2 Guidelines on vulnerability criteria

 2.1 Preface

As described in section 1.2 of part A of the 2008 IS Code, the Administration may for a particular ship or group of ships apply criteria demonstrating that the safety of the ship in waves is sufficient. For this purpose, the criteria for the dynamic stability failure modes in waves have been developed, which address the dead ship condition, excessive acceleration, pure loss of stability, parametric rolling, and surf-riding/broaching failure modes. These criteria should be used for ensuring a uniform international level of safety of ships with respect to these failure modes.

 2.2 Assessment of ship vulnerability to the dead ship condition failure mode

2.2.1 Application

2.2.1.1 The provisions given hereunder apply to all ships, except for ships with an extended low weather deck.^footnote

2.2.1.2 For each loading condition, a ship that:

• .1 meets the standard contained in the criteria contained in 2.2.2 is considered not to be vulnerable to the dead ship condition failure mode; or

• .2 does not meet the standard contained in the criteria contained in 2.2.2 should be subject to more detailed assessment of vulnerability to the dead ship condition failure mode by applying the criteria contained in 2.2.3.

2.2.1.3 Alternatively to the criteria contained in 2.2.2 or 2.2.3, for each loading condition a ship may be subject to either:

• .1 direct stability assessment for the dead ship condition failure mode that is performed according to the Guidelines for direct stability assessment in chapter 3; or

• .2 operational limitations related to operational area or route and season developed in accordance with the Guidelines for operational measures in chapter 4.

2.2.1.4 A detailed assessment of Level 2 vulnerability according to the criteria contained in 2.2.3 may be performed without the requirement to conduct a more simplified assessment in 2.2.2. Similarly, a detailed direct stability assessment as provided in 2.2.1.3.1 may be performed without the requirement to conduct a more simplified assessment in 2.2.2 or 2.2.3.

2.2.1.5 Stability limit information for determining the safe zones as functions of GM, draught and trim is to be provided based on matrix calculations according to the criteria contained in 2.2.2 or 2.2.3, and, if appropriate, direct stability assessment according to the Guidelines for direct stability assessment in chapter 3. If relevant, the stability limit information for determining safe zones should take into account operational limitations related to specific operational areas or routes and specific season according to the Guidelines for operational measures in chapter 4.

2.2.1.6 Reference environmental conditions to be used in the assessment may be modified when introducing operational limitations permitting operation in specific operational areas or routes and, if appropriate, specific season, according to the Guidelines for operational measures in chapter 4.

2.2.1.7 Free surface effects should be accounted for as recommended in chapter 3 of part B of the 2008 IS Code.

2.2.2 Level 1 vulnerability criterion for the dead ship condition

2.2.2.1 A ship is considered not to be vulnerable to the dead ship condition failure mode, if its ability to withstand the combined effects of beam wind and rolling is demonstrated, with reference to figure 2.2.2.1, as follows:

• .1 the ship is subjected to a steady wind pressure acting perpendicular to the ship's centreline which results in a steady wind heeling lever, l_w1;

• .2 from the resultant angle of equilibrium, φ₀, the ship is assumed to roll owing to wave action to an angle of roll, φ₁, to windward; and the angle of heel under action of steady wind, φ₀, 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, l_w2; and

• .4 under these circumstances, area b should be equal to or greater than area a, as indicated in figure 2.2.2.1,

• 

  Figure 2.2.2.1 – Definition of area a and area b

• where the angles in figure 2.2.2.1 are defined as follows:

• φ₀ = angle of heel under action of steady wind (deg)

• φ₁ = angle of roll to windward due to wave action (deg)(see 2.2.2.1.2 and 2.2.2.4)^footnote

• φ₂ = angle of downflooding, φ_f, or 50° or φ_c, whichever is least,

• where:

  φ_f = angle of heel at which openings in the hull, superstructures or deck houses which cannot be closed weathertight immerse. In applying this criterion, small openings through which progressive flooding cannot take place need not be considered as open.

• φ_c = angle of second intercept between wind heeling lever l_w2 and GZ curves.

2.2.2.2 The wind heeling levers l_w1 and l_w2 referred to in 2.2.2.1.1 and 2.2.2.1.3 are constant values at all angles of inclination and should be calculated as follows:

• l_w1 = (m) and

• l_w2 = 1.5 ⋅ l_w1 (m)

where:

• P = wind pressure of 504 (Pa). The value of P used for ships with operational limitations according to 2.2.1.6 may be reduced.

2.2.2.3 Alternative means for determining the wind heeling lever, l_w1, may be used as an equivalent to the calculation in 2.2.2.2. When such alternative tests are carried out, reference should be made to the Guidelines developed by the Organization.^footnote The wind velocity used in the tests should be 26 m/s in full scale with uniform velocity profile. The value of wind velocity used for ships with operational limitations according to 2.2.1.6 may be reduced.

2.2.2.4 The angle of roll, φ₁, referred to in 2.2.2.1 should be calculated as follows:

• φ₁ = 109 ⋅ k ⋅ X₁ ⋅ X₂ ⋅ (deg)

• where:

• X₁ = factor as shown in table 2.2.2.4-1

• X₂ = factor as shown in table 2.2.2.4-2

• k = factor as follows:

  • k = 1.0 for a round-bilged ship having no bilge or bar keels

  • k = 0.7 for a ship having sharp bilges

  • k = as shown in table 2.2.2.4-3 for a ship having bilge keels, a bar keel, or both

• r = 0.73 + 0.6 OG / d, where: OG = KG - d

• s = wave steepness shown in table 2.2.2.4-4

• A_k = total overall area of bilge keels or area of the lateral projection of the bar keel or sum of these areas (m²)

The angle of roll, φ₁, for ships with anti-rolling devices should be determined without taking into account the operation of these devices unless the Administration is satisfied with the proof that the devices are effective even with sudden shutdown of their supplied power.

Note: Intermediate values in these tables should be obtained by linear interpolation.

