o THE SOCIETYOF NAVALARCHITECTSAND MARINE ENGINEERS. ,sc., nC,$

OneWorldTrade Center,Suit. 1369, NewYork, N.Y. 10048#

01!

“+ PaPer,tobePresentedatExtremeLoad,Res$mseSymDosium~8 : Arl,mton,VA,October1>20,1981;$ #“o, &

*H,.S,**

Evaluation of Ultimate Ship Hull StrengthR. S. Dow, R. C. Hugill, J, D. Clark and C. S. Smith,

Admiralty Marine Technology Establishment, Glasgow, Scotland

ABSTRACT

The problem of evaluating the U1timatelongitudinal strength of a ship’s hull is dis-cussed. In addition to behaviour under quasi-static loads, the whipping response of aship’s hull to impulsive loads, eg bow-slamming and underwater explosions, is con-sidered. A method of analysis is described,based on approximate characterization of thestrength of elements of hul1 cross-sectionsunder tensile and compressive loads associatedwith hull-girder bending togetherwith locallateral-pressureeffects. The influence ofimperfections (initial deformations andresidual stresses) is accounted for. Theanalysis is illustrated with reference to atYPiCal warship hull design. Theoreticalresults are correlated with experimental dataderived from collapse tests on a number ofstiffened box-girders.

INTRODUCTION

Assessment of the ultimate longitudinalstrength of a ship’s hull under loads imposedby the sea has traditionally been made by com-paring calculated elastic stresses in the deckor bottom shell with allowable stresses,usually corresponding to prescribed fracticmsof the material yield strength. This dppt.~~~h,applied,in conjunction with a nominal ~stiwteof vertical wave bending moment based onstatic-balanceanalysis, has a certain empiri-cal validity when applied to conventional shipsclosely resembling previous successful hullsdesigned in the same way, but fails to providea true estimate of hull strength and is clearlyunsatisfactorywhen applied to unconventionaldesi ns.

?If the hull structure does not fail

Iota lY, the actual collapse bending rmment mayexceed the wment which nominally causes outer-fibre yield; on the other hand the collapsemoment may be substantially less than thenominal yield moment if local compressive fail-ure of plating or stiffened panels occurs inparts of the cross-section.

A recent trend in the design of ships,offshore platforms and civil engineeringstructures is towards use of limit-state designcriteria, requiring explicit consideration ofultimate strength in relation to statisticallydefined loads. Some effort has consequentlybeen devoted to evaluation of ultimate, elasto-

plastic strength of ships’ hulls (1 to 6)* anda method of estimating hull strength has beenproposed (4) which takes account of local buck-ling failure and po@.-COllapseloss~f10@carrying capacity in elements of the hull cross-section. The purpose of the present paper isto examine the accuracy of this approach bycorrelation of analysis with collapse tests onvarious longitudinally stiffened steel box-girders and to consider extension of theanalysis method to deal with the case ofdynamic loading, in particular whipping of thehull girder caused by bow-slamming or under-water explosions which, as shown by recenttheoretical studies and seakeeping trials, maycause severe midship bending moments.

STATIC ANALYSIS OF HULL STRENGTH

Strength and Stiffness of Plate Elements

Between 6L12iand 80% of a typical hullcross-section is for-redby deck, shell andlongitudinal bulkhead plating: hull strength istherefore likely to be critically influenced bythe stiffness and strength of plate elemnts,particularly under compressive load. Loss ofstiffness in the plating, which may be caused bybuckling or premature yielding, leads directlyto loss of effective moment of inertia andsection modulus in a hull cross-section andalso accelerates yield in the stiffeners andhence elasto-plastic buckling of stiffenedpanels.

In longitudinally framed deck and shellstructure containing closely spaced stringerswith relatively widely spaced transverse fram?s,plates of aspect ratio a/b (length/breadth),1.5 will buckle into one or more half-waves oflength x somewhat less than the plate breadth(0.7 < A/b < 1.0). Various approximate methodsof characterizing the strength and stiffness ofsuch plates, based on large-deflection elasticanalysis or on interpretationof test data, havebeen reviewed by Faulkner (7). A much improvedunderstanding of plate buckling has been ob-tained in recent years from the application ofnonlinear finite element and finite differenceanalysis representing large deflection elasto-plastic behaviour (S to 13): parametric appli-cation of such analysis, accounting for imper-fections (distortions and residual stresses) as

* Denotes references at end of paper,

133

~

P’1.,O

% 0“’ .0

$00, 80

,.,0..

0..

0>

,..

. ..

:7-+.,O4.so

,6 80,0,09.

..

