THE SOCIETY OF NAVAL ARCHITECTS AND MARINE ENGINEERS601PevOnisAvenue,Suite~, JerssyCity, NW Jersey Omx uSA

Paper p_Es.gntdat the Maine s8w5Mral ln~cm Maintemanm and hkmimri.. S.mmir,m...—..--..—.-—,--- ..,-,..-,...=-,..,r_.-...Stsratcm Nalio~ Hornl; Arli~”n,- Wgma, March 1 &19, 1991

Ship LongitudinalAnalysis

Strength Modeling for Reliability

P.A. Frieze, Paul A. Frieze &Associates, Twickenham, United KingdomY-T Lin,. National Taiwan Ocean University, Taiwan, Republic of China

ABSTRACT

A numerical procedure for theevaluation of” ship longitudinal bendingstrength is described. It accountsdirectly for all the relevantdestabil~sing par~eters involved,material properties, geometry, andfabrication defects. Simulations ofexperimental results on steel box girdermodels are used to demonstrate itsaccuracy.

Based on the observation that shipbending strength closely correlates withthe strength of the critical stiffenedpanel of a girder’s” cross-section, asimplified hull strength model isdeveloped. The model accounts explicitlyfar the material, and plate and stiffened.,panel slendernesses, and implicitly forplate and stiffened panel initialdeflections and welding residualstresses. Representative values for thelatter were selected following theanalysis of a range of typical stiffenedplate configurations.

/

‘.

The simplified hull girder model issubject to reliability analysis to helpidentify the important parameters,loading and resistance, affecting hulllifetime reliability.

NOMENCLATURE

aAbBc,d<DEH

IL

Mrs;tvz

ai

P

plate lengthcross-sectional areaplate” widthwidth of angle stiffener tableblock coefficientoverall stiffener depthhull depthYoung’s modulusneutral axis to extreme fibre.”dist.moment of inertiastiffened panel lengthmomentradius of gyrationshape” factorplate thicknessFroude numbersection modulusA~/A

=(blt)(uY/E)l’2=plate slenderness

Y HJD

6. plate initial deflection

A stiffener initial deflection

E strain

h =(L/nr)(UYlE)”2stiffened panel slenderness

qt” width of weld tensile yield zone

0“” stiffened panel strength ratio

x modelling uncertainty

+* ,.combination coefficient to account

for correlation between wave andstill water bending moments

o stress

e heading (180” equates to head seas)

e hull curvature

(3G end of stable curvature range

SuDerscriDts

v normalised w.r.t. yield value

Subscripts

B girder bottomD decke effectiveMC measured v calculatedP plasticQ extreme fibrer residual “stressSw still waters side-shellu ultimatew’ waveY“ yield

INTRODUCTION

Ship rules areexpressed in a form which

traditionallyhas little to

do with their structural strength. Thevery complex loading patterns to whichships are subjected - still water loads,and sea states the effects of which canbe” modified by operator intervention -have had a significant influence on thegrowth of these.

Ill-c-l

With the demand for risk assessmentfor many types of sh5ps now in place, theneed for a formal framework for thesafety evaluation’ “ofships has developed.Structural reliability analysis providesone such frameworkbut requiresthedevelopmentof simple yet accuratemethodologiesfor the ultimate strengthevaluation of: sh”ips structures and theircomponents if it’is to be efficiently ‘andeffectively exploited.

This paper is concerned with thedevelopment of a numerical procedure for:the determination of overall hull girderstrength and which includes considerationof the parameters that contribute to thisthrough buckling and yielding. Thepurpose of this is to provide a basis forthe derivation of a model which alsoaccounts for these “parameters but in asimplified form amenable to reliabilitymodelling.

The basis for the numerical model isdescribed and.its accuracy substantiatedagainst tests on steel box girdersrepresentative of simple ship structures.Following this, the simplified model isestablished and its accuracy quantifiedagainst the experimental data. The modelis then used in the reliability analysisof a ships hull as part of the ISSC’91Committee V.1 work.

HULL GIRDER MODELLING

Background

From the point of view of structuralanalysis, the failure of a ship’s hullgirder subjected to vertical bendingmoment may be due to brittle frac”ture,fatigue fracture, yielding, spreading ofplasticity, instability, or acombinationof these events. It may fail graduallyas in the case of a lengthening fatiguecrack or spreading plasticity, orsuddenly through plastic instability orpropagation of a brittle crack [1],

Although initial yield, which” occursat some points in a structure, does notnecessarily cause direct failure, thespreading of plastic deformation over asubstantial portion of a structure “maylead to structural failure. In the caseof a hull girder, yielding generallycommences in the deck or bottom structureand spreads towards side shells as theapplied vertical bending moment isincreased. Ultimately, a fully plasticmoment is reached when yteld hasdeveloped at every point throughout thegi’tderdepth [2]. “Thismoment representsan upper’’limit of’a girder’s longitudinals%rength, ‘but will rarely be attained duetd the adverse effects of buckling 0$ th”elongitudinal structure between fraines,”ofweld-induced residual stresses, or ofinitial deformations “-resulting fromfdbric’ation. In the practical case,failure is influenced by buckling or

yielding of the compression flange andyielding of the tension flange. ~

,-

When a structural member issubjected to compression, buckling “mayoccur at stress levels well below theyield strength. This type of instabilityfailure is characterised by a relatively-rapid increase.,in deflection for a smallincrease in load as the compressivestress approaches a critical value. Fora hull girder under vertical bendfngmoment, buckling does not immediatelyresult in complete collapse of thegirder. ,“ The post-buckling behaviourdepends on the detailed structuralarrangement. In transversely framedships, the plating buckles between frames.so the reserve strength after bucklingmay be small. In longitudinally framedships , howeveri the plating afterbuckling be”tween longitudinal,redistributes its ltiad to adjacentstiffened panels..and the plate-stiffe-ner.combination can “’”carry“further loadinguntil it buckles between transversesupporting members. As a result, themaximum’” load-carfying capacity O.flongitudinally framed vessels may besignificantly greater than the load atwhich buckling commences.

The possible. collapse modes for” astiffened panel under axial compressionare:(a) Flexural buckling induced by platefailure: this mode involves bucklingtowards the stiffener outstand and isprecipitated” by loss of compressivestrength and stiffness of the plate.(b) Flexural buckling induced bystiffener yield failure: in this casecollapse occurs in ,the direction awayfrom the stiffener outstand.(c) Torsional tripping of the stiffener:this mode can occur in panels withtorsionally weak stiffeners.(d) Overall grillage buckling: thisinvolves buckling of the transverse aswell as longitudinal stiffeners.

Torsional buckling is usuallyavoided by providing sufficiently stockystiffeners or tripping brackets, andsuppression of overall buckling can beachieved by the use of stiff transversesupporting members-. The major problemfacing a designer is interframe flexuralbuckling in modes induced either by plateor stiffener failure.

Discretisation

As indicated, failure. of anindividual structural element; eitherprecipitated by yielding or buckling,does no,tnecessarily imply failure-of theentire girder. Failure of a number ofstructural elements, however, does resultin collapse of the hull. This may occurin two different ways: (1) collapsecaused by a series of failures ofstructural elements, and (2) simultaneous

III-C-2

overall”- instability of the completecross-section.

Fortunately, except for the overallgrillage buckling, the ductile collapseof a ship’s hull girder is”most probablydue to” a sequence rather than acoincidence of failures ‘-of structuralelements. This enables one to divide themidship section into structural elementswhich respond to the imposed loadsindependently, and to concentrate ontheir collapse behaviour. The types ofstructural elements considered in thepresent study””.include the stiffenedpanel, the plate element and the hardcorner, which are now discussed in turn.

Stiffened Panel

This element, which is typicallyfound in longitudinally framedstructures, is comp”osed of a stiffenerand attached plate. The plate rangesfrom the midpoint of a plate panelbetween *WO longitudinal stiffeners tothe midpoint of the adjacent plate panel.The stiffener. may be a flat-bar, aT-section or an angle bar. The behaviourof stiffened panels under compression caneonveni~ntly be represented by load-endshortening curves, i.e. averagestress-strain curves. A computer program[3] was used to generate the load-endshortening curves for stiffened panelswhich make up the cross-section of a hullor box girder. The extent of theanalysis was from mid-span of one’ panelto mid-span of an adjacent panel. Thisapproach by-passed the problem ofidentifying which mode (plate- orstiffener-induced) precipitated failure.The analysis allowed different levels ofinitial stiffener deflection to beconsidered in each half span.

, The parameters which affect ..thecompressive behaviour of stiffened panelsinclude plate slenderness,” weld-inducedresidual stress in the plate, maximuminitial .de.formationin the plate, columnslenderness of the stiffened panel,initial stiffener bow , and materialproperties.. The. effects of. all theseparameters are accounted for by the useof load-end shortening curves. In alongitudinally framed hull girder, themidship section usually comprises fiftyor more stiffened panels, which maydiffer in geometry, material property orlocation in the cross-section. Sinceparts of these panels exhibit virtuallythe same collapse behaviour, it .iauneconomical to generate the averagecompressives tress-strain curve for everystiffened panel forming the cross-section. Stiffened panels are thusdivided into groups of panels with nearlyidentical parameters. A representativepanel within each group is selected,based on engineering judgment, forUeneratinU the load-shortening curve.

Plate Element

This kind of” structural elementcomprises a single plate only and isdefined by its thickness and thecoordinates of the plate edges. Thereare two possible ways in which platepanels may be found in box or hullgirders. Firstly, ‘wide plate’ elementsare created in transversely framedpanels. Secondly, ‘long plate’ elementsare present in longitudinally framedstructures where a plate-stiffenercombination is so stocky that interframeflexural buckling is precluded and onlythe effect of plate buckling has to beallowed for.

The elasto-plastic bucklingbehaviour of wide plate elements underlongitudinal compression is basicallyequivalent to the behaviour of longplates under transverse compression whichcan be appropriately covered by averagestress-strain curves. Numerical studies[4,51 provide a useful basis for theevaluation of the stiffness and strengthof long plates in transverse compression,which are strongly ‘influenced by plateslenderness; aspect ratio and level ofweld-induced imperfections. Typicalaverage stress-strain curves derived from[4] are”illustrated in Fig 1.

1.0

t

D ./b

a 0.96 3.00 —-Qr

i

1.57 3.00?.63 3.00 —.—- .

0.8 3.50 3,00 ————

0.6

t ‘~!y.1, __+__—-_—-L_-— ______

..-/.-

_ __——-— .—--=0.2 ./- _-——————————

,. /.<-----, /---

0.00.0 0.5 1.0

‘+= EIE,:.0

Fig. 1 Average Stress-Strain Curvesfor Plates in Transverse Compression

The compressive behaviour of longplates has similarly been examined usingnumerical procedures [6,7]. To simplifyand generalise the extensive elasto-plas~ic buckling results of [6], [3]developed a parameterised model whichcould efficiently determine the stresslevel” in a. plate given the extent ofcompressive strain, the plateslenderness, and the level of weld-induced imperfections. This same modelwas used in [3] in the derivation ofinterframe buckling stress-strain curves.

The tensile behaviour of plates ismainly influenced by the magnitude of

III-C-3

weld-induced residual stress. Forstress-free plates it is appropriate tosuppose that the plates follow thematerial stress-strain curve. For plateswith weld-induced residual stress,however, initial stiffness is factoreddue to.the yielding tension blocks by theamount:

l.~=l:o,(1)r

If a linear relationship is assumed up tothe point where. .yielding occurs in thepre-compression zone, the non-dimensionalstrain corresponding to this point is:

(2)

If a parabola Ls assumed to represent theresponse over the strain range

15E ’<1+20;

it can reexpressed by1u’ - [-E’2+(2+4U;)E’-1] (~)

4U;(1+O;)

and is connected ,to the neighboringlinear sections with” appropriate slopes’.This form of representation accounts forthe adverse effects” of itiitialdeformation and residual stress.

Hard Corner

At certain locations of a hullcross-section, the local structure isstrengthened by the connecting members in

such a rigid way that it is reasonable toassume that the structure can sustain theimposed compressive load up to the yieldpoint and beyond without suffering anyform of instability. That is, iteffectively follows the materialstress-strain curve over the full loadingrange. These regions, called ‘hardcorners’ [8], include deck stringers,shear strakes, intersections Of deckplating with deep girders andlongitudinal bulkheads, intersections ofshell.plating with. superstructure, bilgekeels, deep -“““girders, longitudinalbulkheads-and keel, etc.

The geometry of hard corners can bedescribed in two different ,ways.Firstly,, a hard corner composed ofseveral interconnecting plates is treatedas a group of .pl”ateelements and definedby the thicknesses and locations of plateele’ment5. Secondly; the area, centroidand moment of. inertia of the ..cross-section” of a hard corner are calculatedmanually and input directly. This”formatallows.a number of interconnecting Platesto be described as a shgle hard cornerelement. ,-.

NUMERICAL PROCEDURE

The assumptions made in predictingthe hull girder response to %xtremevertical bending moment are:1 Plane cross-sections of the hullgirder before bending remain plane afterany load application, thus thedistribution of: strain over the cross-section is linear.2. The midship seotion of the hullgirder is discr,etised into a set ofstructural elements .of which the heightis sufficiently small compared with theship! S depth that a uniform straindistribution can be assumed to act overthe cross-section of each element. Theuniform strain acting on the cross-section of a“structural element 1S takenas the value of the linearly distributedstrain at the centroid of thecross-section (Fig. 2).

Fig. 2 Assumed Cross-SectionStrain Distribution

3. The stress-strain relationship. foreither long or wide plate elements incompression is appropriately representedby average stress-strain curves derivedfrom the large deflection elasto-plasticanalysis” of the “isolated plate panelssimplified as described in [31.4. The elasto-plastic behaviour ofstiffened panels in compression - isdescribed by load-end shortening cu+veswhich are specifically derived using theprocedure developed”in [3].5. The hard corner behaviour is definedby the material stfess-strain curve.6. Since shear forces are generallysmall at the midship section, the effectof shear stress on yielding is neglected.7. Neither fatigue nor ductile fracturemodes of fa”ilure are considered.“8. Since ultimate hull girder collapseoccurs in most cases before outer-fibrestrains reach two times yield strain, a“ndfor shipbuilding steels strain hardeningdoes not occur until strain exceeds yieldstrain by eight ‘to” ten times, thematerial itself is taken to behaveelastic-perfectly plastic in tension andcompression.

AssumDti’ons

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Application of Load Increments

Loading” is applied to the hullgirder in the form of curvature’ insteadof bending moment. Since a linear straindistribution over” the mid-section isassumed, the strain at the centroids ofstructural elements caused by the imposedCurvature can be determined on the basisof the mid-section’s effective neutralaxis. The corresponding stress acting oneach structural element is thencalculated from itsaverage stress-straincurve using interpolation if necessary.Although strain is assumed to bedistributed linearly over the section,stress varies non-linearly due to theeffects of fabrication-induceddistortions, weld-induced residual stressand buckling. Finally, fhe bendingmoment acting on the hull girder iscomputed for the applied curvature bysumming the contributions from all of thestructural elements. After thisprocedure has been repeated forincreasing levels of curvature, pairs ofdata of bending moments and curvaturescan be obtained to plot the bendingmoment-curvature curve which includes>the”maximum bending’ moment and the pre- andpost-collapse behaviour of the hull.girder.

Since “the curvature at whichcollapse of a gi”rder occurs varies to ahigh degree with the magnitude of theship’s height, it is appropriate toexpress the curvature increments in aform non-dimensionalised with respect to

(4)

Location of Effective Neutral Axis

The location of the plane of zerostrain, the ‘effective neutral axis’, foran applied curvature is determined by thestate of stress existing in thestructural elements forming the midshipsection. As “the applied loading isincreased, the effective neutral axisshifts towards the tension flange due toloss of stiffness of the structuralelements in the compression zone. Ateach value of curvature, locat”ionof theeffective neutral axis is determined onthe condition that the sum of the axialforces carried by-all of the structuralelements equals zero.

The state of stress of a structuralelement is connected with a particularvalue of strain through the element’saverage stress-strain curve, while thestrain itself is proportional to thedistance between the centroid of thestructural element “and the effectiveneutral axis. Therefore, the location Ofthe effective neutral axis must beiteratively determined from the condition

of equilibrium. The algorithm employedis described in detail in [3].

Evaluation of Bendina Moment

After the location of the effectiveneutral axis is determined for aparticular curvature increment, thecorresponding incremental bending momentcan be evaluated. This is obtained bysumming the contributions from all theelements that make up the mid-section.The moment and curvature are thennon-dimensionalised with respect to ~ andthe first yield curvature.

The fully plastic moment of a shipgirder, which represents an upper limitof its load-carrying capacity as a beam,is readily calculable based on theplastic hinge concept [2]. Following themethod of classical plasticity, theultimate limit condition is reached whenyield occurs at every point of the cross-section. In this case, the ‘plasticneutral axis’ coincides with theinterface which divides the cross-sectioninto two regions wi-th equal ‘squash’loads-in tensicn and compression.

The first yield curvature and thecorresponding bending moment ~ of aship’s girder can be determined by thestandard section modulus calculation.First the location of the ‘elasticneutral axis’ is determined for the fullyeffective mid-section. Then 0, is given

by eqn (4) and-~ can be evaluated from:

Bendina Moment-Curvature Curve

(5)

Curvature of the hull girder axis isconsidered to be positive when the hullgirder is bent concave upwards andnegative when concave downwards, i.e.positive in sagging and negative inhogging. Similarly the bending moment.is positive when it produces compressionin the deck “and tension in the bottom..

Using the numerical procedurediscussed above, it is therefore possibleto obtain a particular bending momentincrement for any curvature incrementimposed on a girder. If this procedureis “repeatedly carried out forincreasinglevels of curvature, the non-dimensionalbending moment-cu~ature curve “isgenerated. Separate runs are performedfor the hull girder in sagging andhogging to provide the complete historyof the girder under vertical bending.

“Typical moment-curvature curves areshown in Fig. 3. The full range ofbehaviour in either sagg@g or hoggingcan be divided into three zones ofbehaviour. The first depicts stablebehaviour in which the curvature imposed

“III-C-5

1<0

Mq I .-- —-—-—-—

//’,.-

0.5

6“ e.

-2 1ele, 2

:“—3-2* ,.

