Analysis and Design of Ship Structure

18.1 NOMENCLATURE

For specific symbols, refer to the definitions 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 Classifica-

tion SocietiesISSC International Ship & Offshore Structures

CongressISOPE International Offshore and Polar Engi-

neering ConferenceISUM Idealized Structural Unit methodNKK Nippon Kaiji KyokaiPRADS Practical Design of Ships and Mobile

Units,RINA Registro Italiano NavaleSNAME Society of naval Architects and marine

EngineersSSC Ship Structure Committee.a accelerationA areaB breadth of the shipC wave coefficient (Table 18.I)CB hull block coefficientD depth of the shipg gravity acceleration

m(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 a

beamT draft of the shipV(x) shear at section x of a beam

s,w (low case) still water, wave induced component

v,h (low case) vertical, horizontal componentw(x) longitudinal distribution of weightθ roll angleρ densityω angular frequency

18.2 INTRODUCTION

The purpose of this chapter is to present the fundamentalsof direct ship structure analysis based on mechanics andstrength of materials. Such analysis allows a rationally baseddesign that is practical, efficient, 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.

18-1

Chapter 18Analysis and Design of Ship Structure

Philippe Rigo and Enrico Rizzuto

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Analysis 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 defined in terms of stresses,deflections and strength. Then, the estimated response iscompared to the design criteria. Results of this comparisonas well as the objective functions (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, butare absolutely different. Of course design also relates totopology and layout definition.

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 thischapter are Ship Structural Design (3) also published bySNAME and the DNV 99-0394 Technical Report (4).

The present chapter is intimately linked with Chapter11 – Parametric Design, Chapter 17 – Structural Arrange-ment and Component Design and with Chapter 19 – Reli-ability-Based Structural Design. References to thesechapters will be made in order to avoid duplications. In ad-dition, as Chapter 8 deals with classification societies, thepresent chapter will focus mainly on the direct analysismethods 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, 18.5 and 18.6 concern, respectively, the stresses anddeflections (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-formed in Section 18.7. Finally Design Criteria (Section18.8) and Design Procedures (Section 18.9) are discussed.Structural modeling is discussed in Subsection 18.2.2 andmore extensively in Subsection 18.7.2 for finite 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:

The complexities of modern ships and the demand forgreater reliability, efficiency, and economy require a sci-

entific, powerful, and versatile method for their structuraldesign

But, even with the development of numerical techniques,design still remains based on the designer’s 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 atthe preliminary design stage. A 3D Finite ElementMethod (FEM) analysis would be welcome but the timeis not available. For that reason, rule-based design orsimplified 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 efficient. The results from complex analy-ses should be verified by simplified methods to avoid errorsand misinterpretation of results (checks and balances).

• compromise between weight and cost or compromisebetween 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-sign of ship structure. The first one, the oldest, is calledrule-based design. It is mainly based on the rules definedby the classification 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 classification soci-eties. These rules concern the loads, the strength and thedesign criteria and provide simplified and easy-to-use for-mulas for the structural dimensions, or “scantlings” of aship. This approach saves time in the design office and,since the ship must obtain the approval of a classificationsociety, it also saves time in the approval process.

The second school is the Rationally Based StructuralDesign; it is based on direct analysis. Hughes, who couldbe considered as a father of this methodology, (3) furtherstates:

18-2 Ship Design & Construction, Volume 1

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There are several disadvantages to a completely “rulebook”approach to design. First, the modes of structural failureare numerous, complex, and interdependent. With suchsimplified 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-tified. Direct analysis has become the standard procedurein aerospace, civil engineering and partly in offshore in-dustries. In ship design, classification 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 classification societiesinclude, usually as an alternative, a direct analysis procedure(numerical packages based on the finite element method,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. Classification societies are now encouraging andcontributing greatly to the development of direct analysisand 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-jority of specialists together with rapid advances in com-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 methodsusing the latest technologies in order to realize theshipowner’s 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 classification society toclarify the strength that a hull structure should have withrespect to each of the various steps taken in the analysisprocess, 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.

Based on recognition of this need, extensive researchhas been conducted and a careful examination made, 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 flow chart given in Figure 18.1 gives an overviewof the analysis as defined by a major classification 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 difficulty 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-sults suitable for performing buckling, yield, fatigue and

Chapter 18: Analysis and Design of Ship Structure 18-3

Figure 18.1 Direct Structural Analysis Flow Chart

Direct Load Analysis

Design Load

Study on Ocean Waves

Effect on operation Wave Load Response

Response function of wave load

Structural analysis by whole ship model

Stress response function

Short term estimation

Long term estimation

Design Sea State

Design wave Wave impact load

Structural response analysis

Strength Assessment

Yield strength

Nonlinear influence in large waves

Investigation on corrosion

Buckling strength

Ultimate strength

Fatigue strength

Modeling technique Direct structural analysis

Stress Response in Waves

Long term estimation

Short term estimation

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vibration 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 finer 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, specific analyses are requested for fatigue (Subsection18.6.6), collision and grounding (Subsection 18.6.7) andvibration (Subsection 18.6.8). The hydrodynamic loadmodel 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 examined

to ensure that all dimensioning loads are correctly included.A flow 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.).

Preliminary design has the greatest influence on thestructure design and hence is the phase that offers verylarge potential savings. This does not mean that detail de-sign is less important than preliminary design. Each levelis equally important for obtaining an efficient, safe and re-liable ship.

During the detailed design there also are many bene-fits to be gained by applying modern methods of engi-neering science, but the applications are different frompreliminary design and the benefits 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 benefits 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-efficient to attempt to design all of them from first princi-ples. Instead it is generally more efficient to use designcodes 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 thatare published by the various ship classification societiescontain a great deal of useful information on the design oflocal structure, structural connections, and other structuraldetails.

18.3 LOADS

Loads 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 quantification proce-


Figure 18.2 Strength Analysis Flow Chart (4)

Structural modelincluding necessary

load definitions

Hydrodynamic/staticloads

Load transfer tostructural model

Verified structuralmodel

Sub-models to beused in structural

analysisStructural analysis

Verificationof response

Verificationof model/

loads

Yes

No

Transfer ofdisplacements/forces

to sub-model?

Verificationof loadtransfer

Structural drawings,mass description and

loading conditions.

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dures and practical methods for their evaluation will be sum-marized.

18.3.1 Classification 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 specific load con-dition, they have little and very slow variations during avoyage (mainly due to changes in the distribution of con-sumables on board) and they vary significantly only duringloading and unloading operations.

Quasi-static loads: A second class of loads includesthose 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.

Dynamic loads: When studying responses with fre-quency components close to the first 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 classified in the same category.

High frequency loads: Loads at frequencies higher thanthe first 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 specific 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 along time. However, specific and imposing requirementscan arise for particular ships due to the other load cate-gories.

18.3.1.2 Local and global loadsAnother traditional classification of loads is based on thestructural scheme adopted to study the response.

Loads acting on the ship as a whole, considered as abeam (hull girder), are named global or primary loads andthe ship structural response is accordingly termed global orprimary response (see Subsection 18.4.3).

Loads, defined in order to be applied to limited struc-tural models (stiffened panels, single beams, plate panels),generally are termed local loads.

The distinction is purely formal, as the same externalforces 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, representa contribution to the bending moment acting on the hullgirder.

