18.1 NOMENCLATUREFor specic symbols, refer to the denitions
contained inthe various sections.ABS American Bureau of ShippingBEM
Boundary Element MethodBV Bureau VeritasDNV Det Norske VeritasFEA
Finite Element AnalysisFEM Finite Element MethodIACS International
Association of Classica-tion SocietiesISSC International Ship &
Offshore StructuresCongressISOPE International Offshore and Polar
Engi-neering ConferenceISUM Idealized Structural Unit methodNKK
Nippon Kaiji KyokaiPRADS Practical Design of Ships and
MobileUnits,RINA Registro Italiano NavaleSNAME Society of naval
Architects and marineEngineersSSC Ship Structure Committee.a
accelerationA areaB breadth of the shipC wave coefcient (Table
18.I)CBhull block coefcientD depth of the shipg gravity
accelerationm(x) longitudinal distribution of massI(x) geometric
moment of inertia (beam sec-tion x)L length of the shipM(x) bending
moment at section x of a beamMT(x) torque moment at section x of a
beamp pressureq(x) resultant of sectional force acting on abeamT
draft of the shipV(x) shear at section x of a beams,w(low case)
still water, wave induced componentv,h(low case) vertical,
horizontal componentw(x) longitudinal distribution of weight roll
angle density angular frequency18.2 INTRODUCTIONThe purpose of this
chapter is to present the
fundamentalsofdirectshipstructureanalysisbasedonmechanicsandstrength
of materials. Such analysis allows a rationally baseddesign that is
practical, efcient, and versatile, and that hasalready been
implemented in a computer program, tested,and proven.Analysis and
Design are two words that are very oftenassociated. Sometimes they
are used indifferently one forthe other even if there are some
important differences be-tween performing a design and completing
an analysis.18-1Chapter 18Analysis and Design of Ship
StructurePhilippe Rigo and Enrico RizzutoMASTER SETSDC 18.qxd Page
18-1 4/28/03 1:30 PMAnalysis refers to stress and strength
assessment of thestructure. Analysis requires information on loads
and needsan initial structural scantling design. Output of the
structuralanalysis is the structural response dened in terms of
stresses,deectionsandstrength. Then, theestimatedresponseiscompared
to the design criteria. Results of this
comparisonaswellastheobjectivefunctions(weight, cost, etc.)willshow
if updated (improved) scantlings are required.Design for structure
refers to the process followed to se-lect the initial structural
scantlings and to update these scant-lings from the early design
stage (bidding) to the detaileddesign stage (construction). To
perform analysis, initial de-sign is needed and analysis is
required to design. This ex-plains why design and analysis are
intimately linked,
butareabsolutelydifferent.Ofcoursedesignalsorelatestotopology and
layout denition.The organization and framework of this chapter are
basedon the previous edition of the Ship Design and Construction(1)
and on the Chapter IV of Principles of Naval Architec-ture (2).
Standard materials such as beam model, twisting,shear lag, etc.
that are still valid in 2002 are partly duplicatedfrom these 2
books. Other major references used to write
thischapterareShipStructuralDesign(3)alsopublishedbySNAME and the
DNV 99-0394 Technical Report
(4).ThepresentchapterisintimatelylinkedwithChapter11 Parametric
Design, Chapter 17 Structural Arrange-ment and Component Design and
with Chapter 19
Reli-ability-BasedStructuralDesign.Referencestothesechapters will
be made in order to avoid duplications. In ad-dition, as Chapter 8
deals with classication societies,
thepresentchapterwillfocusmainlyonthedirectanalysismethods
available to perform a rationally based structuraldesign, even if
mention is made to standard formulationsfrom Rules to quantify
design loads.In the following sections of this chapter, steps of a
globalanalysis are presented. Section 18.3 concerns the loads
thatare necessary to perform a structure analysis. Then,
Sections18.4, 18.5and18.6concern, respectively,
thestressesanddeections (basic ship responses), the limit states,
and the fail-ures modes and associated structural capacity. A
review ofthe available Numerical Analysis for Structural Design is
per-formedinSection18.7.FinallyDesignCriteria (Section18.8) and
Design Procedures (Section 18.9) are discussed.Structuralmodeling
isdiscussedinSubsection18.2.2andmore extensively in Subsection
18.7.2 for nite element analy-sis. Optimization is treated in
Subsections 18.7.6 and 18.9.4.Ship structural design is a
challenging activity. HenceHughes (3)
states:Thecomplexitiesofmodernshipsandthedemandforgreater
reliability, efciency, and economy require a sci-entic, powerful,
and versatile method for their structuraldesignBut, even with the
development of numerical techniques,design still remains based on
the designers experience andon previous designs. There are many
designs that satisfy thestrength criteria, but there is only one
that is the optimumsolution (least cost, weight, etc.).Ship
structural analysis and design is a matter of com-promises:
compromise between accuracy and the available time toperform the
design. This is particularly challenging
atthepreliminarydesignstage.A3DFiniteElementMethod (FEM) analysis
would be welcome but the timeisnotavailable.Forthatreason,
rule-baseddesignorsimplied numerical analysis has to be performed.
to limit uncertainty and reduce conservatism in design, itis
important that the design methods are accurate. On theother hand,
simplicity is necessary to make repeated de-sign analyses efcient.
The results from complex analy-ses should be veried by simplied
methods to avoid errorsand misinterpretation of results (checks and
balances). compromisebetweenweightandcostorcompromisebetween least
construction cost, and global owner livecycle cost (including
operational cost, maintenance, etc.),and builder optimum design may
be different from the owneroptimum design.18.2.1 Rationally Based
Structural Design versus Rules-Based DesignThere are basically two
schools to perform analysis and de-signofshipstructure. Therstone,
theoldest, iscalledrule-based design. It is mainly based on the
rules denedby the classication societies. Hughes (3) states:In the
past, ship structural design has been largely empir-ical, based on
accumulated experience and ship perform-ance, and expressed in the
form of structural design codesor rules published by the various
ship classication soci-eties. These rules concern the loads, the
strength and thedesign criteria and provide simplied and
easy-to-use for-mulas for the structural dimensions, or scantlings
of aship. Thisapproachsavestimeinthedesignofceand,since the ship
must obtain the approval of a classicationsociety, it also saves
time in the approval
process.ThesecondschoolistheRationallyBasedStructuralDesign; it is
based on direct analysis. Hughes, who couldbe considered as a
father of this methodology, (3) furtherstates:18-2 Ship Design
& Construction, Volume 1MASTER SETSDC 18.qxd Page 18-2 4/28/03
1:30 PMThere are several disadvantages to a completely
rulebookapproach to design. First, the modes of structural
failurearenumerous, complex, andinterdependent. Withsuchsimplied
formulas the margin against failure remains un-known; thus one
cannot distinguish between structural ad-equacy and over-adequacy.
Second, and most important,these formulas involve a number of
simplifying assump-tions and can be used only within certain
limits. Outsideof this range they may be inaccurate. For these
reasons there is a general trend toward directstructural
analysis.Even if direct calculation has always been
performed,design based on direct analysis only became popular
whennumerical analysis methods became available and were cer-tied.
Direct analysis has become the standard procedureinaerospace,
civilengineeringandpartlyinoffshorein-dustries. In ship design,
classication societies preferred tooffer updated rules resulting
from numerical analysis cali-bration. For the designer, even if the
rules were continuouslychanging, the design remained rule-based.