1. sTm mlm

2. TJW61TJM Zmb!

9. LWOMIW ~L1tM

—-— FLfLLYEA5T1C— UEELINC, EFFEC7S INCLUDED

—- —. —-.-.-’ .1.0

Fig. 3 Typical Hull Bending Moment-Curvature Behaviour

on the hull girder is less than acritical value 8= which is the smaller of

the curvatures to first yield or to thebuckling of some major ,components. Ifthe effects of neither buckling noryielding are significant, the curve inthis region is “virtually linear. Theslope of the bending moment-curvaturecurve , --however, ,commences to decreasenoticeably from the curvature e= and

decreases until it eventually approacheszero at the curvature BU at which peak

moment occufs. The second zone, i.e. thetransition zone, ranges from e= to eu.

The thi”rd“zone occurs beyond eUand is

characterised by negative slopes, i.e.the load-carrying capacity of the girderreduces with increasing curvature.

Collapse behaviour of hull girdersmainly differs in the last two zones andis influenced by such factors as materialproperties, geometric configurations,initial imperfections, degree ofstructural redundancy, etc. For a girderwith little redundancy, collapse isusually precipitated by failure of asignificant portion .of the structure,leading to a sudden drop in bendingmoment capacity after reaching thecr%.tical curvature.” The second zone,therefore”, lasts for a relatively shortrange in this case. In contrast, for agirder with. greater redundancy, collapseoccurs gradually as a result .af thesuccessive failure of smaller portions ofthe structure. That is, the load shedbysome components due to their failure. canbe carried by the reserve load-carryingcapacity of neighboring elements. Thus ,final collapse of the whole mid-sectionis delayed.by this load redistribution,which results in. a longer transitionzone.

When all the structural elementsforming ‘The mid-section of a hull girderare a“ssumed to follow the materialstress-strain curve both in tension and

compression, a bending moment-curvaturecurve can be generated for the “fullyeffec!?ive cross-section (Fig. 3). Sinceeffk~zs of structtiral instability areexcluded in this idealised case, bendingfailure can occur through materialyielding only. As the accumulativecurvature is incgeased, yielding beginsat the outermost-,fibre of the deck orbottom structure, and then spreadsgradually down or up the side shells.The curve thus approaches asymptoticallya horizontal line determined by.the fullyplastic moment. Comparing “this fullyeffective curve with a realistic bendingmoment-curvature curve, the differencethat, is attributable,to buckling effectscanclearly be identified.

HULL GIRDER STUDY

Introduction

Using the above procedure: theresponse of a number of box and hullgirder sections has been examined [3].This concentrated on. sections which hadbeen tested experimentally and simulatednumerically in order.to substantiate theprocedure but also .Lncluded additionalanalyses to investigate aspects such asstiffening configurations. These wereexecuted so that the results could begeneralised and simplified for use inreliability modelling as described later.The results considered concern:-

* Box girders Models 2 and 4, whic~were tested at Imperial College underpure bending conditions and failed bybuckling of the ‘stiffened compressionflange panels [9,101. The” effects ofinitial stiffener deflections, behaviourof hard corners and residual stressescontained in the tension flange on thebehaviour of the girders werespecifically examined.

* Box girders-Models 23 and”31, testedby Reckling- [111for which variation ofthe ultimate strengths .of the girderswith differing welding residual stressesand initial imperfections wereconsidered.

* Two transversely framed warships,T.B.D. COBRA and T.B.D. WOLF based onnumerical results obtained by Faulkner etal [12].

Box Girder Models 2 and 4 [9,101

Both models were subjected to”purebending and failed by buckling of thestiffened compression flange panels.Model 8 of the same series was alsotested in pure -bending. It. sufferedorthotropic buckling .-ofthe compressionflange, ,howe,ver, S“O was not readilyamenable ,forcomparison using the presentapproach.

-._.,

III-C-6

-, ...”...

2 2 2,,------., ,--------- -------- .J--.,

../ \./>

‘Q

\)

‘ ---------- ‘. “ ‘.-------.’ . .,/ .------ -__-”

‘\,!

‘1 H

d

HARD CORNER. ENCLOSED BY SOLID L JNE

PANEL. ENCLOSED BY BROKEN L] NE. LABEL No, ACA!N5T I T CORRE$PONOING

3;TO AN EFFE[.T ] YE STR555-5TOAIN C~R”E

I .;

‘\ ‘‘. ,’

\

2 2 22

Fig. 4 Discretisation of Model 2 [13,14]

Model 2. The cross-section of Model2 [13,14] was-discretised into structuralelements, i.e. stiffened panels, plateelements and hard corners, as shown inFig.” 4. The component dimensions andmaterial properties are as listed inTable 1. The hard corners were assumedto have an elastic-perfectly plasticrelationship in both tension andcompression.

In the first test on Model 2 (Test 2A),collapse was precipitated by platebuckling at one end of the box probablydue to the high transverse residualstress occurring near the diaphragm [14].After stiffening of the end bays the boxwas tested again (Test 2B) and ,collapseoccurred at an internal section, but themaximum load was -exactly the same as inTest 2A. Residual stresses and initialimperfections were measured before eachof these two tests, those for the secondtest clearly including those resultingfrom the’firs,t test. These are listed inTable 2 and were used in the presentstudy to analyse the overall behaviour ofthe girder. The average stress-straincurves for plate panels in thecompression flange and webs were derivedby the simplified method and are shown in

Table 1Component Dimensions and Material

Properties of Models 2 and 4 [9,101 andModels 23 and 31 [11]

Model 2 Model 4 Model 23 Model 31Compression Flanae Plateb 241.30 120.65 85.71 120.00t 4.864 5.017 2.50 2.50q 297.3 221.0 246.0 255.0

E 208.500 207.000 210,000 210,000Tension Flanae Plateb 241.30 120.65 85.71 120.00t 4.864 4.943

~Y 297.3 215.6

E 208,500 208,700Web Plate

b 273.05 98.425114.30

111.125t 3.;66 4.943

UY211.9 280.6

E 216,200 214,100Stiffener (anqle )d 50.80 50.80B 15.875 15.875t 4.763 4.763

% 276.2 287.9

E 191,50”0 199,200

2.50246.0

210,000

100.00

2.50246.0

210,000

30.0020.002.50246.0

210,000

2.50255.0

210,000

133.33

2.50255.”0

210,000

30.0020.002.50255.0

210,000Stiffener (flat)d 50.80 30.00 30.00t 6.35 2.50 2.50Oy 303.8 246.0 255.0

E 206.200 210.000 210.000Mid-section Particulars

A 21554 29144 6875.0 6250.0I 3411.8 4331.6 197.44 180.35H~ 465.10 468.8 200.00 200.00M 2243.3 2626.9 268.20 253.65

Kits: Length in nun;Area in mm2; Inertiain m2 mm2; Stress in N/mm2; Moment in kNm.

Fig. 5 for Test 2A. These curves wereused in the large deflection elasto-plastic analysis of the double-spanstiffened panels with residual stressesand initial imperfections as listed inTable 2. The Test 2A computed load-

Table 2Initial Imperfections and Comparisons between Numerical and Test Results

for Girder”Models 2 and 4 [9,10]

Model 2 Model 4Initial Stiff. Deflns Corner Model. Res.Stress

Test “2A Test 2B Mode 1 Mode 2 Mode 3 TvDe A TvDe B in Tens.Fl.0; 0..176 0.176 0.562 0.562 0.562 0.562 0.562 0.562

60/b 1/400 1/100 1/800 1/800 i/800 1/800 1/800 1/800

AIL -1/1450 1(580 -1/1050”,-1/660 -1/510 -1/510 -1/510 -1/510

1/2280 1/1430 1/1950 1/4920 1/510 1/510 1/510 1/510Mu (kNzn)(num.) 1620 1471 2421 2418 2344 2322 2324 2421Mu/

%(num. ) 0.722 0.”656 0.921 0.920 0.892 0.884 0.885 0.882

~ ( Nm)(expt) 1543 1543 2212 2212 2212 2212 2212 2212

Ml% (exDt ) 0.688 0.688 0.842 0.842 0.842 0.842 0.842 0.842, Notes 1. Corner behaviour is represented by material stress-strain cuwe....

2. Unless noted, effects of residual stresses in tension flange are ignored.

III-C-7

shortening curves for the stiffenedpanels in the compression flange and thewebs are shown in Fig. 6.

1.0

1S,/b. 1/40g

a— 0<1760. =

Ov0.6

V/’

0.2 /’

B1.073 DE CK PLATE —

2.540 W U PLATE. —. —..

0.0 v’0.0 0.5 1,0 1.5 2.0

2“5ElE,3”o

Fig . 5 Plate Average Stress-StrainCurves for Model 2 - Test 2A

7.0

t

@ A~ 1.873 0.644 .—Ov

I

2.540 0.530 —.—- .