This terminology is typical of simplified structural analy-ses, in which responses of the two classes of componentsare evaluated separately and later summed up to providethe 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 loadsStructural verifications are always based on a limit stateequation and on a design operational time.

Main aspects of reliability-based structural design andanalysis are (see Chapter 19):

• the state of the structure is identified 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.

This general time-variant problem is simplified into atime-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 corrosionand other degradation effects). The simplification lies inconsidering 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-terized through the probability associated to the extremevalue in the reference (design) time.

In fatigue phenomena, in principle all stress cycles con-tribute (to a different extent, depending on the range) to


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damage 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 simplification is rep-resented by the adoption of characteristic load values inplace of statistical distributions. This usually is done, forexample, when calibrating a Partial Safety Factor format forstructural checks. Such adoption implies the definition of asingle reference load value as representative of a wholeprobability 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 first occurrence.

Characteristic values for ultimate state analysis are typ-ically represented by loads associated to an exceeding prob-ability of 10–8. This corresponds to a wave load occurring,on the average, once every 108 cycles, that is, with a returnperiod of the same order of the ship lifetime. In first yield-ing analyses, characteristic loads are associated to a higherexceeding probability, usually in the range 10–4 to 10–6. Infatigue analyses (see Subsection 18.6.6.2), reference loadsare often set with an exceeding probability in the range 10–3

to 10–5, 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 Definition 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.

Actions on the beam are described, as usual with thisscheme, 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): a

resultant 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 qH and amoment mT about the x axis. All these actions are distrib-uted along the longitudinal axis x.

Five main load components are accordingly generatedalong the beam, related to sectional forces and momentthrough 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, qH and mT.

[6]

Global loads for the verification of the hull girder are ob-tained with a linear superimposition of still water and wave-induced global loads.

They are used, with different characteristic values, indifferent types of analyses, such as ultimate state, first yield-ing, and fatigue.

18.3.3 Still Water Global LoadsStill water loads act on the ship floating 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) 0

V (0) V (L) M (0) M (L) 0

M (0) M (L) 0

V V V V

H H H H

T T

= = = == = = == =

M (x) m ( ) dT T0

x

= ∫ ξ ξ

M (x) V ( ) dH H0

x

= ∫ ξ ξ

V (x) q ) dH H0

x

= ∫ (ξ ξ

M (x) V ( ) dV V0

x

= ∫ ξ ξ

V (x) q ( ) dV V0

x

= ∫ ξ ξ


Figure 18.3 Sectional Forces and Moment

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cal 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 can

be 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 definedonly at the end of construction. Statistical formulations, cal-ibrated on similar ships, are often used in the design de-velopment to provide an approximate quantification ofweight items and their longitudinal distribution on board.The resulting approximated weight distribution, 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 ClassificationSocieties (equation 8).

[8]

where C = wave parameter (Table 18.I).The formulations in equation 8 are sometimes explicitly

reported in Rules, but they can anyway be indirectly de-rived from prescriptions contained in (6, 7). The first re-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 loadsClassification 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 afloat, plus allother conditions that are relevant to the specific ship (non-homogeneous loading at maximum draft, light load at lessthan maximum draft, short voyage or harbor condition, bal-last exchange at sea, etc.).

The direct evaluation procedure requires, for a givenloading condition, a derivation, section by section, of ver-tical resultants of gravitational (weight) and buoyancyforces, 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 imposing

M N mC L B 122.5 15 C (hogging)

C L B 45.5 65 C (sagging)s

2B

2B

⋅−( )+

[ ] = ( )


Figure 18.4 Sectional Resultant Forces in Still Water

Figure 18.5 Examples of Reference Still Water Bending Moment Distribution

(10). (a) oil tankers, bulk carriers, ore carriers, and (b) other ship types

TABLE 18.I Wave Coefficient Versus Length

Ship Length L Wave Coefficient C

90 ≤ L <300 m 10.75 – [(300 – L)/100]3/2

300 ≤ L <350 m 10.75

350 ≤ L 10.75 – [(300 – L)/150]3/2

(a)

(b)

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the correspondence of area and barycenter of the trapezoidrespectively to the total weight and center of gravity of theconsidered ship portion.

The procedure is usually applied separately for differ-ent types of weight items, grouping together the weights ofthe ship in lightweight conditions (always present on board)and those (cargo, ballast, consumables) typical of a load-ing condition (Figure 18.6).

18.3.3.3 Uncertainties in the evaluationA significant contribution to uncertainties in the evaluationof still water loads comes from the inputs to the procedure,in particular those related to quantification and location onboard of weight items.

This lack of precision regards the weight distribution for

the ship in lightweight condition (hull structure, machin-ery, outfitting) but also the distribution of the various com-ponents of the deadweight (cargo, ballast, consumables).

Ship types like bulk carriers are more exposed to uncer-tainties on the actual distribution of cargo weight than, forexample, container ships, where actual weights of singlecontainers are kept under close control during operation.

In addition, model uncertainties arise from neglecting thelongitudinal components of the hydrostatic pressure (Fig-ure 18.7), which generate an axial compressive force on thehull girder.

As the resultant of such components is generally belowthe neutral axis of the hull girder, it leads also to an addi-tional hogging moment, which can reach up to 10% of thetotal bending moment. On the other hand, in some vessels(in particular tankers) such action can be locally counter-balanced by internal axial pressures, causing hull saggingmoments.

All these compression and bending effects are neglectedin the hull beam model, which accounts only for forces andmoments acting in the transverse plane. This represents asource of uncertainties.

Another approximation is represented by the fact thatbuoyancy and weight are assumed in a direction normal tothe horizontal longitudinal axis, while they are actually ori-ented along the true vertical.

This implies neglecting the static trim angle and to consideran approximate equilibrium position, which often creates theneed for a few iterative corrections to the load curve qsv(x) inorder to satisfy boundary conditions at ends (equations 6).

18.3.3.4 Other still water global loadsIn a vessel with a multihull configuration, in addition toconventional still water loads acting on each hull consid-ered as a single longitudinal beam, also loads in the trans-versal direction can be significant, giving rise to shear,bending and torque in a transversal direction (see the sim-plified scheme of Figure 18.8, where S, B, and Q stand forshear, bending and torque; and L, T apply respectively tolongitudinal and transversal beams).

18.3.4 Wave Induced Global LoadsThe prediction of the behaviour of the ship in waves repre-sents a key point in the quantification of both global andlocal loads acting on the ship. The solution of the seakeep-ing problem yields the loads directly generated by externalpressures, but also provides ship motions and accelerations.The latter are directly connected to the quantification of in-ertial loads and provide inputs for the evaluation of othertypes of loads, like slamming and sloshing.


Figure 18.6 Weight Distribution Breakdown for Full Load Condition

Figure 18.7 Longitudinal Component of Pressure

Figure 18.8 Multi-hull Additional Still Water Loads (sketch)

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In particular, as regards global effects, the action of wavesmodifies the pressure distribution along the wet hull sur-face; the differential pressure between the situation in wavesand in still water generates, on the transverse section, ver-tical and horizontal resultant forces (bWV and bWH) and amoment component mTb.

Analogous components come from the sectional result-ants of inertial forces and moments induced on the sectionby ship’s motions (Figure 18.9).

The total vertical and horizontal wave induced forces onthe section, as well as the total torsional component, arefound summing up the components in the same direction(equations 9).