2.2.2.5 For ships subject to operational limitations according to 2.2.1.6, the wave steepness, s, in table 2.2.2.4-4 may be modified.

2.2.2.6 For any ship, the angle of roll, φ₁, may also be determined by alternative means on the basis of the Guidelines developed by the Organization.^footnote

2.2.3 Level 2 vulnerability criterion for the dead ship condition

2.2.3.1 A ship is considered not to be vulnerable to the dead ship condition failure mode if:

• C ≤ R_DS0

  where:

  R_DS0 = 0.06;

  C = long-term probability index that measures the vulnerability of the ship to a stability failure in the dead ship condition based on the probability of occurrence of short-term environmental conditions, as specified according to 2.2.3.2.

2.2.3.2 The value of C is calculated as a weighted average from a set of short-term environmental conditions, as follows:

• 

  where:

  • W_i = weighting factor for the short-term environmental condition, as specified in 2.7.2;

    c_S,i = short-term dead ship stability failure index for the short-term environmental condition under consideration, calculated as specified in 2.2.3.2.1;

    N = total number of short-term environmental conditions, according to 2.7.2.

2.2.3.2.1 The short-term dead ship stability failure index, C_s,i, for the short-term environmental condition under consideration, is a measure of the probability that the ship will exceed specified heel angles at least once in the exposure time considered, taking into account an effective relative angle between the ship and the waves. Each index, C_s,i, is calculated according to the following formula:

• C_s,i = 1, if either:

  • • .1 the mean wind heeling lever (according to 2.2.3.2.2) exceeds the righting lever, GZ, at each angle of heel to leeward, or

    • .2 the stable heel angle under the action of steady wind, φ_S,, is greater than the angle of failure to leeward, φ_fail,+,; and

  • = 1 – exp(–r_EA T_exp), otherwise;

• where:

  Heel angles are to be taken as positive to leeward and negative to windward;

  T_exp = exposure time, to be taken as equal to 3600 s;

• r_EA = (1/s);

• ;

• ;

• T_z,CS = reference average zero-crossing period of the effective relative roll motion under the action of wind and waves determined according to 2.2.3.2.3 (s);

  σ_CS = standard deviation of the effective relative roll motion under the action of wind and waves determined according to 2.2.3.2.3 (rad);

  δφ_res,EA+ = range of residual stability to the leeward equivalent area limit angle, to be calculated as

  • φ_EA+ – φ_S (rad);

• δφ_res,EA– = range of residual stability to the windward equivalent area limit angle, to be calculated as

  • φ_S – φ_EA– (rad);

• φ_EA+ = equivalent area virtual limit angle to leeward, to be calculated as

  • (rad);

• φ_EA- = equivalent area virtual limit angle to windward, to be calculated as

  • (rad);

• φ_S = stable heel angle due to the mean wind heeling lever, , determined according to 2.2.3.2.2 (rad);

  A_res,+ = area under the residual righting lever curve (i.e., GZ - ) from φ_S to φ_fail,+ (m rad);

  A_res,- = area under the residual righting lever curve (i.e., GZ - ) from φ_fail,- to φ_S (m rad) ;

  GM_res = residual metacentric height, to be taken as the slope of the residual righting lever curve (i.e., GZ - ) at φ_S (m);

  φ_fail,+ = angle of failure to leeward, to be taken as min{φ_VW,+ , φ_crit,+ } (rad);

  φ_fail,– = angle of failure to windward, to be taken as max{φ_VW,- , φ_crit,- } (rad);

  φ_VW,+ = angle of second intercept to leeward between the mean wind heeling lever and the GZ curve;

  φ_VW,– = angle of second intercept to windward between the mean wind heeling lever and the GZ curve;

  φ_crit,+ = critical angle to leeward, to be taken as min{φ_f,+,50deg} (rad);

  φ_{crit, –} = critical angle to windward, to be taken as max{φ_f,-,-50deg} (rad);

  φ_f,+, φ_{f, –} = angles of downflooding to leeward and windward, respectively, in accordance with the definition of "angle of downflooding" in the 2008 IS Code, Part A, 2.3.1 (rad);

2.2.3.2.2 The mean wind heeling lever l_{wind, tot} is a constant value at all angles of heel and is calculated according to the following formula:

• (m)

  where:

  = mean wind heeling moment, to be calculated as:

  • (N m);

• U_w = mean wind speed, to be calculated as:

  • (m/s);

    Different expressions can be used when considering alternative environmental conditions, in accordance with 2.2.1.6;

• C_whm = wind heeling moment coefficient, to be taken as equal to 1.22 or as determined by other methods;

  H_S = significant wave height for the short-term environmental condition under consideration, according to 2.7.2.

2.2.3.2.3 For the short-term environmental condition under consideration, the reference average zero-crossing period of the effective relative roll motion, T_Z,CS, and the corresponding standard deviation, σ_CS, to be used in the calculation of the short-term dead ship stability failure index, C_s,i, are determined using the spectrum of the effective relative roll motion under to the action of wind and waves, in accordance with the following formulae:

• σ_Cs = (m₀)^1/2 (rad)

• T_z,Cs = 2π · (m₀ / m₂)^1/2 (s)

• where:

• m_o = area under the spectrum S(ω) (rad²);

• m₂ = area under the function of ω² · S(ω) (rad⁴/s²);

• S(ω) = spectrum of the effective relative roll angle, to be calculated as follows:

  • (rad²/(rad/s))

• 

• 

• S_aa,c(ω) = spectrum of the effective wave slope, to be calculated as

  • r²(ω) · S_aa(ω) (rad²/(rad/s))

• S_aa(ω) = spectrum of the wave slope, to be calculated as

  • (rad²/(rad/s))

• S_zz(ω) = sea wave elevation energy spectrum (m²/(rad/s)). The standard expression for S_zz(ω) is defined in 2.7.2.1.1.

