Strength

THE SOCIETYOF NAVALARCHITECTSAND MARINE ENGINEERSOneWorldTradeCenter,Suite1369,NewYork,N.Y.10048PWeIStobeP,emntedatExtremeloadsResw.%eSyrwodum

Arlington,VA,October19-20,1981

of Ship Hull Girders under Moment,Shear and TorqueAlexis Ostapenko, Lehigh University, Bethlehem, PA

ASSTRACT

A method was developed for thedetermination of the ultimate strengthof longitudinally stiffened ship hullgirder segments of rectangularsingle-cell cross section, subjected tobending, shear and torsion.principal features are:

bet.!’!!Compatibility is enforcedindividual nonl hear components of hullcross section ; (2 ) compression flange istreated as if it coneisted of individualbeam-columns each composed of plate-stiffener combination --- pre- andpostbuckling, large deformations andplastification are taken into account:(3) Sides (webs) are analyzed bY amultiple tension-field approachconsidering redistribution of normal andshearing stresses between platesubpanels. COrnpar i son of the methodwith tbe results of three tests on asmal L hul 1 girder specimen showed thatthe method is acceptably accurate forthe loading case of moment and shear butneeds additional work for the generalloading caee of moment, shear andtorque.

NcW3WCLATEJAS

A

a

b

bc

c

c1

C2

a

area of cross section

length of test segment

width of teat segment

spacing of longitudinal stiffenersin compression flange

proportional ity factor between thecroes-sectional shear and the loadparameter W

bending inome”t-shear ratio

torque-shear ratio

d@pth of test segment

di

E

e

‘bci

Fcci

Fvc i

FY

Fys

G

I

L

M

N

P

Pu

q

r

T

t

-v

‘bi

vtfi

spacing of longitudinal in webs

Young’ a modulus of elasticity

eccentricity of the load

buckling bending stress of thei-th web subpanel for pure bending

buckling compressive stress of thei-th web subpanel for pure axialcompression

buckling shear stress for the i-thweb subpanel for pure shear stress

yield stress

static yield stress at zero strainrate

shear modulus

moment of inertia

computed length of the beam-column

bending moment

axial force, resulting from theStreSSeS in the box section

axial force in the beam-column

ultimate axial force for thebeam-column of length ~ax

lateral loading per unit length ofbeam-column

radius of gyration = 11A

torque

plate thickness

shear force

bucklina strenath of the i-th websubpanei

tension fieldweb subpane 1

strength of the i-th

‘*149

. . -—

Vw

‘JWu

w

w

a

‘min

A

&c

AP

Apu

‘Cr

‘Y

Y

‘ci

‘ui

oave

‘bci

‘cr

‘cci

‘ti

T

‘tfi

‘cci

‘$0

shear force in

ultimate shear

web

capacity of web

load parameter equivalentconcentrated transverseacting on a simply supported

>ultimate load parameter

aspect ratio ( =a/d)

to aload

beam

aspect ratio of the widest webs“bpanel (=a/dimax)

tots1 axial shortening for thebeam-colmn or compression flange

axial shortening due to c“rvat”re

axial shortening due to axialstrain

axial shortening which existsunder the ultimate force p“

plate buckling strain

yield strain

shearing deformation

critical shearing deformation atFeint of buckling of the i-thsubpanel

ultimate shearing deformation ofthe i-th subpanel

avera9e Stress for the plate ofthe compre.esio” flange

bending stress which causeabuckling of the i-th web subpanelwhen acting together withcompression and shearing stresses

plate buckling stress of thecompression flange

pure compression stress whichcauses buckling of the i-th webs“bpanel when acting together withshearing and bending stresees

tension field stress at theultimate condition for the i-thweb subpa”el

shearing stress

equivalent shearing stress in thei-th web subpanel d“e to tensionfield action

shearing stress which causesbuckling of the i-th web s“bpanelWhen acting together withcompression and bending stresseii

assumed mid-span curvature

IUTRODUCTIOW

Background and Related Remearch—— —

A need for developing a reliablemethod of evaluating the maximumstrength of ship hulls has been becomingmore and more i“portant with the growingknowledge of ship loads . Although thetraditional methods of ship design asevolved through the years of p=act. ica~

experience give adequately safe shipstructures, it has been shown that themechanism of failure is often verydifferent from the mechanism predictedby these methods (1). Also, the rapidintroduction of novel ship types (largetankers, container, LNG, special navyships) required a more rational approachto ship design than the semi-empiricaltraditional methods.

Caldwell prowsed to obtain theultimate bending strength of a hullgirder by assuming a fully plastifiedcross section. The pastbucklingresponse of the plate components was tobe incorporated by means of theeffeCtlVe width at the maximum platecapacity and the longitudinals wereassumed “Ot to buckle (2). The approachby Smith for the bending strengthdirectly included the nonlinear ree~”seof the compression flange plating (3).

Much research has been conducted onthe ultimate strength behavior ofindiv iduaL ship hull components:individual plates (4, 5, 6) , stiffenedplates and grillages(7, 8, 9, 10, 11,12, 13, 14, 15, 16) and plate girdersunder shear and bending( 17, 18, 19, 20,21, 22, 23, 24, 25, 26, 27). However,knowledge of the behavior of i“divid”alcomponents i* not sufficient foraccurately predicting the ultimatestrength of a ship hull girder since thecomponents reach their ultimate strengthat different levels of deformation.Some segments may be already in thepost-ultimate range of reduced capacitywhen others just attain their maximumstrength. Also , the distribution of theinternal forces to the componentschanges as the Compne”t s be.omenonlinear.PurpO ae and Scope

The main purpoee of this researchwas to develop an analytical method fordetermining the ultimate strength oflongitudinally and transverselystiffened box girders subjected to thecorabi”ed effects of bending, shear andtorque. Although a typical ship hul 1girder would normallly be subjected to

I

I150 +

relatively small shear and torque valuesIn comparison to the bendir,q moment and,thus , the degradation of the momentcapacity may not be of practicallycritical importance for such ships, boxgirders are used in a wider range ofmarine construction, such as in thecross structure of catamarans, specialpurpose ships, crane frames, etc .Therefore, it was considered importantto devel=p a tool applicable to all boxgirder type structures under a generalcase of loading.