0.8

0.6 ,-

-----0.4

0.2A / L . .1/1450

6, / b . 1/400 — ST. PL. IN DECK—-— 5T, PL. IN MEBS

0.0 v0.0 0.5 .1.0 1.5 2.0 •l~v 2.5

Fig.. 6 Stiffened Plate Load-ShorteningCurves for Model 2 -.Test 2A”

The pti~e bending tests on Model 2were carried out in the saggingcondition. Therefore, the bendingmoment-curvature relationships werecomputed using the:present method for thesagging condition only. They are plottedin Fig. 7 together with the .sxperiinentalresult [15] and the fully plasticresponse in -both sagging and hogging.Comparisons between the predicted andexperimental results are also presented

~~ 1AM .—.—-—.—.—

0.9q

ljL 6,/b /--------—

0.6 .--1/1L54 .Im ,,hw _ /.1/ w .1/1432 ,/, M --—- ,,

/

0.3 ,’

/.2.5 -2.0 -1. $ -1.0 -0.5 0 0.5 !.0 ,.5 .?.0 2.5

/

,}.3 -@/e,

Fig. 7“ Beridin@Moment Curvature.Response for Model 2

in Table 2. The agreement is seen to besatisfactory with the predicted maximumbending moment being 4.9% higher than thecollapse moment in Test 2A and 4.6% lowerthan in Test 2B.

...—Model 4. Box girder Model 4 [16,17]

had the same overall dimensions as Model2 but more closely spaced stiffeners.Similarly to Model 2, the cross-sectionof Model 4 was discretised intostructural elements including stiffenedpanels, plate elements and hard corners.Component dimensions and materialproperties are listed in Table 1.

As illustrated” in Fig. 6, thestiffness and load-carrying capacity ofstiffened panels are dependent upon themagnitudes of the weld-induced residualstress and initial imperfections, inaddition to such parameters as plateslenderness and column slendernesss.Consequently, maximum bending moment andcollapse behaviour of box girdersincorporating such meders are similarlyinfluenced. Therefore, where initialimperfections and residual stresses inpanels vary, it is necessary when usingone stiffened panel load-shortening curveto represent the panel behaviour thatthis should be representative of theentire panel, and not just the maximum,for example.

Thus three combinations of initialimperfections were selected from themeasurements [17] for analysis: they arelisted in T3ble 2. One average residualstress level in the plate and one maximuminitial plate deflection were usedthroughout, while three different levelsof maximum initial stiffener deflectionin two consecutive spans- were adopted.The deflections correspond to the mostseverely defotied (Mode 3) condition andto two slightly deformed shapes (Modes 1and 2). The load-shortening curves forthe stiffened panels in both thecompression flange and the webs with”thethree combinations initialimperfections were ‘comput% [3]”. Byincorporating these load-shorteningcurves as the effective stress-straincurves in the present incremental moment-curvature analysis., ,the Sagging bending“moment-curvature relationships wereobtained.

\

Mode 3 represented the largeatmeasured values of the initial stiffenerdeflections tdwards (+)and away (-) from”the stiffener outstand. Comparing thepredicted peak bending moment in thiscase with the experimental result [18]shows that the pre”sent method over-estimates khe collapse. bending moment by5.9%. The bending moment-c’urvaturecurves for Modes 1 and 2 almostcoincided. The computed peak bendingmoments for Modes 1.and .2were9.4%“and9.3% higherthanthe collapsebending -~....,moment.

III-C-8

The peak bending moments computedfor these three modes of initialimperfections and the experimentalcollapse bending moment are suiimarisedinTable 2. Changing the initial stiffenerdeflection mode from (-1/1050, +.1/1950)to (-1/510, 1/510) results in a 3.3%decrease in maximum bending moment.

The hard corners in the presentstudy are assumed to be elastic-petiec~ly plastic in both tension andcompression.- To exam”ine the effects ofalternative characteristics for these,hard corners were introduced relative tothe Mode-3 specification as follows:-

Type A -the load-shortening curve ofthe stiffened panel adjacent to thecorner; .!

Type B -the average compressivestress-strain curve ~or the plate panelalone.

Based on these assumptions, thecorresponding sagging bending moment-curvature curves for the box girder werecomputed. “The corresponding predictedmaximum bending moments are listed inTable 2. Since both the stiffngss andthe maximum load-carrying capacity of theMode 3 corner are greater than those ofType A, the peak bending moment iscorrespondingly greater. However, thedifference between the two is very small(0.9%) because the plates and thestiffeners in the compression flange and

/ webs are very stocky ( ~=0.786, k=O.490).

Behaviour of “the model incorporating theType B corner lies between these two asshown in Table 2.

In most cases examined “[3],behaviour of the tension elements wasassumed to be elastic-perfectly plastic.To study the effects “of differentelemental. tensile behaviour on thebending moment-curvature relationship,the. elements in the tension flange wereassumed to have an initial stiffnessreduced due to tension yielding blocksinduced by welding as given by gqn (1).

This stiffness remains COnStant up toyield after which it follows theperfectly plastic curve.

The corresponding peak bendingmoment is listed in the last column ofTable 2. The effect of residual stressesin the tension flange is to decrease theinitial stiffness. However, it makeslittle difference (1.2%) as far as themaximum bending moments are concerned.

Box Girder Models 23 and 31 [111

A. Series of seven fabricated steelbox girders subjected to pure bending wastested at the Technical University ofBerlin by Reckling. These girders wereorthogonally stiffened in a similarmanner to ship’s hulls. Of these, Model23 and Model 31 were chosen for analysis.Tests on Model 31 showed that thecollapse was delayed by the restrainingeffect of.the side walls, the ‘box girdereffect’ as described by F!eclding, inspite of earlier buckling of deck panels.Model 23 had the same overall dimensionsas Model 31 but had more longitudinalstiffeners in the deck and side walls.It was observed in the tests thatcollapse of Model 23 occurred by bucklingof the plate panels between longitudinalstiffeners in the deck coinciding withfailure of the whole deck.

Model 23. Component dimensions andmaterial properties for. Model 23 aregiven in Table 1. Its cross-section wasdiscretised as previously described.Since only a single value of yield stresswas given for the entire girder [11],this value was assumed for both theplates and the stiffeners. The weldingresidual strekses and initialimperfections used in the derivation ofthe load-shortening curves are listed inTable 3. The stiffened panels in tensionwere assumed to follow the materialstress-strain curve, as were the hardcorners in both tension and compression.These effective stress-strain curves ofthe structural elements were then used togenerate the bending moment-curvaturecurves for the following cases:-

Table 3Initial Imperfections and Comparisons between Numerical and Te=t Result=

for Girder Models 23 and 31 [11]

Model 23 Model 31.Corner Res.Stress Initial Stiffener Res.Stresses Initial Plate

Modelling in Tens.Fl. Deflections in Comp.Fl. DeflectionsCase 1 Case 2 Case 3 Case 4 Case 1 Case 2 Case 3 Case 4

u, 0.20 0.20 0.20Case 5

0.20Case 6 Case 7

0.20 0.20. 0.20 0.05 0.35 0.20 0.20601b 0.25 0.25 0.25 0.25 0.55 0.55 0.55 0.55 0:55 0.22 0.86AIL -1/1000 i/looo -1/1000-1/1000 1“/1000 1/2000 1/500 1/1000 1/1000 1/1000 1/1000

\_ .

1/1000 1/1000 1/1000 1/1000 -1/1000-1/2000-1/500-1/1000 -1/1000 .1/1000-1/1000Mu(kNm)(num)237.9 239.9 232.8 234.6 203.0 204.3 199.7 200.1 203.0M./MP (num)0.887

205.9 200.30.895 o“.868 0.875 0.800 0.806 0.788

Mu(kNm)(exp)249.40.789

249.40.800 0.812 0.790

249.4” 249.4 215.9 215.9 215.9 215.9&/M. (exD-).0.930 0.930

215.9 215.9 215.90.930 0.930 0.851 0.851 0.851 0.851 0.851

Notes 1. Cornerbehaviouri: represented by material stress.strain Cuwe.0.851 0.851

2. Unlessnoted, effeCts of residual $tresses in ten$ion flamge are not considered.

III-C-9

1. ‘ the fleck plate panels neighboringthe deck-side wall intersections weretreated as hard corners;2. additional to Case 1, the side-wallplate panels adjacent to the corners weretreated as hard corners.

The predicted peak and test COll&psebending moments are listed in Table 3.It can:.:be keen that correlation between*he numerical and experimental results”issatisfactory as far as the maximumbending moments are concerned. Thedifference in the.maximum values is 4.6%for Case.1”,and .3~”8%for Case 2. Despitethis agreement, ‘the computed bendingmoment-”cuqvature curves were found ‘todiffer from the experimental curveparticular~y initially. Since “theinclusion of welding r“esidualstresses inthe tension flange usually leads to adecrease in the %nitial stiffness of abox girder, two further cases wereexamine,d, viz.” y

3. the same ai” Case 1 except thatresidual stresses in structural elementsin tension were considered;

4. the” same as,.Case 2 except thatresidual. stresses in structural ;lernentsin tension were considered.

The resulting”peak predicted bendingmoments are listed in Table 3. Thedifference in maximum bending “momentbetween -the numerical “and experimentalresults is .6.6% for Case 3 and 5.9% forCase 4, slightly” greater than those forCases 1 and” 2; but the predicted initialstiffnesses in Ca&es “3 and 4 correlatedsignifica”nt$ly better with theexperimental results [3].

Model 31. Component dimensions andmaterial”” properties” for Model 31 aregiven in Table” “1. The model wasdiscretised into structural elements andthe stiffened “ panel load-shorteningcurves computed for, the deck and side-walls having”.the residual stresses andin”itia$ imperfections “listed for Case 1in Table 3. .These curves were then inputto perform the ultimate strength analysisof the girder cross-section in which hardcorners were assumed to follow thematerial stress-strain relationship. A6.0% difference was found between thepredicted and obeerved ultimate bendingmoments (Table 3) although the initialstiffnesses were similar.