There really weretwo different methodologies.Hopefully, in 2002
this is no longer true. The advantagesof direct analysis are so
obvious that classication societiesinclude, usually as an
alternative, a direct analysis
procedure(numericalpackagesbasedontheniteelementmethod,see Table
18.VIII, Subsection 18.7.5.2). In addition, for newvessel types or
non-standard dimension, such direct proce-dure is the only way to
assess the structural safety. There-fore it seems that the two
schools have started a long mergingprocedure. Classication
societies are now encouraging
andcontributinggreatlytothedevelopmentofdirectanalysisand
rationally based methods. Ships are very complex struc-tures
compared with other types of structures. They are sub-ject to a
very wide range of loads in the harsh environmentof the sea.
Progress in technologies related to ship designand construction is
being made daily, at an unprecedentedpace. A notable example is the
fact that the efforts of a
ma-jorityofspecialiststogetherwithrapidadvancesincom-puter and
software technology have now made it possible toanalyze complex
ship structures in a practical manner usingstructural analysis
techniques centering on FEM analysis.The majority of ship designers
strive to develop rational andoptimal designs based on direct
strength analysis
methodsusingthelatesttechnologiesinordertorealizetheshipowners
requirements in the best possible way.When carrying out direct
strength analysis in order toverify the equivalence of structural
strength with rule re-quirements, it is necessary for the
classication society toclarify the strength that a hull structure
should have
withrespecttoeachofthevariousstepstakenintheanalysisprocess, from
load estimation through to strength evalua-tion. In addition, in
order to make this a practical and ef-fective method of analysis,
it is necessary to give carefulconsideration to more rational and
accurate methods of di-rect strength
analysis.Basedonrecognitionofthisneed,
extensiveresearchhasbeenconductedandacarefulexaminationmade,
re-garding the strength evaluation of hull structures. The re-sults
of this work have been presented in papers and reportsregarding
direct strength evaluation of hull structures (4,5).The ow chart
given in Figure 18.1 gives an overviewof the analysis as dened by a
major classication society.Note that a rationally based design
procedure requiresthat all design decisions (objectives, criteria,
priorities, con-straints) must be made before the design starts.
This is amajor difculty of this approach.18.2.2 Modeling and
AnalysisGeneral guidance on the modeling necessary for the
struc-tural analysis is that the structural model shall provide
re-sultssuitableforperformingbuckling, yield, fatigueandChapter 18:
Analysis and Design of Ship Structure 18-3Figure 18.1 Direct
Structural Analysis Flow ChartDirect Load AnalysisDesign LoadStudy
on Ocean WavesEffect onoperation Wave Load ResponseResponse
function of wave loadStructural analysis bywhole ship modelStress
response function Short term estimation Long term estimation Design
Sea State Design waveWave impact loadStructural response
analysisStrength Assessment Yield strength Nonlinear influencein
large wavesInvestigation on corrosion Bucklingstrength Ultimate
strength Fatigue strength Modeling techniqueDirect structural
analysisStress Response in WavesLong term estimation Short term
estimation MASTER SETSDC 18.qxd Page 18-3 4/28/03 1:30 PMvibration
assessment of the relevant parts of the vessel. Thisis done by
using a 3D model of the whole ship, supportedby one or more levels
of sub models.Several approaches may be applied such as a
detailed3D model of the entire ship or coarse meshed 3D model
sup-ported by ner meshed sub models.Coarse mesh can be used for
determining stress resultssuited for yielding and buckling control
but also to obtainthe displacements to apply as boundary conditions
for submodels with the purpose of determining the stress level
inmore detail.Strength analysis covers yield (allowable stress),
buck-ling strength and ultimate strength checks of the ship. In
ad-dition, specic analyses are requested for fatigue
(Subsection18.6.6),
collisionandgrounding(Subsection18.6.7)andvibration(Subsection18.6.8).Thehydrodynamicloadmodel
must give a good representation of the wetted sur-face of the ship,
both with respect to geometry descriptionand with respect to
hydrodynamic requirements. The massmodel, which is part of the
hydrodynamic load model, mustensure a proper description of local
and global moments ofinertia around the global ship axes.Ultimate
hydrodynamic loads from the hydrodynamicanalysis should be combined
with static loads in order toform the basis for the yield, buckling
and ultimate strengthchecks. All the relevant load conditions
should be examinedto ensure that all dimensioning loads are
correctly included.A ow chart of strength analysis of global model
and submodels is shown in Figure 18.2.18.2.3 Preliminary Design
versus Detailed DesignFor a ship structure, structural design
consists of two dis-tinct levels: the Preliminary Design and the
Detailed De-sign about which Hughes (3) states:The preliminary
determines the location, spacing, and scant-lings of the principal
structural members. The detailed de-sign determines the geometry
and scantlings of local structure(brackets, connections, cutouts,
reinforcements, etc.).
Preliminarydesignhasthegreatestinuenceonthestructuredesignandhenceisthephasethatoffersverylarge
potential savings. This does not mean that detail de-sign is less
important than preliminary design. Each levelis equally important
for obtaining an efcient, safe and re-liable ship. During the
detailed design there also are many
bene-tstobegainedbyapplyingmodernmethodsofengi-neeringscience,
buttheapplicationsaredifferentfrompreliminary design and the benets
are likewise different. Since the items being designed are much
smaller it ispossible to perform full-scale testing, and since they
aremore repetitive it is possible to obtain the benets of
massproduction, standardization and so on. In fact,
productionaspects are of primary importance in detail design.Also,
most of the structural items that come under de-tail design are
similar from ship to ship, and so in-serviceexperience provides a
sound basis for their design. In fact,because of the large number
of such items it would be in-efcient to attempt to design all of
them from rst
princi-ples.Insteaditisgenerallymoreefcienttousedesigncodes and
standard designs that have been proven by ex-perience. In other
words, detail design is an area where arule-based approach is very
appropriate, and the rules
thatarepublishedbythevariousshipclassicationsocietiescontain a
great deal of useful information on the design oflocal structure,
structural connections, and other structuraldetails.18.3 LOADSLoads
acting on a ship structure are quite varied and pecu-liar, in
comparison to those of static structures and also ofother vehicles.
In the following an attempt will be made toreview the main
typologies of loads: physical origins, gen-eral interpretation
schemes, available quantication proce-18-4 Ship Design &
Construction, Volume 1Figure 18.2 Strength Analysis Flow Chart
(4)Structural modelincluding necessaryload
definitionsHydrodynamic/staticloadsLoad transfer tostructural
modelVerified structuralmodelSub-models to beused in
structuralanalysisStructural analysisVerificationof
responseVerificationof model/loadsYesNoTransfer
ofdisplacements/forces to sub-model?Verificationof
loadtransferStructural drawings,mass description andloading
conditions.MASTER SETSDC 18.qxd Page 18-4 4/28/03 1:30 PMdures and
practical methods for their evaluation will be sum-marized.18.3.1
Classication of Loads18.3.1.1 Time DurationStatic loads: These are
the loads experienced by the ship instill water. They act with time
duration well above the rangeof sea wave periods. Being related to
a specic load con-dition,
theyhavelittleandveryslowvariationsduringavoyage (mainly due to
changes in the distribution of con-sumables on board) and they vary
signicantly only duringloading and unloading
operations.Quasi-staticloads: Asecondclassofloadsincludesthose with
a period corresponding to wave actions (3 to15 seconds). Falling in
this category are loads directly in-duced by waves, but also those
generated in the same fre-quency range by motions of the ship
(inertial forces). Theseloads can be termed quasi-static because
the structural re-sponse is studied with static
models.Dynamicloads: Whenstudyingresponseswithfre-quency components
close to the rst structural resonancemodes, the dynamic properties
of the structure have to beconsidered. This applies to a few types
of periodic loads,generated by wave actions in particular
situations (spring-ing) or by mechanical excitation (main engine,
propeller).Also transient impulsive loads that excite free
structural vi-brations (slamming, and in some cases sloshing loads)
canbe classied in the same category.High frequency loads: Loads at
frequencies higher thanthe rst resonance modes (> 10-20 Hz) also
are present onships: this kind of excitation, however, involves
more thestudy of noise propagation on board than structural
design.Other loads: All other loads that do not fall in the
abovementioned categories and need specic models can be gen-erally
grouped in this class. Among them are thermal andaccidental loads.A
large part of ship design is performed on the basis ofstatic and
quasi-static loads, whose prediction proceduresare quite well
established, having been investigated for alongtime.However,
specicandimposingrequirementscanariseforparticularshipsduetotheotherloadcate-gories.18.3.1.2
Local and global loadsAnother traditional classication of loads is
based on thestructural scheme adopted to study the
response.Loadsactingontheshipasawhole, consideredasabeam (hull
girder), are named global or primary loads andthe ship structural
response is accordingly termed global orprimary response (see
Subsection 18.4.3).Loads, dened in order to be applied to limited
struc-tural models (stiffened panels, single beams, plate
panels),generally are termed local
loads.Thedistinctionispurelyformal, asthesameexternalforces can in
fact be interpreted as global or local loads. Forinstance, wave
dynamic actions on a portion of the hull, ifdescribed in terms of a
bi-dimensional distribution of pres-sures over the wet surface,
represent a local load for the hullpanel, while, if integrated over
the same surface,
representacontributiontothebendingmomentactingonthehullgirder.This
terminology is typical of simplied structural analy-ses, in which
responses of the two classes of
componentsareevaluatedseparatelyandlatersummeduptoprovidethe total
stress in selected positions of the structure.In a complete 3D
model of the whole ship, forces on thestructure are applied
directly in their actual position and theresult is a total stress
distribution, which does not need tobe decomposed.18.3.1.3
Characteristic values for
loadsStructuralvericationsarealwaysbasedonalimitstateequation and
on a design operational time.Main aspects of reliability-based
structural design andanalysis are (see Chapter 19): the state of
the structure is identied by state variablesassociated to loads and
structural capacity, state variables are stochastically distributed
as a func-tion of time, and the probability of exceeding the limit
state surface in thedesign time (probability of crisis) is the
element subjectto evaluation.The situation to be considered is in
principle the worstcombination of state variables that occurs
within the designtime. The probability that such situation
corresponds to anout crossing of the limit state surface is
compared to a (low)target probability to assess the safety of the
structure.Thisgeneraltime-variantproblemissimpliedintoatime-invariant
one. This is done by taking into account inthe analysis the worst
situations as regards loads, and, sep-arately, as regards capacity
(reduced because of corrosionandotherdegradationeffects).
Thesimplicationliesinconsidering these two situations as
contemporary, which ingeneral is not the case.When dealing with
strength analysis, the worst load sit-uation corresponds to the
highest load cycle and is
charac-terizedthroughtheprobabilityassociatedtotheextremevalue in
the reference (design) time.In fatigue phenomena, in principle all
stress cycles con-tribute(toadifferentextent,
dependingontherange)toChapter 18: Analysis and Design of Ship
Structure 18-5MASTER SETSDC 18.qxd Page 18-5 4/28/03 1:30 PMdamage
accumulation. The analysis, therefore, does not re-gard the
magnitude of a single extreme load application, butthe number of
cycles and the shape of the probability dis-tribution of all stress
ranges in the design time.A further step towards the problem
simplication is
rep-resentedbytheadoptionofcharacteristicloadvaluesinplace of
statistical distributions. This usually is done, forexample, when
calibrating a Partial Safety Factor format forstructural checks.
Such adoption implies the denition of
asinglereferenceloadvalueasrepresentativeofawholeprobability
distribution. This step is often performed by as-signing an
exceeding probability (or a return period) to eachvariable and
selecting the correspondent value from the sta-tistical
distribution.The exceeding probability for a stochastic variable
hasthe meaning of probability for the variable to overcome agiven
value, while the return period indicates the mean timeto the rst
occurrence.Characteristic values for ultimate state analysis are
typ-ically represented by loads associated to an exceeding
prob-ability of 108. This corresponds to a wave load occurring,on
the average, once every 108cycles, that is, with a returnperiod of
the same order of the ship lifetime. In rst yield-ing analyses,
characteristic loads are associated to a higherexceeding
probability, usually in the range 104 to 106. Infatigue analyses
(see Subsection 18.6.6.2), reference loadsare often set with an
exceeding probability in the range 103to 105, corresponding to load
cycles which, by effect of bothamplitude and frequency of
occurrence, contribute more tothe accumulation of fatigue damage in
the structure.On the basis of this, all design loads for structural
analy-ses are explicitly or implicitly related to a low
exceedingprobability.18.3.2 Denition of Global Hull Girder LoadsThe
global structural response of the ship is studied withreference to
a beam scheme (hull girder), that is, a mono-dimensional structural
element with sectional characteris-tics distributed along a
longitudinal axis.Actionsonthebeamaredescribed,
asusualwiththisscheme, only in terms of forces and moments acting
in thetransverse sections and applied on the longitudinal
axis.Three components act on each section (Figure 18.3): aresultant
force along the vertical axis of the section (con-tained in the
plane of symmetry), indicated as vertical re-sultant force qV;
another force in the normal direction, (localhorizontal axis),
termed horizontal resultant force qHand amoment mTabout the x axis.
All these actions are distrib-uted along the longitudinal axis
x.Five main load components are accordingly generatedalongthebeam,
relatedtosectionalforcesandmomentthrough equation 1 to
5:[1][2][3][4][5]Due to total equilibrium, for a beam in free-free
condi-tions (no constraints at ends) all load characteristics
havezero values at ends (equations 6).These conditions impose
constraints on the distributionsof qV, qHand mT.[6]Global loads for
the verication of the hull girder are ob-tained with a linear
superimposition of still water and wave-induced global
loads.Theyareused, withdifferentcharacteristicvalues, indifferent
types of analyses, such as ultimate state, rst yield-ing, and
fatigue.18.3.3 Still Water Global LoadsStill water loads act on the
ship oating in calm water, usu-ally with the plane of symmetry
normal to the still watersurface. In this condition, only a
symmetric distribution ofhydrostatic pressure acts on each section,
together with ver-tical gravitational forces.If the latter ones are
not symmetric, a sectional torquemTg(x) is generated (Figure 18.4),
in addition to the verti-V (0) V (L) M (0) M (L) 0V (0) V (L) M (0)
M (L) 0M (0) M (L) 0V V V VH H H HT T M (x) m ( )dT T0x M (x) V (
)dH H0x V (x) q )dH H0x( M (x) V ( )dV V0x V (x) q ( )dV V0x 18-6
Ship Design & Construction, Volume 1Figure 18.3 Sectional
Forces and MomentMASTER SETSDC 18.qxd Page 18-6 4/28/03 1:30 PMcal
load qSV(x), obtained as a difference between buoyancyb(x) and
weight w(x), as shown in equation 7 (2).[7]where AI= transversal
immersed area.Components of vertical shear and vertical bending
canbe derived according to equations 1 and 2. There are no
hor-izontal components of sectional forces in equation 3 and
ac-cordingly no components of horizontal shear and bendingmoment.
