1 Calculation Formula

1.1 General

1.1.1 This appendix describes the procedures of direct calculation of shear flow which is working along a ship cross section due to hull girder vertical shear force. Shear flow q_v, at each location in the cross section, is calculated where considering the cross section is subjected to a unit vertical shear force, 1 N, in the direction of z coordinate.

The unit shear flow per mm, q_v in N/mm, can be considered equal to:

q_v = q_D + q_I

where:

q_D : Determinate shear flow, as defined in [1.2].

q_l : Indeterminate shear flow which circulates around the closed cells, as defined in [1.3].

In the calculation of the unit shear flow, q_v, the longitudinal stiffeners are to be taken into account.

1.2 Determinate shear flow

1.2.1 The determinate shear flow, q_D in N/mm, at each location in the cross section can be obtained from the following line integration:

where:

s : Coordinate value of running coordinate along the cross section, in m.

I_y-n50 : Moment inertia of the cross section, in m⁴.

t_n50 : Net thickness of plating, in mm, or equivalent net thickness of corrugated plate as defined in Ch 5, Sec 1, [3.4.6].

1.2.2 Assuming the cross section is composed of line segments as shown in Figure 1, the determinate shear flow can be calculated by the following equation.

where:

q_Dk, q_Di : Determinate shear flow at node k and node i respectively, in N/mm.

: Length of line segments, in m.

z_k, z_i : Z coordinate of the end point of line segment, in m, as defined in Figure 1.

1.2.3 Where the cross section includes closed cells, the closed cell are to be cut with virtual slits, as shown in Figure 2 in order to obtain the determinate shear flow.

However, the virtual slits must not be located at the walls by which the other closed cell is also bounded.

1.2.4 Calculations of the determinate shear flow at bifurcation points can be calculated such as water flow calculations as shown in Figure 2.

Figure 1 : Definition of line segment

Figure 2 : Calculation of determinate shear flow at bifurcation

1.3 Indeterminate shear flow

1.3.1 The indeterminate shear flow is working around the closed cells and can be considered as a constant value within the same closed cell. The following system of equation for determination of indeterminate shear flows can be developed. In the equations, contour integrations of several parameters around all closed cells are performed.

where:

q_Ik, q_li : Indeterminate shear flow around the closed cell k and i respectively, in N/mm.

1.3.2 With assuming assembly of line segments shown in Figure 1, the equations in [1.3.1] can be expressed as follows:

where:

q_Di : Determinate shear flow, in N/mm, calculated according to [1.2.2].

The difference in the directions of running coordinates specified in [1.2] and this sub-article is to be considered.

Figure 3 : Closed cells and common wall

1.4 Computation of several properties of the cross section

1.4.1 Properties of the cross section can be obtained by the following formulae where the cross section is assumed as the assembly of line segments:

where:

a_n50, A_n50: Area of the line segment and the cross section respectively, in m².

s_y-n50, S_y-n50: First moment of the line segment and the cross section about the baseline, in m³.

i_y0-n50, I_y0-n50: Moment inertia of the line segment and the cross section about the baseline, in m⁴.

1.4.2 The height of horizontal neutral axis, z_n in m, can be obtained as follows:

1.4.3 Inertia moment about the horizontal neutral axis, in m⁴, can be calculated as follows: