage and are converted to metres before SSV calculation.

    Angle of Vanishing Stability approximately equals 110 + ( 400 / (SSV-10) )

    Adapted from: K. Adlard Coles and Peter Bruce (eds.) Adlard Coles’ Heavy Weather Sailing, revised 3rd edition, chapter 2: ‘Stability of Yachts in large breaking waves’, pp11-23, International Marine, Camden, Maine 1980.

    Capsize Screening Formula:
    The following is used to calculate the Capsize Screening Formula:
    Capsize Screening Formula = Beam / (Displacement / 64)1/3

    Displacement is in Pounds
    Beam is measured in Feet

    Adapted from: John Rousmaniere, The Annapolis Book of Seamanship, chapter 1: ‘Boat Selection’, p35, Simon & Schuster, New York (edition uncited).

    [Editor’s note: Adlard Coles’ classic Adlard Coles’ Heavy Weather Sailing, revised 3rd edition is still available second-hand via Amazon and AbeBooks; the 6th edition was published in the UK in 2008]

    Frank Scott

    For many reasons yachting stability formulae such as those that you quote have no relevance at all to the stability of large sailing ships of any era.
    In order to operate within her normal heeling angle range the traditional commercial sailing ship relied entirely on the skill and experience of the master as to how the ship was loaded and trimmed. If he got it wrong, and the weather was against them, the ship would almost certainly be lost, it was brutally simple.
    In naval ships the command had no cargo to worry about, but could still alter the ships performance significantly by how weight was distributed. Although ‘seat of the pants’ rather than calculated, this was not simple stuff, hence the reason that a change of captain could transform a ship’s performance and reputation from mediocre to outstanding.
    Inclining experiments and calculations of range of stability did not come in until the mid-nineteenth century, so concepts such as ‘vanishing angle’ did not exist prior to that time. Moreover our understanding of the problems of large sailing ship stability had to be almost re-discovered after some tragic sail training accidents between the 1960s and 1980s.
    Chapter 14 of my book A Square Rig Handbook : Operations, Safety, Training, Equipment, Nautical Institute, London, revised 2nd edition 2001, provides a good guide to our current view of the issue. However, it must be emphasised that in an era when masts could be cut away, and there was no world-wide rescue service, ships were operated to the limit, and many survived what we would now regard as catastrophic damage.
    DV1/3 is a simple mathematical formula of Displacement Volume to the power of 1/3. However, I am not sure how to explain exponentiation to those who have not studied such advanced mathematics, and it is in any case not relevant to large sailing ships.

    J. W. M

    I passed your questions to an ex-seafaring colleague, holder of an Extra Master’s Certificate. He suggests that the formula DV1/3, would be one third of the diplaced volume in pounds converted to cubic metres by the formula given:
    (Weight in Pounds/64)*0.0283168.

    He invited me to contact his friend and sailing colleague, the co-author of the book mentioned [in the original post], Commander Peter Bruce RN. However, you may wish to contact him directly on, mentioning the name of my old colleague, David Arnold. Incidentally, the latter survived a 130degree roll, without capsizing aboard Morning Cloud, during the 1979 Fastnet race.

    P. H

    Yes. Modern stability calculation applies equally well back into history. A basic primer is J F Kemp and P Young, Ship Stability Notes and Examples, Sevenoaks 1971. An intermediate text is Dan Derret, Ship Stability for Masters and Mates [sixth edition revised C B Barrass, 2006]. Somewhat advanced is K J Rawson and E C Tupper, Basic Ship Theory,/i> [4th edition 2001]. Naturally, they make metric calculations.

    Frank Scott


    Standard stability abbreviations:

    A Silhouette area (as in CE) sails, hull and superstructure (disregards overlaps)
    B Centre of buoyancy of the underwater body (varies with heel angle)
    G Position of the centre of gravity
    GM Metacentric height (positive when ‘M’ is above ‘G’) for small angles of heel (< 12o )
    GZ Righting lever
    K Keel Datum Point
    KN Distance used for initial calculation of GZ (‘G’ moves with load state)

    M Point about which the vessel pivots – Valid for small angles of heel (< 12o)
    N Point horizontally opposite ‘K’
    Z Point horizontally opposite ‘G’
    W Displacement
    WL Waterline

    Special abbreviations for sailing ship stability:
    CE Geometric centre of above water silhouette (‘A’)
    CLR Geometric centre of underwater hull profile (CP is another abbreviation)
    Dhwl Wind heeling lever mathematically derived from WLO
    GZf Righting lever at θf or 60o (whichever is less)
    H Vertical distance between CE and CLR
    P Wind pressure in kilograms per square metre
    V Wind velocity in knots
    WLO Wind heeling moment to heel ship from zero to θf or 60o (whichever is less)
    ρ Air density

    θ Heel Angle
    θd Angle at which Dhwl intersects GZ curve (must exceed 15o)
    θf Downflooding angle

    Useful formulae:
    Heeling Moment = PAH cos1.3θ kg.m (Wolfson formula)
    Righting Lever = KN – KG sineθ
    Dhwl = ½WLO cos1.3θ
    WLO = GZf ÷cos1.3θf
    Righting Moment = GZ x W
    Wind Pressure = ½ ρV2 kg/sq metre (architects’ figure)

    = 1/50 V2 kg/sq metre (working figure)

    Basic Stability Requirements:

    To provide the optimum balance between resistance to capsize, and comfortable living and working conditions, a modern square rigger should have a roll rate of at least eight seconds, combined with a stability range exceeding 90o, and this is not difficult to achieve at the design stage.