This model wqs alSO used to examinethe effects of initial stiffenerdeflections, weld-induced residual stressand tnitial plate deformation on thecollapse behaviour of a girder. Firstthe effects of initial ‘stiffenerdeflection were considered. For this, inaddition to Case 1 ( A/L=l/1000, 60/t=

0.55, a~=0.20)” ultimate strength analyses

were performed for the following cases:

2 A/L=l/200Clr 60/t=0.55, u~=0.20;

3“ A/L=l/500, ~o/t-0.55, u~=0.20.’

The web stiffened panel”swere found to bemore sensitive to initial stiffenerdistortion than the flange Panels [3].

The maximum bending moments a?elisted in Table 3. tiotunexpectedly, theultimate strenath o“fModel 31 is “seentodecreasestiffenerdecrease,stiffener

1/2000 to

Theresidualanalysing

wi<h increasing initialdeflection magnitude. Thehowever, as the initial

defldktion varies from A /L=

A/L=l/500 is only 2.2%... ..

infltience of weld-i”nduced:stresses was examined bythe following cases:

4 A/L=l/1000, &O/t=0.55, a:=O.05;

5 A/L=l~1000,6Jt=0.55, 0:=0.35...,-

Althoug~, the ef~ect of residual stresseson the stiffness and strength of thestiffened panels bet”weenCases 1, 4 and 5was found to be significant [3], “the,peakbending “mom”ents ofily” changed by 1.5%

(Table 3).

.Finally, Model”’ 31was analysed forthe following cases:

6- A/L=l/1000, 50/t=0..22, u~=0.20;

7 A/L=l/1000, 6a/t=C!.88,CI~=O.20.

to examine the effect of initial platedeflections. The maximum predictedbending moments are. listed in Table 3from which it “is seen that the ultimatestrength of the b’oxgirder decreases withincreases in the initial “platedeflectionmagnitude. In particular, when theinitial plate deflection varies from 6Jt

=0.22 to 6Jt=0.88r the computed maximum

bending moment decreases by 2.7%.

Of the cases examined above, Case 6corresponds most closely to the testedgirder so the present procedure under-estimates the experimental value by 4.6%.

.!

T.B.D.COBRA and wOLF [121

The Torpedo-Boat Destroyer COBRAcollapsed by breaking her-back and sankin rough seas on her first journey in1901. The disaster was re-examined inthe light of 1980’s technology [12]leading. to the conclusion that sinceCOBRA’s hull was transversely framed andwas structurally too weak even towithstand. moderate sea conditions, theCOBRA may have been lost due to bucklingnat being properly. considered in herdesign. Hull strength assessments ‘weremade of T.B.D.” COBRA and a similarly

- .-

\. .__.

“\,,’

Ill-c-lo

constructed vessel T.B.D. WOLF In thesagging and hogging conditions us”ing theanalysis procedures ‘reported in [19].The results are compared below with thoseobtained by the present incrementalmoment-curvature approach.

T.B.D. COBRA. The midship cross-section of COBRA was obtained from [12]and was discretised into structuralelements. The effective stress-straincumes for -wide plates in compressionwere also taken from [12]. The ultimatestrength of COBRA ‘S hull was thenevaluated using the present analysis.

Bending moment-curvature cuties werederived- for the sagging and hoggingconditions. Compared with the maximumbending moment results of [12], a 1.3%difference was found for the saggingcondition and 2.0% difference for thehogging condition. In both cases the,present analysis gave the lower result.

T.B.D. WOLF. The midship cross-section of WOLF was obtained from [12].Differences of 3.6% in maximum saggingbending moment were obtained and 1.0% inmaximum hogging bending moment. Thepresent analysis gave a lower result inthe case of sagging but higher in thecase of hogging.

In [3], the effect of introducinglongitudinal stiffening into the hull ofCOBRA was examined. The results showedthat the introduction of 10”longitudinalstiffeners to the existing hull led to anincrease in.ultimate strength of 29.6% insagging and 21.7% in hogging. Where theframe space was increased by a factor.ofthree and 24 stiffeners introduced, thisled to a 33.3% increase in saggingultimate strength and a 51.7% increase inhogging. For this frame space but 44stiffeners, incresses in the maximumbending moments were 65.2% and 65.1% forsagging and hogging respectively.

SIMPLIFIED SHIP BENDING STRENGTH MODEL

Introduction

The conventional approach to shiplongitudinal strength calculation isbased on. linear bending theory. Sincethis makes no distinction between thetension -and compression flanges, it isvalid only when the compression flange isdesigned to resist buckling. To allowfor.such effects, attempts have been.madeto modifyelementary beam theory.

As demonstrated, the present methodis capable of acctirately predicting theultimate longitudinal strength of aship’s hull girder. However, it isimpractical to perform such an ultimatestrength analysis on a ship, particularlyat the stage of preliminary design orwhen assessing safety levels. From thesepoints of view, a simple expression for

the ultimate moment capacity is morehelpful in assessing the margin of safetybetween the load-carrying capacity of thehull girder as a beam and the maximumbending moment acting on the ship.

In this section formulae are derivedfor predicting the ultimate bendingmoment capacity of a sh$ps ‘ hulls.Firstly, however, existing ultimatestrength approaches are briefly reviewed.Simple expressions “for the ultimatemoment capacity are obtained from thedata relating to the maximum load-carrying capacity of stiffened panels andthe ultimate bending moment strength ofhull girders. A design predictionprocedure is then suggested and thestrength formulae compared with numericaland experimental results.

Existina Strength Formulae

Excludina bucklinq. In [20], it wasproposed that once yield stress isexceeded in either flange, the resultingexcessive strain will overload theadjacent structure and hence triggerultimate failure of the hull girder.Based on this limiting condition andlinear bending theory, the ultimatebending moment ~ is given by:

For the purposes of derivingexplicit expressions for the ultimstemoment capacity, it is convenient torepresent the midship cross-section by anequivalent lumped area similar to thatshown in Fig. 4 but with the stiffenersaligned with the plating. Assuming alinear stress distribution with a maximumvalue of OY in the deck, MU for this

simple box girder is [20]

Following the method of classicalplasticity, a fully plastic condition isreached when yie~d stress has developedat every point throughout the depth of agirder [2]. Using the same lumped areaapproach, the ultimate bending momentcorresponding to this limiting condition,i.e. fully plastic moment is [2]:

Mu-MP-u@.D[~y+2a~(112-y+Y2)+~(1-y)l(8)

This moment represents an upper limit ofa girder’s longitudinal strength, but israrely obtained due to the adverseeffects of buckling and initialimperfections as demonstrated above.

Includina bucklina. In discussingthe experimental results relating tofull-scale ship structural tests, Vasta[21] suggested that the limiting bendingmoment for a longitudinally framed hullcan be approximated by the product of its

III-C-II

elastic se”ctjon modulus and its criticalpanel strength. That is, the ultimatebending moment .canbe taken as:

Although the use of this simpleexpression was supported by the ISSC[22],. it has: been criticised as.,beingpessimistic in regard to predictingultimate etrength [23] by ignoringplastic hinge capacity, and as beingoptimistic in respect of girder stiffness-since it ignores the loss of stiffnesswhkch. arises due to buckling of platesand stiffened panels. Dwight [2-4j hassuggested that eqn (9) can be used as alower limit for ultimate moment capacity:this is discussed later.

For the bending stress distributioncorresponding to the limiting conditionsuggested by Vasta [21], the ultimatebending moment for a simple box girder is

MU-6#D[~y+2a~(l/3y-l+y )+(lo)

q(l/y-2+y)]

To take buckling effects intoaccount, Caldwell [2] considered thefollowing limiting .condition existed whenthe girder had reached its ultimatebending moment. In the bottom structureand the side-shells below the neutralaxis, the position of which wasdetermined- ly equating tension andcompression zone areas, yield stress intension was assumed to have fullydeveloped. On the compression side,stmctural” instability factors,~~ and $,

were introduced for the “deck and theside-shells above the neutral axis tomake allowance for buckling. By usingthe lumped area representation of themidship cross-section, the resultingultimate bending moment is:

K-u#J[@.a.Y+2%(l/2-Y+Y2(l+0,)/2)+

%(1-Y)](11)

As indicated in [25] it was implicitin Caldwell’s approach that once anelement reached its maximum load, i.tcontinued -to carry that load underincreasing strain. As has been seen,this is rarely the case in practice. Ittherefore appears to follow that eqn (11)will produce optimistic predictions ofthe ultimate bending moment.

In.considering the buckling of platepanels under compressive loads, Oakley[26] suggested a..practical approach byusing the concept of an effective widthof”~ plating associated with eachlongitudinal stiffener. Since theeffective width varies with the” appliedstress, this approach requires aniterative process to calculate “theeffective section modulus. Two or threeiterations are usually needed to reach a

convergent solution [26] which gives aneffective longi.tudin”al stressdistribution. The resulting. ultimatebending moment is given by:

Mu = 2,a, “.(12)

This .equation satisfies both theequilibrium the deformation conditionassociated with the basic assumption.thatplane sections remain :plane,but does notallow for residual strength after thebuckling of plate panels.