  • Different expressions can be used when considering alternative environmental conditions, in accordance with 2.2.1.6;

• S_{SMwind ,tot}(ω) = spectrum of moment due to the action of the gust, to be calculated as

  • [ρ_air · U_w · C_whm · A_L · Z ]² · χ²(ω) · S_ν (ω) ((N m)²/(rad/s))

• χ(ω) = standard aerodynamic admittance function, to be taken as a constant equal to 1.0;

• S_ν(ω) = gustiness spectrum. The standard expression for S_ν(ω) is as follows:

  • ((m/s)²/(rad/s))

    with K = 0.003 and X_D = 600 ⋅ ω/(π ⋅ U_w). Different expressions can be used when considering alternative environmental conditions in accordance with 2.2.1.6;

• μ_e = equivalent linear roll damping coefficient (1/s), calculated according to the stochastic linearization method. This coefficient depends on linear and nonlinear roll damping coefficients and on the specific roll velocity standard deviation in the considered short-term environmental conditions;

• ω_0,e(φ_S) = modified roll natural frequency close to the heel angle, φ_S, to be calculated as:

  • (rad/s);

• ω₀ = upright natural roll frequency = 2π/T_r (rad/s);

• r(ω) = effective wave slope function determined according to 2.2.3.2.4;

and other variables as defined in 2.2.3.2.1 and 2.2.3.2.2.

2.2.3.2.4 The effective wave slope function, r(ω), should be specified using a reliable method, based on computations or derived from experimental data,^footnote and accepted by the Administration.

2.2.3.2.5 In the absence of sufficient information, the recommended methodology for the estimation of the effective wave slope function should be used, which is based on the following assumptions and approximations:

• .1 The underwater part of each transverse section of the ship is substituted by an "equivalent underwater section" having, in general, the same breadth at waterline and the same underwater sectional area of the original section;

  • However:

    .1 sections having zero breadth at waterline, such as those in the region of the bulbous bow, are neglected; and

    .2 the draught of the "equivalent underwater section" is limited to the ship sectional draught.

• .2 The effective wave slope coefficient for each wave frequency is determined by using the "equivalent underwater sections" considering only the undisturbed linear wave pressure; and

• .3 For each section a formula is applied which is exact for rectangles.

2.2.3.2.6 The recommended methodology is applied considering the actual trim of the ship. The recommended methodology for the estimation of the effective wave slope is applicable only to monohull ships. For a ship that does not fall in this category, alternative prediction methods should be applied.

 2.3 Assessment of ship vulnerability to the excessive acceleration failure mode

2.3.1 Application

2.3.1.1 The provisions given hereunder apply to each ship in each loading condition provided that:

• .1 the distance from the waterline to the highest location along the length of the ship where passengers or crew may be present exceeds 70% of the breadth of the ship; and

• .2 the metacentric height exceeds 8% of the breadth of the ship.

2.3.1.2 For each loading condition and location along the length of the ship where passengers or crew may be present, a ship that:

• .1 meets the standard contained in the criteria contained in 2.3.2 is considered not to be vulnerable to the excessive acceleration failure mode; and

• .2 does not meet the standard contained in the criteria contained in 2.3.2 should be subject to more detailed assessment of vulnerability to the excessive acceleration failure mode by applying the criteria contained in 2.3.3.

2.3.1.3 Alternatively to the criteria contained in 2.3.2 or 2.3.3, for each loading condition a ship may be subject to either:

• .1 direct stability assessment for the excessive acceleration failure mode that is performed in accordance with chapter 3; or

• .2 operational measures developed in accordance with chapter 4.

2.3.1.4 A detailed assessment of Level 2 vulnerability according to the criteria contained in 2.3.3 may be performed without the requirement to perform a more simplified assessment in 2.3.2. Similarly, a detailed direct stability assessment as provided in 2.3.1.3.1 may be performed without the requirement to perform a more simplified assessment in 2.3.2 or 2.3.3.

2.3.1.5 Stability limit information for determining the safe zones as functions of GM, draught and trim is to be provided based on matrix calculations according to the criteria contained in sections 2.3.2 or 2.3.3 and, if appropriate, direct stability assessment according to the provisions in chapter 3 on direct stability assessment. If relevant, the stability limit information for determining safe zones should take into account operational measures or operational guidance according to the provisions in chapter 4 on operational measures.

2.3.1.6 Reference environmental conditions to be used in the assessment may be modified, according to the Guidelines for operational measures in chapter 4.

2.3.1.7 Free surface corrections should not be applied.

2.3.2 Level 1 vulnerability criterion for the excessive acceleration failure mode

2.3.2.1 A ship is considered not to be vulnerable to the excessive acceleration stability failure mode if, for each loading condition and location along the length of the ship where passengers or crew may be present,

• 

• where:

• R_EA1 = 4.64 (m/s²)

• φ = characteristic roll amplitude (rad) = 4.43 r s / δ_φ^0.5;

• k_L = factor taking into account simultaneous action of roll, yaw and pitch motions,

  • =	1.125 – 0.625 x /L,	if x < 0.2 L,
    =	1.0,	if 0.2 L ≤ x ≤ 0.65 L,
    =	0.527 + 0.727 x /L,	if x > 0.65 L;

• x = longitudinal distance (m) of the location where passengers or crew may be present from the aft end of L;

• h_r = height above the assumed roll axis of the location where passengers or crew may be present (m), for which definition, the roll axis may be assumed to be located at the midpoint between the waterline and the vertical centre of gravity;

• r = effective wave slope coefficient = ;

• K₁ = g β T_r²(τ + τ - 1 /) / (4π²);

• K₂ = g τ T_r²(β-) / (4π²);

• OG = KG – d;

• F = β (τ – 1 / );

• β = ;

• τ = exp(- / ;

• = 2 π² B / (g T_r²);

• = 4 π² C_B d / (g T_r²);

• s = wave steepness as a function of the natural roll period T_r (see 2.7.1), as determined from table 2.3.2.1; and

• δ_φ = non-dimensional logarithmic decrement of roll decay.

