Reply To: Angle of Vanishing Stability and Capsize Screening Formulae

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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)