The basic individual comp0nent5 ofa hull girder cross section aresubjected primarily to uniform axialcompression or tension with or withoutlateral pressure (bottom or deckplating ), or to variable axial a“d shearforces (side plating) . A typical crosssection is shown in Fig. 1. In the

Fig. L TYPical Mid-Ship Cross Section

development of the analytical model, amethodology was evolved for determiningthe relationship between theforces (moment, shear and torque) on thecross section and the axial deformationof the fLange and side plating. Fulladvantage was take” of the researchpreviously done on the strength ofindividual components. of particularimportance were the methods and computerprograms developed at Lehigh Universityand elsewhere for the analysis of theultimate strength of ship bottomplating (8, 10, 11, 12) and of plategirders (21, 22, 2s). Three tests wereconducted 0“ a model hull girder inorder to verify the soundness of somesimplifying assumptions which had to bemade in developing the theory and topoint out the areas a“d considerationswhich should be included to make thetheory more accurate.

METHoD OF ANALYSIS

Introduction

The thin-walled beam theory can beused for analyzing box girders when theybehave linearly. However, this theoryis no longer valid after the platecomponents buckle or behave nonlinearly.

The method proposed here considersthe overall nonlinear behavior of a boxsection by taking into account thecompatibility of deformations betweenthe individual nonlinear components.Some of the novel features of the methodare the consideration of strain reversalin the compression flange a“d the use ofdifferent materials for the plates ofsides and flanges and the stiffeners.

The compression flange is treatedas an assembly of be.am-columns eachanalyzed by considering the pre- andpostbuckling behavior of the plate andthe large deformations of theplate-stiffener combi”atio”. The effectof residual stresses ie taken intoacco””t . The tension flange is assumedto be linearly elastic-perfectlyplastic . ‘Ihe side-plating is analyzedby considering the redistribution ofshearing a“d axial forces between theplate subpanels, and the ultimatestrength is obtained as the sum of thecontrib”ticms of individual subpanels .

Analysis is performed on a hullgirder segment defined as thelongitudinal portion of the girderbetween tWo adjacent transversestiffener rings or bulkheads . Theforces on a segment (moment M, shear vand torque T) are expressed in terms ofa load parameter W which is equivalentto a concentrated transverse load actingo“ a simply supported beam as show” inFig. 2. These forces are assumed to bevalid for the full length of theSeglment.

vI ,,

Fig . 2 Forces i“ Segment of HullGirder Test specimen

The following generalare used to make the problem

assumptionsmanageable:

151

I

1.

2.

3.

4.

5.

6.

7.

8.

Girder is straight and prismatic .

Cross section has a single cellrectangular shape and is aynnnetricalabout its vertical centroidal axis .

A section plane before deformationremains plane after deformation.(This assumption was later modifiedto allow warping. )

Material has bilinearelastic-plastic stress-strainrelationship. (However, nonl ine.armaterials can be also considered bydefining the stress-strainrelationship with a series ofpoints. )

Transverses are sufficiently rigidto provide unyielding support to theflange and web plating.Rotationally, this support can bepinned or fixed.

The effect of shear lag isnegligible.

Transverse in-plane loads on theflanges and sides are negligible.

Stresses due to the deformation ofthe shape of the cross section arenegligible.

Some additional assumptions aremade in the discussion of pareic”laritems.

Forces on Conuwnents

In the linearly-ela=elc range ofloading, the forces on the compnents ofthe crOas SeCtlOn (flanges and sides)due to M, v, and T can be readilycomputed by using the ordinary beamtheory. In this, essentially of torqueT is carried by pure (St. Venant )torsion with a constant shear flow inall components even when the closed boxsection is restrained from warping (29).

After some plate components buckle,the mechanism of distribution of thecomponent forces +s modified by assumingthat after buckllng the side subpanelscannot carry any additional normalstresses and that shearing stresses areuniform in each particular subpanel butthey have no effect on the ultimatestrength of the flanges.

Before discussing the interactionbetween Section Compc.ne”ts, the methodsof analyzing the sides and flangeplating are presented .

Behavior of Side Plating— ._ (Webs)

Sides (webs) of hull girder* havethe same basic geometry and are

subjected to similar types of forces,bending and shear, as the webs ofordinary plate girders. Thus , it isprudent to take advantage of theresearch conducted cm plate girders (17.1s, 19, 20, 21, 28, 30, 29). The onlysignificant difference is in therelative size of the flanges and theirability to influence the ~stb”ckli”gstrength of the web plate since the thinflange plate of a hull girder provideslittle in-plane support to the web platein comparison with the large flanges ofa typical plate girder (2S).

Cme of the simpler plate girdermethods was selected and adjusted forthe use in the box girder analysis (21,30). L!p to the buckling of one of thesubpanels, the web is assumed to behavelinearly with the shear img and normalstresses in a constant proportion. Thenthe postbuckling strength of thissubpanel is assumed to developindependently from the behavior of othersubpanels.

The maximum shear capacity of thewhole web is give” by the s“m of theultimate strengths of the compenentsubpanels.

n

i~l(Vbi+vtfi )~wu= (1)

where

Vbi= ~ciditw = buckling strength of thel-th subpanel (2)

V fi=~tfiditw = tension-field strengtho~ the l-th subpanel (3)

In this analytical model the directcontribution of the flanges andlongitudinal stiffeners to the tots1shear is neglected.

The critical shearing stress, Tci,of Eq . 2 for each i-th subpanel is foundfrom the followinq interaction eciuation:

[+)’+(?3+(%’)’1;“)Here the bending and norms 1

stresses are in known proportions to theshearing stress. The reference bucklingstresses Fvci. ‘bci, and Fcci arecomputed by assuming the plate subpanelsto be simply supported at all four edgesand subjected to a respective stressacting alone (2S) . A transition betweenthe yield level and the elastic range istaken into account.

W .due

The equivalent shearing stress of3 reflects the postbuckling strength

Ito the tension field and is given by

(5)

152

I

I

where(6)

‘ti = ‘y -/0.25 (0cci-Otii)~ ‘3Tci

is the tension field stress at theultimate condition for the i-thsubpanel, and

cJmin=a/~i ~ax (7)

the aspect ratio of the widest~~bpanel (28, 30] .