As regards equation 5, only mTg, if present, is tobe accounted for,
to obtain the torque.18.3.3.1 Standard still water bending
momentsWhile buoyancy distribution is known from an early stageof
the ship design, weight distribution is completely denedonly at the
end of construction. Statistical formulations,
cal-ibratedonsimilarships,
areoftenusedinthedesignde-velopmenttoprovideanapproximatequanticationofweight
items and their longitudinal distribution on
board.Theresultingapproximatedweightdistribution, togetherwith the
buoyancy distribution, allows computing shear andbending moment.q
(x) b(x) w(x) gA (x) m(x)gSV I At an even earlier stage of design,
parametric formula-tions can be used to derive directly reference
values for stillwater hull girder loads.Common reference values for
still water bending mo-ment at mid-ship are provided by the major
ClassicationSocieties (equation 8).[8]where C = wave parameter
(Table 18.I).The formulations in equation 8 are sometimes
explicitlyreportedinRules,
buttheycananywaybeindirectlyde-rivedfromprescriptionscontainedin(6,
7). Therstre-quirement (6) regards the minimum longitudinal
strengthmodulus and provides implicitly a value for the total
bend-ing moment; the second one (7), regards the wave
inducedcomponent of bending moment.Longitudinal distributions,
depending on the ship type,are provided also. They can slightly
differ among Class So-cieties, (Figure 18.5).18.3.3.2 Direct
evaluation of still water global loadsClassication Societies
require in general a direct analysisof these types of load in the
main loading conditions of theship, such as homogenous loading
condition at maximumdraft, ballast conditions, docking conditions
aoat, plus allother conditions that are relevant to the specic ship
(non-homogeneous loading at maximum draft, light load at lessthan
maximum draft, short voyage or harbor condition, bal-last exchange
at sea, etc.).Thedirectevaluationprocedurerequires,
foragivenloading condition, a derivation, section by section, of
ver-ticalresultantsofgravitational(weight)andbuoyancyforces,
applied along the longitudinal axis x of the beam.To obtain the
weight distribution w(x), the ship length issubdivided into
portions: for each of them, the total weightand center of gravity
is determined summing up contributionsfrom all items present on
board between the two boundingsections. The distribution for w(x)
is then usually approxi-mated by a linear (trapezoidal) curve
obtained by imposingM NmCLB 122.5 15 C (hogging)CLB 45.5 65 C
(sagging)s2B2B ( )+[ ] ( )Chapter 18: Analysis and Design of Ship
Structure 18-7Figure 18.4 Sectional Resultant Forces in Still
WaterFigure 18.5 Examples of Reference Still Water Bending Moment
Distribution(10). (a) oil tankers, bulk carriers, ore carriers, and
(b) other ship typesTABLE 18.I Wave Coefcient Versus LengthShip
Length L Wave Coefcient C90 L '0 5140 5..B=
criticalbucklingstrength(thatis, Bforshear stress)F= Yfor normal
stress= Y 43 for shear stressY= material yield stressIn ship rules
and books, equation 47 may appear withsomewhat different constants
depending on the structuralproportional limit assumed. The above
form assumes a struc-tural proportional limit of a half the
applicable yield value.Foraxialtensileloading, thecritical
strengthmaybeconsidered to equal the material yield stress
(Y).Under single types of loads, the critical plate
bucklingstrengthmustbegreaterthanthecorrespondingappliedstresscomponentwiththerelevantmarginofsafety.Forcombined
biaxial compression/tension and edge shear, thefollowing type of
critical buckling strength interaction cri-terion would need to be
satised, for example:[48]where:B= usage factor for buckling
strength, which is typicallythe inverse of the conventional partial
safety factor.B= 1.0 is often taken for direct strength
calculation, whileit is taken less than 1.0 for practical design in
accor-dance with classication society
rules.Compressivestressistakenasnegativewhiletensilestress is taken
as positive and = 0 if both xavand yavarecompressive, and = 1 if
either xavor yavor both are ten-sile. The constant c is often taken
as c = 2.Figure 18.40 shows a typical example of the axial
mem-brane stress distribution inside a plate element under
pre-dominantlylongitudinalcompressiveloadingbeforeandafter buckling
occurs. It is noted that the membrane stressdistribution in the
loading (x) direction can become non-uniform as the plate element
deforms. The membrane stressdistributioninthey
directionmayalsobecomenon-uni-form with the unloaded plate edges
remaining straight, whileno membrane stresses will develop in the y
direction if theunloaded plate edges are free to move in plane. As
evident,the maximum compressive membrane stresses are
developedaround the plate edges that remain straight, while the
min-imum membrane stresses occur in the middle of the plateelement
where a membrane tension eld is formed by theplate deection since
the plate edges remain straight.With increase in the deection of
the plate keeping theedges straight, the upper and/or lower bers
inside the mid-dle of the plate element will initially yield by the
action ofbending. However, as long as it is possible to
redistribute xavxBcxavxByavyByavyBcavBcB
_, +
_,
+
_, Chapter 18: Analysis and Design of Ship Structure 18-39Figure
18.39 A Simply Supported Rectangular Plate Subject to
BiaxialCompression/tension, Edge Shear and Lateral Pressure
LoadsMASTER SETSDC 18.qxd Page 18-39 4/28/03 1:30
PMtheappliedloadstothestraightplateboundariesbythemembrane action,
the plate element will not collapse. Col-lapse will then occur when
the most stressed boundary lo-cationsyield,
sincetheplateelementcannotkeeptheboundaries straight any further,
resulting in a rapid increaseof lateral plate deection (33).
Because of the nature of ap-pliedaxialcompressiveloading,
thepossibleyieldloca-tions are longitudinal mid-edges for
longitudinal uniaxialcompressive loads and transverse mid-edges for
transverseuniaxial compressive loads, as shown in Figure 18.41.The
occurrence of yielding can be assessed by using thevon Mises yield
criterion (equation 45). The following con-ditions for the most
probable yield locations will then befound.(a) Yielding at
longitudinal edges:[49a](b) Yielding at transverse edges:[49b]The
maximum and minimum membrane stresses of
equa-tions49aand49bcanbeexpressedintermsofappliedstresses, lateral
pressure loads and fabrication related ini-tial imperfections, by
solving the nonlinear governing dif-ferentialequationsofplating,
basedonequilibriumandcompatibility equations. Note that equation 44
is the lineardifferential equation.On the other hand, the plate
ultimate edge shear strength,u, is often taken u=B(equation 47,
with B instead of B).Also, an empirical formula obtained by curve
tting basedon nonlinear nite element solutions may be utilized
(33).The effect of lateral pressure loads on the plate ultimate
edgeshear strength may in some cases need to be accounted for. x
minx min y max y max2 2 2 + Y x max x max y miny min2 2 2 + Y18-40
Ship Design & Construction, Volume 1Figure 18.40 Membrane
Stress Distribution Inside the Plate Element underPredomianntly