0,

Fig. 1 Load-St?orteni ng Curves far Square PI tes

[1under Uniaxial Compression ~= $~

measured in surveys of ships and welded steelbox-girder bridges (13, 14) has led to datacurves of the type shown in Figure 1 (13)which have the msrit of defining not onlymaximum load-carrying capacity but also platestiffness at any point in the strain range.The curves shown in Figure 1 have been derived

from analysis of simply su ported square orl’”nearly square plates with ongltudinal edges

constrained to remain straight but free to nwvebodily in the plane of the plating. Over thestrain range preceding plate fai1ure thesecurves may be applied without modification tolong rectangular plates on the assumption thatbuckling occurs in approximately square half-waves and that compressive strain occursuniformly over the ful1 length of each plate:in the post-collapse range, where end-shorteningstrain may be confined to a single, approxim-ately square CO1lapse zone with virtually zeroincrease of strain over the ramainder of theplate, a simple correction dependent on theaspect ratio a/b may be applied to the strainscale.

In the case of transversely framed deckand shell structure, as commonly employed inthe fore and after regions of a ship’s hull,failure of wide plate panels (a/b .< 1) underlongitudinal compression is 1ikely to occur bybuckling into single lobes over the full widthof each plata. Recent numerical studies (15,16, 17) have provided some information aboutthe stiffness, strength and imperfection sensi-tivity of such plating. It has been shown thatloss of stiffness and strength depends criti-callY on the form of initial deformation, asshown in Figure 2. Some strength curves forvery wide plates (b/a + -) having initialdeformations of symmetric and antisynnetricform (17), together with curves derived fromReferences 15 and 16 for wide plates of finiteaspect ratio (1 6 b/a & 5) having variouslevels of antisjnanetricdistortion, are shownin Figura 3.

.mwwa%rm brsfl ,.,1’.1.3S,awmT I

015

0-F3

01 -_. —._._- &=05&,-&=o

005-

Fig. 2

134

1 2 3 E/e ~ ~

Load-Shortening Curves for TransverselyCompressed Long Plates (a/b . -) withSynmetric, Antisymetric and Inter-mediate Forms of Distortion

IL

Fig. 3

“’k0.2 0.4~,t

*I

1----‘.

--:----.3 -

---ox --

-----_5 _

. . .---

---.--m-

I------------- ~

O.e

1----.---------5-

0.!.

‘---------- -m

(!,

Strength Curves for Wide P]ates UnderCompression in Shorter Direction

Strength and Stiffness of Stiffened Panels

It has been shown (18) that the most1ikely form of failure qn flat panels oforthogonallY stiffened plating forming bottom

shel1 and deck structures in ships is column-1ike interframe f1exural buckling of 1ongi-tudinal stiffeners with attached plating underlongitudinal compression induced by hull bend-ing: this form of failure, which may be strong-ly influenced by loss of compressive stiffnessof the plating, will in most practical casesprecede other collapse modes including lateral-torsional buckling of girders and overall buck-ling involving bending of transverse frames.

I !-/2 1 L/2

a) Subdivision of stiffened panelinto elements showing assumedinitial deformation

pm tm.ltd.35siqlelibrewithMfmssWiwd by*e d

v

agpwde s!m-sttinmrw(smfig1)

1 H ,

b) Subdivision of cross section intofibres

Fig. 4 Idealisation of Stiffened Panels

For the purpose of examining the CO1umn-1ike interframe flexural buckling of closelyspaced longitudinal stiffeners with attachedP1sting under 1ongitudinal compression,analysis may be confined to a single stiffenerwith attached strip of plating treated as abeam column with conditions of simple orelastic support at positions of transverseframes as shown in Figure 4a. In order toexamine the large deflection, inelastic buck-ling behaviour of such structures, a computerprogram has been developed (18, 19) for thenonlinear analysis of plane frames of generalgeometry, including straight or initially de-formed beam CO1umns as a special case. Themain features of the analysis method are asfollows:

(i) Frames of arbitrary polygonal orcurved geometry are represented asassemblies of straight beam elements,Polygonal representation of curved

135

(ii)

(iii)

(iv)

frames wil1 give satisfactory resultsprovided that a sufficient number ofstraight 1ine elements are includedin the idealisation. In accordancewith normal beam theory the elementemployed has a cubic representationof flexural deformation (w) and a1inear representation of extensionaldeformation (u). F“]1 account istaken of the coupled flexural andextensional deformations of a frame.Plane sections are assumed to remainplane under conditions of elastic andinelastic bending and extension.

Provision is made for finite separa-tion between node points at the ~nd~of elements, account thus being takenof changes in the position of theelement elastic neutral axes (causedby progressive yielding or localbuckling of cross sections) and con-sequent eccentricity of element axialforces.

Initial deformations may have anyform, being defined for analysispurposes as part of the initialstructural geometry.

Sectional properties are assumed tobe constant in each element but mayvary from one element to another.Cross sections are divided intoelemental areas corresponding toframe “fibres” (Figure 4b); residualstresses, which may have any distri-bution over a f‘camesection, arerepresented as initialstressesatfibre ce”troidal pOSitjO”~.