Since the individual subpanels Ofthe web in general have different depthsdi and are subjected to differentcombinations of bending and norms 1stresses, their buckling and theattainment of the ultimate condition donot occur simultaneously and arestaggered in the course of the overalldeformation of the web. The lower plotof Fig. 3 shows the shear ingdeformations of the three subpanels of asample web shown in the upper sketch ofthe figure. The conditions of bucklingand ultimate strength are labeled forsubpanel 3, and they are seen to be atdifferent levels of the overall shearingdeformation than for the other twosubpanels (21) .

Deformation of each subpanel up tothe point of buckling is linear and isdefined by

v ,,“, ,.,

Fig. 3 ~-~ Relationship for EachSubpanel

On the other hand, the postbucklingdeformation cannot be accurately

established. In Figure 3, it isapproximated by a straight line

connecting the buckling deformation withthe ultimate deformat ion which is

assumed to be reached when a diagonalfiber in the subpanel yields due to the

racking deformation of the edge linesassumed to retain their original lengths(21). Thus,

Yui = -;y-( ai + -;~-) (9)

where

ai = a/di (lo)

The application of Eqs . 1, 2, and 3

shea~~~fo~~at~~ ~~~~ra~i”~~ ~fg.th~at

results in a relationship between thetotal shear ‘JW and the overalldeformation for the whole web. In theprocess of computing this relationshipit is important to keep i“ mind that,whereas the shear on a subpanel canincrease after buckling, the norms 1stresses are assumed to remain constantand, thus . the additional momentcorresponding to the increase in thetotal web shear must be redistributed tothe flanges, the longitudinal webstiffeners and to the yet unbuckled websubpanels. With the assumption of “theplane section remaining plane”, thisredistribution process gives acorresponding relationship between thetotal shear V and moment ~ acting onthe web and t~e normal strains at thetop and bottom edges where thecompatibility of strains is enforcedbetween the webs and flanges.

In the present formulation, it isassumed that longitudinal stiffeners arelinearly elastic up to yielding, but thisass”inptio” can be modified once thecriteria for theix premature failure cmnonlinear behavior are established .

Behavior g Longitudinally StiffenedCompression =

Introduction. ‘rbe compressionflange of a hull girder section (thedeck for the sagging a“d the botto” forthe hogging moment ) consists of alongitudinally stiffened plate subjectedto axial compression and, for thebottom, lateral loading. The flangeplating is assumed to be either eimplysupported or fixed at the transverses.The side edges (junctions to webs) areassumed to be free to rotate anddisplace in the plane of the plate (13).The nonlinearity of the axial behaviorof such a plating arises from thewelding residual stresses. buckling andpostbuckling response of the platecompo”e”ts, initial imperfections andlateral loading. The method previouslydeveloped to consider these effects byreplacing the analysis of alongitudinally stiffened plate panelwith a large-deflection a“alysia of abean-column was adapted for the presentresearch (10, 11) .

L... -153

The simplifying assumptions of themethod are the following :

1. The plate is very flexible incompariivan with the relatively largelongitudi”als and therefore theinteraction between thelongit.dinals through the plate maybe neglected. Then, eachlongitudinal with its tributaryportion of the plate may beconsidered as a“ independentsubstitute beam-column a.bjected toaxial and lateral loads.

2. The response of the plate componentof the beam-column cross sectioncorresponds to the behavior of along plate with the width equal tothe spacing of the longit”dinals.The side edges are assumed to besimply supported, b“ t they nulstremain straight although they mayhave in-plane motion.

3. The effect of lateral loading on theplate behavior is negligible sinceit has been found to have littleeffect 0“ the buckling andpostb”ckling behavior (10), and thebendi”.g stresses (in the platespanning between longitudinal ) maybe treated as a tertiary conditionand checked separately. Then, thedistributed lateral loading isapplied as a line load q on thebeam-column as shown in Fig. 4.

~-+p

‘~

-~-_=-------------

Fig . 4 Beam-Column Idealization

Since the original computer program gaveonly the length of . pin- or fixed-endedbeam-column which was in equil~briumunder the given axial and lateral loadsfor an assumed mid-span curvature,several supplementary operations had tobe developed to obtain a complete axialload versus deformation relationship fora zero lateral loading and a specifiedlength. Frincipal operations of thefinal program are briefly describedhere.

Behavior PlateOf __ under

COmpr- Tbe axial behavior of astiffened plate under compression isdescribed by a relationship between theaverac!e stress and the overall strainwhich- is also the strain at the edges.Such a relationship can be supplied tothe program by a SerleS of pointsobtained, for example, from a test, or

by a computational procedure . In thecomputational procedure used, it isassumed that the plate is perfectly flatand the effects of the shearing stresseson the axial buckling stress and thepostb”ckling behavior are(31).

negligible

The three ranges of the plateresponse when the plate is subject tobuckling are shown in Fig. 5 on theaverage stress vs . strain curve .

t

AverageStress

=YP

P

,/, Elr3.Ytlc

Test.4\

r~g%

Koifer’s

m+” ‘q””

r Edge Stroin

‘YP

Fig. 5 Average Stress vs. Edge StrainRelationship for Plates underCompression with WeldingResidual Stresses

1.

2.

* linearly (or nonlinearly)elastic preb”ckll~ =. The—stres5 IS uniform and the end of therange is limited by the bucklingstress (the buckling coefficient isconservatively taken to be k= 4.0) .

Elastic postbucklinq ~ me- pestbuckling relationship isdescribed bv the Koiter equationwhich gives - the average stress interms of the overall (edge) .trai.(4).

3.

The stress pattern is nonuniform,and the average stress vs. strainrelati.onsbip is noticeably flatterthan the material stress-straincurve .