[9]

where IR(x) is the rotational inertia of section x.The longitudinal distributions along the hull girder of hor-

izontal and vertical components of shear, bending momentand torque can then be derived by integration (equations 1to 5).

Such results are in principle obtained for each instanta-neous wave pressure distribution, depending therefore, ontime, on type and direction of sea encountered and on theship geometrical and operational characteristics.

In regular (sinusoidal) waves, vertical bending momentstend to be maximized in head waves with length close tothe ship length, while horizontal bending and torque com-ponents are larger for oblique wave systems.

18.3.4.1 Statistical formulae for global wave loadsSimplified, first approximation, formulations are availablefor the main wave load components, developed mainly onthe basis of past experience.

Vertical wave-induced bending moment: IACS classifi-

q (x) b (x) m(x)a (x)

q (x) b (x) m(x)a (x)

m (x) m (x) I (x)

WV WV V

WH WH H

TW Tb R

= −= −= − θ

cation societies provide a statistically based reference valuesfor the vertical component of wave-induced bending momentMWV, expressed as a function of main ship dimensions.

Such reference values for the midlength section of a shipwith unrestricted navigation are yielded by equation 10 forhog and sag cases (7) and corresponds to an extreme valuewith a return period of about 20 years or an exceeding prob-ability of about 10–8 (once in the ship lifetime).

[10]

Horizontal Wave-induced Bending Moment: Similar for-mulations are available for reference values of horizontalwave induced bending moment, even though they are notas uniform among different Societies as for the main verti-cal component.

In Table 18.II, examples are reported of reference val-ues of horizontal bending moment at mid-length for shipswith unrestricted navigation. Simplified curves for the dis-tribution in the longitudinal direction are also provided.

Wave-induced Torque: A few reference formulations aregiven also for reference wave torque at midship (see ex-amples in Table 18.III) and for the inherent longitudinaldistributions.

18.3.4.2 Static Wave analysis of global wave loadsA traditional analysis adopted in the past for evaluation ofwave-induced loads was represented by a quasi-static waveapproach. The ship is positioned on a freezed wave of givencharacteristics in a condition of equilibrium between weightand static buoyancy. The scheme is analogous to the one de-scribed for still water loads, with the difference that the wa-terline upper boundary of the immersed part of the hull isno longer a plane but it is a curved (cylindrical) surface. Bydefinition, this procedure neglects all types of dynamic ef-fects. Due to its limitations, it is rarely used to quantify waveloads. Sometimes, however, the concept of equivalent staticwave is adopted to associate a longitudinal distribution of

M N m C L B C

C L B C .

(hog)

(sag)WVB

B⋅[ ] =

− +( )190

110 0 7

2

2


Figure 18.9 Sectional Forces and Moments in Waves

TABLE 18.II Reference Horizontal Bending Moments

Class Society MWH [N ⋅ m]

ABS (8) 180 C1L2DCB

BV (9) RINA (10) 1600 L2.1 TCB

DNV (11) 220 L9/4(T + 0.3B)CB

NKK (12) 320 L2C T L L− 35 /

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pressures to extreme wave loads, derived, for example, fromlong term predictions based on other methods.

18.3.4.3 Linear methods for wave loadsThe most popular approach to the evaluation of wave loadsis represented by solutions of a linearized potential flowproblem based on the so-called strip theory in the frequencydomain (13).

The theoretical background of this class of proceduresis discussed in detail in PNA Vol. III (2).

Here only the key assumptions of the method are pre-sented:

• inviscid, incompressible and homogeneous fluid in irro-tational flow: Laplace equation 11

∇ 2Φ = 0 [11]

where Φ = velocity potential• 2-dimensional solution of the problem• linearized boundary conditions: the quadratic compo-

nent of velocity in the Bernoulli Equation is reformu-lated in linear terms to express boundary conditions:

— on free surface: considered as a plane correspondingto still water: fluid velocity normal to the free surfaceequal to velocity of the surface itself (kinematic con-dition); zero pressure,

— on the hull: considered as a static surface, corre-sponding to the mean position of the hull: the com-ponent of the fluid velocity normal to the hull surfaceis zero (impermeability condition), and

• linear decomposition into additive independent compo-nents, separately solved for and later summed up (equa-tion 12).

Φ = Φs + ΦFK + Φd + Φr [12]

where:

Φs = stationary component due to ship advancing in calmwater

Φr = radiation component due to the ship motions in calmwater

ΦFK = excitation component, due to the incident wave(undisturbed by the presence of the ship): Froude-Krylov

Φd = diffraction component, due to disturbance in the wavepotential generated by the hull

This subdivision also enables the de-coupling of the ex-citation components from the response ones, thus avoidinga non-linear feedback between the two.

Other key properties of linear systems that are used inthe analysis are:

• linear relation between the input and output amplitudes,and

• superposition of effects (sum of inputs corresponds tosum of outputs).

When using linear methods in the frequency domain,the input wave system is decomposed into sinusoidal com-ponents and a response is found for each of them in termsof amplitude and phase.

The input to the procedure is represented by a spectralrepresentation of the sea encountered by the ship. Responses,for a ship in a given condition, depend on the input sea char-acteristics (spectrum and spatial distribution respect to theship course).

The output consists of response spectra of point pres-sures on the hull and of the other derived responses, suchas global loads and ship motions. Output spectra can beused to derive short and long-term predictions for the prob-ability distributions of the responses and of their extremevalues (see Subsection 18.3.4.5).

Despite the numerous and demanding simplifications atthe basis of the procedure, strip theory methods, developedsince the early 60s, have been validated over time in sev-eral contexts and are extensively used for predictions ofwave loads.

In principle, the base assumptions of the method are


TABLE 18.III Examples of Reference Values for Wave Torque

Class Society Qw [N . m] (at mid-ship)

ABS (bulk carrier)

(e = vertical position of shear center)

BV RINA 190 8 13250 0 7

1252 2

3LB C .

. LW − −

2700 0 5 0 1 0 130 142 2

0 5LB T C . . .

eD

.TW

.−( ) +[ ] −

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valid only for small wave excitations, small motion re-sponses and low speed of the ship.

In practice, the field of successful applications extendsfar beyond the limits suggested by the preservation of re-alism in the base assumptions: the method is actually usedextensively to study even extreme loads and for fast ves-sels.

18.3.4.4 Limits of linear methods for wave loadsDue to the simplifications adopted on boundary conditionsto linearize the problem of ship response in waves, resultsin terms of hydrodynamic pressures are given always up tothe still water level, while in reality the pressure distribu-tion extends over the actual wetted surface. This representsa major problem when dealing with local loads in the sideregion close to the waterline.

Another effect of basic assumptions is that all responsesat a given frequency are represented by sinusoidal fluctua-tions (symmetric with respect to a zero mean value). A con-sequence is that all the derived global wave loads also havethe same characteristics, while, for example, actual valuesof vertical bending moment show marked differences be-tween the hogging and sagging conditions. Corrections toaccount for this effect are often used, based on statisticaldata (7) or on more advanced non-linear methods.

A third implication of linearization regards the super-imposition of static and dynamic loads. Dynamic loads areevaluated separately from the static ones and later summedup: this results in an un-physical situation, in which weightforces (included only in static loads) are considered as act-ing always along the vertical axis of the ship reference sys-tem (as in still water). Actually, in a seaway, weight forcesare directed along the true vertical direction, which dependson roll and pitch angles, having therefore also componentsin the longitudinal and lateral direction of the ship.