~or a complex section, eqn’ (12’)involves tedious iterations to determinethe effective section modulus. Analternative expression has been suggestedfor longitudinally f~amed ships in thesagging condition by Mansour and Faulkner[23] :

MU-ZUu( l+JnJ)-(ZUy) (l+kV)aU/aY

(.13)

-M@(l+kv)=M.#(l+kv)/SF

where k v is a function of the ratio of

OGe side-shell area ‘tothe deck area andwas taken as’ 0.1 for a frigate crdss-

section in [27].

It is apparent that the ult?matebending strength of a ship’s hull girderis influenced by’thepost-peak behaviourof its compressed members, in addition totheir maximum-load-carrying capacity. Toinclude the effect o~ load-s,heddingrWong[281 proposed two patterns of bending.stress distribution. These were termedType 1 and Type 2 for which ultimate :bending moments were:

Mu=o~[~$,(y.+yti)*a,$,[(y~/3)(1+2~) ““(14)

+yAy~(l+<)+2/3Y~l+2/3a$Y~+a~Ycl

and

K-a@[a,(@,/3+2/3$s~s+l )Y’+(15.)

7(&L.4b-2a,-%)+a.+~1

respectively where y~=a/D, Y~-b/D and Y==

c/D, and a, b and c relate.to extents ofthe”assumed stress distributions. Forthese solutions, “it is necessary todetermine the load-shedding factors ( ~

t CD, ~,) to f=ntiie the ‘ultimate strength

of a girder to” be “calculated. It wassuggested by Wong [28] that a value of0.2 UU load shedding, i.e. ~ =0.8, might

.....

be used on the ‘safe side in connectionwith the limit state design approach.

Th= ship; strength model chosenthe mean ultimate moment by FaulknerSadden [27],was: -

.,..

forand

.. -.

III-C-12

where a=l+~=l+ systematic errors in yield

strength (subscript Y), in usingidealised design codes to evaluate panelstrength (c)j and an allowance forredistribution following firstcompression failure (S). By using theS.Y.Stem&tiCerrors assumed in [27], eqn(16) becomes:

RU=tiy2(-ti.1+1.4465@0.3465$2)1.15

(17)=MP1.15(-0.1+1.4465@-0.3465@2)/s,

Ultimate Moment Expression

It can be seen from the existingstrength formulae, eqns (9,11,13,14,15,17) that the ultimate moment capacity ofa hull under vertical bending isconsidered to be dominated by thestrength of the compression flange.Regarding this strength, Vasta [21]suggested the use of the averagerepresentative panel strength of thecompression flange. This was supportedby Caldwell [2] in considering the effectof structural instability. In Oakley’seffective width approach [261, the Panelefficiency factor is associated withplate effectiveness but makes noallowance for the strength oflongitudinal stiffeners. Faulkner andSadden [27] used the average ultimatecompression strength of the criticalstiffened panel.

As shown above,.the ultimate bendingstrength of Iong;tudinally framed hullsclosely correlates with the maximum load-carrying capacity of their criticalstiffened panels. There”fore, it isproposed to adopt Faulkner and Saddensdefinition of panel efficiency in thederivation of a simple expression forultimate hull strength.

The ultimate strength of a stiffenedpanel is primarily a function of columnslenderness, plate slenderness, initialplate deformation, initial stiffenerdeflection, and weld-induced residualstress. However, by selecting fordesign, suitable values for the plate andstiffener initial deflections andresidual stresses, the number “ofindependent variables reduces to two.From Table 4-6 of [3], app~opriate valuesfor the initial imperfections seem to he”:

&O=b/200, A=0.0015L and a, - 0.2 a,.

Therefore, the ultimate strength can beexpressed as:

~=ou/oy.f(a,p)(18)

‘To fit a function to the numericaldata contained in Table 4-6 of [3], aleast-squares method was adopted. After

some preliminary investigations, eqn (18)was assumed to take the following form:

@=(c,+c2a2+c#2+c,~’P2+c5a4)-0”5(19)

for which the least-squares solution gavethe-following results [3]:

C1=0.960, CZ=0.765, C~=0.176, C4~0.131 andC~=l.046.

The accuracy of the predictions of eqn(19) compared with the numerical resultsin Table 4.6 of [3] has been assessed.The average ratio is 1.004 while the COVis 0.029.

Equation (19) is plotted in the formof ~ - kcurves in Fig. 8, and in the form

of ~ - ~ curves in Fig. 9.

1.0

pk0.00 —1.00 ---–

*U. J -------- 2.00 —.—

OY# 3,00 ––-

-.+O.e

14.oo —-—~., 5.00 ——

--------., ‘,

‘.0.6 —-—----

“1. .,\ .,

-.= ‘---- . .-—----.~~ ‘.. 1. ‘. EULE2 CURVE

0.4 ——--__:’-=.\

~. =..-.,._---.:’-..:< .

--------- -:=-.

0.2::.s-~

---a

0.00,0 0.2 0.4 0.6 0.0 1.0 1+2 1.4 1.6 ),B~ 2.0

Fig. 8 Stiffened Panel Ultimate Strengthversus Stiffened Panel Slenderness

0.8

0.6

0.!4

0.2

0.00 —-.+- 0.20 --–-

-.+ O.m —-——--- -. 0.60 ———

--.,-. k O.BO —--

—.- -1’.’-. 1.00 ——--- —.-

A:&:

-—.---.=+

——_

__--.:-----—---—____ ----.—_ _

——- .—- —--- ----=-’=:

——_~ —---7 —-— — — -—<,

+.1 / (o. ?.o.a.7h5h’ a.17. P’.0,131 A’B’.l .o**i”) ‘.*

0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 L.s

B5.0

Fig. “9 Stiffened Panel Ultimate Strengthversus Plate Slenderness

It can be seen in these figures thateqn (19) gives 1.021 when A and @ equal

“zero. Theoretically it should be unity,but if an allowance is made for thedifference between tensile andcompressive yield, and yield stress isdetermined from tensile tests whilecompression failure is the mode underconsideration, this would be the form ofmodification required. Thus , since eqn(19) provides a good estimate for allother L and ~ , it seems reasonable to

III-C-13

retain the equation as derived since itmakes “a small concession to the extrastrength demonstrated by compressiveyielding.

The ultimate moment capacity of‘longitudinally framed hulls under

vertical bending closely correlates withthe ultimate strength of the criticalcompression panel, as demonstrated inFig. 10 and Table 4 for girders insagging, and in.Fig. 11 and Table 4 forgirders in hogging. Since many hardcorners. are pcovided by the keel plates,deep..girders and bilge keels which existin the compressed part of a vessel inhogging, it is necessary to considersagging anti hogging conditions separatelyin deriving simple expressions for theultimate moment capacity.

qM,

0.0

0.4

0.k

1DATA WINT9 - OBTAINEONUMERICALLY

0.2 CURVE- OBTAINED BY U51NGSTRENGTH FORMULA llu/Hp=-O. 172+1 .548+ -0.368~7

0.0 I0.3 ~~ 0.4 0.5 0.6 0.7 O.e 0.9 1.0

+-0” 10,

Fig. 10 U“ltirnate Bending Moment v

Critical Stiffened Panel 5trength

for Hulls in Sagging

In the sagging. condition, all of thelongitudinally framed girders analysedabove and in [3] are included in Fig. 10and Table 4 except for HULL B (composite ---girder), Type 1 COBRA (mixed framing),Type 2 COBRA (mixed ffaming) and TYPE 14 .+._.class (relatively small deck area). Thenumerical results for the TYPE 81 andLEANDER Class hulls were derived withoutsuperstructures.

The” relationship between ly21iultimate moment and stiffenedstrength

panelwas assumed to take the

following form:

MU/k$=d1!dz$+d~$2(20)

where panel strength”is obtained Eromeqn(19). USin9 least-squares [3], the

I “0.4

t OATA POINT5 - 00TAINEO ;

I NUMERICALLYCURVE -. OBTAINED BY USlhG

0.2 STRENGTH FORMULA Mw/Hp=+0.003.1 .&59+ -0+&~I+2

0.00,3 0.4 0.5 , 0.6 0.7 O.e 0.9

+-0”10,Fig. 11 Ultimate Bending homent vCritical Stiffened Panel Strength

for Hulls in HOgging ,

Table 4Comparison of Numerical, Experimental anti.Predicted”Bending Strengths

for Girders in Sagging and HOgging

Critical Stiffened Panel Ultimate Bending Morn”. ~/Flp Ratios

Ship or 4 MU / MP eqn(*) eqn(*)

Girder Model L B eqn(19) num. exut. can(*) num. eXDt.