The method of analysis employed isthe 1inearised incremental finiteelement process, in which loads (ordisplacements) are appljed jn’re-mentally, a linear solution beingobtained for each incremental loadaPPlication by the usual matrixdisplacement method, ie by solvingthe incremental matrix equation:

[q {M) = (AP} (1)

where KT -

$ -

‘G -

is the tangent stiff-ness matrix(KT = KE + KG)

is the conventionalelastic stiffnessmatrix representingthe axial and flexuralstiffness of framee]ements

is the geometric stiff-ness matrix represent-ing the destabilizinginfluence of axialforces in frame ele-ments

(v)

(vi)

136

M - is the column vector ofincremental nodal dis-placements

AP - is the COIUm” vectOr Ofincremental nodal loads

The above linearised incremental pro-cess is coupled with a generalisedNewton Raphson iterative equilibriumcorrection scheme based on the matrixequation.:

(6}j+, = (61j +

[K~j’[{pext}- {pint}](2)

The expression in parenthesis repre-sents the unbalanced forces arisingfrom the application of the 1inear-ised incremental procedure. Thegradient for the iterations is up-dated for every iteration cycle.The v@ctor {Pext} is the total

aPPlied 10ad vector and (Pint} is thetotal internal load vector calculatedfrom integration of the total fibrestresses. The equilibrium iterationsare terminated when the unbalancedforces have converged to a specifiedpercentage of the total appliedloads.

Following each incremental solution,cumulative values of nodal displace-ments, together with stresses,strains and destabilising forces inthe frame elements, are updated.The state of stress of each fibre ofeach element is examined and wherethe total average stress (includinginitial residual stress) in thefibre exceeds the yield stress, thefibre is taken either

(a)

(b)

to contribute no stiffness i“the next incremental step, anelastic-perfectly plasticmaterial stress-straincw.vebeing assumed, or

to have a reduced modulus de-rived from a numerical v definedstress-strain curve of ;ny pres-cribed shape.

Option (a) has been adopted for steelstructures defined in the presentpaper. The element elastic sectionproperties, element stiffness co-efficients and hence the stiffnessmatrix for the complete frame aremodified accordingly before enteringthe next incremental solution.

klIowance is made for elastic un-loading of yielded fjbt.es:ie whereelastic unloading is found to have

—

(vii)

(viii)

(ix)

occurred in a yielded fibre followingan incremental cycle, the fibre isassumed to recover its elasticsectionproperties for the next incrementalstep.

Where part of a frame cross-sectionis formed by a strip of slenderplating it may be necessary to allowfor loss of stiffness caused by localplate buckling: provision has beenmade in the analysis for treatment ofthe plate as a single fibre whoseincremental stiffness is derived as afunction of average compressivestrain from the slopes of the load-shortening curves shown in Figure1.

The effect of large deflections isallowed for by updating the elementco-ordinate system and structuralgeometry,followingeach incrementalCYC1e (UpdatedLagrangie.nFormula-tion). This approachallowsthe postCO11apse behaviour of certainstruct-ures to be examined, including par-ticularly that of beam CO1umns underprescribed end shortening.

~o?d application is normally divided.lnro seversI ranges, large Incrementsbeing taken in the first range wherethe behaviour ia elastic reducingprogressively to small increments asthe collapse load is approached. Arestart facility has been built intothe program: this enables the user tochange the solution strategy andincrement size as the calculationprogresses, thus making the solutionprocess mare efficient especially forlarge problems where 1ittle informa-tion about the nonlinear behaviour isknown in advance.

Hull-Girder Strength

Once the Ioad-shortening curves forelements of the hul1 cross-section have beenobtained, using the method described above:these data can be used to compute the vertlcalmoment-curvature relationship, up to and beyondthe collapse moment, for the hul1 section.

This stage of the analysis is carried outusing a simple procedure (4) In which:

(i) vertical curvature of the hul1 isassumed to occur incremental1y:correspond ng incremental elementstrains are calculated on the assump-tion that plane sections remain planeand that bending occurs about theinstantaneous elastic neutral axia ofthe cross-section; combined verticaland horizontal bending of a hull canbe acconmadated provided that theirrelative magnitude and phasing areknown and can be represented byincrementing vertical and horizontalcurvature;

(ii)

(iii)

element incremental stresses arederived from the incremental strainsusing the slopes of the stress-straincurves (effective tangent moduli)derived from interframe bucklinganalysis in the case of longitudin-allY framed structure (in the case oftransversely framed structure use maybe made of the load-shortening curvesfor wide plates as illustrated inFigure 2);

element stress increments are inte-grated over the cross-section to6btain bending moment increments,these together with incrementalcurvatures being summed to providecumulative ‘#alues of bending momentM and curvature o;

Shear forces and shear lag effects inthe midships region of a ship’s hullare usuallY negligible: the effectsof shear and transverse direct stresscan however be accounted for, if sig-nificant, by modifying the elamentload-shortening curves in accordancewith data for P1ates under combinedload (15, 20).

HULL-STRENGTH UNDER OYNANIC LOAO

Whipping of a ship’s hull due to slatmning(Figure 5) or explosive loading (Figure 6) cancause large additional hull girder bendingmoments, but, because of their transient nature,these loadings may be 1ess effective in pro-ducing plastic CO1lapse than the more slowlyvarying bending moments due to buoyancy effects.The ultimate hull girder bending analysismethod described above is being extended to‘!ncludedynamic loading in order to investigatehow significant whipping stresse$ are in pro-ducing hull failure. This project is still inthe development phase but some preliminaryresults which may be of general interest arereported here.

