4 Load Application

4.1 General

4.1.1 Structural weight

Effect of the weight of hull structure is to be included in static loads, but is not to be included in dynamic loads. Density of steel is to be taken as given in Ch 4, Sec 6.

4.1.2 Sign convention

Unless otherwise mentioned in this Section, the sign of moments and shear force is to be in accordance with the sign convention defined in Ch 4, Sec 1.

4.2 External and internal loads

4.2.1 External pressure

External pressure is to be calculated for each load case in accordance with Ch 4, Sec 5. External pressures include static sea pressure, wave pressure and green sea pressure.

The forces applied on the hatch cover by the green sea pressure are to be distributed along the top of the corresponding hatch coamings. The total force acting on the hatch cover is determined by integrating the hatch cover green sea pressure as defined in Ch 4, Sec 5, [5]. Then the total force is to be distributed to the total length of the hatch coamings using the average line load. The effect of the hatch cover self weight is to be ignored in the loads applied to the ship structure.

4.2.2 Internal pressure

Internal pressure is to be calculated for each load case in accordance with Ch 4, Sec 6 for design load scenarios given in Ch 4, Sec 7, Table 1. Internal pressures include static dry and liquid cargo, ballast and other liquid pressure, setting pressure on relief valve and dynamic pressure of dry and liquid cargo, ballast and other liquid pressure due to acceleration.

4.2.3 Pressure application on FE element

Constant pressure, calculated at the element’s centroid, is applied to the shell element of the loaded surfaces, e.g. outer shell and deck for external pressure and tank/hold boundaries for internal pressure. Alternately, pressure can be calculated at element nodes applying linear pressure distribution within elements.

4.3 Hull girder loads

4.3.1 General

Each loading condition is to be associated with its corresponding hull girder loads which is to be applied to the model according to the procedure described in [4.4] for shear force and bending moment and in [4.5] for torsional moment. The hull girder loads are the combinations of still water hull girder loads and wave induced hull girder loads as specified in Ch 4, Sec 8. For each required FE load combination, the wave induced hull girder loads are to be calculated with the Load Combination Factors (LCFs), specified in Ch 4, Sec 2.

4.3.2 Target hull girder vertical bending moment

The target hull girder vertical bending moment, M_v-targ, in kNm, at a longitudinal position for a given FE load combination is taken as:

M_v–targ = C_BM–LC M_sw + M_{wv – LC}

where:

C_BM-LC : Percentage of permissible still water bending moment applied for the load combination under consideration as given in Ch 4, Sec 8,

M_sw : Permissible still water bending moments in kNm, at the considered longitudinal position for seagoing and harbour conditions as defined in Ch 4, Sec 4, [2.2.2] and Ch 4, Sec 4, [2.2.3] respectively.

M_wv-LC : Vertical wave bending moment in kNm, for the dynamic load case under consideration, calculated in accordance with Ch 4, Sec 4, [3.5.2].

Both C_BM-LC M_sw and M_wv-LC are either in sagging or in hogging condition according to the FE load combinations given in the tables of Ch 4, Sec 8.

4.3.3 Target hull girder shear force

where:

Q_fwd, Q_aft: Vertical shear forces, in kN, due to the local loads respectively at the forward and aft bulkhead position of the mid-hold, as defined in [4.4.6].

C_SF-LC : Percentage of permissible still water shear force as given in Ch 4, Sec 8, for the FE load combination under consideration.

Q_sw-pos, Q_sw-neg: Positive and negative permissible still water shear forces, in kN, at any longitudinal position for seagoing and harbour conditions as defined in Ch 4, Sec 4, [2.3.3] and Ch 4, Sec 4, [2.3.4] respectively.

f_β : Wave heading factor, as given in Ch 4, Sec 4.

C_QW : Load combination factor for vertical wave shear force, as given in Ch 4, Sec 2.

Q_wv-pos, Q_{wv_neg}: Positive and negative vertical wave shear force, in kN, as defined in Ch 4, Sec 4, [3.2.1].

The values of Q_targ-aft and Q_targ-fwd are to be taken at after and forward transverse bulkheads of the mid-hold under consideration.

