Ultimate Longitudinal Strength-Based Safety and ... - J-STAGE

(Read at the Spring Meeting of The Society of Naval Architects of Japan, May 1991) 403

Ultimate Longitudinal Strength-Based Safety

and Reliability Assessment of Ship's Hull Girder (2nd Report)

Stiffened Hull Structure

by Jeom K. Paik*, Member

Summary

In order to make a system safety and reliability assessment of ship structures in the longitudinal strength point of view, it is essential to analyze ultimate longitudinal strength of ship's hull girder under sagging or hogging bending moment. In the previous report"), the author formulated an idealized plate element using idealized structural unit method and the developed method was then applied to ultimate longitudinal strength analysis of unstiffened hull structure as an example, concluding that the method is useful for ultimate collapse strength analysis of plate structure.

In this study, an attempt is made to analyze ultimate longitudinal strength of stiffened hull structure by using idealized structural unit method. For this purpose, an idealized stiffened plate element subjected to biaxial load is developed taking account of the influence of initial imperfections as well as the interaction effect between local and global failure in the structure. The developed method is then applied to ultimate longitudinal strength analysis of Suezmax-sized double skin hull girder as an example for the intact and the damaged condition in the event of accidents of grounding and collision. A new deterministic measure of safety based on the absorbed energy capacity is proposed. Based on the obtained results, ultimate longitudinal strength-based safety and reliability assessment is made using the proposed measure of safety in addition to the conventional one, concluding that the objective hull structure has relatively sufficient safety and the present procedure is useful when ultimate longitu-dinal strength-based safety assessment or optimization is made in the structural design stage for a new type of hull structure.

1. Introduction

Recently, in order to minimize the risk of ocean

pollution due to cargo oil leakage in the event of accidents such as grounding or collision, the needs for

construction of a double skin type of hull structure

based on a new design concept are increasing in tanker

industry1),2). Also, for achieving more reliable and

economical design of structures, system safety assess-

ment which refers to the redundancy of the overall

structure becomes of great interest3). In general, the

procedure of system safety assessment of structures is split into two categories : one is a deterministic

approach and another is probabilistic one. The former

measures deterministically the margin between the

ultimate capacity of the overall structure and the

applied loads, or the ratio of the ultimate collapse load

between the intact and the damaged condition. In the

latter approach, reliability or probability of failure of

the structure is evaluated, in which each design parame-

ter is treated as a random variable. For both

approaches, however, the main task is to obtain the

ultimate collapse strength of the overall structure under external loading. Here, in order to determine a more reasonable layout of each structural member, structural designer is preferable to have understanding of detail

progressive collapse history as well as ultimate collpase capacity itself.

In usual, ship's strength for the overall structure is classified in three categories such as longitudinal strength, transverse strength and torsional strength. Thus if we are going to make a system safety assess-ment of ship structure in the longitudinal strength point of view, then we should perform the ultimate longitudi-nal bending strength analysis of ship's hull girder sub-

jected to sagging or hogging bending moment. With the increasing of the external bending moment,

however, ship's hull girder exhibits highly complicated nonlinear behaviour associated with geometric and material nonlinearity. Also since ship structures are fabricated by heating process such as welding, flame cutting and so on, initial imperfections in the form of initial deflection and residual stress are always present in the structural members and these make structural behaviour more complex one.

For the nonlinear analysis of structures, finite element method (FEM) is one of the most powerful approach. However, a direct application of the conventional FEM

* Department of Naval Architecture , Pusan National University, Korea

404Journal of The Society of Naval Architects of TaDan. Vol. 169

to the ultimate longitudinal strength analysis of ship's hull girder which is very large and complex structure is considered to be nearly impossible because a tremen-dous amount of human labor and computational efforts is required. Therefore, many structural designers or analyzers have a great interest about how to reduce the computing time needed in the nonlinear analysis of structures while gaining the reliable solution.