The elastic response of ships to bowslansning has been studied by many investi-gators. Extension of strip theory methods toinclude slamming loads have been reported byKawakami et al (21), Vamamoto et al (22),Chuang et al (23), Meyerhoff and Schlachter(24) and recently Bishop et al (25) have re-ported a method of predicting slamming responsebased on superposition of dry hull modes. Amethod of analysing the elastic whipping ofships produced by explosive 1oads has beendeveloped by Hicks (26).

Numerous measurements of slamming stressesin ships at sea have been reported (27 to 31).Figure 7 shows examples of longitudinal deckstresses measured in a frigate during a severeweather sea trial (31). In this case, as inmast of the measurements made over the midshipregion, the whipping was predominantly of thetwo node mode which has a frequency of 1.7 Hzin this ship (LBP . 110 m). The whippingstress transient is superimposed on that due towave bending which varies at the wave encounterfrequency (=zo.2 Hz in this case).

137

I‘i

SHIP‘X LONGITUDINALDECKSTRESS22K..,.HeadS..’

Fig. 5 Slamning of a Warship In Severe Seas

Fig. 6 ~~~~ping of a Warship under Explosive

Thereare two main reasonswhy higherfrequencytransientshave 1ess effectin pro-ducing plastic CO1lapse. Firstly there is adynamic enhancement of the yield stress athigh rates of loading, and secondly the inertiaof the structure limits the deformation whichcan occur within the time of the transient.The first of these is not very significant atthe frequencies associated with whipping sincethe strain rates are too low. A transient toa maximum strain of say 0.2% at 1.7 Hz would

maximum strain rate of about 0.02 sec .!~~~dium and high strength steels, Harding-’(41) has shown that the increase in yield

stress is negligible until rates of 10 sec-1

and above are reached. Mi1d steel is marerate sensitive. Campbel1 (42) shows that theincrease in the yield strength is proportionalto the log of the strain rate between 0.1 and

10’se,-’ (Increasing from 264 to 467 MPa overthis range) b!: the increase in yield strength

below 0.1 sec is small. Since the two nodefrequency (and hence the strain rate) isapproximately inversely proportional to the

138

D;,,,”..f,c.m FP,9.LPP)

!8--—

Fig. 7 Measured Slam Induced WhippingStresses

square root of ship length, dynamic enhance-ment of yield is only likely to be significanttn the higher modes of vibration of shortships.

The main factor to consider in assessingwhipping response is therefore the inertiaeffect. This can be represented by consider-ing the ship as a series of lumped masses con-nected by elements whose nonlinear bendingStiffness is given by the moment/curvaturerelationships derived frmn the analysis des-cribed above. It can be assumed that themoment/curvature relationship is not affectedby inertia effacts et whipping frequenciesbecause these frequencies are much lower thanthe natural vibration frequencies of thestiffened panels included in the analysis.

The hull loading occurring duringslamning is mainly over the forward region andcan be divided into two types, “bottom impact”which produces high 1oads but for very shortdurations (typically.+ 0.1 sees) and “momentumtransfer” or “bow flare slamming” which pro-duces lower forces but of much longer duration.Theoretical analysis by Bishop et al (25) sug-gests that momentum transfer is more imvortsntfn warships because the resulting bendingmments ere higher.

—-

800 rBending

Moment ’00

(MNm)400 b-Sagging

200

-6 ‘4 ‘2 2 4 6X10-4Curvature (m-r)

Hogging

Fig. 8 Destroyer Midships Bending Moment/Curvature Relationship used in DynamicAnalysis

In a preliminary investigation of theeffect of whipping induced plastic CO1lapse,the response of a destroyer to simDlifiedrepresentationsof both types of siamuninghasbeen analysed. The ship was represented by21 lumped masses connected by beam elements.The masses included the added water massassociated with the two node whipping mode,although values derived for higher modes werenot greatly different. Four elements near thecentre, where plastic collapse was expected tooccur, were assumed ta follow the nmment/curvature relationship shown in Figure 8 whichwas calculated for the midship section usingthe method described above. All other beamswere assumed to behave elastically.

Figure 9a shows a comparison between thebending moment at midships derived in this wayand that derived assuming purely elasticbehaviour for a short (0.05 see) impulseapplied near the bow. This was intended to bea simplified representation of “bottom impact”slamming without the complication of theadditional mere slowly varying loads. In bothanalyses high frequency modes have beensuppressed by numerical damping.

In the absence of a dynamic plasticitycalculation failure would have to be assumedto occur at an impulse which produced anelastically calculated moment equal to themaximum value of the moment curvature relation-ship (640 t44m). The impulse applied for thecase shown tn Figure 9a was in fact 70% higher

Iwo

800

800

400

200

0

a) Response to 0.05 sec impulse repre-senting bottom impact slamming

I800 h“dinil

MWMIM ,’”’..,stlclc m?+,;.