Ultimate stress condition is assumed~— reached when the maximum(edge) stress of the nonuniform

154

--

pattern reaches yield stress level.This assumption has been confirmedbY numerOus teste and sometheoretical analyses (6, 7, 10).Compression of the plate beyond thispoint generally shows a reduction ofthe average str.ees as indicated inFig. 5 by the curve portion labeled“True” (6, 7). However, samplecomputations have demonstrated thatin stiffened plating of theproportions typical for shipstructures, ultimate strength of theplating is reached at the platestrains which do “ot significantlyexceed the ultimate strain and, whenthey do, the effect is negligible,It is thus reasonable to assumethat, as shown in Fig. 5, theaverage stress remains constant fordefcmmat ions beyond the ultimatecondition (10) .

The effect of the welding residualstresses was included in this method(lo).

Beam-Co lumn Analysis. Thebean-column to be analyzed is shown inFig. 4. It is subjected to an axialload P, end moments M and a line loadinaq. The cross section consists of th;plate with the stress-straincharacteristics established above andthe longitudinal stiffener with astrees-strain response given by thematerial .

The purpose of the analysis is toestablish tbe relationship between theaxial load and axial deformation. Theprocess requires several interdependentsteps.

The first step is to develop aseries of relationships between themid-span curvature and the length of thebeam-column of the given cross sectionwhen the line load is kept constant anddifferent axial load is applied. Apreviously developed computer program isused for this purpose (10, 11, 32, 33) .The resultant curves are shown in Fig.6a. Each time an iterative numericalintegration is involved.

The next step is to transform thelength L vs. $0 mid-span curvaturerelationships into the length vs. axialshortening relationships by utilizingthe length shortenings computed in thefirst step. The resultant L vs. bcurves are shown for different axialloads P in Fig. bb.

since the given beam-column has aspecific length a, the relationshipbetween P and b is obtained by passing ahorizontal Line in Fig. bb for tia andtaking the A-values corresponding toeach value of P. The results are thencombined into a P vs. A curve valid for

,.,

(m

,0

1A, .!

DP ---— . . . . . . . . .

~, +

Fig. 6 Procedure of Obtaining P VB .Relationship for Given q

a specified line loading q and lengthL-a as shown in Fig. 6C . The peak ofthe curve gives the ultimate axialstrength of the beain-column.

Effect g Strain Reversal. Aspecial correction for the effect ofstrain reversal had to be made in thepost-ultimate range of the P-Arelationship. me need for this arosefrom the fact that the proceduredescribed above for obtaining the L vs .

‘$0 and B vs. curves is based onformulating an equilibrium condition o“a member deformed to the configurationconsidered. This is equivalent toobtaining each point of the P-A curve asif the path of deformation followed astraight line from the origin asindicated by the dashed line in Fig. 7.Whereas the pre-ultimate range of theP-A curve which for an increasing valueof P is not affected by this procedure,the post-ultimate range becomes verydistorted . This is shown in Fig. 7 bythe dotted z-shaped curve defined by

t,,00“p?

BL.m,tikw, S,,,ati

0.80 ,.!, ““d,, ,.”’,,.,s..

,.0.3 ,Ml”,w k. ,!.s!.o.m

,.-, m),,.*

.= g.0.0,

040 / / %%.: ~m~,,m~

,0.m P ~pbnn ?

.%’ ,..0”.,,..0, %,,,,’ Ld.i-’

(M)*,o 0.20 0,0 0.60 . ..0 ,,CU ,,,. ,.40 ,,ec

1

Fig . 7 Load “S . Axial ShorteningAdjusted for Strain Reversal .,

!

\

crosses . In this case, the nonlinearand plastic deformations which had takenplace under the higher past load andsubsequent elastic relaxation are nottaken into account.

In order to correct the anomaly of

the reduction of the deformationindicated by the dashed curve, the truedeformation path including the strainreversal resulting from the drop in theaxial load in the post-ultimate rangeWas approximateti by modifying theA-value as follows. Since the totalshortening consists of two parts,

A=AP + Ac (12)

A = axial shortening due to axialstrain p(effect on P)

AC= axial shortening d“e tocurvature,

it was assumed that in the post-ultimaterange 4 remains constant and equal to

tk~ir them;~,imate load P“ (at thethat is, the value which existed

peak) . the shortening in thepost-ultimate range becomes

A= A+ACpu

(13)

The result of this adjustment is shownin Fig. 7.

Axial Behavior. The proceduredescribed above requires that thelateral line loading q be non-zero andthus the procedure is not directlyapplicable to the analysis of ship deckplating. To obtain the pure axial loadvs . shortening behavior, a set of P vs .A relationships are computed fordecreasing values of q a“d the P-valuesfor q = O are extrapolated.

Two examples of graphicalextrapolation are shown in Fig. 8 forthe ultimate capacity of the compressionflange of the test specimen. The topplot is for the original designdimensions a“d the “cxninal yield stress .The bottom plot is for the dimensions

?.w..,,,.5> ,9

till,V%

o ..,0 0.m o.m 0..0

Fig . 8 Extrapolatio” of Pure AxialStrf?”qth from Beam-columnStrengths

156

,/,,,.0, F =

v! (./011.,1 1 I 1 1

0 0,20 .40 ,,, ,.,0 ,m ,.,0 ,.40

Fig. 9 Load vs. Shortenin~Stiffened Plate underCompression and Normal Loading

ofAxial

and the yield stress as they weremeasured in the fabricated specimen.Another example isthe COln@ete P ,S. ~l%;”.%”:;:;:

of initial curves for various values ofq is shown. Usually three values of qbetween 0.03 a“d O.10 were sufficient .

For comparison, the response of atension flange, corresponding to thematerial stress-strain diagram, is alsoshown in Fig. 9. Also, for greaterconvenience, the axial load isnondimensionalized to PIP = PI (AFY) and

7the axial shortening to A a Zy.

Consideration ~ InitialImperfections. Initial deflections dueto fabrication were not considered inthe procedure described above. However,a modification can be readily made bytransforming the initial deflectionpatterns into a curvature diagram andthen adding the corresponding curvaturevalues at each segment in theintegration process. Since theintegration length L may be longer thanthe actual length of the beam-column a,the initial curvature diagram should beextended by, for example, making itconstant and equal to the value ofcurvature at the end or to zero.