Table 2.3.2.1 – Values of wave steepness, _s

(Intermediate values in the table should be obtained by linear interpolation)

2.3.3 Level 2 vulnerability criterion for the excessive acceleration failure mode

2.3.3.1 A ship in a loading condition is considered not to be vulnerable to the excessive acceleration stability failure mode if, for each location along the length of the ship where passengers or crew may be present:

• C ≤ R_EA2

• where:

• R_EA2 = 0.00039;
• C = long-term probability index that measures the vulnerability of the ship to a stability failure due to excessive acceleration for the loading condition and location under consideration based on the probability of occurrence of short-term environmental conditions, as specified according to 2.3.3.2.

2.3.3.2 The value of C is calculated as a weighted average from a set of short-term environmental conditions, as follows:

• 

• where:

• W_i = weighting factor for the short-term environmental condition, as specified in 2.7.2;

• C_{S ,i} = short-term excessive acceleration failure index for the short-term environmental condition under consideration, calculated as specified in 2.3.3.2.1; and

• N = total number of short-term environmental conditions, according to 2.7.2.

2.3.3.2.1 The short-term excessive acceleration failure index, C_S,i, for the loading condition, location and for the short-term environmental condition under consideration is a measure of the probability that the ship will exceed a specified lateral acceleration, calculated according to the following formula:

• C_S,i = exp(-R₂² / (2 σ_LAi²));

• where:

• R₂ = 9.81 (m/s²);
• σ_LAi = standard deviation of the lateral acceleration at zero speed and in a beam seaway determined according to 2.3.3.2.2 (m/s²).

2.3.3.2.2 The standard deviation of the lateral acceleration at zero speed and in a beam seaway, σ_LAi, is determined using the spectrum of roll motion due to the action of waves. The square of this standard deviation is calculated according to the following formula:

• 

• where:

  • Δω = interval of wave frequency (rad/s) = (ω₂ – ω₁) / N (rad/s);

  • ω₂ = upper frequency limit of the wave spectrum in the evaluation range = min((25 / T_r),2.0) (rad/s);

  • ω₁ = lower frequency limit of the wave spectrum in the evaluation range = max((0.5 / T_r),0.2) (rad/s);

  • N = number of intervals of wave frequency in the evaluation range, not to be taken less than 100;

  • ω_j = wave frequency at the mid-point of the considered frequency interval = ω₁ + ((2j – 1) / 2) Δω (rad/s);

  • S_zz(ω_j) = sea wave elevation spectrum (m²/(rad/s)). The standard expression for S_zz(ω) is defined in 2.7.2.1.1;

  • a_y(ω_j) = lateral acceleration = k_L(g+h_r · ω²_j) · φ_a(ω_j) per unit wave amplitude ((m/s²)/m);

  • k_L, h_r = as defined in 2.3.2.1;

  • φ_a(ω_j) = roll amplitude in regular beam waves of unit amplitude and circular frequency ω_j at zero speed, = (φ_r(ω_j)² + φ_i(ω_j)²)^0.5 (rad/m);

  • φ_r(ω_j) = (rad/m);

  • φ_i(ω_j) = (rad/m);

  • a, b = cosine and sine components, respectively, of the Froude-Krylov roll moment in regular beam waves of unit amplitude (kN·m/m), calculated directly or using an appropriate approximation;

  • B_e = equivalent linear roll damping factor (kN m s), with B_e = 2J_{T ,roll} μ_e where μ_e (1/s) is the equivalent linear roll damping coefficient;

  • J_{T ,roll} = roll moment of inertia comprising added inertia = (t·m²)

Other suitable formulations for the numerical integration in the range from ω1 to ω2 can be used as an alternative.

 2.4 Assessment of ship vulnerability to the pure loss of stability failure mode

2.4.1 Application

2.4.1.1 The provisions given hereunder apply to all ships, except for ships with an extended low weather deck,^footnote for which the Froude number, Fn, corresponding to the service speed exceeds 0.24.

2.4.1.2 For each loading condition, a ship that:

• .1 meets the standard contained in the criteria contained in 2.4.2 is considered not to be vulnerable to the pure loss of stability failure mode; and

• .2 does not meet the standard contained in the criteria contained in 2.4.2 should be subject to more detailed assessment of vulnerability to the pure loss of stability failure mode by applying the criteria contained in 2.4.3.

2.4.1.3 Alternatively to the criteria contained in 2.4.2 or 2.4.3, for each loading condition a ship may be subject to either:

• .1 direct stability assessment for the pure loss of stability failure mode that is performed according to the Guidelines for direct stability assessment in chapter 3; or

• .2 operational measures according to the Guidelines for operational measures in chapter 4.

2.4.1.4 A detailed assessment of Level 2 vulnerability according to the criteria contained in 2.4.3 may be performed without the requirement to perform a more simplified assessment in 2.4.2. Similarly, a detailed direct stability assessment, as provided in 2.4.1.3.1, may be performed without the requirement to perform a more simplified assessment in 2.4.2 or 2.4.3.

2.4.1.5 Stability limit information for determining the safe zones as functions of GM, draught and trim is to be provided based on matrix calculations according to the criteria contained in sections 2.4.2 or 2.4.3 and, if appropriate, direct stability assessment according to the provisions in chapter 3 on direct stability assessment. If relevant, the stability limit information for determining safe zones should take into account operational measures according to the provisions in chapter 4.

2.4.1.6 Reference environmental conditions to be used in the assessment may be modified, according to the Guidelines for operational measures in chapter 4.

2.4.1.7 Free surface effect should be accounted for as recommended in chapter 3 of part B of the 2008 IS Code.

2.4.2 Level 1 vulnerability criterion for the pure loss of stability failure mode

2.4.2.1 A ship is considered not to be vulnerable to the pure loss of stability failure mode, if:

• GM_mm ≧ R_PLA and

• where:

• R_PLA = 0.05 (m); and

• GM_mm = minimum value of the metacentric height (m) calculated as provided in 2.4.2.2.