    Maximum angles of steady heel:

    The Wolfson Unit at Southampton University (UK) developed a maximum recommended angle of steady heel (irrespective of sail set) which allows for gust response without heeling to downflooding angle. Vessels must be able to sail at a minimum of 15o of heel while meeting that standard. There is a further limiting angle for squalls, and these maximum angles are intended to set limits rather than encourage people to sail at as great an angle as possible:
    Maximum angle of steady heel to withstand gusts: This gives a fixed angle of heel that should never be exceeded. This angle is extrapolated and as long as the ship is sailed at no more than that angle of heel she will enjoy at least a 1.4 protection factor against gusts, regardless of the steady wind strength.
    Maximum angle of steady heel to withstand squalls: These curves develop the above concept further, and are intended to provide a ready method of calculating a squall protection factor. However, unlike the fixed simple gust protection angle, the result is variable, and the greater the strength of the squall that you wish to protect against, the lower the maximum angle of heel, and thus the more you will have to reduce sail.

    One Wolfson example showed a sailing vessel in 20 knots of wind and at 11o of heel that was already on the limit for countering a 45 knot squall, and obviously even more vulnerable to a 60 knot squall. However, her maximum (limiting) steady angle to counter simple gusts was 27o.

    Downflooding & intact freeboard:
    Given that circumstances can result in knockdown for even the best vessel, maintaining her intact freeboard and minimising the risk of resultant downflooding is critical to her ability to recover. There are many features that can improve ‘intact’ stability at large angles of heel, such as watertight deckhouses. Downflooding creates free surface effect, and thus transforms a difficult situation into an irrecoverable one. A good range of stability is indeed essential, but it must be combined with maximum resistance to downflooding.
    The size of opening that will cause critical downflooding is calculated from a formula related to the ship’s displacement. The angle at which these openings submerge is called the downflooding angle, and it is an important figure for sailing ship stability. UK regulations preclude submergence at less than 40o of any such openings. A downflood angle of more than 65o is both achievable and regarded as best practice.
    Many certifying authorities require that the shipside should not be pierced for opening scuttles, in order to protect intact freeboard. Excessive water trapped on deck is a related potential problem. The modern recommendation is that ships should be able to free their decks of water in a time close to (better still – less than) their period of roll.

    References for Sailing Ship Stability:
    Cleary, Daidola & Reyling. Sailing ship intact stability criteria, Marine Technology, Vol 33 (July 1996)
    Deakin. B. The development of stability standards for UK sailing vessels, Royal Institute of Naval Architects, paper 4 (Spring 1990)
    DTP. Model Stability Booklet for Sail Training Ships under 24 metres, (HMSO, 1990)
    DTP. The Auxiliary Barque Marques DTP Report of Court Number 8073, (HMSO, 1987)
    Kriegsmarine. Niobe: Havarie-Untersuchungs-Akte der Marine, (Berlin, 1932)
    MSA. The Code of Practice for Large Commercial Sailing & Motor Vessels, (HMSO, 1997)
    Marean. P.E. and Long. R.W. Survey of sailing ship stability leading to Modified Regulations, New England Section, Society of Naval Architects and Marine Engineers (1985)
    Tsai.N.T. and Haciski. E.C. Stability of large sailing vessels, Marine Technology vol 23 (1986)
    Scott. F. J. M. A Square Rig Handbook: Operations – Safety – Training – Equipment, (London, 2nd edition 2001)
    University of Southampton, Wolfson Unit. Sail Training Vessel Stability – DTP Report Number 798, (Southampton, 1987)
    NTSB. Marine Accident Report: Capsizing & sinking of the US Sailing Vessel Pride of Baltimore, (Washington, 1987)
    White. W.H. Manual of Naval Architecture, (London, 1894)

    Wayne Tripp

    I apologize for my late arrival to this discussion. In terms of applicability of any modern stability methods to 18th and early 19th century vessels, the answer is that yes, we can apply them from a modern view to understand the possible conditions of a period vessel. The answer is also no, it is not appropriate to apply modern formulations to that era.

    Why Yes? We can, with appropriate information concerning the hull form, ballast, displacement and so on, estimate various stability measures for historic vessels. These measures can help in understanding the performance as reported for a vessel, such as whether a vessel handled well to windward, pitched or rolled excessively &c.

    Why No? There are several reasons for the no, but perhaps the most important is that for the period in question, shipwrights of many nations were struggling with the scientific developments that brought a theory to naval architecture. It remained very rare that a ship builder in Britain accurately predicted the draught of water for a vessel when fully burdened before the ship was afloat (that is, the floating level when carrying the intended weight in stores, cargo, crew, weapons, provisions and so on was determined before the vessel was built, and then verified when launched, rather than the common practice of finding the capacity after launch by loading the vessel until it reached the desired draught of water).

    The French, Swedish and other nations were somewhat more advanced in the use of scientific theory rather than rule of thumb, and indeed even into the mid 18th century we see the persistence of a parabolic method of design and construction rather than a due consideration of hull form and shape in British shipbuilding.

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