Behavior and Ultimate Strength of HU1l‘eg=nt

——

Once the load-de formation behaviorof the individual components is defined,the analysis of the entire hull girdersegment proceeds by enforcing thecompatibility between these componentsas the load is incremented. Thefollowing load-deformation relationshipsof the components are involved :

-~ve. y relationships for theindividual subpanels of the sides(webs) (Fig. 3).

- P/Py vs. (tia)/zy relationship forthe longitudinal of the compressionflange, each with its share of theplate (Fig.9 ).

- 0 vs. E relationship (material curve)for the tension flange, and thestiffeners and the unbuck Ledsubpanels of the webs.

The internal forces acting on themid-segment section (moment, shear andtorque) are related to loading parameterw as shown i“ Fig. 2 and can beexpressed by the following equations:

V=cw (14)

M= Cld V= CC1d W (15)

T= C2d V= CC2d W (16)

where: C= proportionality factor betweenthe cross-section shear and W,

c1 moment-shear ratio = M/(Vd),

C2 torque-shear ratio = T/(Vd), and

d = depth of the cross section

The deformation parameter to beused aqai”st W in the computer programwas chosen to be the average strain inthe junction line between the web andthe compression flange. This straincorresponds to the average shortening ofthe compression flange, A/a. An exampleof the resultant curve for theload-deformation relationship is shownin Fig. 10. This curve im for Test Lbut using the initial design dimensionsand a somewhat different testarrangement than in the actual test.

1,”,(k]

ID,0 ‘(”S’”‘-h ,.1.!

F,m*”,3

,00

?

.,5

5010

, . . .

0 0 0., ..4 0.. 0.8 ,.0 ,L2 ,.. ,.6

(./.1 /.,

Fig . 10 Load Distribution between Websand Flanges for Box Sectionunder Shear and Bending

?lIe procedure for obtaining the Wvs. (A/a)/&relationship is according

Fto the follo lng basic steps:

L. An initial value of the web (side)-to-compression flange junctionstrain is as.s”med and the strain atthe other edge of the web isiterated until the total axial forceon the cross section is equal tozero (N=o) . Since the cross sectionis assumed to remain plane, the

strain distribution -k eachiteration is used for computing thenorms 1 stresses and then thereeultant axial force and thebending moment. Gnce the axialforce is equal to zero (N=O), therelationships of Eqs. 14 to 16provide a means for computing W, V,and T and the resultant shearingstresses . If there is no bucklingin the web subpanels, the assumedjunction strain and the w-valueProvide one point for the CU-W=.Then, the junction strain isincremented and the process isrepeated for additional peints ,

2. If the shearing stress in aparticular web subpanel becomeshigher than the critical (orultimate) value indicated in Fig. 3,the junction strain must be reduceda“d the procedure repeated until avalue acceptably close to thecritical ( or ultimate) shearingstress is found, At each iterationafter subpanel buckling, theredistribution of shearing stressesbetween i“divid”al subpanels takesplace to maintain shear deformationccinpatibi1ity as shown by the curvesin Fig. 3. Thus, the operations ofthis step must be repeated at eachkink of these curves.

3. As the junction strain values areincreased and the .orrespondi”gvalues of W are computed, a completew vs. junction strain is obtained,including the pre- and P.astultirn.ateranges.

In the example of Fig. 10, the j“nctio”strain is non-dimensional ized withrespect to the yield strain.Contributions of the webs and flanges tothe total load are shown by separatecurves. The share for each was taken tobe proportional to the percentage of themoment carried by the respectivecomponent.

Modified Computer Program

The computer program, based on theprocedure described above, correlatedwell with the test result on thehull-girder Segment subjected to shearand bending (Test 1), b“t it was toooptimistic when torque was also applied(Tests 2 and 3) .

The computer program was thenmodified not to require that the sectionremain plane and two corner strains wereintroduced into the iterative processwith two other corner strains graduallyincremented. The variation of the axialdeformations (axial strain) across thewidth of each co”ponent (flange or web)was assumed to be linear. Thus, each

I

‘&--157

longitudinal of the compression flangemade a different contribution inaccordance with the P .?s. Arelationship. The criteria used for theconvergence of the two iterated strainswere that the axial force and the momentabout the vertical axis be equal to zero(N=O and ~= o).

The results from the modifiedprogram give a better correlation withtests, but the program needs furtherwork ( ae of May 1981) .

TEST SPECIMEN

A test specimen had bee” designedto conduct three tests under differentcombinations of moment, shear andtorque. For each test, a particularsegment was tested to failure while theother two segments were reinforced.Figure 11 shows the test arrangement foreach of the tests and the correspondingcombinations of moment, shear and torquedefined in terms of the jack load W.

d T.,! I MOmm!. m..,

w

l*i ....6... (,...,v.o.m,w ,,,,,.0.,94 .(,...,

b, 7.s, , .0.$., . ,,,., .,.,,”,

e,,.,! , .0., ”! . ,“8., . ,.,,.,

Fig. 11 Test Segments a“d FOrCe S

Test l(Fig ha) : Bending momentshear .

and

Test 2( Fig. llb) : Bending moment shearand torque.

Test 3(Fig. llc) : Another combinationof be”di”g Incmle”t,shear and torque.

lle scantlings of the test specimenwere selected to model the relativeproportions of the Conlpcxlents of atypical hull girder . ?WO views and theprincipal cross sections are shown in

““tilt!ir’’”p’iii“+* +“ .s6’”

e e

Fig. 12 Test Specimen scantlings

Fig. 12. The spating of thelongitudinal and the thickness of theplate in the test portions were selected50 that plate instability would occurbefore reaching the ultimate capacity.The sca”tli”gs of the fabricatedspecimen were slightly different thwthe design scantli”gs show” i“ Fig. 12.The most significant change was in theplate thickness from 1.59 mm (1/16 in.)to 1.85 mm (0.073 in.). The teetspecimen was fabricated from ASTM A36steel plate with a nominal yield stressof 250 MPa (36 ksi) and the actualstatic yield stress in the longitudinaldirection of 237 MPa (34.34 ksi ) a“d thedynamic vield stress at the strain rateo: 1042 >m/m/sec (ASTM strain rite) of280 MPa (40.55ksi) (34).

Although residual stresses d“e tothe yielding process were not measured,some reasonable levels of intensity wereassumed in the computer analysis .