400

200

0 0.5

-200

’400

b) Response to 1 sec impulse represent-ing bow flare slamning

Fig. 9 Calculated Bending Moment at MidshipsInduced by Impulsive Loading

than this value. A similar calculation for animpulse 95% higher tuacamenumerically unstableand it must be assumed that the structure wouldhave collapsed under this load. It thereforeaPP@ars,that for short impulses approximatingbottom Impact slansning,the hull can withstandloads 70-95% higher than the value which pro-duces elastically calculated rrxnentsequal tothe static collapse value.

As a simplified representation of‘rmmentumtransfer’ slansning,calculationshave also been carried out for a 1 sec longsinusoidal (half wave) impulse applied to fivenodes adjacent to the bow. The resultantbending moment near midships is shown inFigure 9b. The case shown is for an impulse20% higher than that which gives an elastically

139

calculated bending nwment equal to the staticcollapse value. A value 30% higher producedCO1lapse. As expected therefore, the dynamicenhancement of failure load is greater forimpulses short compared with the naturalvibration period of the hull.

These simple representations of the slamloading give an indication of the magnitude ofthe enhancement of failure load when plasticityeffects aw taken intO account. Further workis now being carried out to analyse morerealistic cases where there are superim~sedloads due to still water and wave inducedbending, and the slam impulse is more accurate-ly represented. Calculations will be based onoutput from the Bishop & Price analysis method(25) using bottom impact forces calculated frommethods derived by Ochi and Mtter (32) andStovovy and Chuang (33).

ULTIt4ATEHULL STRENGTH - CORRELATION OF THEORYAND EXPERIt4ENT

The experimental data available on theultimate strength of a ship’s hul1 girder whensubjected to vertical bending are 1imited tothree important cases. These were the testson the American destroyers PRESTON and BRUCE(34, 35, 37) in 1930, and those on the Britishdestroyer ALBUERA (36, 37) in 1950, all ofwhich were riveted ships. In the light of thisfact the emphasis of the comparisons betweentheory and experimental data has been placed onwelded steel box girder models tested atImperial College, London and the TechnicalUniversity, Berlin. Thesa comparisons havebeen carried out with two main objectives inview:

(a) to validate the theoretical approachfor evaluating the ultimate strengthof a ship’s hul1 girder subjected tovertical bending;

(b) to investigate the effects of the“hard spots” in the cross-sectionwhere buckling is resisted locallyby transverse bending stiffness,for example at deck edges and inter-sections of decks with longitudinalbulkheads or superstructure.

Comparisons of theory and experiment havebeen carried out for the following cases:

(i)

(ii)

(iii)

two steel box girders, tested byCowling et al (38, 39), under con-ditions of pure bending;

a steel box girder, tested byReckling (40), under conditions ofpure bending;

comparison with U1timate longitud-inal strength tests carried out onthe destroyer AL8UEPA.

(i) hdel Tests by Oowlin~

Dowling (38, 39) has tested a number ofsteel box girder models subjected to differentload conditions. Two of these models, both of

140

which were subjected to pure bending, have beenchosen to compare with the theoretical analvsistechnique.

Experimentally the models chosen bothfailed by compression flange buckling.

The principal dimensions and properties ofthe mdels are shown in Table 1(a). The ct.oss-sections were subdivided into stiffened panelelements as shown in the figures in Table I(a).The effective stress-strain curves for theseelements were established using the approachoutlined previously, residual stress andinitial imperfection data used in the analysisare shown in Table II.

I05

0-23

o&-

03-

02

01.

r’-)

a.

10.~

‘Yoa

06.

04.

m 10 20 +63

Compressive stress/strain curves forstiffened panels

Ms&%Eia

h(D]

1

(A)02 B)

(cl

L00 01 02

“3 @tc. ‘“ x ‘0-4

b)

Fig, 10

Mid-span mement/curvaturerelationship

Oowling’s Box Girder Model No. 2

—-

TABLE 1. PRINCIPAL DIMENSIONS OF MOOELS ANO PROPERTIES OF NATERIALS

(a) hwling’sflodels

Mcdel Cross section of ~OdelNo. Cmponent sizesandwterial properties

dimensions,~Ccmpmnt NomM~ size, f d

‘mm N h% N k%

t12192

,,, ,, CF 476

211=1

488 2980 mm1778’ .s ---

794.-TF 4.76 488 2980 2C85CCI

2 w 318 338 2116 262034572

,,, LS W8X15%48 L - 2765 1915CK,,

L=7874 !4=5 TS X2XS08X66L - 3104 1=

12f92 CF 4.76 503 2209 Zu6m,., .,,

/m:

.,, , ,,,.TF 476 495 2162 K!fiW3

4572

4 w 476 4.98 2811 Zf6200

457.2 LSwH4w) 508x159x48L - 2873 1S9.2U0,,, ,,,!,. ,, LS(TF) 508X64FLAT - W3 1992J0L=7874 N= 5 TS m 6x63%64 L - 30’+3 2U6%0

(b) Reckling’s Models

Model Cross section of WelM.

