2 Incremental-Iterative Method

2.1 Assumptions

2.1.1 In applying the procedure described in [2.2], the following assumptions are generally to be made:

The ultimate strength is calculated at hull transverse sections between two adjacent transverse webs.
The hull girder transverse section remains plane during each curvature increment.
The hull material has an elasto-plastic behaviour.
The hull girder transverse section is divided into a set of elements, which are considered to act independently.

These elements are:

Transversely framed plating panels and/or stiffeners with attached plating, whose structural behaviour is described in [2.3.1].
Hard corners, constituted by plating crossing, whose structural behaviour is described in [2.3.2].
According to the iterative procedure, the bending moment M_i acting on the transverse section at each curvature value χ_i is obtained by summing the contribution given by the stress σ acting on each element. The stress σ corresponding to the element strain, ε is to be obtained for each curvature increment from the non-linear load-end shortening curves σ-ε of the element.

These curves are to be calculated, for the failure mechanisms of the element, from the formulae specified in [2.2]. The stress, σ is selected as the lowest among the values obtained from each of the considered load-end shortening curves σ-ε.

The procedure is to be repeated until the value of the imposed curvature reaches the value χ_F in m^-1, in hogging and sagging condition, obtained from the following formula:

where:

M_Y : Lesser of the values M_Y1 and M_Y2, in kNm.

M_Y1 = 10³ R_eH Z_B-n50.
M_Y2 =10³ R_eH Z_D-n50.

If the value χ_F is not sufficient to evaluate the peaks of the curve M-χ, the procedure is to be repeated until the value of the imposed curvature permits the calculation of the maximum bending moments of the curve.

2.2 Procedure

2.2.1 General

The curve M-χ is to be obtained by means of an incremental-iterative approach, summarised in the flow chart in Figure 1.

In this procedure, the ultimate hull girder bending moment capacity, M_U is defined as the peak value of the curve with vertical bending moment M versus the curvature χ of the ship cross section as shown in Figure 1. The curve is to be obtained through an incremental-iterative approach.

Each step of the incremental procedure is represented by the calculation of the bending moment M_i which acts on the hull transverse section as the effect of an imposed curvature χ_i.

For each step, the value χ_i is to be obtained by summing an increment of curvature, Δχ to the value relevant to the previous step χ_i-1. This increment of curvature corresponds to an increment of the rotation angle of the hull girder transverse section around its horizontal neutral axis.

This rotation increment induces axial strains, ε in each hull structural element, whose value depends on the position of the element. In hogging condition, the structural elements above the neutral axis are lengthened, while the elements below the neutral axis are shortened, and vice-versa in sagging condition.

The stress σ induced in each structural element by the strain ε is to be obtained from the load-end shortening curve σ-ε of the element, which takes into account the behaviour of the element in the non-linear elasto-plastic domain.

The distribution of the stresses induced in all the elements composing the hull transverse section determines, for each step, a variation of the neutral axis position, since the relationship σ-ε is non-linear. The new position of the neutral axis relevant to the step considered is to be obtained by means of an iterative process, imposing the equilibrium among the stresses acting in all the hull elements.

Once the position of the neutral axis is known and the relevant stress distribution in the section structural elements is obtained, the bending moment of the section M_i around the new position of the neutral axis, which corresponds to the curvature χ_i imposed in the step considered, is to be obtained by summing the contribution given by each element stress.

The main steps of the incremental-iterative approach described above are summarised as follows (see also Figure 1):

a) Step 1: Divide the transverse section of hull into stiffened plate elements.
b) Step 2: Define stress-strain relationships for all elements as shown in Table 1.
c) Step 3: Initialise curvature χ₁ and neutral axis for the first incremental step with the value of incremental curvature (i.e. curvature that induces a stress equal to 1% of yield strength in strength deck) as:
where:
z_D : Z coordinate, in m, of strength deck at side, with respect to reference coordinate defined in Ch 1, Sec 4, [3.6]
d) Step 4: Calculate for each element the corresponding strain, ε_i = χ (z_i - z_n) and the corresponding stress σ_i.
e) Step 5: Determine the neutral axis z_{NA_cur} at each incremental step by establishing force equilibrium over the whole transverse section as:
ΣA_i-n50 σ_i = ΣA_j-n50 σ_j (i-th element is under compression, j-th element under tension).
f) Step 6: Calculate the corresponding moment by summing the contributions of all elements as:
g) Step 7: Compare the moment in the current incremental step with the moment in the previous incremental step. If the slope in M-χ relationship is less than a negative fixed value, terminate the process and define the peak value of M_U. Otherwise, increase the curvature by the amount of Δχ and go to Step 4.

2.2.2 Modelling of the hull girder cross section

Hull girder transverse sections are to be considered as being constituted by the members contributing to the hull girder ultimate strength.

Sniped stiffeners are also to be modelled, taking account that they do not contribute to the hull girder strength.

The structural members are categorised into a stiffener element, a stiffened plate element or a hard corner element.