This aspect represents one of the intrinsic non-lineari-ties in the actual system, as the direction of an external inputforce (weight) depends on the response of the system itself(roll and pitch angles).

This effect is often neglected in the practice, where lin-ear superposition of still water and wave loads is largely fol-lowed.

18.3.4.5 Wave loads probabilistic characterizationThe most widely adopted method to characterize the loadsin the probability domain is the so-called spectral method,used in conjunction with linear frequency-domain methodsfor the solution of the ship-wave interaction problem.

From the frequency domain analysis response spectraSy(ω) are derived, which can be integrated to obtain spec-tral moments mn of order n (equation 13).

[13]

This information is the basis of the spectral method,whose theoretical framework (main hypotheses, assump-tions and steps) is recalled in the following.

If the stochastic process representing the wave input tothe ship system is modeled as a stationary and ergodicGaussian process with zero mean, the response of the sys-tem (load) can be modeled as a process having the same char-acteristics.

The Parseval theorem and the ergodicity property es-tablish a correspondence between the area of the responsespectrum (spectral moment of order 0: m0Y) and the vari-ance of its Gaussian probability distribution (14). This al-lows expressing the density probability distribution of theGaussian response y in terms of m0Y (equation 14).

[14]

Equation 14 expresses the distribution of the fluctuatingresponse y at a generic time instant.

From a structural point of view, more interesting dataare represented by:

• the probability distribution of the response at selectedtime instants, corresponding to the highest values in eachzero-crossing period (peaks: variable p),

• the probability distribution of the excursions betweenthe highest and the lowest value in each zero-crossingperiod (range: variable r), and

• the probability distribution of the highest value in thewhole stationary period of the phenomenon (extremevalue in period Ts, variable extrTsy).

The aforementioned distributions can be derived fromthe underlying Gaussian distribution of the response (equa-tion 14) in the additional hypotheses of narrow band re-sponse process and of independence between peaks. The firsttwo probability distributions take the form of equations 15and 16 respectively, both Rayleigh density distributions (see14).

The distribution in equation 16 is particularly interest-ing for fatigue checks, as it can be adopted to describe stressranges of fatigue cycles.

[15]

[16]f rr

mrmR ( ) = −

4 80

2

0exp

f pp

mpmP ( ) = −

0

2

02exp

f (y)m

eYY

y m Y= −( )1

2 0

2202

π/

m S ( )dnyn

y= ∫∞

ω ω ω0


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The distribution for the extreme value in the stationaryperiod Ts (short term extreme) can be modeled by a Pois-son distribution (in equation 17: expression of the cumula-tive distribution) or other equivalent distributions derivedfrom the statistics of extremes.

[17]

Figure 18.10 summarizes the various short-term distri-butions.

It is interesting to note that all the mentioned distribu-tions are expressed in terms of spectral moments of the re-sponse, which are available from a frequency domainsolution of the ship motions problem.

The results mentioned previously are derived for theperiod Ts in which the input wave system can be consid-ered as stationary (sea state: typically, a period of a fewhours). The derived distributions (short-term predictions)are conditioned to the occurrence of a particular sea state,which is identified by the sea spectrum, its angular distri-bution around the main wave direction (spreading func-tion) and the encounter angle formed with ship advancedirection.

To obtain a long-term prediction, relative to the ship life(or any other design period Td which can be described as aseries of stationary periods), the conditional hypothesis isto be removed from short-term distributions. In other words,the probability of a certain response is to be weighed by theprobability of occurrence of the generating sea state (equa-tion18).

[18]

where:

F(y) = probability for the response to be less than valuey (unconditioned).

F(y Si) = probability for the response to be less than valuey, conditioned to occurrence of sea state Si (shortterm prediction).

P(Si) = probability associated to the i-th sea state.n = total number of sea states, covering all combi-

nations.

Probability P(Si) can be derived from collections of sea databased on visual observations from commercial ships and/oron surveys by buoys.

One of the most typical formats is the one contained in(15), where sea states probabilities are organized in bi-di-mensional histograms (scatter diagrams), containing classes

F y F y S P(S )i ii

n

( ) = ( ) ⋅=∑

1

F pmm

pm

TextrTss( ) = − −

exp exp1

2 22

0

2

0∂

of significant wave heights and mean periods. Such scatterdiagrams are catalogued according to sea zones, such asshown in Figure 18.11 (the subdivision of the world atlas),and main wave direction. Seasonal characteristics are alsoavailable.

The process described in equation 18 can be termed de-conditioning (that is removing the conditioning hypothesis).The same procedure can be applied to any of the variablesstudied in the short term and it does not change the natureof the variable itself. If a range distribution is processed, along-term distribution for ranges of single oscillations isobtained (useful data for a fatigue analysis).

If the distribution of variable extrTsy is de-conditioned, aweighed average of the highest peak in time Ts is achieved.In this case the result is further processed to get the distri-bution of the extreme value in the design time Td. This isdone with an additional application of the concept of sta-tistics of extremes.

In the hypothesis that the extremes of the various seastates are independent from each other, the extreme on timeTd is given by equation 19:

[19]

where F(extrTdy) is the cumulative probability distributionfor the highest response peak in time Td (long-term extremedistribution in time Td).

18.3.4.6 Uncertainties in long-term predictionsThe theoretical framework of the above presented spectralmethod, coupled to linear frequency domain methodolo-gies like those summarized in Subsection 18.3.4.3, allowsthe characterization, in the probability domain, of all thewave induced load variables of interest both for strengthand fatigue checks.

The results of this linear prediction procedure are af-fected by numerous sources of uncertainties, such as:

F y F yextrTd extrTs Td/Ts( ) = ( )[ ]


Figure 18.10 Short-term Distributions

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• sea description: as above mentioned, scatter diagramsare derived from direct observations on the field, whichare affected by a certain degree of indetermination.

In addition, simplified sea spectral shapes are adopted,based on a limited number of parameters (generally, bi-parametric formulations based on significant wave andmean wave period),

• model for the ship’s response: as briefly outlined in Sub-section 18.3.4.3, the model is greatly simplified, partic-ularly as regards fluid characteristics and boundaryconditions.

Numerical algorithms and specific procedures adoptedfor the solution also influence results, creating differenceseven between theoretically equivalent methods, and

• the de-conditioning procedure adopted to derive longterm predictions from short term ones can add furtheruncertainties.

18.3.5 Local LoadsAs previously stated, local loads are applied to individualstructural members like panels and beams (stiffeners or pri-mary supporting members).

They are once again traditionally divided into static anddynamic loads, referred respectively to the situation in stillwater and in a seaway.

Contrary to strength verifications of the hull girder, whichare nowadays largely based on ultimate limit states (for ex-ample, in longitudinal strength: ultimate bending moment),checks on local structures are still in part implicitly basedon more conservative limit states (yield strength).

In many Rules, reference (characteristic) local loads, aswell as the motions and accelerations on which they arebased, are therefore implicitly calibrated at an exceedingprobability higher than the 10–8 value adopted in global loadstrength verifications.

18.3.6 External Pressure LoadsStatic and dynamic pressures generated on the wet surfaceof the hull belong to external loads. They act as local trans-verse loads for the hull plating and supporting structures.