In this respect, Ueda and Rashed4) proposed an efficient and accurate numerical method for the non-linear analysis of large size structures, named idealized structural unit method (ISUM). The main idea behind this methodology is in modelling the objective structure to be analyzed by using very large size structural ele-ment, named idealized structural element, comparing with the conventional finite element, which results in the considerable reduction of the computational efforts because the number of unknowns in structural stiffness equation is greatly decreased. In general, for the analy-sis of a certain type of structure, various kinds of the idealized structural element formulated in advance are needed like in the conventional FEM. Usually each basical structural member composing the objective structure to be analyzed is chosen as an idealized struc-tural element. By idealizing and formulating the non-linear behaviour of the actual structural member, an idealized structural element is developed. Through many application examples of ISUM to various types of steel structures such as transverse rings4),5), upper decks6)-7), double bottom structures8)-10), hull girder11)-13) and tubular offshore structures14)-21), it is well known that ISUM is a very powerful and useful methodology for the nonlinear analysis of large size steel structures.

In the previous report13), the author formulated an idealized plate element on the basis of idealized struc-tural unit method and the developed method was then applied to ultimate longitudinal strength analysis of unstiffened hull structure as an example, concluding that the method is useful for the ultimate collapse strength analysis of plate structure. In this study, an attempt is made to analyze ultimate longitudinal strength of stiffened hull structure. For this purpose, an idealized stiffend plate element subjected to biaxial load is formulated taking account of geometric and material nonlinearity. The interaction effect between local and global buckling in the structure is taken into consideration. Also, the influence of initial imperfec-tions in the form of initial deflection and residual stress on strength and stiffness of the structure is considered.

A computer program, ALPS/ISUM22) is completed based on the present theory. The program is then applied to ultimate longitudinal strength analysis of Suezmax-sized double skin hull structure as an exam-

ple. Series analysis is performed for the intact and the damaged condition caused by accidents of grounding and collision. A new deterministic measure of system safety based on the absorbed energy capacity is

proposed. System safety assessment of the objective hull girder is then made by using the proposed measure of safety in addtion to the conventional measure.

2. Measure of System Safety

Unitil now, three groups of measure of system safety which refers to redundancy of the overall structure have been used, denoting by reserve strength factor (RSF) , residual resistance factor (RRF) and safey index (SI). RSF and RRF quantifying the safety for the intact and the damaged structure, respectively are deterministic measures, while SI are given by the probabilistic analy-sis, in which Cornel's second moment method"' is applied in this study. Here, in addition to these conven-tional measures, a new deterministic measure ( AEF) denoting by the ratio of the absorbed energy capacity between the intact and the damaged structure is

proposed. In order to evaluate AEF (absorbed energy factor) , the detail collapse history of the overall struc-

ture even after its ultimate collapse strength should be known.

Then each measure of system safety is defined as :

(1.a)

(1.b)

(1.c)

(1.d)

where, Mu, Mud: mean of ultimate longitudinal bend-

ing moment in the intact and the

damaged condition, respectively

Me : mean of external bending moment

(= Ms+ MƒÖ)

Ms : mean of still water bending moment

MƒÖ : mean of wave-induced bending moment

Ei, Ed absorbed energy capacity in the intact and

the damaged condition, respectively

Su, Ss, SƒÖ : standard deviation of ultimate longitudinal

bending moment, still water bending moment

and wave-induced bending moment, respec-

tively

In the above equation, Mu and Mud are directly

obtained by applying the present method described in

section 3 and also Ms and MƒÖ are estimated by [24]

( 2 )

where, Ms and MƒÖ indicate the still water and the

wave-induced bending moment, respectively which are

assumed by using the existing rule of a classification

society in this study.

Also, Ei and Ed can be evaluated by integrating the

area below the bending moment-curvature curve for the

intact and the damaged structure, respectively.

