3 Hull Girder Shear Strength Assessment

3.1 General

3.1.1 The hull girder shear strength requirements apply along the full length of the hull girder, from AE to FE.

3.2 Hull girder shear capacity

3.2.1 The total vertical hull girder shear capacity, Q_R in kN, is the minimum of the calculated values for all plates i contributing to the hull girder shear of the considered transverse section and is to be taken as:

where:

t_i-n50 : Net thickness of plate i, in mm. For longitudinal bulkheads between cargo tanks of oil tankers, t_i-n50 is to be taken as t_sfi-n50 (see [3.4.1]) and t_sti-k-n50 (see [3.5.1]) as appropriate.

q_vi : Contribution ratio for hull girder shear force per mm, in mm^-1, for the plate i based on net scantlings with deduction of 0.5 t_c, which is equal to the unit shear flow per mm, in N/mm, obtained from a numerical calculation based on thin-walled beam theory according to Ch 5, App 1.

τ_i-perm : Permissible shear stress, in N/mm², as given in Table 4, for plate i.

Table 4 : Permissible hull girder shear stress

Operation	Design load	Permissible hull girder shear, τ_i-perm
Seagoing	(S+D)	120/k
Harbour/sheltered water	(S)	105/k
Flooded condition at sea of bulk carriers having a length L of 150 m or above	(A:S+D)	120/k

3.3 Acceptance criteria

3.3.1 Permissible vertical shear force

The positive and negative permissible vertical shear forces are to comply with the following criteria:

For seagoing operation:
For harbour/sheltered water operation:
For flooded condition at sea of bulk carriers having a length L of 150 m or above:

where:

Q_R : Total vertical hull girder shear capacity, in kN, as defined in [3.2.1].

The shear force Q_wv, used in 2 above criteria is to be taken with the same sign as the considered shear forces Q_sw, and Q_sw-f respectively.

3.3.2 Vertical still water shear force

The vertical still water shear forces, in kN, for all loading conditions are to comply with the following criteria:

For seagoing operation:
For harbour/sheltered water operation:
For flooded condition at sea of bulk carriers having a length L of 150 m or above:

where:

ΔQ_mdf : Shear force correction at the transverse section considered, in kN, taken as:

For bulk carriers, the value defined in [3.6.1].
For oil tankers, ΔQ_mdf = 0.

The permissible shear forces Q_sw, Q_sw-p and Q_sw-f are to be taken with the same sign as the considered shear forces Q_sw-Lcd, Q_sw-Lcd-p and Q_sw-Lcd-f respectively.

3.4 Effective net thickness for longitudinal bulkheads between cargo tanks of oil tankers

3.4.1 For longitudinal bulkheads between cargo tanks, the net thickness of the plating above the inner bottom, t_sfi-n50 for plate i, in mm, is given by:

t_{sfi – n}₅₀ = t_i–n₅₀ – t_Δi

where:

t_Δi : Thickness deduction for plate i, in mm, as defined in [3.4.2].

3.4.2 The vertical distribution of thickness reduction for shear force correction is to be triangular as indicated in Figure 5. The thickness deduction, t_Δi in mm, to account for shear force correction on the plate i, is to be taken as:

where:

δQ₃ : Shear force correction for longitudinal bulkhead as defined in [3.4.3] and [3.4.4] for ships with one or two longitudinal bulkheads respectively, in kN.

: Length of cargo tank, in m.

h_blk : Height of longitudinal bulkhead, in m, defined as the distance from inner bottom to the deck at the top of the bulkhead, as shown in Figure 5.

x_blk : Minimum longitudinal distance from section considered to the nearest cargo tank transverse bulkhead, in m. To be taken positive and not greater than 0.5 .

z_p : Vertical distance from the lower edge of plate i to the base line, in m, but not taken less than h_db.

h_db : Height of double bottom, in m, as shown in Figure 5.

τ_i-perm : Permissible hull girder shear stress, in N/mm², for plate i:

τ_i-perm = 120/k.

Figure 5 : Shear force correction for longitudinal bulkheads

3.4.3 Shear force correction for a ship with a centreline longitudinal bulkhead

For ships with a centreline longitudinal bulkhead, the shear force correction in way of transverse bulkhead, δQ₃, in kN, is to be obtained from the following formula:

δQ₃ = 0.5 K₃ F_db

where:

F_db : Maximum resulting force on the double bottom in a tank, in kN, as defined in [3.4.5].

K₃ : Correction factor, to be taken equal to:

n : Number of floors between transverse bulkheads.

f₃ : Shear force distribution factor, as defined in Table 7.

3.4.4 Shear force correction for a ship with two longitudinal bulkheads between the cargo tanks

For ships with two longitudinal bulkheads between the cargo tanks, the shear force correction, δQ₃ in kN, is to be obtained from the following formula:

δQ₃ = 0.5 K₃ F_db

where:

F_db : Maximum resulting force on the double bottom in a tank, in kN, as defined in [3.4.5].