Saaaina ean (*) = eqn (21)Model 2 0.644 1.873 0.664 0.722 0.689 ‘0.69”4 0’.961 1.007Model 4 0.490 0.786 0.866 0.892 0.842 0.893 1.001 1.060Model 23 0.465 1.173 0.829 0.887 “0.930 0.859 0.968Model 31 0.39.6 1.673 0.777 0.800 .0.851

0.9240.809 1.011

Hull A 0.598 2.204 0.6390.951

0.680 - 0.667 0.981 -Cobra - Type 3 0.769 2.073 0.590 0.591 - 0.613 1’.037 -Whitby Class 0.466 3.666 0.505 0.514 - 0:516 1.004 -Roth@say. Class .0.466 3.666 0.505 0.524 - “ 0;516” 0.985 -Type 81 Class 0.509 2.376 0.643 0.6-19 - 0.671 1.084 -Leander Class 0.765 1.925 0.532 0.564 - 0.549 0.971 ‘-

Mean 1.000 0.986

T...--

Hoaaina m (*) = eqn (22) Cov 0.037 0.060Cobra - Typee3 0.650. 2.073. 0.637 “0.751 - 0.7,45 0.992 -Type 14-Class ,. 0.548 1.582 0.741 0.825 .- 0.831” “- 1.007 -Whitby Claas ,. 0.408 1.633 0.780 0.863 - 0.860 0.996 -Rothesay Glass 0.408 1.633 0.780. 0.870 - 0.860Type 81Class 0.385 2.232

0.988 -“0.695 0.784 - 0.788 1.005 -

Leander.Class 0.413 1.481 0.802 0.875 - 0.877 1.002 ,-Mean . 0.998Cov 0.008

III-C-14

,-

constants in eqn (20) were found as:

MU/MP=-0.172+1.548~-0.368@2

(21)

In the hogging condition, all boxgirder models, HULL A (relatively weakbottom structure) and HULL E (composite

girder) were excluded from the derivationand hence are not included in Fig. 11 orTable 4. The TYPE.81 and LEANDER Classhulls were again treated as hulls withoutsuperstructure in deriving the numericalresults. By similarly using least-squaree the following expression wasobtained. [31:

MU/~=0.003+1.459$-0.461@2(22)

Equations (21) and (22) @demonstratethat the ultimate bending moment is afunction of.the fully’plastic moment andthe ultimate strength of the criticalstiffened panel. TO predict the ultimatelongitudinal strength for a hull girder.or box girder, “the’following procedure issuggested:

1. Determine” the plastic neutral axisposition, i.e. the interface that dividesthe cross-section into two regions withequal squash loads in compression andtension.

2. Calculate the fully plastic momentof the fully effective midship section.

.:-,3. Identify the critical stiffenedpanel. InitLally, this can be selectedas the panel appearing most frequently inthe Compression flange of the girder ineither the sagging or hogging condition.More rigorously,” the strength of severalpanels can be evaluated and the weakest~n “identified.

4. Compute ‘the values of the columnslenderness kand the” plate slenderness p,.for the critical panel.

5. Calculate the ultimate strength ~ of

the critical panel by using eqn (19).

6. Calculate the ultimate bendingmoment for the girder by using eqn (21)or (22).

Comparisons with Numerical and ExistinqPredictions

The results of applying the aboveprocedure “to the vessels underconsideration ,are cornp”ared with thoseobtained numerically in Fig.10 (sagging)and Fig. 11 (hogging): Table 4 summarisesthese results: The agreement is seen tobe satisfactory, with a mean of”l.000 andCOV = 0.037 in sagging and a ?ean of0.998 and COV = 0.008 in hogging beingobtained for the ratio of the predictionsto the numerical results. Even thepredictions of the experimental resultsare good as shown ii Table 4 ,where the

mean for the girders in sagging is 0.986with a COV of 0.060.

The predictions of the existingstrength formulae for ultimate momentcapacity reviewed above are compared withthe numerical results in Table 5 for boththe sagging and hogging conditions.Since the effective width approachrequires an iterative process tocalculate the effective section modulus,it is excluded from the comparisons. Inview of the uncertainty concerning theload-shedding factor in Wong’s formulae,these are also excluded.

It was suggested in the review thatvasta’s approach would be pessimisticwhen -predicting ultimate hull strengthsince it ignored plastic hinge capacity.This seems to be confirmed in Table 5where it is seen that a mean of 0.815 anda COV = 5.7% for sagging and a mean of0.776 and a COV = 1.7% for hogging areobtained for the ratio of Vasta’sprediction to the numerical result.

The review also suggested that sincethe effect of load-shedding is notallowed for in Caldwell’s approach, eqn(11) is likely to produce optimisticpredictions of the ultimate bendingmoment. This is clearly demonstrated inTable 5.

Since Mansour and Faulkner’sexpression eqn (13) gives a mean comparedwith the numerical result of 0.897 andCOV = 0.057 (sagging) and a mean of 0.854and COV = 0.017 ‘(hogging), this approachachievee an improvement over that ofVasta but with the same COV.

Faulkner and Sadden’s expression,eqn (17), is seen to improve both themean (0.995) and the .COV (0.052) insagging and to improve the mean (0.941)with a small COV (0.020) in hogging, andclearly is the best of the existingformulae. Compared with the presentapproach, it still generates largervalues of COV (0.052 and 0.020) comparedwith 0.037 and 0.008 for hulls in saggingand hogging respectively while stillneeding more calculations including someestimates of systematic errors.

RELIABILITY MODELLING AND ANALYSIS

Background

The work of ISSC’91 Committee V.1involves the reliability analysis of afloating production vessel and theattendant wave load and structuralanalyses [29]. Both conventional linearand more innovative non-linear solutionsto the wave-induced moments weredetermined. Comparisons withmeasurements on the vessel concernedenabled the accuracy of these results tobe quantified stochastically for use inthe reliability analysis. The following

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Table 5Comparison of Present Numerical and Predicted Bending Strengths

for Girders in Sagging and Hogging

Ultimate Bending MomentMu/MP M. / % Ratios

Ship or Present Vasta CaldwellMansour Faulkner m m’.~~Girder Model Num. [2J [2] [23] [g] (1) (1)

(1) (3) (4)SaaainqModel 2 0.722 0.626 0.840 0.689 0.768 0.867 1.163 0.954 1.064Model “4 0.892 0.722 0>924 0.794 0.856 0.809 1..036 0.890 0.960Model 23 0.887 0.747 0.940 0.822 0.892 0.842 1.060 0.927 1.006Model 31 0.800 0.700 0.880 .O.77Q 0.844 0.875 1.100 0.963 1.055Hull A 0.680 0.586 0.758 0.645 0.720 0.862 1.115 0.949 1.059Cobra - Type 3 0.591 0.464 0.760 0.511 0.573 0.785 1.286 .0.865 0.970Whitby”class 0.514 0.402 0.666 0.443 0.497 0.782 1.296 0.862 0.967Rothesay Class 0.524 0.401 0.670 0.441 0.495 0:76,5 1.279 0.842 0.945Type 81-Class 0.619 0’.510 0.768 0.560 0.626Leander Class

0.824” 1.241 0.905 1.0110.564 0.418 0.727 0.459 0.516 0.741 1.289 0:814 0.915

Mean 0.815 1.186 0.8.97 0.995HoqqinqCobra - .Type 3 0.751

Cov 0.057- 0.087 0.057 0.0520.574 0.850 0.631 0.705 0.764 1.132 0.840 0.939

Type 14 Class 0.825 0.640 0.877 0.704 0.777 0.776 1.063 0.853 0.942Whitby Class 0.863 0.663 0.914 0.729 0.799 0.768 1.059 0.845 0.926Rothesay Class 0.870 0.666 0.920 0.733 0.803 0.766 1.057 0.842 0.923Twe 81 Class 0.784 0.626 0.827 0.689 0.765 0.798 1.055 0.879 0.976L~~nder Class 0.875 0.685 0.933 0.754 0.823 0.783 1.066 0.862 0.941

is a summary of this activity emphasizingthose parts relevant to ths simple hullgirder modelling discussed above.

For the reliability analysis, asecond-order procedure was used. Thiswas described in the ISSC’88 Report ofthis Committee [30] but it has since beenenhanced w“ith an 1989 release [31] inwhich it is possible to, evaluate theparametric sensitivity of the reliabilityindex to any parameter” at the “failurepoint” in the basic variable ‘space. Thisoption is used to assess the influence ofthe model uncertainty parameters snd ofother parameters contained in the failureequations. This particular software alsohas the capability to generate an improvesecond-order solution through simulationabout the failure point in theindependent standard normal space.

Mean 0.776 -1.0720.854, Q.941Cov 0.017 0.028 0.017 0,020

L, t, b geometric variables, modelledas in [30];- ~ usingnon-lineart heory (see Chapter2 of [29]), the adopted extreme valuedistribution is Gumbel, the parametersfor which ”have been derived on the basisof time simulations of “long-termequivalents” to short-term seas states;

M.w. the Rayleigh distribution closelyfits the still water extreme bendingmoment distribution as demonstrated in[32] which present data pertaining to %heanalysed vessel. The most important non-linear effect in the response is that ofthe buoyaticy force which manifests itselfas the difference between hogging andsagging.- x. this value is assessed by means of

experimental - calculation comparisons..,

Failure eauation and stochastic modellinq

For this case, the failure equationis

X. M.- Xn.MW-V,X,.MmZ O(23)

The adopted stochastic”modelling involves12 variables which are listed togetherwith their distribution functions inTable 6.

The g.tochasticmodeliing for each ofthe variables was chosen as follows:

- Oy derived from DnV material

certificates which gave good agreementwith ex~sting data;

E in. accordance with previous ISSCwork [301;

...