4.3.4 Target hull girder horizontal bending moment

The target hull girder horizontal bending moment, M_h-targ, in kNm, for a given FE load combination is taken as:

M_h–targ = M_wh–LC

where:

M_wh-LC : Horizontal wave bending moment, in kNm, for the dynamic load case under consideration, calculated in accordance with Ch 4, Sec 4, [3.5.4].

4.3.5 Target hull girder torsional moment

For bulk carriers only, the target hull girder torsional moment, M_wt-targ, in kNm, for the dynamic load cases OST and OSA is the value at the target location taken as:

M_wt–targ = M_wt–LC (x_targ)

where:

M_wt-LC (x): Wave torsional moment, in kNm, for the dynamic load case OST and OSA, defined in Ch 4, Sec 4, [3.5.5], calculated at x position.

x_mid : X-coordinate, in m, of the mid-hold centre.

x_bhd : X-coordinate, in m, of the after or forward transverse bulkhead of mid-hold.

For dynamic load cases of bulk carriers other than OST and OSA and for all dynamic load cases of oil tankers, hull girder torsional moment M_wt-targ, at middle of mid-hold is to be adjusted to zero.

4.4 Procedure to adjust hull girder shear forces and bending moments

4.4.1 General

The procedure given in this sub-article [4.4] describes how to adjust the hull girder horizontal bending moment, vertical force and vertical bending moment distribution on the three cargo hold FE model to achieve the required target values at required locations. The hull girder load target values are specified in [4.3].

The target locations for hull girder shear force are at the transverse bulkheads of the mid-hold. The final adjusted hull girder shear force at the target location should not exceed the target hull girder shear force.

The target location for hull girder bending moment is, in general, located at the centre of the mid-hold. If the maximum value of bending moment is not located at the centre of the mid-hold, the final adjusted maximum bending moment within the mid-hold is not to exceed the target hull girder bending moment.

4.4.2 Local load distribution

With the above local loads applied to the FE model, the FE nodal forces are obtained through FE loading procedure. The 3D nodal forces will then be lumped to each longitudinal station to generate the one dimension local load distribution. The longitudinal stations are located at transverse bulkheads/frames and typical longitudinal FE model nodal locations in between the frames according to the cargo hold model mesh size requirement. Any intermediate nodes created for modelling structural details are not treated as the longitudinal stations for the purpose of local load distribution. The nodal forces within half of forward and half of afterward of longitudinal station spacing are lumped to that station. The lumping process will be done for vertical and horizontal nodal forces separately to obtain the lumped vertical and horizontal local loads, f_vi and f_hi, at the longitudinal station i.

4.4.3 Hull girder forces and bending moment due to local loads

With the local load distribution, the hull girder load longitudinal distributions are obtained by assuming the model is simply supported at model ends. The reaction forces at both ends of the model and longitudinal distributions of hull girder shear forces and bending moments induced by local loads at any longitudinal station are determined by the following formulae:

when x_i < x_j

when x_i < x_j

when x_i < x_j

when x_i < x_j

where:

R_{V_aft}, R_{V_fore}, R_{H_aft}, R_{H_fore} : Vertical and horizontal reaction forces at the aft and fore ends, in kN.

x_aft : X-coordinate of the aft end support, in m.

x_fore : X-coordinate of the fore end support, in m.

f_vi : Lumped vertical local load at longitudinal station i as defined in [4.4.2], in kN.

f_hi : Lumped horizontal local load at longitudinal station i as defined in [4.4.2], in kN.

F_l : Total net longitudinal force of the model, in kN.

f_li : Lumped longitudinal local load at longitudinal station i as defined in [4.4.2], in kN.

x_j : X-coordinate, in m, of considered longitudinal station j.

x_i : X-coordinate, in m, of longitudinal station i.

Q_{V_FEM} (x_j), Q_{H_FEM} (x_j), M_{V_FEM} (x_j), M_{H_FEM} (x_j) : Vertical and horizontal shear forces, in kN, and bending moments, in kNm, at longitudinal station x_j created by the local loads applied on the FE model. The sign convention for reaction forces is that a positive creates a positive shear force.

4.4.4 Longitudinal unbalanced force

In case total net longitudinal force of the model, F_l, is not equal to zero, the counter longitudinal force, (F_x)_j, is to be applied at one end of the model, where the translation on X-direction, δ_x, is fixed, by distributing longitudinal axial nodal forces to all hull girder bending effective longitudinal elements, as follows:

where:

(F_x)_j : Axial force applied to a node of the j-th element, in kN.