Fig . 13 Initial Imperfection, in theCompression Flange (alldimensions in mm)

15s

Initial deflections were measuredfor the plates of the webs and ~h.compression flange. For the webs, thernximum offsets were of the order Of*4.6 ~ (Q.~9 in.) , i.e. , ~ppxoxima~eyy2.6 times the plate thickness. Initial~rfectiO,ns of the compression flangeare shown In Fig. 13 by a contour map.Ibst of the offsets are in the range of=.54 mm (*0.1 in. ) with the maximumwalues of the order of(*0.2 in.), i.e. ,

*5. o&unl2.7 times the plate

thickness.

TS4T PRCCEOURE ANO RRSULTS

The test setup is shown in Fig. 14.me specimen is positioned on tWosupport pedestals, and a Concentratedload is apPlied by means of a jackattached to a transverse beam of thetest frame. ‘Ihe load is transmitted tothe test specimen by a spreader bean,set transversely on two plates welded tothe transverse stiffeners of the webs,as shown in Section A-A. For Test 1,the cross section was loadedsymmetrically (section A-A ) and forTests 2 and 3 with an eccentricity.

There were three paints of supportf0x the specimen. The X-Y rollerbeariny at one end consisted of anarrangement of two mutuallyperpendicular rollers separated by aplate so that rotation and translationwere possible in the longitudinal andtransverse directions. The other e“dhad two X-roller bearings, one on eachside of the cross section, whichpermitted free rotation. In Tests 2 and3, one of these two supports was alsoanchored down to prevent uplift of thesupport due to torsion.

[ w

SEcm. . ...,,,! 1) ,6,740. A-NT- *I

Fig. 14 Test Setup

To accomplishsame soecimen. the

several tests on theseaments ad iacent to

the t:st segment we”re reinf~rced bymm11 steel bacs cee-claItQed to thelongitudinal stiffeners, corner anglesat the web-to-compression flangejunctions and by pieces of wood o“ thecompression flange. For the yetuntested segments these reinforcementswere tight ly wedged between thetransferees. For the already testedsegments, the reinforcement bars and thecorner angles were tack welded to thecompression flange. and the webs werereinforced with steel bars welded to thetransverse stiffeners in the directionof the tension diagonal.

Instrumentation

The instrumentation consieted ofboth mechanical dial gages and electricresistance strain gages. The dial gageswere used to measure the verticaldeflections of the specimen along bothwebs. The approximately fifty linearand three-branch-rosette strain gagesprovided information about the stressdistribution over the cross eectio” andabout the tension field pattern whichdeveloped after theoretical b“ckl ing ofthe webs.

Diagonal deformations of the sidesof the tested segments were measured bymeans of a portable variable-lengthextensometer. This exte”50meter wasalso used at other pointe to measure thavariation of the segment length betweentransverse stiffeners . Distortion ofthe cross section at the ends of thetest seyme”t was measured by means ofelectrical extensometers placeddiagonally from corner to corner (Test3).

Teet Results

In all three tests, the ultimatecapacity was limited by the failure ofthe compreseio” flange characterized bylarge out-of-plane deformations asshown, for example, in Fig. 15 for Test1.

Vertical deflection of the heavierloaded web at the transverse under theload was used as the principal indicatorof the overall deformations of thetested segment. Figures 16, 17, and 18show the plote of the applied load W vs.this deflection, respectively, for Tests1, 2 and 3. The points plotted near theultimate load and in the pest-ultimaterange correspond to the maximum loadrecorded for that deflection and, thus,they were affected by the loading rate(actually,the straining rate) . Thezig-zag pattern in the right-handportion of Fig. 18 illustrates the

L..159

i

—

Fig. 15 Deformation of CompressionFlange and Web 1 (Test 1)

,“ “,.

.

.

..

lam

,~~ ,.

,. -,. ,

*P’,,’

*n/’,,’ bf_,

‘-i-’sl.

Fig. 16 Load v. . Deflection Curve forTest 1

reduction of the maximum load to thestabilized load after closing the valveof the testing machine.

‘IIIeoverall behavior of the testsegment, as shown in Figures 16, 17 and1S, is characterized for all three testsby the following four portions :

1. Linear ; to 0.5-0.6 of(ultimate ~~ad) .

Wu

2. Gradually curving : up to theultimate load Wu,. The deviation ofthe curve from llnearity appeared tobe mainly due to the increase of the

3.

4.

out-of-plane deflections of thecompress ion flange and the upper websubpanels and due to local yielding.

Post-ultimate drop-dcmu” portion.After reaching the ultimatecapacity, the load suddenly (Test1), or gradually (Tests 2, and 3),dropped a“d then stabilized. Whenthe machine valve was opened again,the load climbed somewhat and thendropped further to a lower stablelevel.

Unloading portion. ?.fter obtainingthe post- ”ltiinate range the girderwas, unloaded to zero in two or moresteps .

,. .,s

. ..0 /’”

. -my

IKa. L

.0

,.

.

.. 0.-. . -m!. 4..03 . ..m

‘oCwl.,cm, 8

Fig . 17 Load v.. Deflection Curve forTest 2

mm

,.. .2 ., ., .. ,.0 N ).4 >.0 M ,.0,,

OfFLECTION

Fig. 18 Load v.. Deflection Curve forTest 3

Diagonal Deformations of the Web———

A good representation of thebehavior of a test segment is given by aplot of defcnnuationa (changes in length)of the compression and tension diagonalsof the web versus the applied loadW. For example, Fig. 19 shows the

160L..-

;

L A,: ToldwOrmdi;nd ..,o”.,,

-m a 4“”Afc ““” ‘0x.-.

Fig. 19 Diagonal Deformations of webs 1and 2 (Test 2 )

relative diagonal deformations,respectively shortening and elongation,of the compression and tension diagonalsof the two webs of Segment 2 (Test 2) .Other segments behaved similar ly.