On the other hand, each standard deviation is

assumed by using coefficient of variation as

Ultimate Longitudinal Strength-Based Safety and Reliability Assessment of Shin's Hull Girder (2nd Report) 405

( 3 )

where, CO Vu, COVs and GOVƒÖ denote coefficients of

variation for the ultimate longitudinal bending moment,

the still water bending moment and the wave-induced

bending moment, respectively. In this study, each COV

is prescribed in advance.

3. Theory of Ultimate Longitudinal Strength

Analysis Using ISUM

3. 1 General Aspects

In usual, the hull structure is composed of stiffened

panels having many stiffeners in the longitudinal and

transverse direction. When ISUM is applied a stiffened

panel composing the hull structure is chosen as an

idealized stiffened plate element.

Fig. 1 represents a typical configuration of a stiffened

plate element. Many one-sided stiffeners are attached

on the plate in the longitudinal and transverse direction.

The stiffener space is usually even and also the geome-

try and material property of each stiffener arranged in

a certain panel are assumed to be same in each direc-

tion. In actual ship panels, the bending stiffness of

stiffeners is sufficiently large such that no occurence of

the overall buckling in the panel is expected. It is very

paractical that the boundary of the element is assumed

to have simply supported condition, in which the edge

always keeps straight even after lateral deflection is

occured").

The stiffener and the plate between each stiffener

have initial imperfections in the form of initial

deflection and residual stress. The initially deflected

plate does not show the clear buckling phenomenon.

The geometric configuration of initial deflection exist-

ing in the plate between stiffeners takes the shape of so

called hungry horse's back according to the measured

results'"). This shape of initial deflection can be

expressed by using Fourier series function having a

number of terms. It is well known that the large

deflection behaviour of the plate is governed by the

buckling mode component among these terms")-"). Also, due to heating process such as welding, the residual stress is developed in the panel. Along the welding line, the tensile residual stress is remained and in order to keep an equilibrium state the compressive residual stress is developed in the middle portion of the plate. The large deflection behaviour of the panel is seriously affected by the compressive residual stress. On the other hand, the collapse strength of stiffener is greatly decreased if initial imperfections are present.

In general, the local panels composing ship's hull structure are subjected to combined in-plane and lateral loads. For in-plane load, however, since hull girder under consideration is subjected to sagging or hogging bending moment, shearing load is very small and biax-ial load is dominant. The lateral load is due to the action of water pressure or cargo loading which is accounted for the estimation of the external sagging or hogging bending moment but the influence of the lateral load affecting the behaviour of local panel itself is not considered in this study.

3. 2 Formulation of an Idealized Stiffened Plate Element Under Biaxial Load

( 1 ) Nodal Force and Nodal Displacement Vector The idealized stiffened plate element has four corner

nodal points as shown in Fig. 2. As described in the later, since the deflected panel is treated by an equiva-lent flat one that has no deflection in this study, the rotational degree of freedom in the element becomes to

be removed. Thus the total degree of freedom at each nodal point is just three axial displacements which are u, v an w in x, y and z direction, respectively. Then nodal force vector (RI and nodal displacement vector

{U} are defined by

(4.a)

(4.b)

Fig. 1 General configuration of a stiffened panel Fig. 2 An idealized stifffened plate element

406 Journal of The Society of Naval Architects of Japan, Vol. 169

( 2 ) Strain-Displacement Relation

The relationship between the membrane strain and

displacement taking account of the large deflection

effect of the element reads

( 5 )

where, ƒÃx, ƒÃy and ƒÁxy indicate the membrane strain

components for a plane stress state.

The incremental matrix form of Eq. ( 5 ) is taken by

( 6 )

where,

( 3 ) Stress-Strain Relation of the Deflected Element

in the Elastic Range

Since the actual stiffened panel always has the initial

deflection, the in-plane stiffness decreases gradually

from the beginning as the axial compression increases.

In this study, the deflected panel is replaced by an

equivalent flat element with no deflection. For this

purpose, the reduced in-plane stiffness due to the exis-

tence of lateral deflection is expressed in the form of the

relationship between the average stress and the average

strain.