F_l : Total net longitudinal force of the model, as defined in [4.4.3], in kN.

A_j-n50 : Net cross sectional area of the j-th element, in m².

n_j : Number of nodal points of j-th element on the cross section, n_j = 1 for beam element, n_j = 2 for 4- node shell element.

4.4.5 Hull girder shear force adjustment procedure

The hull girder shear force adjustment procedure defined in this requirement applies to all FE load combinations given in Ch 4, Sec 8. The FE load combinations not directly covered by the load combination tables of Ch 4, Sec 8 are to be considered on a case by case basis.

The “maximum shear force load combinations“ are marked as “Max SFLC“ in the load combination tables of Ch 4, Sec 8. The “other shear force load combinations“ are those which are not the maximum shear force load combinations. They are not marked in the load combination tables of Ch 4, Sec 8.

4.4.6 Method 1 for shear force adjustment at one bulkhead

where:

M_{Y_aft}, M_{Y_fore}: Vertical bending moment, in kNm, to be applied at the aft and fore ends in accordance with [4.4.10], to enforce the hull girder vertical shear force adjustment as shown in Table 5. The sign convention is that of the FE model axis.

4.4.7 Method 2 for vertical shear force adjustment at both bulkheads

Table 6 shows examples of the shear adjustment application due to the vertical bending moments and to vertical loads.

where:

M_{Y_aft}, M_{Y_fore}: Vertical bending moment, in kNm, to be applied at the aft and fore ends in accordance with [4.4.10], to enforce the hull girder vertical shear force adjustment. The sign convention is that of the FE model axis.

ΔQ_aft : Adjustment of shear force, in kN, at aft bulkhead of mid-hold.

ΔQ_fwd : Adjustment of shear force, in kN, at fore bulkhead of mid-hold.

The above adjustments in shear forces, ΔQ_aft and ΔQ_fwd, at the transverse bulkhead positions are to be generated by applying vertical loads at the transverse frame positions as shown in Table 7. For bulk carriers, the transverse frame positions correspond to the floors. Vertical correction loads are not to be applied to any transverse tight bulkheads, any frames forward of the forward cargo hold and any frames aft of the aft cargo hold of the FE model.

The vertical loads to be applied to each transverse frame to generate the increase/decrease in shear force at the bulkheads may be calculated as shown in Table 7. In case of uniform frame spacing, the amount of vertical force to be distributed at each transverse frame may be calculated in accordance with Table 8.

If non-uniform frame spacing is used within each cargo hold, the average frame spacing is used to calculate the average distributed frame loads δw_av-i, according to Table 8, where i = 1, 2, 3 for each hold.

Then δw_av-i is redistributed to the non-uniform frame as follows:

k = 1, 2, ..., n_i − 1 , for each frame in cargo hold i, i = 1, 2, 3

where:

: Average frame spacing, in m, calculated as _i /n_i, in cargo hold i with i = 1, 2, 3.

: Length, in m, of the cargo hold i with i = 1, 2, 3 as defined in Table 8.

n_i : Number of frame spacing in cargo hold i with i = 1, 2, 3 as defined in Table 8.

δw_av-i : Average uniform frame spacing, in m, distributed force calculated according to Table 8 with the average frame spacing _av-i in cargo hold i with i = 1, 2, 3.

δw_i^k : Distributed load, in kN, for non-uniform frame k in cargo hold i.

: Frame spacing, in m, between the frame k - 1 and k in the cargo hold i:

The required vertical load δw_i for a uniform frame spacing or δw_i^k for non-uniform frame spacing, are to be applied by following the shear flow distribution at the considered cross section, as described in Ch 5, App 1. For a frame section under vertical load δw_i, the shear flow, qf, at the middle point of the element is calculated as:

where:

q_f-k : Shear flow calculated at the middle of the k-th element of the transverse frame, in N/mm.

δw_i : Distributed load at each transverse frame location for i-th cargo hold, i = 1, 2, 3, as defined in Table 8, in N.

I_y-n50 : Moment of inertia of the hull girder cross section, in mm⁴.

z_neu : Vertical distance from the integral point, s, to the vertical neutral axis.

t_n50 : Net thickness, in mm, of the plate at the integral point of the cross section.