Ca’npcnenfsizes and materialpropertiesdimension, rntm

COmpcfvmt F&xnh21size,mm cm” N rni+N/~k

600

n

,,, CF 25 246 2m,,, ,,, ,

TF 25 246 21KKYJ23 400

w 25 246 zm

,,, ,., . L~F) 30x20x2.5 L - 246 210XIO,,, ,

L= 500 L51W) 30.25 FLAT - 246 2Im

TF tensionfia~ ,LS Iongitwhl StiNener,LS(CF)Iongitudiwdstiffeneron ccmprmsion flange.CF compresswn Narge, TS tmnsvem stiffener,LS(TF) Iorgitudtistiffww on tensionflange.W web ,t* mm.suml thick-, LS(W) Icrrjtudtistiffen+yon web,

N numb ofbays along s~n of mcdel

L lengthbetween bays

L.141

TABLE II.

I 1 I

INITIAL IMPERFECTIONS OF TEST MODELS

I I I

Madel Maximum stiffener Maximum plate AvwagF Icmgiludnd Avemge tmmwerseNo. initialdefksction ~ .cmel initial residml strain t residualstrain

deflection (~ strain) (u stmin)

.L/22802

- L/ 1450b/ 330 -250

+L/690L

- U510-600

maximum initia[deformation23 of deck = L/400 o

* .irdicotesdeflection towards stiffener,- towards plate

T residual strain in plate panels (negativesgn indicates com~ession)

tidel No. 2. This model had only fourstiffeners on he compression flange; becauseof this comparatively wide stiffener spacing(50 t) it was assumed that the corner “hardspot” was renwte enough from the stiffenersfor there to be no stiffening effect on thestiffened panel behaviour. The “hard spot”was therefore assumed to be localised to thecorner area only. The compressive effectivestress-strain curves for the stiffened vanelelements of the cross-section are shown’inFigure 10(a), where:

(1) is the effective stress-strain curvefor the stiffened plate elements ofthe compression and tension flangesin compression;

(2) is the effective stress-strain curvefor the stiffened plate elemsnts ofthe webs in compression.

The “hard spats” were assumed to have anelastic-perfectly plastic stress-strain curvecorresponding to that of the material in bothtension and compression as were the stiffenedpanel elements in tension. From these stress-strain curves the sagging moment/curvaturerelationshipswere computed for the box girder,these ara shown in Figure 10(b), where:

(A)

(B)

(c)

represents the nkmsnt/curvaturerelationship for the box girder withthe “hard spots” as described aboveand shown in Table I(a);

same as curve (A) but the areas ofthe “hard spots” have been reducedby fifty per cent;

same as curve (B) but the “hardspots” on the cross-section havebeen removed completely;

(0) is the experimental result obtainedby Lbwling et al.

The bending moment/curvature relationshipsshown in these and subsequent Figures have beennon-dimsnsionalised by dividing the mnsnt bythe fully plastic value (Mp) and multiplyingthe curvature by the thickness of the com-pression flange (tCF).

It can be seen from Figure 10(b) thatthese curves bound the experimental values andgive a very good prediction of the collapsebending mment for this particular box girder.

These results suggest that for a stfffenerspacing of approxim+.tely 50 t the “hard spat”has only a localised effect at the corner anddoes not spread into the stiffened compressionf1ange beyond the first stringer.

Nedel No. 4. This model was of the samedimensions as ffadel No. 2 but had many morsstiffeners on both the compression and tensionflanges than did Model No. 2. It was assumed,because the stiffeners were closely spaced andthe end stiffener was very close to the cornerof the box girder, that one stiffener WOU1d beincluded in the local “hard spot” at thecorner of the box girder, A second analysiswhich included two stiffeners in the “hard spot”was also cart-fedout for comparison. The com-pressive effective stress-strain curves for thestiffened panel elements of the cross-sectionare shown in Figure n(a), where:

Curve (1) represent$ the behaviour of boththe compression flange and webelements in compression;

Curve (2) represents the behaviour of thetension flange in compression.

142

Compressive stress/strain curves forstiffened panels

r00 01 02 03 04 .[0-+

@t.,

b) Mid-span nnment/curvature

Fig, 11 Oowling’s Box Girder Model No. 4

The “hard spots” are again assumed tohave an elastic-perfectly plastic stress-straincurve in both tension and compression as do thestiffened panel elements which are in tension.

The sagging moment/curvature relation-ships for the box girder were computed usingthese stress-strain curves and are shown inFigure n(b), where:

(A)

(B)

(c)

nnmentlcurvature relationship forbox girder where one stiffener isincluded in local “hard spot” atcorner of box girder:

e?m?nt/curvature relationship forbox girder where a second stiffeneris included in the “hard 6pot”;

is the experimental moment/curvaturerelationship for the box girderobtained by Mwling et al.