Fig. 3 shows a typical membrane stress distribution of

an axially compressed stiffened panel having initial

deflection when the aspect ratio of the plate between

stiffeners in both directions is nearly unity. In actual

ship panels, since the bending stiffness of each stiffener

is considered to be sufficiently large such that the

overall elastic buckling is not occured, it is assumed

that the stiffness of each stiffener is fully effective until

the panel collapses as a whole. Thus, the in-plane

stiffness of stiffened panel in the elastic range can be

evaluated by dividing into two parts6),7) : One is the

reduced in-plane stiffness of deflected plate between

stiffeners and another is fully effective stiffness of each

stiffener.

The reduced in-plane stiffness of deflected plate

between stiffeners can be measured by the same manner

for the unstiffened plate element described in Ref. 13).

Then, the following relationship between the average

stress and the average strain is emerged by the in-

cremental expression.

(7)

where,

[Dpi] E =the reduced matrix between the average stress and the average strain for plate part between stiffeners in the elastic range which has the following form because of the symmetry

On the other hand, the stress-strain relation of the

fully effective stiffener reads

( 8 )

where, ƒ¢ƒÐxaƒÒ,st, ƒ¢ƒÐyaƒÒ,st : average stress of stiffener

attached in x and y direction,

respectively

Asx, Asp : total cross-sectional area of stiffeners in

the stiffened panel in x and y direction,

respectively

a, b, t : length, breadth and plate thickness of the

stiffened panel, respectively

E : Young's modulus

Accordingly, the relationship between average stress

and average strain of the stiffened panel is expressed by

summing up Eqs. ( 7 ) and ( 8 ) .

( 9 )

where,

[D]E =the reduced matrix between the average stress and the average strain for combined

plate and stiffener in the elastic range

( 4 ) Displacement Function

Since the element behaviour is depicted by using only

axial displacements for both in-plane and out-of-plane,

Fig. 3 A typical membrane stress distribution in a

stiffened panel

Ultimate Longitudinal Strength-Based Safety and

Reliability Assessment of Ship's Hull Girder (2nd Report) 407

displacements take the following linear function.

(10)

where, coefficients a., a2, •c are the unknown constants

which are expressed in terms of the nodal displace-

ments.

( 5 ) Tangential Elastic Stiffness Matrix

By applying the principle of the virtual work and

updated Lagrangian approach the tangential elastic

stiffness equation is derived in the same manner with

the case of idealized unstiffened plate element described

in Ref.13).

(11)

where,

[K]ƒÃ: the tangential elastic stiffness matrix of the

element

, [D]' Eq. ( 9 )

( 6 ) Ultimate Strength Criterion The stiffened panel will collapse in two kinds of

failure mode according to the magnitude of the bending stiffness ratio between the stiffener and the plate between each stiffener. One of failure modes is local collapse of plate between stiffeners, in which the stiffener still remains straight and another failure mode is overall collapse that the stiffener with the reduced breadth of flange fails as a column.

In this respect, ultimate strength criterion is described in two failure modes as follows,

( a ) Local Collapse In this case, the collapse of only plate part between

stiffeners is considered. Thus the ultimate strength criterion has the same expression with the case of the unstiffened plate element described in Ref13) as.

(12)

where,

σxmax, σymax　　　: maximum membrane stress compo-

nents

σxmin, σymin　　　:minimum membrane stress compo-

nents

τ(=Zxya　υ)　　　 : average shearing stress

σO　　　　　　　: yield stress of the materialIn the above equation, maximum and minimum

membrane stresses are expressed as functions of aver-

age membrane stresses taking account of the initial

imperfections13). If any one criterion in Eq. (12) is

satisfied then the element reaches the ultimate limit

state.

( b ) Overall Collapse

In this case, the column strength formula is used

considering the compressed stiffener having the

reduced breadth of flange that corresponds to the

effective width of the plate between stiffeners. The

Perry-Robertson formula25 which approximately takes

into account the influence of initial eccentricity and

welding residual stress is employed in this study.

