Khasiat Fadhilat (Isnad) Doa Haikal - Majmu' Syarif

Hull girder ultimate strength check

1.1	Introduction

1.1.1

Ch 6, Sec 3, [2] defines the criteria for calculating the ultimate bending moment capacities in hogging condition M_UH and sagging condition M_US of a hull girder transverse section.
As specified in Ch 6, Sec 3, [2] , the ultimate bending moment capacities are defined as the maximum values of the curve of bending moment capacity M versus the curvature χ of the transverse section considered (see Fig 1 ).

1.1.2

This Appendix provides the criteria for obtaining the curve M-χ.

1.2	Criteria for the calculation of the curve M-χ

1.2.1

Procedure

The curve M-χ is to be obtained by means of an incremental-iterative approach, summarised in the flow chart in Fig 2 .

Figure 1 - Curve bending moment capacity M
versus curvature χ

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 Dχ 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 e 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. Vice-versa in sagging condition.
The stress s induced in each structural element by the strain e is to be obtained from the load-end shortening curve s-e 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 s-e 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.

Figure 2 - Flow chart of the procedure for the evaluation of the curve M-χ

1.2.2

Assumption

In applying the procedure described in [1.2.1] , the following assumptions are generally to be made:

• The ultimate strength is calculated at hull transverse sections between two adjacent reinforced rings.
• 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 ordinary stiffeners with attached plating, whose structural behaviour is described in [1.3.1]
  • hard corners, constituted by plating crossing, whose structural behaviour is described in [1.3.2] .
• 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 s acting on each element. The stress s, corresponding to the element strain e, is to be obtained for each curvature increment from the non-linear load-end shortening curves s-e of the element.
  These curves are to be calculated, for the failure mechanisms of the element, from the formulae specified in [1.3] . The stress s is selected as the lowest among the values obtained from each of the considered load-end shortening curves s-e.
• The procedure is to be repeated for each step, 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:

  MY

  :

  The lesser of the values MY1 and MY2 , in kN.m: MY1 = 10-3 ReH ZAB MY2 = 10-3 ReH ZAD

  
  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.

1.3	Load-end shortening curves s-e

1.3.1

Plating panels and ordinary stiffeners

Plating panels and ordinary stiffeners composing the hull girder transverse sections may collapse following one of the modes of failure specified in Tab 1 .

1.3.2

Hard corners

Hard corners are sturdier elements composing the hull girder transverse section, which collapse mainly according to an elasto-plastic mode of failure. The relevant load-end shortening curve s-e is to be obtained for lengthened and shortened hard corners according to [1.3.3] .

Table 1 - Modes of failure of plating panels
and ordinary stiffeners

Element	Mode of failure	Curve s-e defined in
Lengthened transversely framed plating panel or ordinary stiffeners	Elasto-plastic collapse	[1.3.3]
Shortened ordinary stiffeners	Beam column buckling	[1.3.4]
	Torsional buckling	[1.3.5]
	Web local buckling of flanged profiles	[1.3.6]
	Web local buckling of flat bars	[1.3.7]
Shortened transversely framed plating panel	Plate buckling	[1.3.8]

1.3.3

Elasto-plastic collapse

The equation describing the load-end shortening curve s-e 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 Fig 3 ):
s = F R_eH
where:

F

:

Edge function: F = -1 for e < -1 F = e for -1 < e < 1 F = 1 for e > 1

e

:

Relative strain:

eE

:

Element strain

eY

:

Strain inducing yield stress in the element:

Figure 3 - Load-end shortening curve σ-ϵfor elasto-plastic collapse

1.3.4

Beam column buckling

The equation describing the load-end shortening curve s_CR1-e for the beam column buckling of ordinary stiffeners composing the hull girder transverse section is to be obtained from the following formula (see Fig 4 ):

where:

F

:

Edge function defined in [1.3.3]

sC1

:

Critical stress, in N/mm2:

e

:

Relative strain defined in [1.3.3]

sE1

:

Euler column buckling stress, in N/mm2:

IE

:

Net moment of inertia of ordinary stiffeners, in cm4, with attached shell plating of width bE1

bE1

:

Width, in m, of the attached shell plating:

AE

:

Net sectional area, in cm2, of ordinary stiffeners with attached shell plating of width bE

bE

:

Width, in m, of the attached shell plating:

Figure 4 - Load-end shortening curve s_CR1-e
for beam column buckling

1.3.5

Torsional buckling

The equation describing the load-end shortening curve s_CR2-e for the lateral-flexural buckling of ordinary stiffeners composing the hull girder transverse section is to be obtained according to the following formula (see Fig 5 ):

where:

F

:

Edge function defined in [1.3.3]

sC2

:

Critical stress, in N/mm2:

sE2

:

Euler torsional buckling stress, in N/mm2, defined in Ch 7, Sec 2, [4.3.3]

e

:

Relative strain defined in [1.3.3]

sCP

:

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

βE

:

Coefficient defined in [1.3.4] .

Figure 5 - Load-end shortening curve σCR2-ϵfor flexural-torsional buckling

1.3.6

Web local buckling of flanged ordinary stiffeners

The equation describing the load-end shortening curve s_CR3-e for the web local buckling of flanged ordinary stiffeners composing the hull girder transverse section is to be obtained from the following formula:

where:

F

:

Edge function defined in [1.3.3]

bE

:

Width, in m, of the attached shell plating, defined in [1.3.4]

hWE

:

Effective height, in mm, of the web:

e

:

Relative strain defined in [1.3.3] .

1.3.7

Web local buckling of flat bar ordinary stiffeners

The equation describing the load-end shortening curve s_CR4-e for the web local buckling of flat bar ordinary stiffeners composing the hull girder transverse section is to be obtained from the following formula (see Fig 6 ):

where:

F

:

Edge function defined in [1.3.3]

sCP

:

Buckling stress of the attached plating, in N/mm2, defined in [1.3.5]

sC4

:

Critical stress, in N/mm2:

sE4

:

Local Euler buckling stress, in N/mm2:

e

:

Relative strain defined in [1.3.3] .

Figure 6 - Load-end shortening curve σCR4-ϵfor web local buckling of flat bars

1.3.8

Plate buckling

The equation describing the load-end shortening curve s_CR5-e for the buckling of transversely stiffened panels composing the hull girder transverse section is to be obtained from the following formula:

where:

βE

:

Coefficient defined in [1.3.4] .