18.3.6.1 Static external pressuresHydrostatic pressure is related through equation 20 to thevertical distance between the free surface and the load point(static head hS).

pS = ρghS [20]

In the case of the external pressure on the hull, hS cor-responds to the local draft of the load point (reference ismade to design waterline).


Figure 18.11 Map of Sea Zones of the World (15)

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18.3.6.2 Dynamic pressuresThe pressure distribution, as well as the wet portion of thehull, is modified for a ship in a seaway with respect to thestill water (Figure 18.9). Pressures and areas of applicationare in principle obtained solving the general problem ofship motions in a seaway.

Approximate distributions of the wave external pressure,to be added to the hydrostatic one, are adopted in Classifi-cation Rules for the ship in various load cases (Figure 18.12).

18.3.7 Internal Loads—Liquid in TanksLiquid cargoes generate normal pressures on the walls ofthe containing tank. Such pressures represent a local trans-versal load for plate, stiffeners and primary supporting mem-bers of the tank walls.

18.3.7.1 Static internal pressureFor a ship in still water, gravitation acceleration g gener-ates a hydrostatic pressure, varying again according to equa-tion 20. The static head hS corresponds here to the verticaldistance from the load point to the highest part of the tank,increased to account for the vertical extension over thatpoint of air pipes (that can be occasionally filled with liq-uid) or, if applicable, for the ullage space pressure (the pres-sure present at the free surface, corresponding for exampleto the setting pressure of outlet valves).

18.3.7.2 Dynamic internal pressureWhen the ship advances in waves, different types of mo-tions are generated in the liquid contained in a tank on-board, depending on the period of the ship motions and onthe filling level: the internal pressure distribution varies ac-cordingly.

In a completely full tank, fluid internal velocities rela-tive to the tank walls are small and the acceleration in thefluid is considered as corresponding to the global ship ac-celeration aw.

The total pressure (equation 21) can be evaluated in termsof the total acceleration aT, obtained summing aw to grav-ity g.

The gravitational acceleration g is directed according tothe true vertical. This means that its components in the shipreference system depend on roll and pitch angles (in Fig-ure 18.13 on roll angle θr).

pf = ρaThT [21]

In equation 21, hT is the distance between the load pointand the highest point of the tank in the direction of the totalacceleration vector aT (Figure 18.13)

If the tank is only partially filled, significant fluid inter-

nal velocities can arise in the longitudinal and/or transver-sal directions, producing additional pressure loads (slosh-ing loads).

If pitch or roll frequencies are close to the tank reso-nance frequency in the inherent direction (which can beevaluated on the basis of geometrical parameters and fill-ing ratio), kinetic energy tends to concentrate in the fluidand sloshing phenomena are enhanced.

The resulting pressure field can be quite complicatedand specific simulations are needed for a detailed quantifi-cation. Experimental techniques as well as 2D and 3D pro-cedures have been developed for the purpose. For moredetails see references 16 and 17.

A further type of excitation is represented by impacts thatcan occur on horizontal or sub-horizontal plates of the upperpart of the tank walls for high filling ratios and, at low fill-ing levels, in vertical or sub-vertical plates of the lower partof the tank.

Impact loads are very difficult to characterize, being re-lated to a number of effects, such as: local shape and ve-locity of the free surface, air trapping in the fluid andresponse of the structure. A complete model of the phe-nomenon would require a very detailed two-phase schemefor the fluid and a dynamic model for the structure includ-ing hydro-elasticity effects.

Simplified distributions of sloshing and/or impact pres-sures are often provided by Classification Societies for struc-tural verification (Figure 18.14).


Figure 18.13 Internal Fluid Pressure (full tank)

Figure 18.12 Example of Simplified Distribution of External Pressure (10)

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18.3.7.3 Dry bulk cargoIn the case of a dry bulk cargo, internal friction forces arisewithin the cargo itself and between the cargo and the wallsof the hold. As a result, the component normal to the wallhas a different distribution from the load corresponding toa liquid cargo of the same density; also additional tangen-tial components are present.

18.3.8 Inertial Loads—Dry CargoTo account for this effect, distributions for the componentsof cargo load are approximated with empirical formulationsbased on the material frictional characteristics, usually ex-pressed by the angle of repose for the bulk cargo, and onthe slope of the wall. Such formulations cover both the staticand the dynamic cases.

18.3.8.1 Unit cargoIn the case of a unit cargo (container, pallet, vehicle or other)the local translational accelerations at the centre of gravityare applied to the mass to obtain a distribution of inertialforces. Such forces are transferred to the structure in dif-ferent ways, depending on the number and extension of con-tact areas and on typology and geometry of the lashing orsupporting systems.

Generally, this kind of load is modelled by one or moreconcentrated forces (Figure 18.15) or by a uniform load ap-plied on the contact area with the structure.

The latter case applies, for example, to the inertial loadstransmitted by tyred vehicles when modelling the responseof the deck plate between stiffeners: in this case the load isdistributed uniformly on the tyre print.

18.3.9 Dynamic Loads18.3.9.1 Slamming and bow flare loadsWhen sailing in heavy seas, the ship can experience suchlarge heave motions that the forebody emerges completelyfrom the water. In the following downward fall, the bottomof the ship can hit the water surface, thus generating con-siderable impact pressures.

The phenomenon occurs in flat areas of the forward partof the ship and it is strongly correlated to loading condi-tions with a low forward draft.

It affects both local structures (bottom panels) and theglobal bending behaviour of the hull girder with generationalso of free vibrations at the first vertical flexural modes forthe hull (whipping).

A full description of the slamming phenomenon involvesa number of parameters: amplitude and velocity of ship mo-tions relative to water, local angle formed at impact between

the flat part of the hull and the water free surface, presenceand extension of air trapped between fluid and ship bottomand structural dynamic behavior (18,19).

While slamming probability of occurrence can be stud-ied on the basis only of predictions of ship relative motions(which should in principle include non-linear effects due toextreme motions), a quantification of slamming pressureinvolves necessarily all the other mentioned phenomenaand is very difficult to attain, both from a theoretical andexperimental point of view (18,19).

From a practical point of view, Class Societies prescribe,for ships with loading conditions corresponding to a low fore


Figure 18.14 Example of Simplified Distributions of Sloshing and Impact

Pressures (11)

Figure 18.15 Scheme of Local Forces Transmitted by a Container to the

Support System (8)

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draft, local structural checks based on an additional exter-nal pressure.

Such additional pressure is formulated as a function ofship main characteristics, of local geometry of the ship(width of flat bottom, local draft) and, in some cases, of thefirst natural frequency of flexural vibration of the hull girder.

The influence on global loads is accounted for by an ad-ditional term for the vertical wave-induced bending mo-ment, which can produce a significant increase (15% andmore) in the design value.

A phenomenon quite similar to bottom slamming canoccur also on the forebody of ships with a large bow flare.In this case dynamic and (to a lesser extent) impulsive pres-sures are generated on the sides of V-shaped fore sections.

The phenomenon is likely to occur quite frequently onships prone to it, but with lower pressures than in bottomslamming. The incremental effect on vertical bending mo-ment can however be significant.

A quantification of bow flare effects implies taking intoaccount the variation of the local breadth of the section asa function of draft. It represents a typical non-linear effect(non-linearity due to hull geometry).

Slamming can also occur in the rear part of the ship,when the flat part of the stern counter is close to surface.