For Compression

(13.a)

For Ten si on

(13.b)

where,

αx, αy　　　　:initial eccentricity factor of stiffener in x and

y direction, respectively which are set to be 0.0035 in this study25)

Isx,　 Isy　　　　　: moment of inertia of each stiffener in x and

y direction, respectively

( 7 ) Post-Ultimate Stiffness Matrix If the stiffened panel reaches the ultimate limit state

with the overall collapse mode then the behaviour of the element follows that of the orthotropic plate in the

post-ultimate range. Also even though local collapse mode is occured, it is considered that the stiffener does not keep straight so far against the further increasing of the external load and fails in a moment. Therefore, the collapsed panel is assumed to follow the behaviour of orthotropic plate element regardless of local collapse mode in this study.

In this case, since the deflected form of the element may become to be changed comparing with that of

pre-ultimate range, the half-wave number of the ele-ment should be newly recalculated just after collapse, in which the same method with the case of the unstiffened

plate element described in Ref.") is applied. It is assumed that the magnitude of maximum men-

brane stress components of the element after collapse is not changed so seriously and then remains constant in the subsequent loading step.

Then by applying the concept of effective width30'31 the average membrane stress components in the post-ultimate range are expressed by

(14)

where, be and ae represent the effective widths in x and


y direction, respectively. Also, superscript u indicates the value just after collapse of the element.

With the increasing of the deformation, the effective widths decrease continueously even in the post-ultimate range. It is assumed that elastic effective widths can be applied even after collapse"). Thus, the following expression of the elastic effective widths is used even after collapse of the element.

(15)

where ƒÐxaƒÒ*, ƒÐyaƒÒ* : virtual average membrane stres-

ses

cxmax*, ƒÐymax*: virtual maximum membrane

stresses

Since the influence of initial deflection on effective

widths at loads far in excess of collapse is known to be

negligible30)'31), for simplicity the initial deflection is not

accounted for evaluation of the effective widths in this

range. Then, the maximum membrane stress compo-

nents in Eq. (15) can be derived analytically by solving

governing equation of orthotropic plate subjected to

biaxial load as :

(16)

where,

nSX,　 nsy　　　　:numbers of stiffeners in x and y direction,

respectively

υ　　　　:Poisson's ratio

m, n : half-wave numbers of orthotropic plate in x

and y direction, respectively

σγx,σγy　　　　　:compressive residual stresses in x and y

direction, respectively

σγex,σγey　　　　　:effective compressive residual stresses in

x and y direction, respectively which are

defined by

Since the edge of the element is considered to keep

straight even in the post-ultimate range, the virtual

maximum membrane stress components in Eq. (15) are

also estimated in terms of average strain components.

(17)

where,

Using Eqs. (16) and (17), we can derive virtual aver-

age stress components as functions of average strain

components. Thus, substituting Eq. (17) and virtual

membrane stress components expressed in terms of

average strain components into Eq. (15) , the effective

widths are also expressed as functions of average strain

components.

Accordingly, substituting Eq. (15) expressed as func-

tions of average strain components into Eq. (14), the

following average membrane stress-average membrane

strain relation is emerged.

(18)

On the other hand, the magnitude of shearing stress

acting on the element is very small in the problem under

consideration such that the shearing modulus of the

element is considered to be fully effective even in the

post-ultimate range.(19)

Taking Eqs. (18) and (19) into the incremental form,

the following relation between average stress and aver-

age strain in the post-ultimate range is emerged.

(20)

where, [D]" is the average stress-average strain matrix in the post-ultimate range.

Finally, the post-ultimate stiffness matrix of the ele-ment is obtained by replacing [D]E in Eq. (11) with

[D]" in Eq. (20), that is,

(21)

where, [K]" denotes the post-ultimate stiffness matrix of the element.