It can be seen that the first assumptionabout the “hard corner” effects, Figure 11(b)Curve (A), yields a collapse load 6% lowerthan the experimental CO1lapse bending momentfor the box girder; the second analysis inwhich an additional stiffener was included inthe corner “hard spot” gives a collapse load4% higher than the experimental value.

Ignoring other possible inaccuracies inthe theoretical mdel, this result suggeststhat the stiffening effect of the corner on thecompression flange extands between one and twoframe spacings into the compression flange forthis particular model (stiffener spacing ’25t).

~ii ) Ik3delTests by Recklins

Reckling (40) reported an a series ofseven steel box girder tests, al1 loaded inpure bending, carried out at the TechnicalUniversity, Berlin. Wodel No. 23, which hadthe largest number of stringers and was in thisrespect most similar to a ship hull, was chosento compare with the theoretical analysis tech-nique.

Experimentally, Model No. 23 showed adelayed collapse mode where the deck, at first,did not buckle owing to the strengtheningeffect of the side walls (referred to byReckling as the “box girder effect”) althoughits plastic buckling strength had been ex-ceeded. Finally, collapse occurred by nearlysimultaneous buckling between the stiffenersand of the whole compression flange.

Model No. 23, having principal dimensionsand properties shown in Table I(b), was sub-divided into stiffenad panel elements as shownin the figure of Table I(b). Again, because ofthe reasonably close stiffener spacing (34 t),the corner “hard spot” was chosen to includeone stiffener in its area. The effectivestress-strain curves for these elements, beingcomputed using the previously outlined approachwith the residual stress and initial imperfec-tion data shown i“ Table II are $hOwn inFigure 12(a), where:

(1)

(2)

compressive stress-strain curve forthe tension and ,compressionflangeelements;

compressive stress-strain curve forthe web elements.

143

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06.

04.

0.0 102“0 6/6 ~

a)

10.~

mPo8.

06.

04.

02.

Compressive stress/strain curvesfor stiffened panels

b)

Fig. 12

Mid-span mnment/curvaturerelationship

Reckling’s Box Girder Model No. 23

The “hard spots” were again assumed tohave an elastic-perfectly plastic stress-straincurve in both tension and compression as werethe stiffened panel elements in the tensionflange.

Using these effective stress-strain curvesthe sagging bending Mment/curvature relation-ships were computed and are shown in Figure12(b), where:

(A) corresponds to the box girder withone stiffener included in the “hardspot” at each corner;

(B) represents the case where anadditional stiffener is includedin the “hard spot”;

(C) corresponds to the experimentalresult obtained by Reckling.

The original assumptions about the “hardcorner” effects again produce an estimate ofcollapse moment 5% lower than the experimentalvalue. Including another stiffener in.the areaof the hard corner produces alnmst exactly theexperimental collapse moment.

It can also be seen from the moment/curvature graph that the initial stiffness ofthe box girder observed experimentally is lessthan that predicted theoretically; possiblereasons for this are:

(1) inadequate allowance for initialdeformations (because of the lack ofdetailed information about the plateand stiffener imperfections on thermdel);

(2) the influence of residual stresses,reported to be zero in the nndel andignored in the analysis.

(iii) ALBUERA

Of the three ultimate nment tests carriedout on ships, all of which were riveted, it wasdecided that an attempt would be made to tryand predict the collapse moment for thedestroyer ALBUERA. This ship was chosen forthe following reasons:

(1)

(2)

The tests on the ALBUERA wereoriginally carried out at NCREDunfennline and hence the structuraldetai1s and experimental data werereadily available;

The ALBUERA was longitudinallystiffened and was therefore morelike a modern warship, where theAmerican destroyers PRESTON andBRUCE had a transverse framinasystem.

The midship cross-section of ALBUERA, givingstructural details,is shown in Figure 13. Theelemsnt subdivision is shown in Figure 14.The effective stress-strain curves for thestiffened panel elements were computed in theusual manner and are shown in Figure 15(a);the numbering of the elements in Figure 14 isconsistent with the effective stress-straincurves of Figure 15(a). Unnumbered elementscorrespond to “hard spots” on the cross-sectionand are assumed to have an elastic-perfectlyplastic stress-strain curve in tension andcompression. Since the test was carried outwith the ship in hogging no compressive stress-strain curve was computed for the deck elements:these were assumed to follow an elastic-perfectly plastic stress-strain curve undertensile load. No information was available ondistortion and residual ‘stresslevels i“ the

144 lJ-?.-

-

ALBuERA structure: plate imperfectionsweretherefore assumed rather arbitrarily to be“nmderate” (as represented in Figure 1(b)) andstringer deformations were assumed to have theform of uniform curvature of amplitude 0.001 L“’betweentransverse frames.

172’D’635 3/

172‘D’ 14’D’ mX ~

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1127x635x635.79

“F%#e10I6xEc.9x509xw,

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106x5J8@x64

~127x635@W9

Fig. 13 ALBUER4 Midship Section StructuralDetai1s

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ii.-

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Fig. 14 ALBUERA Midship Section ShowingElemant Subdivision

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‘-6 nunberinginfigureM‘Y @g.