Longitudinal Compressive Loads; (a) Before buckling, (b)
Afterbuckling, unloaded edges move freely in plane, (c) After
buckling, unloadededges kept straightFigure 18.41 Possible
Locations for the Initial Plastic Yield at the Plate Edges(Expected
yield locations, T: Tension, C: Compression); (a) Yield at
longitudinalmid-edges under longitudinal uniaxial compression, (b)
Yield at transversemid-edges under transverse uniaxial
compression)MASTER SETSDC 18.qxd Page 18-40 4/28/03 1:31 PMFor
combined biaxial compression/tension, edge shearand lateral
pressure loads, the last being usually
regardedasagivenconstantsecondaryload, theplateultimatestrength
interaction criterion may also be given by an
ex-pressionsimilartoequation48,
butreplacingthecriticalbucklingstrengthcomponentsbythecorrespondingulti-mate
strength components, as follows:[50]where: and c = variables dened
in equation 48u= usage factors for the ultimate limit statexuand
yu= solutions of equation 49a with regard to xavand equation 49b
with regard to yav, respec-tively18.6.3.2 Simplied modelsIn the
interest of simplicity, the elastic plate buckling
strengthcomponents under single types of loads may sometimes
becalculated by neglecting the effects of in-plane bending
orlateral pressure loads. Without considering the effect of
lat-eral pressure, the resulting elastic buckling strength
predic-tion would be pessimistic. While the plate edges are
oftensupposed to be simply supported, that is, without
rotationalrestraints along the plate/stiffener junctions, the real
elasticbuckling strength with rotational restraints would of
coursebe increased by a certain percentages, particularly for
heavystiffeners. This arises from the increased torsional
restraintprovided at the plate edges in such cases.The theoretical
solution for critical buckling stress, B,in the elastic range has
been found for a number of casesof interest. For rectangular plate
subject to compressive in-plane stress in one direction:[51]Here
kcis a function of the plate aspect ratio, = a/ b,the boundary
conditions on the plate edges and the type ofloading. If the load
is applied uniformly to a pair of oppo-site edges only, and if all
four edges are simply supported,then kc is given by:[52]where m is
the number of half-waves of the deected platein the longitudinal
direction, which is taken as an integersatisfying the conditionFor
long plate in m(m + 1).kmmc +_, 2B ckE tb_,22212 1 ( )
xavxucxavxuyavyuyavyucavucu
_, +
_,
+
_, compression (a > b), kc = 4,and for wide plate (a b)
incompression, kc = (1 + b2/ a2)2, for simply supported edges.For
shear force, the critical buckling shear stress, B, canalso be
obtain by equation 51 and the buckling coefcientfor simply
supported edges is:kc = 5.34 + 4(b/a)2[53]Figure 18.42 presents,
kc, versus the aspect ratio, a/b, fordifferent congurations of
rectangular plates in compression.For the simplied prediction of
the plate ultimate strengthunder uniaxial compressive loads, one of
the most common ap-proaches is to assume that the plate will
collapse if the maxi-mum compressive stress at the plate corner
reaches the materialyield stress, namely x max = Y for xavor y max
= Y for yav.Thisassumptionisrelevantwhentheunloadededgesmove freely
in plane as that shown in Figure 40(b). Anotherapproximate method
is to use the plate effective width con-cept, which provides the
plate ultimate strength componentsChapter 18: Analysis and Design
of Ship Structure 18-41Figure 18.42 Compressive Buckling Coefcient
for Plates in Compression; for5 Congurations (2) (A, B, C, D and E)
where Boundary Conditions of UnloadedEdges are: SS: Simply
Supported, C: Clamped, and F: FreeMASTER SETSDC 18.qxd Page 18-41
4/28/03 1:31 PMunder uniaxial compressive stresses (xuand yu), as
fol-low:[54]where aeuand beuare the plate effective length and
width atthe ultimate limit state, respectively.While a number of
the plate effective width expressionshave been developed, a typical
approach is exemplied byFaulkner, who suggests an empirical
effective width (beu /b)formula for simply supported steel plates,
as follows, for longitudinal axial compression (34),[55a] for
transverse axial compression (35),[55b]where: =is the plate
slendernessE = the Youngs modulust = the plate thicknessc1, c2=
typically taken as c1 = 2 and c2 = 1The plate ultimate strength
components under uniaxialcompressive loads are therefore predicted
by substitutingthe plate effective width formulae (equation 55a)
into equa-tion 54.Morechartsandformulationsareavailableinmanybooks,
forexample, Bleich(36), ECCS-56(37), Hughes(3) and Lewis (2). In
addition, the design strength of plate(unstiffened panels) is
detailed in Chapter 19, Subsection19.5.4.1, including an example of
reliability-based designand alternative equations to equations 56
and 57.18.6.3.3 Design criteriaWhen a single load component is
involved, the buckling orultimate strength must be greater than the
corresponding
ap-pliedstresscomponentwithanappropriatetargetpartialsafety factor.
In a multiple load component case, the struc-tural safety check is
made with equation 48 against
buck-lingandequation50againstultimatelimitstate beingsatised.To
ensure that the possible worst condition is met (buck-ling and
yield) for the ship, several stress combination mustbe considered,
as the maximum longitudinal and transversebt EYaabaeu +
_,
0 9 1 910 92 2. . .bbforc cforeu< '1 111 22xuYeuyuYeubbaa
andcompressiondonotoccursimultaneously.Forinstance,DNV (4)
recommends: maximumcompression, x, inaplateeldandphaseangle
associated with y, (buckling control), maximumcompression, y,
inaplateeldandphaseangle associated with x, (buckling control),
absolute maximum shear stress, , in a plate eld andphase angle
associated with x, y(buckling control),and maximum equivalent von
Mises stress, e,at given po-sitions (yield control).In order to get
xand y, the following stress compo-nents may normally be considered
for the buckling control:1= stress from primary response, and2=
stress from secondary response (that is, doublebottom
bending).Asthelateralbendingeffectsshouldbenormallyin-cluded in the
buckling strength formulation, stresses fromlocal bending of
stiffeners (secondary response), 2*, andlocal bending of plate
(tertiary response), 3, must there-fore not to be included in the
buckling control. If FE-analy-sis is performed the local plate
bending stress, 3, can easilybe excluded using membrane
stresses.18.6.4 Buckling and Ultimate Strength of
StiffenedPanelsFor the structural capacity analysis of stiffened
panels, it ispresumed that the main support members including
longi-tudinalgirders, transversewebsanddeepbeamsarede-signed with
proper proportions and stiffening systems sothat their instability
is prevented prior to the failure of thestiffened panels they
support.In many ship stiffened panels, the stiffeners are
usuallyattached in one direction alone, but for generality, the
de-sign criteria often consider that the panel can have stiffen-ers
in one direction and webs or girders in the other, thisarrangement
corresponds to a typical ship stiffened panels(Figure18.43a).