2.4.2.2 As provided by 2.4.2.1, GM_min should be determined according to:

• GM_mm =

where:

• I_TL = transverse moment of inertia of the waterplane at the draft d_L (m⁴);

• d_L d - δd_L (m);

• δd_L = min(d -0.25d_full,) (m);

  • and d – 0.25d_full should not be taken less than zero; and

• s_W = 0.0334.

2.4.2.3 The use of the simplified conservative estimation of GM_min described in 2.4.2.2 without initial trim effect can be applied for ships having non-even keel condition.

2.4.3 Level 2 vulnerability criteria for the pure loss of stability failure mode

2.4.3.1 A ship is considered not to be vulnerable to the pure loss of stability failure mode if, when underway at the service speed, V_S,

• max(CR₁, CR₂ ) ≤ R_PL0

• where:

• R_PL0 = 0.06; and

• CR₁, CR₂ = criteria calculated according to 2.4.3.2.

2.4.3.2 Each of the two criteria, CR₁ and CR₂ in 2.4.3.1, represents a weighted average of certain stability parameters for a ship considered to be statically positioned in waves of defined height, H_i, and length, λ_i, obtained according to 2.4.3.2.2. CR₁ and CR₂ are calculated as follows:

• CR₁ =

  CR₂ =

• where:

• CR₁ = weighted criterion 1, computed using Criterion 1, C1_i, as evaluated according to 2.4.3.3;

• CR₂ = weighted criterion 2, computed using Criterion 2, C2_i, as evaluated according to 2.4.3.4;

• W _i = weighting factor for the short-term environmental condition, as specified in 2.4.3.2.2; i

  N = total number of wave cases for which C1_i and C2_i are evaluated, according to 2.4.3.2.2.

2.4.3.2.1 For calculating the restoring moment in waves, the following wavelength and wave heights should be used:

• Length λ = L; and

• Height h =0.01⋅ iL i = 0,1,…,10 .

The index for the two criteria, based on φ_v and φ_s, should be calculated according to the formulations given in 2.4.3.3 and 2.4.3.4, respectively. This is undertaken for the loading condition under consideration and the ship assumed to be balanced in sinkage and trim in a series of waves with the characteristics as described above.

In these waves to be studied, the wave crest is to be centred amidships, and at 0.1L, 0.2L, 0.3L, 0.4L and 0.5L forward and 0.1L, 0.2L, 0.3L and 0.4L aft thereof.

2.4.3.2.2 For each combination of H_s and T_z specified in 2.7.2, W_i is obtained as the value in table 2.7.2.1.2 divided by the amount of observations given in this table, which is associated with a H_i as calculated in 2.4.3.2.3 below and λ_i is taken as equal to L. The indices for each H_i should be linearly interpolated from the relationship between h used in 2.4.3.2.1 and the indices obtained in 2.4.3.2.1 above.

2.4.3.2.3 The 3% largest effective wave height, H_i, for use in the evaluation of the requirements is calculated by filtering waves within the ship length. For this purpose, an appropriate wave spectrum shape should be assumed.

2.4.3.3 Criterion 1

Criterion 1, C1_i, is a criterion based on the calculation of the angle of vanishing stability, φ_V , as provided in the following formula:

• 

where:

• K_PL1 = 30 (deg)

The angle of vanishing stability, φ_V, should be determined as the minimum value calculated, as provided in 2.4.3.2.1, 2.4.3.2.2 and 2.4.3.2.3 for the ship without consideration of the angle of downflooding.

2.4.3.4 Criterion 2

Criterion 2, C2_i, is a criterion based on the calculation of the angle of heel, φ_sw, under action of heeling lever specified by l_PL2 as provided in the following formula:

• 

where:

• K_PL2 = 15 degrees for passenger ships; and

  • = 25 degrees for all other ship types

• l_PL2 = 8(H_i/λ) dFn² (m);

• H_i = as provided in 2.4.3.2.2 and 2.4.3.2.3;

• λ = as provided in 2.4.3.2.2;

The angle of heel, φ_sw, should be determined as the maximum value calculated as provided in 2.4.3.2.1, 2.4.3.2.2 and 2.4.3.2.3, for the ship without consideration of the angle of downflooding.

 2.5 Assessment of ship vulnerability to the parametric rolling failure mode

2.5.1 Application

2.5.1.1 For each loading condition, a ship that:

• .1 meets the standard contained in the criteria contained in 2.5.2 is considered not to be vulnerable to the parametric rolling failure mode;

• .2 does not meet the standard contained in the criteria contained in 2.5.2 should be subject to more detailed assessment of vulnerability to the parametric rolling failure mode by applying the criteria contained in 2.5.3.

2.5.1.2 Alternatively to the criteria contained in 2.5.2 or 2.5.3, for each loading condition a ship may be subject to either:

• .1 a direct stability assessment for the parametric rolling failure mode that is performed according to the Guidelines for direct stability assessment in chapter 3; or

• .2 operational measures for the parametric rolling failure mode according to the Guidelines for operational measures in chapter 4.

2.5.1.3 A detailed assessment of Level 2 vulnerability according to the criteria contained in 2.5.3 may be performed without the requirement to perform a more simplified assessment in 2.5.2. Similarly, a detailed direct stability assessment as provided in 2.5.1.3.1 may be performed without the requirement to perform a more simplified assessment in 2.5.2 or 2.5.3.

2.5.1.4 Stability limit information for determining the safe zones as functions of GM, draught and trim is to be provided based on matrix calculations according to the criteria contained in 2.5.2 or 2.5.3 and, if appropriate, direct stability assessment according to the provisions in chapter 3 on direct stability assessment. If relevant, the stability limit information for determining safe zones should take into account operational measures according to the provisions in chapter 4.