3. 3 Characteristic of the Present Element In this section, the characteristic of the idealized

stiffened plate element formulated here is observed with varying several parameters.

First, Fig. 4 compares the present solution with the result obtained by the elasto-plastic large deflection finite element analysis for a simply supported stiffened

panel. The element has an one-sided stiffener and is subjected to uniform displacement in the longitudinal direction. In the same figure, the result of unstiffened

plate is also compared. In the present analysis, only one idealized element is employed, while 10 x 5 mesh using triangular plate element is used in FE analysis. Also,

Fig. 5 and 6 represent the load-shortening curve for the same panel with varying the initial deflection and welding residual stress existing in the plate part, respec-tively. Fig. 7 shows the number of stiffeners on the nonlinear behaviour of an stiffened panel.

Through the above results, it is considered that the

present idealized stiffened plate element has been for-mulated satisfactorily because a reasonable solution is

given.

4. Application to a Suezmax-Sized Double Skin Hull Girder

4. 1 Outline of the objective Hull Structure As an example, a Suezmax-sized double skin hull

Fig. 4 A comparison between the present solution and

FEM result for a stiffened panel under uniaxial

displacement

Fig. 5 Average stress-average strain curve for a

stiffened panel under uniaxial displacement

with varying the magnitude of initial deflection



with varying the magnitude of residual stress



with varying the number of stiffeners


structure is considered. The ship has nine cargo tanks

having a hull form of double-skinned bottom and side as

shown in Fig. 8. The transverse bulkhead between

cargo tanks has also a double-skinned structure. The

deck panel has a single skin and is supported by heavy

deck girders. The principal particulars of the ship is

indicated in Table 1. The detail midship section of the

ship is shown in Fig. 9. The length of a standard cargo

tank is 22. 24 m. Transverse space is 5. 56 m in deck

part and 2. 78 m in bottom and side structure. The deck and bottom structure are built of 36 HT high-tensile steel whose yield stress is 36 kg/mm2, while for side shell the material of 32 HT that the yield stress is 32 kg/mm2 is used. The scantling of each stiffener is determined such that no overall buckling takes place in every stiffened panels. The section modulus of the

present scantling at midship section is checked being greater than the required one from the existing rule of classification societies.

4. 2 Structural Modelling Based on the theory described above, a computer

program, ALPS/ISUM22) is completed. By using the present idealized stiffened plate element the objective Suezmax-sized double skin hull girder is modelled as shown in Fig. 10. No. 6 cargo oil tank located at midship is chosen as the extent of the analysis. Because

Fig. 8 Profile and midship section of a new Suezmax-

sized double skin tanker

Table 1 Principal particulars of the Suezmax-sized

double skin tanker

Fig. 9 Detail scantling of midship section

Fig. 10 ISUM modelling for the objective hull girder

Fig. 11 Outline of boundary and loading condition

Ultimate Longitudinal Strength-Based Safety and Reliability Assessment of Ship's Hull Girder (2nd Report) 411

of the symmetry with regard to the center line, a half

side of one cargo tank is modelled. The number of the

idealized plate elements and nodal points used in the

modelling is 40 and 60, respectively. Fig. 11 represents

outline of the boundary and loading condition. The

longitudinal displacement along the left-end of the tank

is restrained since the double-skinned transverse bulk-

head is considered to be sufficiently stiff. The symmet-

ric condition is introduced at the center line. The

bottom at the loading position is supported toward the

depth direction. Series analysis of 10 cases denoted in

Table 2 is carried out including the intact and the

damaged condition of the hull structure. The magni-

tude of initial deflection and welding residual stress

existing in the plate between stiffeners is assumed to be

wo/t =0.1 and ƒÐƒÁx/ƒÐ0=0.1, where ƒÖo is the amplitude of

initial deflection for buckling mode component and a,

is the compressive residual stress in the longitudinal

direction. Also, as the initial eccentricity of stiffener, ax

and ay in Eq. (13) are set to be all O. 0035 and the

residual stress in stiffener is assumed to be not existed.