Thestiffenersandwebs/girdersareat-tached to only one side of the
panel.The number of load components acting on stiffened steelpanels
are generally of four types, namely biaxial loads, thatis
compression or tension, edge shear, biaxial in-plane bend-ing and
lateral pressure, as shown in Figure 18.43. When thepanel size is
relatively small compared to the entire structure,the inuence of
in-plane bending effects may be negligible.However, for a large
stiffened panel such as that in sideshellofships,
theeffectofin-planebendingmaynotbenegligible, since the panel may
collapse by failure of stiff-18-42 Ship Design & Construction,
Volume 1MASTER SETSDC 18.qxd Page 18-42 4/28/03 1:31
PMenerswhichareloadedbylargestaddedportionofaxialcompression due to
in-plane bending
moments.Whenthestiffenersarerelativelysmallsothattheybuckle
together with the plating, the stiffened panel typi-cally behaves
as an orthotropic plate. In this case, the av-erage values of the
applied axial stresses may be used byneglecting the inuence of
in-plane bending. When the stiff-eners are relatively stiff so that
the plating between stiffen-ersbucklesbeforefailureofthestiffeners,
theultimatestrength is eventually reached by failure of the most
highlystressed stiffeners. In this case, the largest values of the
axialcompressive or tensile stresses applied at the location of
thestiffeners are used for the failure analysis of the
stiffeners.In stiffened panels of ship structures, material
properties ofthe stiffeners including the yield stress are in some
casesdifferent from that of the plate. It is therefore necessary
totake into account this effect in the structural capacity
for-mulations, at least approximately.For analysis of the ultimate
strength capacity of stiffenedpanels which are supported by
longitudinal girders, trans-versewebsanddeepbeams,
itisoftenassumedthatthepanel edges are simply supported, with zero
deection andzero rotational restraints along four edges, with all
edgeskept
straight.Thisidealizationmayprovidesomewhatpessimistic,but adequate
predictions of the ultimate strength of stiffenedpanels supported
by heavy longitudinal girders, transversewebs and deep beams (or
bulkheads).Today, directnon-linearstrengthassessmentmethodsusing
recognized programs is usual (38). The model shouldChapter 18:
Analysis and Design of Ship Structure 18-43Figure 18.43 A Stiffened
Steel Panel Under Biaxial Compression/Tension,Biaxial In-plane
Bending, Edge Shear and Lateral Pressure Loads. (a)
StiffenedPanelLongitudinals and Frames (4), and (b) A Generic
Stiffened Panel (38).(a)(b)Figure 18.44 Modes of Failures by
Buckling of a Stiffened Panel (2). (a) Elastic buckling of plating
between stiffeners (serviceability limit state).(b) Flexural
buckling of stiffeners including plating (plate-stiffener
combination,mode III).(c) Lateral-torsional buckling of stiffeners
(trippingmode V).(d) Overall stiffened panel buckling (grillage or
gross panel bucklingmode I).(a)(b)(c)(d)MASTER SETSDC 18.qxd Page
18-43 4/28/03 1:31
PMbecapableofcapturingallrelevantbucklingmodesanddetrimental
interactions between them. The fabrication re-lated initial
imperfections in the form of initial deections(plates, stiffeners)
and residual stresses can in some casessignicantly affect (usually
reduce) the ultimate strength ofthe panel so that they should be
taken into account in thestrength computations as parameters of
inuence.18.6.4.1 Direct analysisThe primary modes for the ultimate
limit state of a stiffenedpanel subject to predominantly axial
compressive loads maybe categorized as follows (Figure 18.44): Mode
I: Overall collapse after overall buckling, Mode II: Plate induced
failureyielding of the plate-stiffener combination at panel edges,
Mode III: Plate induced failureexural buckling fol-lowed by
yielding of the plate-stiffener combination atmid-span, Mode IV:
Stiffener induced failurelocal buckling ofstiffener web, Mode V:
Stiffener induced failuretripping of stiffener,and Mode VI: Gross
yielding.Calculation of the ultimate strength of the stiffened
panelunder combined loads taking into account all of the possi-ble
failure modes noted above is not straightforward, be-cause of the
interplay of the various factors previously notedsuch as geometric
and material properties, loading,
fabri-cationrelatedinitialimperfections(initialdeectionandwelding
induced residual stresses) and boundary conditions.As an
approximation, the collapse of stiffened panels is thenusually
postulated to occur at the lowest value among thevarious ultimate
loads calculated for each of the above col-lapse patterns.This
leads to the easier alternative wherein one calcu-lates the
ultimate strengths for all collapse modes mentionedabove separately
and then compares them to nd the min-imum value which is then taken
to correspond to the realpanel ultimate strength. The failure mode
of stiffened pan-els is a broad topic that cannot be covered
totally within
thischapter.Manysimplieddesignmethodshaveofcoursebeen previously
developed to estimate the panel ultimatestrength,
consideringoneormoreofthefailuremodesamong those mentioned above.
Some of those methods havebeen reviewed by the ISSC2000 (39). On
the other hand,a few authors provide a complete set of formulations
thatcover all the feasible failure modes noted previously,
namely,Dowling et al (40), Hughes (3), Mansour et al (41,42),
andmore recently Paik
(38).Assessmentofdifferentformulationsbycomparisonwith experimental
and/or FE analysis are available (43-45).An example of
reliability-based assessment of the stiff-ened panel strength is
presented in Chapter 19. Formula-tions of Herzog, Hughes and
Adamchack are also discussed.18.6.4.2 Simplied
modelsExistingsimpliedmethodsforpredictingtheultimatestrength of
stiffened panels typically use one or more of thefollowing
approaches: orthotropic plate approach, plate-stiffener combination
approach (or beam-columnapproach), and grillage approach.These
approaches are similar to those presented in Sub-section 18.4.4.1
for linear analysis. All have the same back-ground but, here, the
buckling and the ultimate strength isconsidered.In the orthotropic
plate approach, the stiffened panel isidealized as an equivalent
orthotropic plate by smearing thestiffeners into the plating. The
orthotropic plate theory willthen be useful for computation of the
panel ultimate strengthfortheoverallgrillagecollapsemode(ModeI,
Figure18.44d), (31,46,48).Theplate-stiffenercombination approach
(alsocalledbeam-column approach) models the stiffened panel
behav-ior by that of a single beam consisting of a stiffener
to-getherwiththeattachedplating, asrepresentativeofthestiffened
panel (Figure 18.38, level 3b). The beam is con-sidered to be
subjected to axial and lateral line loads. Thetorsional rigidity of
the stiffened panel, the Poisson ratio
ef-fectandtheeffectoftheintersectingbeamsareallneg-lected.Thebeam-columnapproachisusefulforthecomputation
of the panel ultimate strength based on ModeIII, which is usually
an important failure mode that must beconsidered in design. The
degree of accuracy of the beam-column idealization may become an
important considera-tion when the plate stiffness is relatively
large compared
totherigidityofstiffenersand/orundersignicantbiaxialloading.Stiffened
panels are asymmetric in geometry about theplate-plane. This
necessitates strength control for both plateinduced failure and
stiffener-induced failure.Plate induced failure: Deection away from
the plate as-sociated with yielding in compression at the
connection
be-tweenplateandstiffener.Thecharacteristicbucklingstrength for the
plate is to be used.Stiffener induced failure: Deection towards the
plate as-sociated with yielding in compression in top of the
stiffeneror torsional buckling of the stiffener.Various column
strength formulations have been used as18-44 Ship Design &
Construction, Volume 1MASTER SETSDC 18.qxd Page 18-44 4/28/03 1:31
PMthe basis of the beam-column approach, three of the morecommon
types being the following: Johnson-Ostenfeld (or Bleich-Ostenfeld)
formulation, Perry-Robertson formulation, and empirical
formulations obtained by curve tting exper-imental or numerical
data.A stocky panel that has a high elastic buckling strengthwill
not buckle in the elastic regime and will reach the ulti-mate limit
state with a certain degree of plasticity. In
mostdesignrulesofclassicationsocieties,
theso-calledJohn-son-Ostenfeld formulation is used to account for
this behav-ior(equation47).Ontheotherhand,
intheso-calledPerry-Robertsonformulation,
thestrengthexpressionas-sumes that the stiffener with associated
plating will collapseas a beam-column when the maximum compressive
stress inthe extreme ber reaches the yield strength of the
material.In empirical approaches, the ultimate strength
formula-tionsaredevelopedbycurvettingbasedonmechanicalcollapse test
results or numerical solutions. Even if limitedto a range of
applicability (load types, slenderness
ranges,assumedlevelofinitialimperfections, etc.)theyareveryuseful
for preliminary design stage, uncertainty assessmentand as
constraint in optimization package. While a vast
num-berofempiricalformulations(sometimescalledcolumncurves) for
ultimate strength of simple beams in steel framedstructures have
been developed, relevant empirical formu-lae for plate-stiffener
combination models are also available.As an example of the latter
type, Paik and Thayamballi (49)developed an empirical formula for
predicting the ultimatestrength of a plate-stiffener combination
under axial com-pression in terms of both column and plate
slenderness ra-tios, based on existing mechanical collapse test
data for theultimate strength of stiffened panels under axial
compres-sionandwithinitialimperfections(initialdeectionsandresidualstresses)atanaverage
level.Sincetheultimatestrength of columns (u) must be less than the
elastic col-umn buckling strength (E), the Paik-Thayamballi
empiri-cal formula for a plate-stiffener combination is given
by:[56]andwith btYEuYEY 12 uY+ + + 10 995 0 93620 1720 1882 20
0674. . . . .andwhere:r = radius of gyration= 4I / A, (m)I =
inertia, (m4)A = cross section of the plate-stiffener combination
with fullattached plating, (m2)t = plate thickness, (m)a = span of
the stiffeners, (m)b = spacing between 2 longitudinals, (m)Note
that A, I, r, ... refer to the full section of the
plate-stiffenercombination, thatis, withoutconsideringanef-fective
plating.Figure 18.45 compares the Johnson-Ostenfeld
formula(equation 47), the Perry-Robertson formula and the
Paik-Thayamballi empirical formula (equation 56) for on the
col-umnultimatestrengthforaplate-stiffenercombinationvarying the
column slenderness ratios, with selected
initialeccentricityandplateslendernessratios.InusageofthePerry-Robersonformula,
thelowerstrengthasobtainedfrom either plate induced failure or
stiffener-induced
fail-ureisadoptedherein.Interactionbetweenbendingaxial
arYEYEChapter 18: Analysis and Design of Ship Structure 18-45Figure
18.45 A Comparison of the Ultimate Strength Formulations for
Plate-stiffener Combinations under Axial Compression ( relates to
the initial deection)MASTER SETSDC 18.qxd Page 18-45 4/28/03 1:31
PMcompression and lateral pressure can, within the same fail-ure
mode (Flexural BucklingMode III), leads to three-fail-ure scenario:
plate induced failure, stiffener induced failureor a combined
failure of stiffener and plating (see Chapter19 Figure 19.11
).18.6.4.3 Design criteriaThe ultimate strength based design
criteria of stiffened pan-els can also be dened by equation 50, but
using the
corre-spondingstiffenedpanelultimatestrengthandstressparameters.