2.5.1.5 Reference environmental conditions to be used in the assessment may be modified, according to the Guidelines for operational measures in chapter 4.

2.5.1.6 Free surface effects should be accounted for as recommended in chapter 3 of part B of 2008 IS Code.

2.5.2 Level 1 vulnerability criterion for the parametric rolling failure mode

2.5.2.1 A ship is considered not to be vulnerable to the parametric rolling failure mode if

• and

where:

• R_PR = 1.87, if the ship has a sharp bilge; and, otherwise,

  • = 0.17 + 0.425, if C_{m , full} > 0.96;

  • = 0.17 +(10.625 ✕ C_{m , full} -9.775) , if 0.94 ≦ C_{m , full} ≦ 0.96;

  • = 0.17 + 0.2125 , if C_{m , full} < 0.94; and

  • for each formula, should not exceed 4;

• δGM₁ = amplitude of the variation of the metacentric height (m) calculated as provided in 2.5.2.2.

2.5.2.2 As provided by 2.5.2.1, δGM₁ should be determined according to:

• δGM₁ =

• where:

• δd_H = min(D - d,) (m);

• δd_L = min(d - 0.25d_full,) (m);

  • and d - 0.25d_full should not be taken less than zero;

• d_H = d + δd_H (m);

• d_L = d - δd_L (m);

• S_W = 0.0167;

• I_TH = transverse moment of inertia of the waterplane at the draft d_H (m⁴); and

• I_Tl = transverse moment of inertia of the waterplane at the draft d_Hl (m⁴); and

2.5.2.3 The use of the simplified conservative estimation of δGM₁ described in 2.5.2.2, without initial trim effect, can be applied for ships having a non-even keel condition.

2.5.3 Level 2 vulnerability criteria for the parametric rolling failure mode

2.5.3.1 A ship is considered not to be vulnerable to the parametric rolling failure mode, if

• .1 C1 ≤ R_PR1 ; or

• .2 C2 ≤ R_PR2 ;

• where:

• R_PR1 = 0.06

• R_PR2 = 0.025

• C1 = criterion calculated according to 2.5.3.2; and

• C2 = criterion calculated according to 2.5.3.3.

2.5.3.2 The value for C1 is calculated as a weighted average from a set of waves specified in 2.5.3.2.3, as:

• C1 =

• where:

• W_i = weighting factor for the respective wave specified in 2.5.3.2.3;

• C_i = 0, if the requirements of either the variation of GM in waves contained in 2.5.3.2.1 or the ship speed in waves contained in 2.5.3.2.2 is satisfied;

  • = 1, if not;

• N = the number of wave cases evaluated, as specified in 2.5.3.2.3.

2.5.3.2.1 For each wave specified in 2.5.3.2.3, the requirement for the variation of GM in waves is satisfied if:

• GM(H_i, λ_i > 0 and

• where:

• R_PR = as defined in 2.5.2.1;

• δGM(H_i, λ_i) = one-half the difference between the maximum and minimum values of the metacentric height calculated for the ship (m), corresponding to the loading condition under consideration, considering the ship to be balanced in sinkage and trim on a series of waves characterized by a wave height H_i, and a wavelength λ_i;

• GM(H_i, λ_i) = the average value of the metacentric height calculated for the ship (m), corresponding to the loading condition under consideration, considering the ship to be balanced in sinkage and trim on a series of waves characterized by a wave height H_i, and a wavelength λ_i;

• H_i = wave height specified in 2.5.3.2.3 (m); and

• λ_i = wavelength specified in 2.5.3.2.3 (m).

2.5.3.2.2 For each wave specified in 2.5.3.2.3, the requirement for the ship speed in waves is satisfied if:

• V_PRi > V_s

• where:

• V_PRi = the reference ship speed (m/s) corresponding to parametric resonance conditions, when GM(H_i, λ_i)>0:

  • 

• GM(H_i, λ_i) = as defined in 2.5.3.2.1 (m);

• λ_i = wavelength specified in 2.5.3.2.3 (m);

• | | = the absolute value operation.

2.5.3.2.3 The specified wave cases for evaluation of the requirements contained in 2.5.3.2.1 and 2.5.3.2.2 are presented in table 2.5.3.2.3. In table 2.5.3.2.3, W_i, H_i, λ_i are as defined in 2.5.3.2.

Table 2.5.3.2.3 Wave cases for parametric rolling evaluation

2.5.3.2.4 In the calculation of δGM(H_i, λ_i) and GM(H_i, λ_i) in 2.5.3.2.1, the wave crest should be located amidships, and at 0.1 λ_i, 0.2 λ_i, 0.3 λ_i, 0.4 λ_i, and 0.5 λ_i forward and 0.1 λ_i, 0.2 λ_i, 0.3 λ_i, and 0.4 λ_i aft thereof.

2.5.3.3 The value of C2 is calculated as an average of values of C2(Fn_i,β_i), each of which is a weighted average from the set of waves specified in 2.5.3.4.2, for each set of Froude numbers and wave directions specified:

• C2 =

• where:

• C2(Fn_i,β_h) = C2(Fn,β) calculated as specified in 2.5.3.3.1 with the ship proceeding in head waves with a speed equal to V_i;

• C2(Fn_i,β_f) = C2(Fn,β) calculated as specified in 2.5.3.3.1 with the ship proceeding in following waves with a speed equal to V_i;

• Fn_i = , Froude number corresponding to ship speed V_i;

• V_i = V_s·K_i, ship speed (m/s); and

• K_i = as obtained from table 2.5.3.3.

Table 2.5.3.3 Speed factor, K_i

2.5.3.3.1 The weighted criteria C2(Fn_i,β) are calculated as a weighted average of the short-term parametric rolling failure index considering the set of waves specified in 2.5.3.4.2, for a given Froude number and wave direction, as follows:

• C2(Fn_i,β) =

• where:

• W_i = weighting factor for the respective wave cases specified in 2.5.3.4.2;

• C_{S ,i} = 1, if the maximum roll angle evaluated according to 2.5.3.4 exceeds 25 degrees, and

  • = 0, otherwise;

• N = total number of wave cases for which the maximum roll angle is evaluated for a combination of speed and heading.