06

~E

(3)(6)

‘\(1)

04

‘\ “Q

02Q 4)

(2)51

a) Compressive stress/strain curve forstiffened panels

v0604 “’””G‘w~[klingof

02 Shpsti&

00 05 -4“0 q$t

15 X 10

b) Midship moment/curvaturerelationship

Fig. 15 ALBUER4

In two cases (Curves 4 and 5) unstablepost-collapse behaviour was computed; thisbehaviour will be influenced by the elasticstiffness of the surrounding structure, lead-ing to unloading curves as illustrated by thedotted 1ine in Figure 15(e) where the unloadingfrom points P to Q will occur dynamically.Hence the dotted 1ine path is used to representpost-collapse behaviour instead of the ful11ine curve (which corresponds to the computed,notional, static unloading path). L145

Using these effective stress-strain curvesthe hogging mnent/curvature relationship forthe midship cross-section was computed and isshown in Figure 15(b), where:

(A) represents the hogging bendingmoment/curvature relationship forthe midship cross-section;

(B) indicates the experimental‘hoggingbending moment/curvature relation-shipp.

The theoretically predicted ultimatemoment value is 26% in excess of the experi-mental CO11apse bending moment. This differ-ence can be attributed to several factors:

(1)

(2)

(3)

No account has been taken in theanalysis of rivet S1ip or the effectsof eccentricity of loading due to theoverlapping of the P1ate panels.Neglecting rivet S1ip in the analysiswill have a S1ight stiffening effecton the elastic part of the theoreti-cal mom?nt/curvature relationship.

The ALBUER4 was supported on pairsof blocks similar to those wed inthe tests on the American destroyerBRUCE. At failure the section whichbuckled was not the midship section,but franw 72~ which was in fact out-side the test section near one ofthe supports.

The supporting arrangement caused.... . . .. .very nlgn snear rorces, In con-junction with the high bendingmoments, at the section wherefailure finally occurred. In fact,at a reasonably low load 1evelshear buckling of the ship’s sideoccurred thus reducing the stiffnessof the section and P1ayed an import-ant part in reducing the CO1lapsemoment. It is very unlikely thatsea loads could develop a verticalshear force and longitudinal bendingmoment to bring about simultaneouscompression and shear buckling as inALBUERA. No attempt was made toinclude the effect of these highshear forces in the analysis due tothe relatively large amount of workinvolved to analyse an unrealistic1oad case. It should be noted atthis point that the analysis tech-nique is capable of representing thecombination of high shear forces andhigh bending naments if necessary.

Taking these factors into account thetheoretical result is judged to be consistentwith the experimental value although the com-parison is inevitably rather inconclusive.

CONCLUSIONS

The problem of evaluating U1timate bend-ing strength of a ship’s hull-girder has beendiscussed and an approximate nethod of estim-ating static hul1 strength, with allowance for

local buckling failure of plating and stiffen-ers has been outlined. Recent numericalstudies of the buckling and post-bucklingbehaviour of plating and stiffened panels haveprovided data which allow accurate estimates tobe made of the stiffness and post-collapseload-carrying capacity of longitudinally ortransversely framed deck and shel1 platingunder conditions of biaxial in-plane load,shear and lateral pressure: from the nonlinearstiffness characteristics of elements of a hul1section the rament/curvature relationship andhence the CO1lapse strength of a hull-girdermay be found by incremental analysis which canaccount for both vertical and horizontal bend-ing and the presence of shear force and tor-sion.

The analysis of hull strength underdynamic loads, in particular elasto-plastichul1-girder whipping as may be caused byS1ansning or underwater explosions, has beendiscussed and some illustrative results pre-sented. Further work is needed in this area,including particularly development of stablenumerical procedures and experimental evalu-ation of theory.

The proposed method of static hullstrength analysis has been correlated withresults of experiments on welded steel box-girder models and with CO1lapse tests on thedestroyer ALBUERA. Agreement between theoryand experiment for smsll-scale models was satis-factory and gave some guidance on the inf1uenceof “hard corners”, which undoubtedly have asignificant influence on box-girder strength.In the case of ALBUERA, correlation betweentheory and experiment was less satisfactory:this is attributable mainly to unrealisticallyhigh shear forces and premature local failurerwulting frwm the method of load application.It appears, unfortunately, that none of theexisting data from hull collapse experinwntsis capable of providing an accurate check ontheory, relating in al1 cases to riveted and/or transversal.vframed construction with un-known imperfec~ion levels and being influencedin some cases by unrepresentative, prematurelocal failure. It would clearly be worthwhileto obtain hul1 CO1lapse data representative ofnndern, longitudinally framed, welded con-struction, including careful characterizationof imperfections, and it is suggested thatenergetic efforts should be made by the re-search consnunity,in CO1laboration with designand certification authorities, to initiatefurther hul1 CO1lapse tests on ships of approp-riate type, following completion of service andprior to scrapping.

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