Because of the load eccentricity,one web (Web 1) was subjected to ahigher shear than the other (web 2I,and, therefore, the deformations of thetwo “ebs are noticeably different. Theresponse of the tension and compressiondiagonals for Web 2 (smaller shear) andthe tension diagonal for Web 1 areessentially linear “p to 0.95 W“. Asudden change in the slope occure atthis load before the ultimate load isreached. Deformation of the compressiondiagonal of Web 1 was much morepronounced. Although the curve isessentially linear up to 0.54 Wu, theslope is noticeably flatter than for theother diagonals, then, it reducesfurther and after 0.95 Wu becomes almosthorizontal until the curve reaches theultimate load Wu. Actually, the load atwhich the response of the compressiondiagonal of web 1 becomee nonlinearcorresponds to the beginning of thenonlinear portion of the load-deflectioncurve of Fig . 17.

The larger deformation of thecompression diayonal of Web 1 than ofthe tension diagonal demonstrates theextent of the overalldistortion of

shearingthe panel caused by

subpanel buckling and the shortening ofthe compression flange on this side.For Web 2, the deformations of thetension and compression die.go”als areapproximately the same indicating arelatively linear shearing deformation.

Strain Distribution in cross section.— —

Strain readings provided a means ofdetermining stresses in the testedsegment and comparing these with thecomputed stresses. However, since the

gages were placed only on the outsidesurface and the plate components hadimpractically large initial deflections,the readings could not be considered togive accurate values of axialcrOss-.5ecti0nal stresses, especially intbe post-buckling range. Still, thedistribution of the measured strsins inthe cross section gave a good indicationof the bebav ior of the tested segments.

Probably the most significant is acomparison between the straindistributions in a segment subjected tomoment and shear (Testl) and in asegment subjected to moment, shear andtorque (Tests 2 and 3). Figure 20 showsthe strains across the half -”idth of thecompression flange for Test 1 (the otherhalf is approximately symmetrical) .Although there is a significantreduction of str.ai”s (and, thus ,stresses] away from the edge, especiallyat higher load values, the middle

Srml”

Fig . 20 Strain Distribution atMid-Length of CompressionFlange (Test 1)

,.,6 , 6 ., ~ 10 !,,w<,, 1’

bCohw?,ss,o.FLAME

Fig. 21 Strain Distributions in Flangesfor Half-Width (Test 2)

)

1+161

Frtion has a relatively uniformdistribution, In contrast to this, thestrain distrib”ticm i“ Fig. 21, (Test 2)is ba=ically linear except for thevariation between subpanels andlongit”di”als a“d the hcrease at theedge, and is thus analcgous to thetheoretical strains given by the thinlines for an “open Channel,a section.This means that there was a gradualtransition frOM a closed to an opensection as the heavier loaded web (Web1) was weakeninq while the web subpa”elswere going into the post-buckling range .

-..d -, -. -, 0 l..-’Simh DMribmkahWE+1

b

Fig. 22 Strain Distribution in Web 1(Test 2)

Figure 22 shows the straindistribution across Web 1 of Segment 2.7he distribution patterns for theindividual loads are very irregular andcan hardly be considered to support the“plane sectionhypotheses.

remaining plane”However, except for the

last two load increments, the neutralaxis remained at essentially the samelocation, “although below the mid-depthpoint. This indicates that the overallrespo”ee of the cross section wasessentially linearly elastic with thecompression flange being ‘Sweakex,nthanthe tension flange . The cfownwa~d Shi’ftof the neutral axis for the loads over0.68 Wu, was mainly caused by theprogressive failure of the compressionflange and of the top web s“bpanel .

CCMPARISOW OF ANALYTICAL ANDlfXPERIMEWTAL RSSULTS

The tested girder segments wereanalyzed by the method described above.

Since the three segments had thesame dimensions, the response of thelongitudinal in the compression flangeto axial force was the same. lhua, thedifference from test to test was only inthe moment-shear-torque combinations asshown in Fig. 11. Table 1 liste thesecombinations in terms of the

non-dimens ionalized parameters C, C

and C2 (see Eq.. 14, 15, and 16$:Analysis of the test segments or theultimate load by using the proposedmethod gave the values shown in Table 1.The experimental values are also listedin the table.

Table 1. Specimen Parametersand Ultimate Loads

Test 1 2 3Shear/w = C 0.615 0.385 0.538M/Wd = Cl 1.106 1.213 1.211T/Wd = C2 o 0.3S2 0.393

Load Ifu~ (*N) 266.9 164.6 192.4

‘theo (kN) 306.9 2s0.2 276.9

‘theo/wexp 1.15 1.70 1.44

The ratios of the theoretical a“dexperimental ultimate loads, Wt ~o/Wexgiven in the last line of the ta~le sho~that the theory is only I5a toooptimistic for the segment not s“bjectedto torque (Test 1) but 70% a“d 44% forthe segments which are subjected totorque (Tests 2 and 3, respectively) .

The computed load vs. junctionstrain relationships for Teets 1 and 2are plotted in Fig. 23. Both curves arelinear up to the first kink whichcorresponds to the buckling of the topweb subpanel (Webs 1 and 2 for Test 1and Web 1 for Test 2) . There is onemore kink in each curve before theultimate load is reached, and itreflects the buckling of the middle websubpanel. The pcx+t-ultimate rangeexhibits a rapid red”ctio” in strength.It is noteworthy that the ultimatestrength developed at approximately 0.89&y on the abscissa, that is, before thejunction had an average strain equal tothe yield strain. However, thispredict ion could not be checked byexperimental results since the straingages measured not the average but onlythe local strains at a few locations .

(k)

II,00 ‘0

m,s0

50

,00

*-

,,0,0

‘m20

40 ,0h,,.,!!, WI.”,.

,., -’.,.’ ...

,/ ~.,’/ ..-. .

/’/ T.,, 2 w ,,,,,

,/’ w.”.,, (M.“,

/’ w

mm<.,

Load vs. Average Strain inFlange-Web Junction for Tests 1and 2

162

All three tests confirmedqualitatively the analytical predictionshat the strength of the box section waslimited by the capacity of thecompression flange,

The substantially lowerexperimental loads for Tests 2 and 3could be only partially explained by themeasurement * and observations.Apparently, the most significantweakening effect was made by thetransformation (from the analyticalpint of view) of the closad box sectionInto a partially open channel-typesection with the Corresponding shift Ofthe shear center. ‘Ibis-effec~ would becaused by the “softening” of the heavierloaded web as its subpanels went intothe postbuckling range. Suchtra”sformaticm would not only force thesection to carry an increasing portionof the torque by warping torsion (VS .

the pure St. Venant torsion) but also,a. mentioned above, amplify the torqueItself as the shear center shifted awayfr.nnthe weaker web (web 1) toward themore rigid web (Web 2). me nonu”i formdistribution of the strains i“ Fig. 21confirms this conclusion. A similardistribution was also found for Test 3.