18.3.9.2 SpringingAnother phenomenon which involves the dynamic responseof the hull girder is springing. For particular types of ships,a coincidence can occur between the frequency of wave ex-citation and the natural frequency associated to the first(two-node) flexural mode in the vertical plane, thus pro-ducing a resonance for that mode (see also Subsection18.6.8.2).

The phenomenon has been observed in particular on GreatLakes vessels, a category of ships long and flexible, with com-paratively low resonance frequencies (1, Chapter VI).

The exciting action has an origin similar to the case ofquasi-static wave bending moment and can be studied withthe same techniques, but the response in terms of deflec-tion and stresses is magnified by dynamic effects. For re-cent developments of research in the field (see references16 and 17).

18.3.9.3 Propeller induced pressures and forcesDue to the wake generated by the presence of the after partof the hull, the propeller operates in a non-uniform incidentvelocity field.

Blade profiles experience a varying angle of attack dur-ing the revolution and the pressure field generated aroundthe blades fluctuates accordingly.

The dynamic pressure field impinges the hull plating in

the stern region, thus generating an exciting force for thestructure.

A second effect is due to axial and non axial forces andmoments generated by the propeller on the shaft and trans-mitted through the bearings to the hull (bearing forces).

Due to the negative dynamic pressure generated by theincreased angle of attack, the local pressure on the back ofblade profiles can, for any rotation angle, fall below thevapor saturation pressure. In this case, a vapor sheet is gen-erated on the back of the profile (cavitation phenomenon).The vapor filled cavity collapses as soon as the angle of at-tack decreases in the propeller revolution and the local pres-sure rises again over the vapor saturation pressure.

Cavitation further enhances pressure fluctuations, be-cause of the rapid displacement of the surrounding watervolume during the growing phase of the vapor bubble andbecause of the following implosion when conditions for itsexistence are removed.

All of the three mentioned types of excitation have theirmain components at the propeller rotational frequency, atthe blade frequency, and at their first harmonics. In addi-tion to the above frequencies, the cavitation pressure fieldcontains also other components at higher frequency, relatedto the dynamics of the vapor cavity.

Propellers with skewed blades perform better as regardsinduced pressure, because not all the blade sections pass si-multaneously in the region of the stern counter, where dis-turbances in the wake are larger; accordingly, pressurefluctuations are distributed over a longer time period andpeak values are lower.

Bearing forces and pressures induced on the stern counterby cavitating and non cavitating propellers can be calculatedwith dedicated numerical simulations (18).

18.3.9.4 Main engine excitationAnother major source of dynamic excitation for the hullgirder is represented by the main engine. Depending ongeneral arrangement and on number of cylinders, diesel en-gines generate internally unbalanced forces and moments,mainly at the engine revolution frequency, at the cylindersfiring frequency and inherent harmonics (Figure 18.16).

The excitation due to the first harmonics of low speeddiesel engines can be at frequencies close to the first natu-ral hull girder frequencies, thus representing a possible causeof a global resonance.

In addition to frequency coincidence, also direction andlocation of the excitation are important factors: for exam-ple, a vertical excitation in a nodal point of a vertical flex-ural mode has much less effect in exciting that mode thanthe same excitation placed on a point of maximum modaldeflection.


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In addition to low frequency hull vibrations, componentsat higher frequencies from the same sources can give riseto resonance in local structures, which can be predicted bysuitable dynamic structural models (18,19).

18.3.10 Other Loads18.3.10.1 Thermal loadsA ship experiences loads as a result of thermal effects, whichcan be produced by external agents (the sun heating thedeck), or internal ones (heat transfer from/to heated or re-frigerated cargo).

What actually creates stresses is a non-uniform temper-ature distribution, which implies that the warmer part of thestructure tends to expand while the rest opposes to this de-formation. A peculiar aspect of this situation is that the por-tion of the structure in larger elongation is compressed andvice-versa, which is contrary to the normal experience.

It is very difficult to quantify thermal loads, the mainproblems being related to the identification of the temper-ature distribution and in particular to the model for con-straints. Usually these loads are considered only in aqualitative way (1, Chapter VI).

18.3.10.2 Mooring loadsFor a moored vessel, loads are exerted from external actionson the mooring system and from there to the local sup-porting structure. The main contributions come by wind,waves and current.

Wind: The force due to wind action is mainly directed inthe direction of the wind (drag force), even if a limited com-ponent in the orthogonal direction can arise in particular sit-uations. The magnitude depends on the wind speed and onextension and geometry of the exposed part of the ship. Theaction due to wind can be described in terms of two force

components; a longitudinal one FWiL, and a transverse oneFWiT (equation 22), and a moment MWiz about the verticalaxis (equation 23), all applied at the center of gravity.

[22]

[23]

where:

φWi = the angle formed by the direction of the wind rela-tive to the ship

CMz(φWi), CFL(φWi), CFT(φWi) are all coefficients dependingon the shape of exposed part of the ship and on angle φWi

AWi = the reference area for the surface of the ship exposedto wind, (usually the area of the cross section)

VWi = the wind speed

The empirical formulas in equations 22 and 23 accountalso for the tangential force acting on the ship surfaces par-allel to the wind direction.

Current: The current exerts on the immersed part of thehull a similar action to the one of wind on the emerged part(drag force). It can be described through coefficients andvariables analogous to those of equations 22 and 23.

Waves: Linear wave excitation has in principle a sinu-soidal time dependence (whose mean value is by definitionzero). If ship motions in the wave direction are not con-strained (for example, if the anchor chain is not in tension)the ship motion follows the excitation with similar time de-pendence and a small time lag. In this case the action onthe mooring system is very small (a few percent of the otheractions).

If the ship is constrained, significant loads arise on themooring system, whose amplitude can be of the same orderof magnitude of the stationary forces due to the other actions.

In addition to the linear effects discussed above, non-lin-ear wave actions, with an average value different from zero,are also present, due to potential forces of higher order, for-mation of vortices, and viscous effects. These componentscan be significant on off-shore floating structures, whichoften feature also complicated mooring systems: in thosecases the dynamic behavior of the mooring system is to beincluded in the analysis, to solve a specific motion prob-lem. For common ships, non-linear wave effects are usu-ally neglected.

A practical rule-of-thumb for taking into account waveactions for a ship at anchor in non protected waters is to in-crease of 75 to 100% the sum of the other force components.

Once the total force on the ship is quantified, the ten-sion in the mooring system (hawser, rope or chain) can be

M C A L VWiz Mz Wi Wi Wi= ( )1 2 2/ φ φ

F C A VWiL,T F L,T Wi Wi Wi= ( )1 2 2/ φ φ


Figure 18.16 Propeller, Shaft and Engine Induced Actions (20)

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derived by force decomposition, taking into account theangle formed with the external force in the horizontal and/orvertical plane.

18.3.10.3 Launching loadsThe launch is a unique moment in the life of the ship. Fora successful completion of this complex operation, a num-ber of practical, organizational and technical elements areto be kept under control (as general reference see Reference1, Chapter XVII).

Here only the aspect of loads acting on the ship will bediscussed, so, among the various types of launch, only thosewhich present peculiarities as regards ship loads will beconsidered: end launch and side launch.

End Launch: In end launch, resultant forces and motionsare contained in the longitudinal plane of the ship (Figure18.17).