2.5.3.4 The maximum roll angle in head and following waves is evaluated as recommended in 2.5.3.4.1 for each speed, V_i, defined in 2.5.3.3. For each evaluation, the calculation of stability in waves should assume the ship to be balanced in sinkage and trim on a series of waves with the following characteristics:

• wavelength, λ = L ;

• wave height, h_j = 0.01 · jL, where j = 0,1,...,10 .

For each wave height, h_j, the maximum roll angle is evaluated.

2.5.3.4.1 The evaluation of roll angle should be carried out using the time domain simulation method with GZ calculated in waves.

2.5.3.4.2 W_i is obtained as the value in table 2.7.2.1.2 divided by the number of observations given in the table. Each cell of the table corresponds to an average zero-crossing wave period, T_z, and a significant wave height, H_s. With these two values, a representative wave height, Hr_i, should be calculated by filtering waves within the ship length. The maximum roll angle, corresponding to the representative wave height, Hr_i, is obtained by linear interpolation of the maximum roll angles for different wave heights, h_j, obtained in 2.5.3.4. This maximum roll angle should be used for the evaluation of C_S,i in 2.5.3.3.1.

 2.6 Assessment of ship vulnerability to the surf-riding/broaching failure mode

2.6.1 Application

2.6.1.1 For each loading condition, a ship that:

• .1 meets the standard contained in the criteria contained in 2.6.2 is considered not to be vulnerable to the surf-riding/broaching failure mode;

• .2 does not meet the standard contained in the criteria in 2.6.2 should be subject to either:

  • .1 the procedures of ship handling on how to avoid dangerous conditions for surf-riding/broaching, as recommended in section 4.2.1 of the Revised guidance to the master for avoiding dangerous situations in adverse weather and sea conditions (MSC.1/Circ.1228), subject to the approval of the Administration; or

  • .2 more detailed assessment of vulnerability to the surf-riding/broaching failure mode by applying the criteria contained in 2.6.3.

2.6.1.2 Alternatively to the criteria contained in 2.6.2 or 2.6.3, for each loading condition a ship may be subject to either:

• .1 direct stability assessment for the surf-riding/broaching failure mode that is performed according to the Guidelines for direct stability assessment in chapter 3; or

• .2 operational measures based on the Guidelines for operational measures in chapter 4.

2.6.1.3 A detailed assessment of Level 2 vulnerability according to the criteria contained in 2.6.3 may be performed without the requirement to perform a more simplified assessment in 2.6.2. Similarly, a detailed direct stability assessment as provided in 2.6.1.3.1 may be performed without the requirement to conduct a more simplified assessment in 2.6.2 or 2.6.3.

2.6.1.4 For ships that do not meet the standard contained in 2.6.2 and which are not applying MSC.1/Circ.1228 according to 2.6.1.1 above, relevant consistent safety information should be provided according to the criteria contained in either 2.6.3 of these Guidelines, Guidelines for direct stability assessment in chapter 3 or Guidelines for operational measures in chapter 4, as appropriate.

2.6.1.5 Reference environmental conditions to be used in the assessment may be modified according to the Guidelines for operational measures in chapter 4.

2.6.2 Level 1 vulnerability criteria for the surf-riding/broaching failure mode

2.6.2.1 A ship is considered not to be vulnerable to the surf-riding/broaching failure mode if:

• .1 L ≥ 200 m; or

• .2 Fn ≤ 0.3.

2.6.3 Level 2 vulnerability criterion for the surf-riding/broaching failure mode

2.6.3.1 A ship is considered not to be vulnerable to the surf-riding/broaching failure mode if

• C ≤ R_SR

where:

• R_SR = 0.005;

• C = criterion calculated according to 2.6.3.2.

2.6.3.2 The value of C is calculated as

• 

• where:

• W2(H_s, T_z) = weighting factor of short-term sea state specified in 2.7.2.1 as a function of the significant wave height, H_S, and the zero-crossing wave period, T_Z, in which W2(H_s, T_z) is equal to the number of occurrences of the combination divided by the total number of occurrences in the table, and it corresponds to the factor W_i specified in 2.7.2;

• w_ij = statistical weight of a wave specified in 2.6.3.3 with steepness (H/λ)_j and wavelength to ship length ratio (λ /L)_i calculated with the joint distribution of local wave steepness and lengths, which is, with specified discretization N_λ = 80 and N_a = 100; and

• C2_ij = coefficient specified in 2.6.3.4.

2.6.3.3 The value of w_ij should be calculated using the following formula:

• where:

• ν = 0.425;

  T₀₁ = 1.086 T_z;

  s_j = (H/λ)_j = wave steepness varying from 0.03 to 0.15 with increment Δs = 0.0012; and

• r_i = (λ/L)_i = wavelength to ship length ratio varying from 1.0 to 3.0 with increment Δr = 0.025.

2.6.3.4 The value of C2_ij is calculated for each wave, as follows:

• 

• where:

• Fn_cr = critical Froude number corresponding to the threshold of surf-riding (surf-riding occurring under any initial condition) which should be calculated in accordance with 2.6.3.4.1 for the regular wave with steepness s_j and wavelength to ship length ratio r_i.

2.6.3.4.1 The critical Froude number, Fn_cr, is calculated as

• Fn_cr = u_cr /

where the critical nominal ship speed, u_cr (m/s), is determined according to 2.6.3.4.2.