Either all of the six design criteria, that is, againstindividual
collapse modes I to VI noted above, or a single de-sign criterion
in terms of the real (minimum) ultimate strengthcomponents must be
satised. For stiffened panels follow-ing Mode I behavior, the
safety check is similar to a plate,using average applied stress
components. The applied axialstress components for safety
evaluation of the stiffened panelfollowing Modes IIVI behavior will
use the maximum axialstresses at the most highly stressed
stiffeners.18.6.5 Ultimate Bending Moment of Hull GirderUltimate
hull girder strength relates to the maximum loadthat the hull
girder can support before collapse. These loadsinduce vertical and
horizontal bending moment, torsionalmoment, vertical and horizontal
shear forces and axial force.For usual seagoing vessels axial force
can be neglected. Asthe maximun shear forces and maximum bending
momentdo not occur at the same place, ultimate hull girder
strengthshould be evaluated at different locations and for a range
ofbending moments and shear forces.The ultimate bending moment (Mu)
refers to a combinedverticalandhorizontalbendingmoments(Mv,
Mh);thetransverse shear forces (Vv,Vh) not being considered.
Then,the ultimate bending moment only corresponds to one ofthe
feasible loading cases that induce hull girder collapse.Today, Muis
considered as being a relevant design case.Two major references
related to the ultimate strength ofhull girder are, respectively,
for extreme load and ultimatestrength, Jensen et al (24) and Yao et
al (50). Both presentcomprehensive works performed by the Special
Task Com-mittees of ISSC 2000. Yao (51) contains an historical
re-view and a state of art on this
matter.ComputationofMudependscloselyontheultimatestrength of the
structures constituent panels, and particularlyon the ultimate
strength in compressed panels or components.Figure 18.46 shows that
in sagging, the deck is compressed(deck) and reaches the ultimate
limit state when deck= u.On the other hand, the bottom is in
tensile and reaches its ul-timatelimitstateaftercompleteyielding,
bottom=0(0being the yield stress).Basically, there exist two main
approaches to evaluatethe hull girder ultimate strength of a ships
hull under lon-gitudinal bending moments. One, the approximate
analy-sis, istocalculatetheultimatebendingmomentdirectly(Mu, point
C on Figure 18.46), and the other is to performprogressive collapse
analysis on a hull girder and obtain,both, Muand the curves
M-.Therstapproach, approximateanalysis, requiresanassumption on the
longitudinal stress distribution. Figure18.47 shows several
distributions corresponding to differ-ent methods. On the other
hand, the progressive collapseanalysis does not need to know in
advance this distribution.Accordingly, to determine the global
ultimate bendingmoment (Mu), one must know in advance the ultimate
strength of each compressed panel (u), and the average
stress-average strain relationship (), toperform a progressive
collapse analysis.Foranapproximateassessment,
suchastheCaldwellmethod, only the ultimate strength of each
compressed panel(u) is required.18.6.5.1 Direct analysisThe direct
analysis corresponds to the Progressive collapseanalysis. The
methods include the typical numerical analy-18-46 Ship Design &
Construction, Volume 1Figure 18.46 The Moment-Curvature Curve
(M-)Figure 18.47 Typical Stress Distributions Used by Approximate
Methods. (a)First Yield. (b) Sagging Bending Moment (c) Evans (d)
PaikMansour (e)Caldwell Modied (f) Plastic Bending Moment.(a) (b)
(c) (d) (e) (f)MASTER SETSDC 18.qxd Page 18-46 4/28/03 1:31 PMsis
such as Finite Element Method (FEM) and the
IdealizedstructuralElementmethod(ISUM) andSmithsmethod,which is a
simplied procedure to perform progressive col-lapse analysis.FEM:
is the most rational way to evaluate the ultimatehull girder
strength through a progressive collapse analysison a ships hull
girder. Both material and geometrical non-linearities can be
considered.A3Danalysisofaholdorashipssectionisfunda-mentally
possible but very difcult to perform. This is be-cause a ships hull
is too large and complicated for such kindof analysis.
Nevertheless, since 1983 results of FEM analy-ses have been
reported (52). Today, with the developmentof computers, it is
feasible to perform progressive collapseanalysis on a hull girder
subjected to longitudinal bendingwith ne mesh using ordinary
elements. For instance, theinvestigation committee on the causes of
the Nakhodka ca-sualty performed elastoplastic large deection
analysis withnearly 200 000 elements (53).However, the modeling and
analysis of a complete hullgirder using FEM is an enormous task.