K₃ : Correction factor, to be taken equal to:

where:

n : Number of floors between transverse bulkheads.

r : Ratio of the part load carried by the wash bulkheads and floors from longitudinal bulkhead to the double side taken as:

: Length of cargo tank, between transverse bulkheads in the side cargo tank, in m.

b₈₀ : 80% of the distance from longitudinal bulkhead to the inner hull longitudinal bulkhead, in m, at tank mid length.

A_T-n50 : Net shear area of the transverse wash bulkhead, including the double bottom floor directly below, in the side cargo tank, in cm², taken as the smallest area in a vertical section.

A_1-n50, A_2-n50, A_3-n50:Net areas, as defined in Table 7, in m².

f₃ : Shear force distribution factor, as defined in Table 5.

n_S : Number of wash bulkheads in the side cargo tank.

R : Total efficiency of the transverse primary supporting members in the side tank in cm².

A_Q-n50 : Net shear area, in cm², of a transverse primary supporting member in the wing cargo tank, taken as the sum of the net shear areas of floor, cross ties and deck transverse webs. The net shear area is to be calculated at the mid span of the members.

I_psm-n50 : Net moment of inertia for transverse primary supporting members, in cm⁴, in the wing cargo tank, taken as the sum of the moments of inertia of transverses and cross ties. The net moment of inertia is to be calculated at the mid span of the member including an attached plate width equal to the primary supporting member spacing.

3.4.5 Vertical force on double bottom

The maximum vertical resulting force on the double bottom in a tank, F_db is in no case to be less than that given by the minimum conditions given in Table 5.

The maximum resulting force on the double bottom in a tank, F_db in kN, is to be taken as:

where:

W_CT : Weight of cargo, in tonnes, as defined in Table 6.

W_CWBT : Weight of ballast, in tonnes, as defined in Table 6.

b₂ : Breadth, in m, as defined in Table 6.

: Length of cargo tank, in m.

T_mean : Draught at the mid length of the tank for the loading condition considered, in m.

Table 5 : Minimum conditions for double bottom

Structural configuration	Positive/negative force, F_db	Minimum condition
Ships with centreline bulkhead	Max positive net vertical force, F_db+	0.9T_sc and empty cargo tanks and ballast tanks
Ships with centreline bulkhead	Max negative net vertical force, F_db-	0.6T_sc and full cargo tanks and empty ballast tanks
Ships with two longitudinal bulkheads	Max positive net vertical force, F_db+	0.9T_sc and empty cargo tanks and ballast tanks
Ships with two longitudinal bulkheads	Max negative net vertical force, F_db-	0.6T_sc and full centre cargo tank and empty ballast tanks

Table 6 : Desig‘n conditions for double bottom

Structural configuration	W_CT	W_CWBT	b₂
Ships with centreline bulkhead	Weight of cargo in cargo tanks, in tonnes, using a minimum density of 1.025 t/m³.	Weight of ballast between port and starboard inner sides, in t.	Maximum breadth between port and starboard inner sides at mid length of tank, in m, as shown in Figure 6.
Ships with two longitudinal bulkheads	Weight of cargo in the centre tank, in tonnes, using a minimum density of 1.025 t/m³.	Weight of ballast below the centre cargo tank, in t.	Maximum breadth of the centre cargo tank at mid length of tank, in m, as shown in Figure 6.

Figure 6 : Tank breadth b₂

Table 7 : Shear force distribution factor for oil tanker

Hull configuration	f₃ factor
One centreline bulkhead
Two longitudinal bulkheads
where: A_1-n50, A_2-n50, A_3-n50: Net projected area onto the vertical plane based on net thickness, t_n50, of the side shell, inner hull or the longitudinal bulkhead respectively, at one side of the section under consideration. The area A_1-n50 includes the net plating area of the side shell, including the bilge. The area A_2-n50 includes the net plating area of the inner hull, including the hopper side and the outboard girder under. The Area A_3-n50 includesthe net plating area of the longitudinal bulkheads, including the double bottom girders in line. The area A_3-n50 for the centreline bulkhead is not to be reduced for symmetry around the centreline. When the longitudinal bulkhead is made with corrugation, A_3-n50 is to consider the equivalent net thickness of the corrugation as defined in [3.4.6].

3.4.6 Equivalent net thickness of corrugation

The equivalent net thickness, in mm, of the corrugation of vertical and horizontal corrugated bulkheads, t_cor-n50, to be used for the calculation of the effective net shear area and for the unit shear flow, is given as follows:

where:

t_w-gr : Gross corrugation web thickness, in mm.

t_f-gr : Gross corrugation flange thickness, in mm.

s_c : Projected length of one corrugation, in mm, as defined in Ch 3, Sec 6, Figure 21.

c : Breadth of corrugation web, in mm, as defined in Ch 3, Sec 6, Figure 21.

a : Breadth of corrugation flange, in mm, as defined in Ch 3, Sec 6, Figure 21.