The distributed shear force at j-th FE grid of the transverse frame, F_j-grid, is obtained from the shear flow of the connected elements as following:

where:

: Length of the k-th element of the transverse frame connected to the grid j, in mm.

n : Total number of elements connect to the grid j.

The shear flow has direction along the cross section and therefore the distributed force, F_j-grid, is a vector force. For vertical hull girder shear correction, the vertical and horizontal force components calculated with above mentioned shear flow method need to be applied to the cross section.

4.4.8 Procedure to adjust vertical and horizontal bending moments for midship cargo hold region

In case the target vertical bending moment needs to be reached, an additional vertical bending moment is to be applied at both ends of the cargo hold FE model to generate this target value in the mid-hold of the model. This end vertical bending moment is given as follows:

M_v–end = M_v–targ – M_v–peak

where:

M_v-end : Additional vertical bending moment, in kNm, to be applied to both ends of FE model in accordance with [4.4.10].

M_v-targ : Hogging (positive) or sagging (negative) vertical bending moment, in kNm, as specified in [4.3.2].

M_{V_FEM(x)}: Vertical bending moment, in kNm, at position x, due to the local loads as described in [4.4.3].

F : Reaction force, in kN, at model ends due to application of vertical loads to frames as defined in Table 7.

x : X-coordinate, in m, of frame in way of the mid-hold.

In case the target horizontal bending moment needs to be reached, an additional horizontal bending moment is to be applied at the ends of the cargo tank FE model to generate this target value within the mid-hold. The additional horizontal bending moment is to be taken as:

M_h–end = M_h–targ – M_h–peak

where:

M_h-end : Additional horizontal bending moment, in kNm, to be applied to both ends of the FE model according to [4.4.10].

M_h-targ : Horizontal bending moment, as defined in [4.3.4].

M_{H_FEM} (x): Horizontal bending moment, in kNm, at position x, due to the local loads as described in [4.4.3].

The vertical and horizontal bending moments are to be calculated over the length of the mid-hold to identify the position and value of each maximum/minimum bending moment.

4.4.9 Procedure to adjust vertical and horizontal bending moments outside midship cargo hold region

To reach the vertical hull girder target values at each frame and transverse bulkhead position, as defined in [4.3.2], the vertical bending moment adjustments, m_vi, are to be applied at web frames and transverse bulkhead positions of the finite element model, as shown in Figure 19. The vertical bending moment adjustment at each longitudinal location, i, is to be calculated as follows:

where:

i : Index corresponding to the i-th station, starting from i =1 at the aft end section up to n_t

n_t : Total number of longitudinal stations where the vertical bending moment adjustment, m_vi, is applied.

m_vi : Vertical bending moment adjustment, in kNm, to be applied at transverse frame or bulkhead at station i.

m_{v_end} : Vertical bending moment adjustment, in kNm, to be applied, at the fore end section (n_t+1 station).

M_v-targ(i) : Required target vertical bending moment, in kNm, at station i, calculated in accordance with [4.3.2].

M_V-FEM(i): Vertical bending moment distribution, in kNm, at station i due to local loads as given in [4.4.3].

M_lineload(i):Vertical bending moment, in kNm, at station i, due to the line load for the vertical shear force correction as required in [4.4.8].

Figure 19 : Adjustments of bending moments outside midship cargo hold region.

m_hi can be substituted to m_vi in this figure and mi positive bending moment in FE coordinate system

To reach the horizontal hull girder target values at each frame and transverse bulkhead position as defined in [4.3.4], the horizontal bending moment adjustments, m_hi, are to be applied at web frames and transverse bulkhead positions of the finite element model, as shown in Figure 19. The horizontal bending moment adjustment at each longitudinal location, i, is to be calculated as follows:

f(i) = M_h–targ(i)–M_H–FEM(i)

where:

i : Longitudinal location for bending moment adjustments, m_hi.

n_t : Total number of longitudinal stations where the horizontal bending moment adjustment, m_hi, is applied.

m_hi : Horizontal bending moment adjustment, in kNm, to be applied at transverse frame or bulkhead at station i.

m_{h_end} : Horizontal bending moment adjustment, in kNm, to be applied at the fore end section (n_t+1 station).