By using MIPS-M/120 mini-computer, the computing

time required was about 5 minitues, that is considered

to be very small comparing with the case of the conven-

tional FEM.

4. 3 Numerical Results and Discussions

( 1 ) Effect of Move of the Neutral Axis Position

The neutral axis of midship section in the intact

condition indicated in Fig. 9 is positioned at 10. 73 m

above the bottom keel. With the increasing of the

vertical bending moment, if deck panel or botom plat-

ing that is subjected to axial compression is yielded in

the earlier stage than the side shell plating, the position

of the neutral axis is expected to be moved. As a result,

the side shell plating contributes to sustain the further

increasing of the external load even after the deck

panel or bottom plating is collapsed.

(a) Dispi. control (b) Load control

Here, in order to investigate the effect of the change of the neutral axis on the ultimate longitudinal bending strength of the hull girder, two loading conditions that the cross-section of the hull at the loading position keeps the plane state are considered as shown in Fig. 12. One is that the vertical bending moment is generated by the displacement control at corresponding nodal point shown in Fig. 12 ( a ). In this case, the position of the neutral axis is not changed. Another case is the loading control through a rigid body which transfers the exter-nal force to loading direction only and then gives rise to no additional restraint against transverse and shearing deformation is attached in the right-end of the hold shown in Fig. 12 ( b ) . In this case, the neutral axis can be freely moved according to the spread of the plasticity. The rigid body is also composed of a number of rigid

plate elements, which are modelled by the idealized orthotropic plate element with rigid stiffness in the loading direction only, that is, no stiffness for transverse and shearing deformation is present.

Fig. 13 represents the change of the neutral axis of the hull with the increasing of the axial compression at the deck panel and bottom plating, resulting from the action of sagging and hogging bending moment, respec-tively. Also, Fig. 14 shows the stress-rotation curve at deck panel and bottom plating which are subjected to axial compression. Regardless of a change of the neutral axis, its effect on the ultimate collapse strength

Table 2 Result of system safety assessment of the objective hull girder

Note:1) The value in a parenthesis is utimate collapse stress(kg/mm2) at

deck panel in sagging and at bottom plating in hogging.

2) M. and Mn obtained are 360.63•~103(ton-m) and 534.94x103(ton-m)

from ABS rule, respectively

3) Notation of case No. symbol

(Ex.) S-ID

D:Displacement control(unoading behaviour considered) U:Displacement control(unloading behaviour not considered) B: Load control through a rigid body C:Collision damage existed G:Grounding damage existed

I: Intact, D:Damaged S:Sagging,H:Hogging

Fig. 12 Generation of external bending moment by

displacement and load control

Fig. 13 Change of neutral axis of midship section with

the increasing of external load


of the hulll girder is considered to be negligible.

( 2 ) Effect of Unloading Behaviour of Locally Col-

lapsed Panel

After the stiffened panel is collapsed under compres-

sive load, the average internal stress goes down even

though the deformation increases continuously. This

behaviour, so called unloading behaviour, gives rise to

redistribution of the stress in the structure. In particu-

lar, if the residual capacity remaining in the structure

after one or more local panels have failed is relatively

large, then the effect of the unloading behaviour of

earlier collapsed panel on the ultimate collapse strength

is expected to be severe.

Here, for the ultimate longitudinal bending strength

of ship's hull girder, the effect of unloading behaviour of

locally collapsed panel is investigated. For this pur-

pose, two cases are compared using two kinds of ideal-

ized stiffened plate elements representing different

behaviour in the post-ultimate range. One is the case

that the present element formulated above is used such

that the locally collapsed panel follows the unloading

path and another is that the element having zero stiffness in the post-ultimate range is employed.

Fig. 15 compares the moment-curvature curve for two

cases. From this result, the influence of the unloading

behaviour of locally collapsed panel on the ultimate

collapse strength is considered to be small. This is due

that with the increasing of the vertical bending moment

as soon as the deck panel or bottom structure is failed,

the hulll girder reaches the ultimate collapse strength in

a moment.