Table 6Variable Stochastic ’Modelling -

Var. Dist. Mean Cov Ref. :,.TYDe Value

Uy Lognormal 407.5* 0.066

E“ Normal 210,000 0.050 [30]L Normal 3700 0.04

Normal i3.5 0.04: Normal 690 0.04% Gumbel 3693 0.133 “M Rayleigh 32”6.7 0.52 [32]

i“ Normal lio- 0.15

%, Normal 0.90 0.15” [33]

x●W Normal 1.0 0.05

r Normal 79.5 0.04.*’for the plating

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Taking the inverses of the valuespresented in Table 4, the bias is 1.018and COV is 0.061. Because the< tests

./ involved a small number of laboratoryexperiments on steel box girders, thesewere felt to be not necessarily fullyrepresentative of ships girders so theCOV was increased .to 0.15, a compromisebetween the derived value and that of0.20 suggested in [34]-.- X“c this has been evaluated from an

analysis .of full scale data relating to;= Selected”vessel. The factor accounts

the uncertainties arising fromspectral representation and from theshape of the trans,fer function which isusually predicted using linear striptheory. The model uncertainty, normallydistributed -with a bias of “0.90.and COVof 0.15 is in good agreement with theformulation in [33]

Bias--0.0050e-0.42v+0.70cB+i.25

9“oses180(24)

Bias.-0.0.063e+l.22vfO.66c~+0.06

and a standard deviation of 0.12.Equations (24) have been determined by asystematic comparitionbetween theoreticaland experimental. results.- x,wdue to the large amount of available

data [32], it should:’be quite small,however, it is increased because of theRayleigh fit to the extreme valuedistribution.

r taken as geometry variable whileactually a function of”a number of suchentities.

Results and “Comments

The results of the reliabilityanalysis are presented in Table 7.

From Table 7 it is seen that thereliabilities determined by the first-(1) and second-order (11) methods. aresimilar and that simulation (III) doesnot improve on these est$mates despiteevidence [29] that the failure surface ismulti-modal. Modelling of radius ofgyration was introduced to help reducethe impact “’of the multiple failuresurface on the.solution.

In the panel strength modellingapproach [3], the stiffened panel isconsidered as adjacent half panels withalternate modes of initial deformation.Thus streng$h is controlled by thecombined” -failing bf the plate in onepanel and the stiffener in the other.The stiffened panel response is initiallygoverned by loss of stiffness in the

.,’ initially distorted plating but strengthis then usually ltmited by the onset of

Table 7Hull Girder Reliability Analysis

-,Var. Mean Failure Sens. Partial

Point Factors Factors

UY 407.5* 415.1

210000 207750k 3693 4548MSw 326.7 329L 3700 3724

13.5 13.4:. 690 695x“ 1.00 0.810

XMC 0.90 1.055

xE* 1.00 1.00

r 79.5 77.5P, 2.765

k 2.667

%4 2.608

* for the plating

0.101 1.019

-.078 0.989..567 1.232.052 1.007.058 1.006-.065 0.993.064 1.007-.686 0.810

.41”7 1.172

1.000

-.058 0.975

yield at the stiffener tip. In [3], andthus in the present study, it was assumedthat yield stress in both the plate andthe stiffener was the same. The effectsof different yield stresses have not beenexamined. Presumably however as averagevalue might “be more appropriate for thetype of analysis conducted here.

Parametric sensitivities wereevaluated for a limited range of thevariables. Those considered and theresults achieved are listed in Table 8.

Table 8Parametric Sensitivity Factors

SensitivityParameter Factors

UY 0.004

Mean XU 4.574

s.d. XU -8.676

Mean XMC -3.088

s.d. ~c -3.560

Mean x-w 0.095

sad. Xcw -0.001

0, -0.001

thatthe

From the table it is seenmodelling accuracy d~minates, i.e.accuracy of the predictions relative tomeasurements from a strength as well as aloading point of view are most critical.

CONCLUSION

‘A numerical procedure for thedetermination of ship girder longitudinalbending st~ength is d.sscribed. Itaccounts for the primary variablesaffecting strength, namely, yielding and

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..

plate and stiffened panel slendernesses,and the secondary contributors of initialdeflections and welding residualstresses. Comparisons are presented withexperimental results which demonstrateits actiuracy.

For prel~minary design andreliab~lity analysis modelling, asimplified model’ is developed based onthe observation that ship bendingstrength correlates closely with thecritical stiffened panel strength~ Theprocess for identifying the criticalpanel is described and comparisons withthe experimental and numerical “workdemonstrate its accuracy. The model isthen subjected to reliability analysis aspart of the ISSC’91 studies.

ACKNOWLEDGEMENT

The ‘ISSC work reported here wasexecuted by RINa under the direction ofDr M Dogliani.

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

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Lin, Y-T, “Ship LongitudinalStrength Modeling”, PhD Thesis,University of Glasgow, 1985.

Valsgard, S, “Ultimate Capacity ofPlates in Transverse Compression”,Det norske Veritas,’Report No 79-0104, 1979.

Dier, A F and Dowling, P J, “TheStrengtti” of Plates subjected toBiaxial Forces”, in Behaviour ofThin-Walled Structures, Ed. JRhodes and J Spencer ElsevierApplied Science, Londonr 1984.

Frieze, P A, Dowling, P J and Hobbs,R E, “ultimate Load Behaviour ofPlates in Compression”, in SteelPlated Structures, Ed. P JDowling, J E Harding and P AFrr;:, Crosby Lockwood Staples,

, 1977.Little, G H, “Stiffened Steel

Compression Panels - TheoreticalFatlure ‘Analysis’’;”The StructuralEngineer, Vol 54, No 12, Dec.19Z6 .

Smith,. C, S, “Influence of LocalCompressive Failure on UltimateLongitudinal Strength of a Ship’sHull” , International Symposium onPractical Design in Shipbuilding,PRADS, Tokyo, Oct. 1977.

Dowlinu,P J et al. “Experimental andPredicted Co_llapse-Behaviour .ofRectangular Steel BOX Girders”,Proc. International ‘Conf. SteelBox Girder Bridges, ICE, London,Feb. 1973.

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Issc “Report of Committee 11.2 onNon-Linear Structural Response”,7th International Ship”StructuresCongress, Paris, 1979.

Faulkner, J A et al, “The Loss ofHMS COBRA -. “A Reassessment”,Spring Meetings, RINA , London,1984. “’

Guile, P J, D “and Dowling, P J,“Steel Box Girders, Model 2 -Progress Report 1“, EnggStructures: Laboratory, ImperialCollege, Londbn, Nov. 1971.

Frieze, P A and Dowling, P J, “SteelBox Girdersr Model 2 - ProgressReport 2,,, Engg StructuresLaboratory, Imperial College,London, June, 1972.

Chatterjee, S and Dowling, P J,“Steel Box Girders, Model 2 -Progress Report 3“, EnggStructures Laboratory, ImperialCollege, London, May, 1972.

Guile, P J D and Dowling, P J,“Steel Box Girders, Model 4 -Progress Report ~n, EnggStructures Laboratory, ImperialCollege, London, Feb., 1972.

Moolani, F, M and Dowling, P J,“Steel Box Girders; Model 4 -Progress Report 2“, EnggStructures Laboratory, ImperialCollege, London, March, 1972.

Chstterjee, S and Dowling, P J,“Steel Box Girders, Model 4 -Progress Report 3,,, EnggStructures Laboratory,. ImperialCollege, London, June, 1972.

Smith, C S, “’StructuralRedundancyand Damage Tolerance in Relationto Ultimate Ship Hull. Strength”,Proc. Symposium on Design,Inspection and Redundancy,Williamsburg, 1983.

Evans,. J H, .Discussion of ref. 2.Vasta, J, “Lessons Learned from Full

Scale Structural- Tes”ts”, Trans.SNAME, VO1 66, 1958.

Issc, “Report. of Committee onPlastic and. Limit Analysis”,International Ship StructuresCongress, -Tokyo, lg70.

Mansour, A E and Faulkner; D, ‘~OnApplying the S.tatis~ical Approachto Extreme Sea Loads and Ship HullStrength”, Trans. RINA, Vol 115,1973. -

Dwight, J B, “Gollapse of SteelCompression, Panels” , inDevelopments in Bridge Design and.Construction, . Crosby Lockwood,-London, 1971.

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L_.

Ship Structural safety”, Trans.RINA, VOl 121, 1979.-\ 28. Wong, Y L, “Some Considerations inthe Ultimate Strength of Ships”,

L-1 MSC Thesis, University of Glasgow,1977.

29. Issc, “Report of Committee V.1 -ADDlied DeSian”. Internationalship and O?fshore StructuresCongress, Wuxi, 1991.

30. ISSC , “Report of Committee V-l -Applied Design”, InternationalShip and Offshore StructuresCongress, Copenhagen, 1988.

31. Gollwitzer, S, Guers, F andRackwitz, R, “SORM (Second-Order

Reliability Method ) Manual”,Release 1989.

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III-C-19

DISCUSSION

Y.N. Chen P.A. Frieze -,Paul,I just want@ to makea @int of clarification. The The partial factis listed in the pqwr are the paditionalpartialfactors thatyou obtained, aretheyapplicable to the ones extmctedfiorn ~Rh4 techniques. They, therefore, ‘-–-<mean value mare they some kind of &si@ value? representthe “fhilure” or “deiign” point value divided by

the man value of the vsriable concerned.

—.

111-c-20