The plate panel including web plate of girder or side stringer is idealised into either a stiffened plate element, an attached plate of a stiffener element or a hard corner element.

The plate panel is categorised into the following two kinds:

Longitudinally stiffened panel of which the longer side is in the longitudinal direction, and
Transversely stiffened panel of which the longer side is in the perpendicular direction to the longitudinal direction.

a) Hard corner element:
Hard corner elements are sturdier elements composing the hull girder transverse section, which collapse mainly according to an elasto-plastic mode of failure (material yielding); they are generally constituted by two plates not lying in the same plane.
The extent of a hard corner element from the point of intersection of the plates is taken equal to 20 t_n50 on transversely stiffened panel and to 0.5 s on a longitudinally stiffened panel, see Figure 2.
where:
t_n50 : Net offered thickness of the plate, in mm.
s : Spacing of the adjacent longitudinal stiffener, in m.
Bilge, sheer strake-deck stringer elements, girder-deck connections and face plate-web connections on large girders are typical hard corners. Enlarged stiffeners, with or without web stiffening, used for Permanent Means of Access (PMA) are not to be considered as a large girder so the attached plate/web connection is only considered as a hard corner, see Figure 3.
b) Stiffener element:
The stiffener constitutes a stiffener element together with the attached plate.
The attached plate width is in principle:

Equal to the mean spacing of the stiffener when the panels on both sides of the stiffener are longitudinally stiffened, or
Equal to the width of the longitudinally stiffened panel when the panel on one side of the stiffener is longitudinally stiffened and the other panel is of the transversely stiffened, see Figure 2.
Figure 1 : Flow chart of the procedure for the evaluation of the curve M-χ
- c) Stiffened plate element:
- The plate between stiffener elements, between a stiffener element and a hard corner element or between hard corner elements is to be treated as a stiffened plate element, see Figure 2.

Figure 2 : Extension of the breadth of the attached plating and hard corner element

The typical examples of modelling of hull girder section are illustrated in Figure 3 and Figure 4. Notwithstanding the foregoing principle, these figures are to be applied to the modelling in the vicinity of upper deck, sheer strake and hatch side girder.

Figure 3 : Examples of the configuration of stiffened plate elements, stiffener elements and hard corner elements on a hull section

In case of the knuckle point as shown in Figure 5, the plating area adjacent to knuckles in the plating with an angle greater than 30 deg is defined as a hard corner. The extent of one side of the corner is taken equal to 20 t_n50 on transversely framed panels and to 0.5 s on longitudinally framed panels from the knuckle point.
Where the plate members are stiffened by non-continuous longitudinal stiffeners, the non-continuous stiffeners are considered only as dividing a plate into various elementary plate panels.
Where the opening is provided in the stiffened plate element, the openings are to be considered in accordance with Ch 5, Sec 1, [1.2.9].
Where attached plating is made of steels having different thicknesses and/or yield stresses, an average thickness and/or average yield stress obtained from the following formula are to be used for the calculation.
- where R_eHp1, R_eHp2, t_1-n50, t_2-n50, s₁, s₂ and s are shown in Figure 6.

Figure 4 : Extension of the breadth of the attached plating and hard corner element

Figure 5 : Plating with knuckle point

Figure 6 : Element with different thickness and yield strength

2.3 Load-end shortening curves

2.3.1 Stiffened plate element and stiffener element

Stiffened plate element and stiffener element composing the hull girder transverse sections may collapse following one of the modes of failure specified in Table 1.

Where the plate members are stiffened by non-continuous longitudinal stiffeners, the stress of the element is to be obtained in accordance with [2.3.3] to [2.3.8], taking into account the non-continuous longitudinal stiffener. In calculating the total forces for checking the hull girder ultimate strength, the area of non-continuous longitudinal stiffener is to be assumed as zero.
Where the opening is provided in the stiffened plate element, the considered area of the stiffened plate element is to be obtained by deducting the opening area from the plating in calculating the total forces for checking the hull girder ultimate strength. The consideration of the opening is in accordance with the requirement in Ch 5, Sec 1, [1.2.9] to [1.2.13].
For stiffened plate element, the effective width of plate for the load shortening portion of the stress strain curve is to be taken as full plate width, i.e. to the intersection of other plate or longitudinal stiffener – neither from the end of the hard corner element nor from the attached plating of stiffener element, if any. In calculating the total forces for checking the hull girder ultimate strength, the area of the stiffened plate element is to be taken between the hard corner element and the stiffener element or between the hard corner elements, as applicable.

Table 1 : Modes of failure of stiffened plate element and stiffener element

Element	Mode of failure	Curve σ-ε defined in
Lengthened stiffened plate element or stiffener element	Elasto-plastic collapse	[2.3.3]
Shortened stiffener element	Beam column buckling Torsional buckling Web local buckling of flanged profiles Web local buckling of flat bars	[2.3.4] [2.3.5] [2.3.6] [2.3.7]
Shortened stiffened plate element	Plate buckling	[2.3.8]

2.3.2 Hard corner element

The relevant load-end shortening curve σ-ε is to be obtained for lengthened and shortened hard corners according to [2.3.3].