M_h-targ(i) : Required target horizontal bending moment, in kNm, at station i, calculated in accordance with [4.3.4].

M_H-FEM(i): Horizontal bending moment distribution, in kNm, at station i due to local loads as given in [4.4.3].

The vertical and horizontal bending moment adjustments, m_vi and m_hi, are to be applied at all web frames and bulkhead positions of the FE model. The adjustments are to be applied in FE model by distributing longitudinal axial nodal forces to all hull girder bending effective longitudinal elements in accordance with [4.4.10].

4.4.10 Application of bending moment adjustments on the FE model

where:

M_v : Vertical bending moment adjustment, in kNm, to be applied to the considered cross section of the model.

M_h : Horizontal bending moment adjustment, in kNm, to be applied to the considered cross section the ends of the model.

(F_x)_i : Axial force, in kN, applied to a node of the i-th element.

I_y-n50 : Hull girder vertical moment of inertia, in m⁴, of the considered cross section about its horizontal neutral axis.

I_z-n50 : Hull girder horizontal moment of inertia, in m⁴, of the considered cross section about its vertical neutral axis.

Z_i : Vertical distance, in m, from the neutral axis to the centre of the cross sectional area of the i-th element.

Y_i : Horizontal distance, in m, from the neutral axis to the centre of the cross sectional area of the i-th element.

A_i-n50 : Cross sectional area, in m², of the i-th element.

n_i : Number of nodal points of i-th element on the cross section, n_i = 1 for beam element, n_i = 2 for 4- node shell element.

For cross sections other than cross sections at the model end, the average area of the corresponding i-th elements forward and aft of the considered cross section is to be used.

4.5 Procedure to adjust hull girder torsional moments

4.5.1 General

The procedure in this sub-article describes how to adjust the hull girder torsional moment distribution on the cargo hold FE model to achieve the target torsional moment at the target location. The hull girder torsional moment target values are given in [4.3.5].

4.5.2 Torsional moment due to local loads

Torsional moment, in kNm, at longitudinal station i due to local loads, M_T-FEMi in kNm, is determined by the following formula (see Figure 20):

where:

M_T-FEMi : Lumped torsional moment, in kNm, due to local load at longitudinal station i.

f_hik : Horizontal nodal force, in kN, of node k at longitudinal station i.

f_vik : Vertical nodal force, in kN, of node k at longitudinal station i.

y_ik : Y-coordinate, in m, of node k at longitudinal station i.

z_ik : Z-coordinate, in m, of node k at longitudinal station i.

R_{H_fwd} : Horizontal reaction forces, in kN, at the forward end, as defined in [4.4.3].

R_{H_aft} : Horizontal reaction forces, in kN, at the aft end, as defined in [4.4.3].

z_ind : Vertical coordinate, in m, of independent point as defined in [2.5.3].

Figure 20 : Station forces and acting location of torsional moment at section

4.5.3 Hull girder torsional moment

The hull girder torsional moment, M_T-FEM (xj) in kNm, is obtained by accumulating the station torsional moment from the aft end section (forward end for foremost cargo hold model) as follows:

where:

M_T-FEM (x_j): Hull girder torsional moment, in kNm, at longitudinal station x_j.

x_j : X-coordinate, in m, of considered longitudinal station j.

The torsional moment distribution given in [4.5.2], has a step at each longitudinal station.

4.5.4 Procedure to adjust hull girder torsional moment to target value

The torsional moment is to be adjusted by applying a hull girder torsional moment M_T-end in kNm, at the independent point of the aft end section of the model (forward end for foremost cargo hold model), given as follows:

M_{T – end} = M_wt–targ – M_T–FEM (x_targ)

where:

x_targ : X-coordinate, in m, of the target location for hull girder torsional moment, as defined in [4.3.5].

M_wt-targ : Target hull girder torsional moment, in kNm, specified in [4.3.5], to be achieved at the target location.

M_T-FEM (x_targ): Hull girder torsional moment, in kNm, at target location due to local loads.

Due to the step of hull girder torsional moment at each longitudinal station, the hull girder torsional moment is to be selected from the values aft and forward of the target location as follows: Maximum value for positive torsional moment and minimum value for negative torsional moment.

4.6 Summary of hull girder load adjustments

4.6.1 The required methods of hull girder load adjustments for different cargo hold regions are given in Table 9.