The vessel is subjected to vertical sectional forces dis-tributed along the hull girder: weight w(x), buoyancy bL(x)and the sectional force transmitted from the ground way tothe cradle and from the latter to the ship’s bottom (in thefollowing: sectional cradle force fC(x), with resultant FC).

While the weight distribution and its resultant force(weight W) are invariant during launching, the other distri-butions change in shape and resultant: the derivation oflaunching loads is based on the computation of these twodistributions.

Such computation, repeated for various positions of thecradle, is based on the global static equilibrium s (equa-tions 24 and 25, in which dynamic effects are neglected:quasi static approach).

BT + FC – W = 0 [24]

xB BT + xF FC – xW W = 0 [25]

where:

W, BT, FC = (respectively) weight, buoyancy and cradleforce resultants

xW, xB, xF = their longitudinal positions

In a first phase of launching, when the cradle is still incontact for a certain length with the ground way, the buoy-ancy distribution is known and the cradle force resultantand position is derived.

In a second phase, beginning when the cradle starts torotate (pivoting phase: Figure 18.18), the position xF cor-responds steadily to the fore end of the cradle and what isunknown is the magnitude of FC and the actual aft draft ofthe ship (and consequently, the buoyancy distribution).

The total sectional vertical force distribution is found asthe sum of the three components (equation 26) and can be

integrated according to equations 1 and 2 to derive verticalshear and bending moment.

qVL(x) = w(x) – bL(x) – fC(x) [26]

This computation is performed for various intermediatepositions of the cradle during the launching in order to checkall phases. However, the most demanding situation for thehull girder corresponds to the instant when pivoting starts.

In that moment the cradle force is concentrated close tothe bow, at the fore end of the cradle itself (on the fore pop-pet, if one is fitted) and it is at the maximum value.

A considerable sagging moment is present in this situ-ation, whose maximum value is usually lower than the de-sign one, but tends to be located in the fore part of the ship,where bending strength is not as high as at midship.

Furthermore, the ship at launching could still have tem-porary openings or incomplete structures (lower strength)in the area of maximum bending moment.

Another matter of concern is the concentrated force atthe fore end of the cradle, which can reach a significant per-centage of the total weight (typically 20–30%). It representsa strong local load and often requires additional temporaryinternal strengthening structures, to distribute the force ona portion of the structure large enough to sustain it.

Side Launch: In side launch, the main motion compo-nents are directed in the transversal plane of the ship (seeFigure 18.19, reproduced from reference 1, Chapter XVII).

The vertical reaction from ground ways is substituted ina comparatively short time by buoyancy forces when the shiptilts and drops into water.

The kinetic energy gained during the tilting and drop-ping phases makes the ship oscillate around her final posi-


Figure 18.17 End Launch: Sketch

Figure 18.18 Forces during Pivoting

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tion at rest. The amplitude of heave and roll motions andaccelerations governs the magnitude of hull girder loads.Contrary to end launch, trajectory and loads cannot be stud-ied as a sequence of quasi-static equilibrium positions, butneed to be investigated with a dynamic analysis.

The problem is similar to the one regarding ship mo-tions in waves, (Subsection 18.3.4), with the difference thathere motions are due to a free oscillation of the system dueto an unbalanced initial condition and not to an external ex-citation.

Another difference with respect to end launch is thatboth ground reaction (first) and buoyancy forces (later) arealways distributed along the whole length of the ship andare not concentrated in a portion of it.

18.3.10.4 Accidental loadsAccidental loads (collision and grounding) are discussedin more detail by ISSC (21).

Collision: When defining structural loads due to colli-sions, the general approach is to model the dynamics of theaccident itself, in order to define trajectories of the unit(s)involved.

In general terms, the dynamics of collision should beformulated in six degrees of freedom, accounting for a num-ber of forces acting during the event: forces induced by pro-peller, rudder, waves, current, collision forces between theunits, hydrodynamic pressure due to motions.

Normally, theoretical models confine the analysis tocomponents in the horizontal plane (3 degrees of freedom)and to collision forces and motion-induced hydrodynamicpressures. The latter are evaluated with potential methodsof the same type as those adopted for the study of the re-sponse of the ship to waves.

As regards collision forces, they can be described dif-ferently depending on the characteristics of the struck ob-ject (ship, platform, bridge pylon…) with differentcombinations of rigid, elastic or an elastic body models.

Governing equations for the problem are given by con-servation of momentum and of energy. Within this frame-work, time domain simulations can evaluate the magnitudeof contact forces and the energy, which is absorbed by struc-ture deformation: these quantities, together with the responsecharacteristics of the structure (energy absorption capacity),allow an evaluation of the damage penetration (21).

Grounding: In grounding, dominant effects are forces andmotions in the vertical plane.

As regards forces, main components are contact forces,developed at the first impact with the ground, then friction,when the bow slides on the ground, and weight.

From the point of view of energy, the initial kinetic en-ergy is (a) dissipated in the deformation of the lower partof the bow (b) dissipated in friction of the same area againstthe ground, (c) spent in deformation work of the ground (ifsoft: sand, gravel) and (d) converted into gravitational po-tential energy (work done against the weight force, whichresists to the vertical raising of the ship barycenter).

In addition to soil characteristics, key parameters for thedescription are: slope and geometry of the ground, initialspeed and direction of the ship relative to ground, shape ofthe bow (with/without bulb).

The final position (grounded ship) governs the magni-tude of the vertical reaction force and the distribution ofshear and sagging moment that are generated in the hullgirder. Figure 18.20 gives an idea of the magnitude ofgrounding loads for different combinations of ground slopesand coefficients of friction for a 150 000 tanker (results ofsimulations from reference 22).

In addition to numerical simulations, full and modelscale tests are performed to study grounding events (21).


Figure 18.19 Side Launch (1, Chapter XVII) Figure 18.20 Sagging Moments for a Grounded Ship: Simulation Results (22)

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18.3.11 Combination of LoadsWhen dealing with the characterization of a set of loadsacting simultaneously, the interest lies in the definition ofa total loading condition with the required exceeding prob-ability (usually the same of the single components). Thiscannot be obtained by simple superposition of the charac-teristic values of single contributing loads, as the probabil-ity that all design loads occur at the same time is much lowerthan the one associated to the single component.

In the time domain, the combination problem is ex-pressed in terms of time shift between the instants in whichcharacteristic values occur.

In the probability domain, the complete formulation ofthe problem would imply, in principle, the definition of ajoint probability distribution of the various loads, in orderto quantify the distribution for the total load. An approxi-mation would consist in modeling the joint distributionthrough its first and second order moments, that is mean val-ues and covariance matrix (composed by the variances ofthe single variables and by the covariance calculated foreach couple of variables). However, also this level of sta-tistical characterization is difficult to obtain.

As a practical solution to the problem, empirically basedload cases are defined in Rules by means of combinationcoefficients (with values generally ≤ 1) applied to singleloads. Such load cases, each defined by a set of coefficients,represent realistic and, in principle, equally probable com-binations of characteristic values of elementary loads.

Structural checks are performed for all load cases. Theresult of the verification is governed by the one, which turnsout to be the most conservative for the specific structure.This procedure needs a higher number of checks (which, onthe other hand, can be easily automated today), but allowsconsidering various load situations (defined with differentcombinations of the same base loads), without choosing apriori the worst one.