After the computer program was-dified to allow non-planar deformationof the cross section, analysis of theultimate conditicm of Test 3 waspr fo?nned by inputti”g the actualexperimentally measured Strains at threecorners a“d iteratimg the fourth strain:0 converge the total axial load to zero(N=o). The ultimate load was computed

‘0 be ‘theo = 250- I kN, that is, 1.30‘ex rather than 1.40 Wex , and, “ithsom~ additional assumptlo& (i“cl”ding:he setting of the compressive residualstress equal to 20% of the yieldstress) , Wtheo = 214.3 kN, that is, 1.14

‘exp which .gIvesonly 14% deviation.

It appears, thus , that amodification of the program to performiteration of the strains at two cornersand irnposinq the requirement that N=Oand M..- = O - should l-cad to a workabl eprograll.properly

sLNIwARY,

Summary

A

such a program has not bee”tested yet (May 1981 ) .

CONCLUSIONS AND REC~Ef4DATICd4S

theoretical and experimentalstudy was performed on the pre- andpstultimate behavior of longitudinallyand transversely stiffened box girdersof the scantlinys typical for shiphulls . TWO loading conditions wereconsidered: (1) moment a“d shear, a“d

(2) moment, shear and torsion. Threetests were conducted o“ a hull girderspecimen to verify the analyticalmethod.

The principal feature of theanalytical method was the considerationof continuous interaction between thecomponents of a hull girder crosssection through the compatibility ofaxial strains at the junction linesbetween the deck, side a“d battomplatings. This was needed for thefollowiny reasons:

1. The danger of computing the maximumstrength of a hull cross section by

2

adding the maximum strengths of th~individual components rests cm thefact that the segments reach theirmaximum strengths at differentlevels of deformation. Thus, somesegments Ill?.ybe already in thepost-ultimate range of reducedcapacity when some others justattain their maximum strength.

Redistribution of internal forces,specifically, of the bending momentbetween the webs and flanges couldbe considered by mai”taini”gcompatibility of strains at thejunction lines a“d requiring that“plane sections remain plane” orthat the section may warp, but thestrains vary linearly across thewidth of each component.

The behavior and ultimate strength ofindividual coqmnents of the crosssection was established by adapting andextending available methods. Thecompression flange was analyzed as if itwas composed of a series of substitutebeam-columns each consisting of astiffener and a plate and subjected toaxial and lateral loads. Buckling andpost-buckling respOnse of the plate,plastification and large deformationwere considered. The webs were analyzedby using an ultimate strength theorypreviously developed for longitudinallystiffened plate and box girders.

2TE three tests conducted o“separate segments of the hull girderspecimen 1ed to the followingobservations :

~~ (IIKUnent and Shear) : ( 1 ) Theexperirnent~ina~ load was 15% belowth: theoretical load: (2) Theexperimental stress distribution i“ thecompression flange agreed well with thetheoretical “p to 50% of the ultimateload. Then, there was a predictabledeviation, with the stresses at theweb- flange junctionsignificantly higher thanportion of the flange.

becomingin the middle

163

Tests 2 and 3 (moment, shear andme) T ( l_J_Th~ e-tal ulti.STZ+ 70 and 44% SE1OW the predicted

respectively for Tests 2 and 3.(2) ‘ Contrary to the analyticalprediction, the stresses measured in thetiension and compression flanges were notdistributed uniformly or symmetric ally-.Apparently, the rigidity of the websubjected to a greater shear wasdeteriorating much faster thananticipated. The consequent

redistribution of internal forces made-the cross section to behave as if it wasgradually transformed from a closed boxto an open channel section.

Conclusions

Comparison of the theoretical andexperimental results showed that in thecase when only moment and shear ~reacting on the girder segment (Test 1),the original analytical method wasacceptably accurate although it wassomewhat optimistic .

However, the method was founa togive significantly higher ultimateCaPaCitY when the girder segment INaSsubjected to the general loading ofmoment, shear and torque (Tests z and3). Although the values of shear a“dtorque used in this analysis wererelatively hiyher than those encounteredin ordinary ship hulls, the method isimportant since bQx girders are used ina wider range of marine structures, forexample, i“ the cross stxuct”re ofcatamarans.

Recommendation= for Future Research..—

In order to meet the originalobjective of developing a reliablemethod for hull girders under moment,shear and torque the followingimprovements are recommended:

1.

2.

3.

4.

5.

Teste on hull girder segments oflarger dimensions, so that theinitial imperfections would be inthe relative practical range.

Inclusion of the effect of shearingstressee on the strength of flanges,especially of the compressionflange.

Inclusion of the effect of shearlag.

Refinement of the strengthformulation for the webs.

Inclusion of the effect of warpingby the consideration of non”” iformbut linearly varying axia1

6.

shortening (or elongation) acrossthe width of the flanges and sidee(current work) .

Consideration of the deformation ofthe shape of the cross section “hentransverse rings (diaphragms,transverse bulkheads ) are notsufficiently rigid.

ACKNOWLEDGEMENTS

The author is grateful to theMaritime Administration for sponsoringthis re~earch under the MARAD UniversityResearch Program Contr.act Nos .MA-79-SAC-BO019 and MA-8 O-SAC- O1O81 .The help by Mr. F. Seibold, ProgramManager, and Dr. W. M. Maclean,Technical Coordinator, is appreciated .

Mr . A. Vaucher was the principalco-worker in this research and he wasinstrumental in developing the testarrangement and i“ conducting the firsttwo tests, as well as, in their analysisand i“ the DEeDaration of the Derti”e”tdrawings. i.fr.-M. F. !3alley ~onductedthe third test .

‘IManksY. F. Chenassisted inthis paper .

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166