3.5 Effective net thickness for longitudinal bulkheads between cargo tanks of oil tankers - Correction due to loads from transverse bulkhead stringers

3.5.1 In way of transverse bulkhead stringer connections, within areas as specified in Figure 8, the equivalent net thickness of plate, t_sti-k-n50 in mm, where the index k refers to the identification number of the stringer, is not to be taken greater than:

where:

τ_sti-k : Shear stress in plate i, in N/mm², in the longitudinal bulkhead due to the stringer force in way of stringer k, taken as:

t_sfi-n50 : Effective net plating thickness as defined in [3.4.1], in mm, calculated at the transverse bulkhead for the height corresponding to the level of the stringer.

τ_i-perm : Permissible hull girder shear stress, in N/mm², for the plate i:

τ_i-perm = 120/k

: Connection length of stringer k, in m, as defined in Figure 7.

Q_st-k : Shear force on the longitudinal bulkhead from the stringer in loaded condition with tanks abreast full in kN, taken as:

F_st-k : Total stringer supporting force in way of a longitudinal bulkhead, in kN, taken as:

h_db : Double bottom height, in m.

h_blk : Height of bulkhead, in m, defined as the distance from inner bottom to the deck at the top of the bulkhead.

z_st-k : Z coordinate of the stringer k, in m.

P_st-k : Pressure on stringer k, in kN/m², taken as:.

P_st–k = g ρ_L h_tt–k

ρ_L : Density of the liquid in cargo tank, in t/m³, ad defined in Ch 4, Sec 6.

h_tt-k : Height from the top of the tank to the midpoint of the load area between h_k/2 below and h_k-1/2 above the stringer k, in m.

h_k : Vertical distance from the considered stringer k to the stringer k+1 below. For the lowermost stringer, it is to be taken as 80% of the average vertical distance to the inner bottom, in m.

h_k-1 : Vertical distance from the considered stringer k to the stringer k-1 above. For the uppermost stringer, it is to be taken as 80% of the average vertical distance to the upper deck, in m.

b_st-k : Load breadth acting on stringer k, in m, as defined in Figure 9 and Figure 10.

Figure 7 : Effective connection length of stringer

Figure 8 : Region for stringer correction, ti, for ships with 3 stringers

Figure 9 : Load breadth of stringers for ships with a centreline bulkhead

Figure 10 : Load breadth of stringers for ships with 2 inner longitudinal bulkheads

In this figure:

b_wt is the breadth of wing cargo tank, in m.

b_ctr is the breadth of centre cargo tank, in m.

3.5.2 Where reinforcement is provided to meet the above requirement, the reinforced area based on the maximum value of t_sti-k-n50 is to extend longitudinally for the full length of the stringer connection and a minimum of one frame spacing forward and aft of the bulkhead. The reinforced area is to extend vertically from above the stringer level and down to 0.5 h_k below the stringer, where h_k, the vertical distance from the considered stringer to the stringer below is as defined in [3.5.1]. For the lowermost stringer the maximum plate thickness requirement, t_sti-k-n50 is to extend down to the inner bottom, see Figure 8.

3.6 Shear force correction for bulk carriers

3.6.1 When hull girder shear strength assessment is performed in accordance with [3], shear force correction, which takes into account the portion of loads transmitted by the double bottom longitudinal girders to the transverse bulkheads, is to be considered.

For the considered cargo hold, the shear force correction at the considered transverse section is to be obtained, in kN, from the following formula:

where:

C_d : Distribution coefficient taken as:

C_d = -1 at the aft end of the considered cargo hold except for aftmost cargo hold.
C_d = 1 at the fore end of the considered cargo hold except for foremost cargo hold.
C_d = 0 at mid-length of the cargo hold.
C_d = 0 at the aft bulkhead of the aftmost cargo hold.
C_d = 0 at the fore bulkhead of the foremost cargo hold.
C_d : Linearly distributed at other locations.

α : Coefficient taken as:

M : Mass, in t, in the hold in way of the considered transverse section for the considered loading condition. M is to include the mass of ballast water and fuel oil located directly below the flat portion of the inner bottom, if any, excluding the portion under the bulkhead stool.

B_H Breadth of the cargo hold, in m, as defined in Ch 4, Sec 6.

_H : Length of the cargo hold, in m, as defined in Ch 4, Sec 6.

₀, b₀ : Length and breadth, respectively, in m, of the flat portion of the double bottom in way of the hold considered; b0 is to be measured on the hull transverse section at the middle of the hold.

, but not greater than 3.7.

T_LC,mh : Draught, in m, measured vertically on the hull transverse section at the middle of the hold considered, from the moulded baseline to the waterline in the loading condition considered.

ΔQ_CF : Shear force correction for the full hold.

ΔQ_CE : Shear force correction for the empty hold.

Figure 11 : Shear force correction, ΔQ_C