2.6.3.4.2 The critical nominal ship speed, u_cr, is determined by solving the following equation with the critical propulsor revolutions, n_cr:

• T_e(u_cr; n_cr) - R(u_cr) = 0

• where:

• R(u_cr) = calm water resistance (N) of the ship at the ship speed of u_cr, see 2.6.3.4.3;

• T_e (u_cr; n_cr = thrust (N) delivered by the ship's propulsor(s) in calm water determined in accordance with 2.6.3.4.4; and

• n_cr = commanded number of revolutions of propulsor(s) (1/s) corresponding to the threshold of surf-riding (surf-riding occurs under any initial conditions), see 2.6.3.4.6.

2.6.3.4.3 The calm water resistance, R(u), is approximated based on available data with a polynomial fit suitable to represent the characteristics of the resistance for the ship in question. The fit should be appropriate to ensure the resistance is continuously increasing as a function of speed in the appropriate range.

2.6.3.4.4 For a ship using one propeller as the main propulsor, the propulsor thrust, T_e(u;n), in calm water may be approximated using a second degree polynomial:

• T_e(u; n) = (1-t_p)ρn²D_p⁴{K₀ + K₁J + K₂J²} (N)

• where:

• u = speed of the ship (m/s) in calm water;

• n = commanded number of revolutions of propulsor (1/s);

• t_p = approximate thrust deduction factor;

• w_p = approximate wake fraction;

• κ₀, κ₁, κ₂ = approximation coefficients for the approximated propeller thrust coefficient in calm water;

• = advance ratio.

In case of a ship having multiple propellers, the overall thrust can be calculated by summing the effect of the individual propellers calculated as indicated above.

For a ship using a propulsor(s) other than a propeller(s), the propulsor thrust should be evaluated by a method appropriate to the type of propulsor used.

2.6.3.4.5 The amplitude of wave surging force for each wave is calculated as:

• (N)

• where:

• k_i = wave number = (1/m);

• H_ij = wave height = s_jr_iL (m);

• s_j ,r_i = as defined in 2.6.3.3;

• 

  • 

• F_Ci and F_Si are parts of the Froude-Krylov component of the wave surging force (m)

• x_m = longitudinal distance from the midship to a station (m), positive for a bow section;

• δx_m = length of the ship strip associated with station m (m);

• d(x_m) = draft at station m in calm water (m);

• s(x_m) = area of submerged portion of the ship at station m in calm water (m²);

• N = number of stations; and

• m = index of a station.

2.6.3.4.6 The critical number of revolutions of the propulsor corresponding to the surf-riding threshold, n_cr (r_j, s_i), can be determined by solving the following quadratic equation:

• 

• where:

• ;

• ;

• ;

• ;

• ;

• ;

• r₁, r₂, r₃, r₄, r₅ = regression coefficients for the calm water resistance under a fifth degree polynomial approximation R(u) ≈ r₁u + r₂u² + r₃u³ + r₄u⁴ + r₅u⁵.

• M = mass of the ship (kg);

• M_x = added mass of the ship in surge (kg). In absence of ship specific data, M_x may be assumed to be 0.1 M;

• = wave celerity (m/s).

• τ₁ = ĸ₁(1 - t_p) (1 - w_p)ρD_p³

• τ₂ = ĸ₂(1 - t_p) (1 - w_p)² ρD_p²

 2.7 Parameters common to stability failure mode assessments

2.7.1 Inertial properties of a ship and natural period of roll motion

2.7.1.1 In the absence of direct calculations, the roll moment of inertia of the ship comprising the effect of added inertia, J_T,roll, may be estimated as follows:

• (t·m²)

2.7.1.2 The natural roll period, T_r, in a given loading condition, in the absence of sufficient information, direct calculation or measurement, may be approximated using the formulae given in part A, 2.3 of the 2008 IS Code, which is repeated below,

• where C = 0.373 + 0.023 (B / d) - 0.043 (L_WL/100) ,

or its alternatives.

2.7.2 Environmental data

2.7.2.1 A set of standard environmental conditions are assumed. The characterization of the standard environmental conditions refers to both the short-term and the long-term. The short-term characterization is given in terms of the spectrum of sea elevation, known as the spectral density of the sea wave elevation. The long-term characterization is given in terms of a wave scatter table. The standard short-term and long-term characterizations are given in 2.7.2.1.1 and 2.7.2.1.2, respectively.

2.7.2.1.1 The spectral density of sea wave elevation, S_zz(ω), is provided by the Bretschneider wave energy spectrum as a function of the wave frequency, ω, as follows:

• 

2.7.2.1.2 The long-term characterization of the standard environmental conditions (used in unrestricted service) is given by means of a wave scatter table. The wave scatter table contains the number of occurrences W_i within each range of significant wave height H_s and zero crossing wave period T_z in 100,000 observations. The wave scatter table, given in table 2.7.2.1.2, specifies factors W_i as functions of H_s and T_z values which represent the mean values of corresponding ranges.^footnote

Table 2.7.2.1.2 Wave scatter table

2.7.2.2 Alternative environmental conditions can be used for ships subject to operational measures according to chapter 4 and should be accepted by the Administration.

2.7.2.2.1 Such alternative environmental conditions should specify the short-term characteristics of wind and sea state, together with the probability of occurrence of each short-term environmental condition.

2.7.2.2.2 The short-term sea state characteristics should be given in terms of a sea elevation spectrum. The short-term wind state should be given in terms of a mean wind speed and a gustiness spectrum.

2.7.2.2.3 The long-term characterization of the environmental condition should be given in terms of probability of occurrence of each short-term condition. The probability of occurrence of each short-term environmental condition corresponds to the weighting factor, W_i. The set of short-term environmental conditions and corresponding weighting factors should be such that the sum of the weighting factors, i.e. the probabilities of occurrence, is unity.

2.7.3 Other common parameters

2.7.3.1 Active means of motion reduction, such as active anti-roll fins and anti-roll tanks, can significantly reduce roll motions in seaway. However, the safety of the ship should be ensured in cases of failure of such devices, therefore, the vulnerability assessment according to these Interim Guidelines should be conducted with such devices inactive or retracted, if they are retractable.