For this reason theanalysis is more conveniently performed on a
section of thehull that sufciently extends enough in the
longitudinal di-rection to model the characteristic behavior. Thus,
a typi-calanalysismayconcernoneframespacinginawholecompartment
(cargo tank). These analyses have to be sup-plemented by
information on the bending and shear loadsthat act at the fore and
aft transverse loaded sections. SuchFinite Element Analysis (FEA)
has shown that accuracy islimited because of the boundary
conditions along the trans-verse sections where the loading is
applied, the position ofthe neutral axis along the length of the
analyzed section andthe difculty to model the residual
stresses.Idealized Structural Unit Method (ISUM): presented
inSubsection 18.7.3.1, can also be used to perform
progres-sivecollapseanalysis.Itallowscalculatingtheultimatebending
moment through a 3D progressive collapse analy-sis of an entire
cargo hold. For that purpose, new elementsto simulate the actual
collapse of deck and bottom platingare actually
underdevelopment.SmithsMethod(Figure18.48):
Aconvenientalterna-tivetoFEMistheSmithsprogressivecollapseanalysis(54),
which consists of the following three steps (55).Step 1:
Modeling(meshmodelingofthecross-sectioninto elements),Step 2:
Derivation of average stress-average strain
rela-tionshipofeachelement( curve), Figure18.49a.Step 3: To perform
progressive collapse analysis, Figure18.49b.Chapter 18: Analysis
and Design of Ship Structure 18-47Figure 18.49 Inuence of Element
Average Stress-Average Strain Curves () on Progressive Collapse
Behavior. (a) Average stress-average strainrelationships of
element, and (b) moment-curvature relationship of
cross-section.(a)(b)Figure 18.48 The Smiths Progressive Collapse
MethodMASTER SETSDC 18.qxd Page 18-47 4/28/03 1:31 PMIn Step 1, the
cross-section of a hull girder is dividedinto elements composed of
a longitudinal stiffener and at-tached plating. In Step 2, the
average stress-average strainrelationship () of this stiffener
element is derived underthe axial load considering the inuences of
buckling andyielding. Step 3 can be explained as follows: axial
rigidities of individual elements are calculated usingthe average
stress-average strain relationships (), exural rigidity of the
cross-section is evaluated usingthe axial rigidities of elements,
vertical and horizontal curvatures of the hull girder areapplied
incrementally with the assumption that the planecross-section
remains plane and that the bending occursabout the instantaneous
neutral axis of the cross-section,
thecorrespondingincrementalbendingmomentsareevaluated and so the
strain and stress increments in in-dividual elements, and
incrementalcurvaturesandbendingmomentsofthecross-section as well as
incremental strains and stressesof elements are summed up to
provide their cumulativevalues.Figure 18.48 shows that the curves
are used to es-timate the bending moment carried by the complete
trans-verse section (Mi). The contribution of each element
(dM)depends on its location in the section, and specically onits
distance from the current position of the neutral axis (Yi).The
contribution will then also depend on the strain that isapplied to
it, since = y , where is the hull curvatureand y is the distance
from the neutral axis (simple beam as-sumption). The average
stress-average strain curve (-)will then provide an estimate of the
longitudinal stress (i)acting on the section. Individual moments
about the neu-tral axis are then summed to give the total bending
momentfor a particular curvature
i.Theaccuracyofthecalculatedultimatebendingmo-ment depends on the
accuracy of the average stress-aver-agestrain
relationshipsofindividualelements.Maindifcultiesconcernthemodelingofinitialimperfections(deection
and welding residual stress) and the boundaryconditions (multi-span
model, interaction between adjacentelements, etc.).Many
formulations and methods to calculate these
av-eragestress-averagestrain relationshipsareavailable:Adamchack
(56), Beghin et al (57), Dow et al (58), Gordoand Guedes Soares
(59,60) and, Yao and Nikolov (61,62).The FEM can even be used to
get these curves (Smith 54).For most of the methods, typical
element types are: plateelement, beam-column element (stiffener and
attached plate)and hard corner.An interesting well-studied ship
that reached its ultimatebending moment is the Energy Concentration
(63). It fre-quently is used as a reference case (benchmark) by
authorsto validate methods.Figure 18.49 shows typical average
stress-average strainrelationships,
andtheassociatedbendingmoment-curva-ture relationships (M-). Four
typical curves are con-sidered, which are:Case A: Linear
relationship (elastic). The M- relationshipis free from the
inuences of yielding and buck-ling, and is linear.Case B:
Bi-linearrelationship(elastic-perfectlyplastic,without
buckling).Case C: With buckling but without strength reduction
be-yond the ultimate strength.Case D:
Withbucklingandastrengthreductionbeyondthe ultimate strength
(actual behavior).In Case B, where yielding takes place but no
buckling,the deck initially undergoes yielding and then the
bottom.With the increase in curvature, yielded regions spread in
theside shell plating and the longitudinal bulkheads towardsthe
plastic neutral axis.In this case, the maximum bending moment is
the fullyplastic bending moment (Mp) of the cross-section and
itsabsolute value is the same both in the sagging and the hog-ging
conditions.For Cases C and D, the element strength is limited
byplate buckling, stiffener exural buckling, tripping, etc. ForCase
C, it is assumed that the structural components can
con-tinuetocarryloadafterattainingtheirultimatestrength.The
collapse behavior (M- curve) is similar to that of CaseB, but the
ultimate strength is different in the sagging andthe hogging
conditions, since the buckling collapse strengthis different in the
deck and the bottom.Case D is the actual case; the capacity of each
structuralmember decreases beyond its ultimate strength. In this
case,the bending moment shows a peak value for a certain valueof
the curvature. This peak value is dened as the ultimatelongitudinal
bending moment of the hull girder (Mu).Shortcomings and limitations
of the Smiths method re-lates to the fact that a typical analysis
concerns one framespacing of a whole cargo hold and not a complete
3D hold.As simple linear beam theory is used, deviations
suchasshearlag, warpingandrackingarethusignored. Thismethod may be
a little un-conservative if the structure ispredominantly subjected
to lateral pressure loads as well asaxial compression, and if it is
not realized that the trans-verse frames can deect/fail and
signicantly affect the stiff-ened plate structure and hull girder
bending capacity.18-48 Ship Design & Construction, Volume
1MASTER SETSDC 18.qxd Page 18-48 4/28/03 1:31 PM18.6.5.2 Simplied
modelsCaldwell (64) was the rst who tried to theoretically
eval-uate the ultimate hull girder strength of a ship subjected
tolongitudinal bending. He introduced a so-called Plastic De-sign
considering the inuence of buckling and yielding ofstructural
members composing a ships hull (Figure 18.47).He idealised a
stiffened cross-section of a ships hull
toanunstiffenedcross-sectionwithequivalentthickness.Ifbucklingtakesplaceatthecompressionsideofbending,compressive
stress cannot reach the yield stress, and the fullyplastic bending
moment (Mp) cannot be attained. Caldwellintroduced a stress
reduction factor in the compression sideof bending, and the bending
moment produced by the reducedstress was considered as the ultimate
hull girder
strength.SeveralauthorshaveproposedimprovementsfortheCaldwell
formulation (65). Each of them is characterizedby an assumed stress
distribution (Figure 18.47). Such meth-ods aim at providing an
estimate of the ultimate bendingmoment without attempting to
provide an insight into thebehaviour before, and more importantly,
after, collapse ofthe section. The tracing out of a progressive
collapse curveis replaced by the calculation of the ultimate
bending mo-ment for a particular distribution of stresses. The
quality ofthe direct approximate method is directly dependent on
thequality of the stress distribution at collapse. It is
assumedthat at collapse the stresses acting on the members that
arein tension are equal to yield throughout whereas the stressesin
the members that are in compression are equal to the in-dividual
inelastic buckling stresses. On this basis, the
plas-ticneutralaxisisestimatedusingconsiderationsoflongitudinal
equilibrium. The ultimate bending moment isthen the sum of
individual moments of all elements aboutthe plastic neutral axis.In
Caldwells Method, and Caldwell Modied Methods,reduction in the
capacity of structural members beyond theirultimate strength is not
explicitly taken into account.
Thismaycausetheoverestimationoftheultimatestrengthingeneral (Case
C, Figure 18.49).Empirical Formulations: In contrast to all the
previousrationalmethods, therearesomeempiricalformulationsusually
calibrated for a type of specic vessels (66,67). Yaoet al (50),
found that initial yielding strength of the deckcan provide in
general a little higher but reasonably accu-rate estimate of the
ultimate sagging bending moment. Onthe other hand, the initial
buckling strength of the bottomplate gives a little lower but
accurate estimate of the ulti-mate hogging bending moment. These in
effect can providea rst estimate of the ultimate hull girder
moment.Interactions: In order to raise the problem of combinedloads
(vertical and horizontal bending moments and shearforces),
severalauthorshaveproposedempiricalinterac-tion equations to
predict the ultimate strength. Each
loadcomponentissupposedtoactseparately. Thesemethodswere reviewed
by ISSC (68) and are often formulated asequation 57.[57]where:Mvand
Mh= vertical and horizontal bending momentsMvuand Mhu= ultimate
vertical and horizontal bending mo-mentsa, b and = empirical
constantsFor instance, Mansour et al (47) proposes a=1, b=2 and=
0.8 based on analysis on one container, one tanker and2 cruisers,
and Gordo and Soares (60) 1.5