2.3.3 Elasto-plastic collapse of structural elements

The equation describing the load-end shortening curve σ-ε for the elasto-plastic collapse of structural elements composing the hull girder transverse section is to be obtained from the following formula, valid for both positive (shortening) and negative (lengthening) strains, see Figure 7:

σ = Φ R_eHA where:

R_eHA : Equivalent minimum yield stress, in N/mm², of the considered element, obtained by the following formula:

Φ : Edge function, equal to:

Φ = -1 for ε < -1
Φ= ε for -1 ≤ ε ≤ 1
Φ = 1 for ε > 1

ε : Relative strain, equal to:

ε_E : Element strain.

ε_Y : Strain at yield stress in the element, equal to:

Figure 7 : Load-end curve σ-ε for elasto plastic collapse

2.3.4 Beam column buckling

The equation describing the load-end shortening curve σ_CR1-ε for the beam column buckling of stiffeners composing the hull girder transverse section is to be obtained from the following formula, see Figure 8:

where:

Φ : Edge function, as defined in [2.3.3].

σ_C1 : Critical stress, in N/mm², equal to:

R_eHB : Equivalent minimum yield stress, in N/mm², of the considered element, obtained by the following formula:

A_pEl-n50 : Effective area, in cm², equal to:

A_pEl-n50 = 10 b_E1 t_n50

: Distance, in mm, measured from the neutral axis of the stiffener with attached plate of width b_E1 to the bottom of the attached plate.

: Distance, in mm, measured from the neutral axis of the stiffener with attached plating of width b_E1 to the top of the stiffener.

ε : Relative strain, as defined in [2.3.3].

σ_E1 : Euler column buckling stress, in N/mm², equal to:

I_E-n50 : Net moment of inertia of stiffeners, in cm⁴, with attached plating of width b_E1.

A_E-n50 : Net area, in cm², of stiffeners with attached plating of width b_E.

b_E1 : Effective width corrected for relative strain, in m, of the attached plating, equal to:

for β_E > 1.0
b_E1 = s for β_E ≤ 1.0

β_E:

A_pE--n50 : Net sectional area, in cm², of attached plating of width b_E, equal to:

A_pE-n50 = 10 b_E t_n50

b_E : Effective width, in m, of the attached plating, equal to:

for β_E > 1.25
b_E = s for β_E ≤ 1.25

Figure 8 : Load-end shortening curve σ_CR1-ε for beam column buckling

2.3.5 Torsional buckling

The equation describing the load-end shortening curve σ_CR2-ε for the flexural-torsional buckling of stiffeners composing the hull girder transverse section is to be obtained according to the following formula, see Figure 9.

where:

Φ : Edge function, as defined in [2.3.3].

σ_C2 : Critical stress, in N/mm², equal to:

σ_E2 : Euler torsional buckling stress, in N/mm², taken as σ_ET in Ch 8, Sec 5, [2.3.4]

ε : Relative strain, as defined in [2.3.3].

σ_CP : Buckling stress of the attached plating, in N/mm², equal to:

for β_E > 1.25
σ_CP = R_eHp for β_E ≤ 1.25

β_E : Coefficient, as defined in [2.3.4].

Figure 9 : Load-end shortening curve σ_CR2-ε for flexural-torsional buckling

2.3.6 Web local buckling of stiffeners made of flanged profiles

The equation describing the load-end shortening curve σ_CR3-ε for the web local buckling of flanged stiffeners composing the hull girder transverse section is to be obtained from the following formula:

where:

Φ : Edge function, as defined in [2.3.3].

b_E : Effective width, in m, of the attached shell plating, as defined in [2.3.4].

h_we : Effective height, in mm, of the web, equal to:

for β_w ≥ 1.25
h_we = h_w for β_w < 1.25

β_w :

ε : Relative strain, as defined in [2.3.3].

2.3.7 Web local buckling of stiffeners made of flat bars

The equation describing the load-end shortening curve σ_CR4-ε for the web local buckling of flat bar stiffeners composing the hull girder transverse section is to be obtained from the following formula, see Figure 10:

where:

Φ : Edge function, as defined in [2.3.3].

σ_CP : Buckling stress of the attached plating, in N/mm², as defined in [2.3.5].

σ_C4 : Critical stress, in N/mm², equal to:

σ_E4 : Local Euler buckling stress, in N/mm², equal to:

ε : Relative strain, as defined in [2.3.3].

Figure 10 : Load-end shortening curve σ_CR4-ε for web local buckling

2.3.8 Plate buckling

The equation describing the load-end shortening curve σ_CR5-ε for the buckling of transversely stiffened panels composing the hull girder transverse section is to be obtained from the following formula:

where:

Φ : Edge function, as defined in [2.3.3].

β_E :

s : Plate breadth, in m, taken as the spacing between the stiffeners.

: Longer side of the plate, in m.