( 3 ) Effect of Damage Caused by Accidents

The hull girder is anticipated to encounter with many

types of accidents such as grounding, collision and so on

which cause damages in the hull structure. The system

safety of the damaged structure is decreased compared

with that of the intact structure.

Here, bottom structure damage due to grounding and

side shell damage due to collision are considered. In the

event of collision by a passing vessel, the hull structure

is assumed to have a damage in the lower side shell

plating shown in Fig. 16. Also, due to grounding of the

ship, six outer bottom panels and five bottom girders

shown in Fig. 16 become to be ineffective.

In the analysis, the external bending moment is gener-

ated by displacement control such that the position of

neutral axis of midship section is not changed and also

the unloading behaviour of locally collapsed panel is

taken into consideration.

A comparison of the bending moment-curvature

curve between the intact and the damaged structure is

made in Fig. 17. The result of system safety assessment

is indicated in Table 2. The still water and the wave-

induced bending moment are estimated by using ABS

rule"). The absorbed energy capacity of the hull struc-

ture is calculated until the rotation of cross-section at

Fig. 14 Influence of change of neutral axis on stress-

rotation curve

Fig. 15 Influence of unloading behaviour of locally

collapsed panel on bending moment-rotation

curve

Fig. 16 Damage caused by accidents of collision and

grounding



loading position is 5 •~10-3 (rad). For assessment of

safety index (SI), COVu, COVs and COVƒÖ in Eq. ( 3 )

are assumed to be 10 %,10 % and 20 %, respectively.

The ultimate collapse strength of the damaged struc-

ture due to collision, compared with the intact structure

is 98. 2 % in sagging and 97. 3 % in hogging and also in

the event of grounding, the reduced ultimate collapse

strength is 95. 7 % in sagging and 80. 6 % in hogging.

For absorbed energy factor (AEF) , the tendancy is

inverse comparing with S-DC and S-DG case. This is

due that although the hull structure in S-DG case col-

lapsed in the earlier stage, the absorbed energy capacity

after collapse is larger than that in S-DC case. Since

the reserve strength factor (RSF) and safety index (SI)

as well as RRF and AEF are considered to be relatively

large, the hull structure designed here has sufficient

safety in the longitudinal strength point of view.

5. Concluding Remarks and Future Research

In this study, an attempt is made to analyze ultimate

longitudinal strength of stiffened hull structure under

sagging or hogging bending moment by using idealized

structural unit method. For this purpose, an idealized

stiffened plate element is formulated taking account of

the influence of initial imperfections as well as the

interaction effect between local and global buckling in

the structure. The developed method is then applied to

ultimate longitudinal strength analysis of Suezmax-

sized double skin hull girder as an example. The effect

of the change of the neutral aixs of midship section and

of the unloading behaviour of the locally collapsed

panel on the ultimate longitudinal bending moment is

investigated. A new deterministic measure of system

safety based on the absorbed energy capacity between

the intact and the damaged structure is proposed.

Ultimate longitudinal strength analysis of the objective hull structure having two kinds of damages caused by accidents of grounding and collision are carried out. Based on the obtained results, system safety and reli-ability assessment is made using the proposed measure of safety in addition to the conventional one.

It is concluded that the objective hull girder has sufficient system safety in the longitudinal strength point of view because the magnitude of each measure of safety is considered to be relatively large. Also since the computing time required is very small while giving a reasonable result, the present procedure will be useful when ultimate collaspe strength-based safety assess-ment or optimization is made in the structural design stage for a new type of hull girder.

In this study, the idealized stiffened plate element is developed under the loading condition that the biaxial load is dominant. However, since the general loading components acting on the panels composing ship's hull

girder are shearing force and lateral load in addition to the biaxial load, the present element should be extended to take account of general loading conditions.

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