18.3.12 New Trends and Load Non-linearitiesA large part of research efforts is still devoted to a betterdefinition of wave loads. New procedures have been pro-posed in the last decades to improve traditional 2D linearmethods, overcoming some of the simplifications adoptedto treat the problem of ship motions in waves. For a com-plete state of the art of computational methods in the field,reference is made to (23). A very coarse classification ofthe main features of the procedures reported in literature ishere presented (see also reference 24).

18.3.12.1 2D versus 3D modelsThree-dimensional extensions of linear methods are avail-able; some non-linear methods have also 3-D features, whilein other cases an intermediate approach is followed, withboundary conditions formulated part in 2D, part in 3D.

18.3.12.2 Body boundary conditionsIn linear methods, body boundary conditions are set withreference to the mean position of the hull (in still water).Perturbation terms take into account, in the frequency or inthe time domain, first order variations of hydrodynamic andhydrostatic coefficients around the still water line.

Other non-linear methods account for perturbation termsof a higher order. In this case, body boundary conditionsare still linear (mean position of the hull), but second ordervariations of the coefficients are accounted for.

Mixed or blending procedures consist in linear methodsmodified to include non-linear effects in a single compo-nent of the velocity potential (while the other ones are treatedlinearly). In particular, they account for the actual geome-try of wetted hull (non-linear body boundary condition) inthe Froude-Krylov potential only. This effect is believed tohave a major role in the definition of global loads.

More evolved (and complex) methods are able to takeproperly into account the exact body boundary condition(actual wetted surface of the hull).

18.3.12.3 Free surface boundary conditionsBoundary conditions on free surface can be set, dependingon the various methods, with reference to: (a) a free streamat constant velocity, corresponding to ship advance, (b) adouble body flow, accounting for the disturbance inducedby the presence of a fully immersed double body hull onthe uniform flow, (c) the flow corresponding to the steadyadvance of the ship in calm water, considering the free sur-face or (d) the incident wave profile (neglecting the inter-action with the hull).

Works based on fully non-linear formulations of the freesurface conditions have also been published.

18.3.12.4 Fluid characteristicsAll the methods above recalled are based on an inviscidfluid potential scheme.

Some results have been published of viscous flow mod-els based on the solution of Reynolds Averaged NavierStokes (RANS) equations in the time domain. These meth-ods represent the most recent trend in the field of ship mo-tions and loads prediction and their use is limited to a fewresearch groups.


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18.4 STRESSES AND DEFLECTIONS

The reactions of structural components of the ship hull toexternal loads are usually measured by either stresses ordeflections. Structural performance criteria and the associ-ated analyses involving stresses are referred to under the gen-eral term of strength. The strength of a structural componentwould be inadequate if it experiences a loss of load-carry-ing ability through material fracture, yield, buckling, orsome other failure mechanism in response to the appliedloading. Excessive deflection may also limit the structuraleffectiveness of a member, even though material failuredoes not occur, if that deflection results in a misalignmentor other geometric displacement of vital components of theship’s machinery, navigational equipment, etc., thus ren-dering the system ineffective.

The present section deals with the determination of theresponses, in the form of stress and deflection, of structuralmembers to the applied loads. Once these responses areknown it is necessary to determine whether the structure isadequate to withstand the demands placed upon it, and thisrequires consideration of the different failure modes asso-ciated to the limit states, as discussed in Sections 18.5 and18.6

Although longitudinal strength under vertical bendingmoment and vertical shear forces is the first importantstrength consideration in almost all ships, a number of otherstrength considerations must be considered. Prominentamongst these are transverse, torsional and horizontal bend-ing strength, with torsional strength requiring particular at-tention on open ships with large hatches arranged closetogether. All these are briefly presented in this Section. Moredetailed information is available in Lewis (2) and Hughes(3), both published by SNAME, and Rawson (25). Notethat the content of Section 18.4 is influenced mainly fromLewis (2).

18.4.1 Stress and Deflection ComponentsThe structural response of the hull girder and the associ-ated members can be subdivided into three components(Figure 18.21).

Primary response is the response of the entire hull, whenthe ship bends as a beam under the longitudinal distributionof load. The associated primary stresses (σ1) are those, whichare usually called the longitudinal bending stresses, but thegeneral category of primary does not imply a direction.

Secondary response relates to the global bending of stiff-ened panels (for single hull ship) or to the behavior of dou-ble bottom, double sides, etc., for double hull ships:

• Stresses in the plating of stiffened panel under lateralpressure may have different origins (σ2 and σ2*). For astiffened panel, there is the stress (σ2) and deflection ofthe global bending of the orthotropic stiffened panels,for example, the panel of bottom structure contained be-tween two adjacent transverse bulkheads. The stiffenerand the attached plating bend under the lateral load andthe plate develops additional plane stresses since theplate acts as a flange with the stiffeners. In longitudinallyframed ships there is also a second type of secondarystresses: σ2* corresponds to the bending under the hy-drostatic pressure of the longitudinals between trans-verse frames (web frames). For transversally framedpanels, σ2* may also exist and would correspond to thebending of the equally spaced frames between two stifflongitudinal girders.

• A double bottom behaves as box girder but can bend lon-gitudinally, transversally or both. This global bending in-duces stress (σ2) and deflection. In addition, there is also


Figure 18.21 Primary (Hull), Secondary (Double Bottom and Stiffened Panels)

and Tertiary (Plate) Structural Responses (1, 2)

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the σ2* stress that corresponds to the bending of the lon-gitudinals (for example, in the inner and outer bottom)between two transverse elements (floors).

Tertiary response describes the out-of-plane deflectionand associated stress of an individual unstiffened plate panelincluded between 2 longitudinals and 2 transverse webframes. The boundaries are formed by these components(Figure 18.22).

Primary and secondary responses induce in-plane mem-brane stresses, nearly uniformly distributed through the platethickness. Tertiary stresses, which result from the bendingof the plate member itself vary through the thickness, butmay contain a membrane component if the out-of-plane de-flections are large compared to the plate thickness.

In many instances, there is little or no interaction be-tween the three (primary, secondary, tertiary) componentstresses or deflections, and each component may be com-puted by methods and considerations entirely independentof the other two. The resultant stress, in such a case, is thenobtained by a simple superposition of the three componentstresses (Subsection 18.4.7). An exception is the case ofplate (tertiary) deflections, which are large compared to thethickness of plate.

In plating, each response induces longitudinal stresses,transverse stresses and shear stresses. This is due to thePoisson’s Ratio. Both primary and secondary stresses arebending stresses but in plating these stresses look like mem-brane stresses.

In stiffeners, only primary and secondary responses in-duce stresses in the direction of the members and shearstresses. Tertiary response has no effect on the stiffeners.

In Figure 18.21 (see also Figure 18.37) the three types of re-sponse are shown with their associated stresses (σ1, σ2, σ2*and σ3). These considerations point to the inherent sim-plicity of the underlying theory. The structural naval archi-

tect deals principally with beam theory, plate theory, andcombinations of both.

18.4.2 Basic Structural ComponentsStructural components are extensively discussed in Chap-ter 17 – Structure Arrangement Component Design. In thissection, only the basic structural component used exten-sively is presented. It is basically a stiffened panel.

The global ship structure is usually referred to as beinga box girder or hull girder. Modeling of this hull girder isthe first task of the designer. It is usually done by model-ing the hull girder with a series of stiffened panels.

Stiffened panels are the main components of a ship. Al-most any part of the ship can be modeled as stiffened pan-els (plane or cylindrical).

This means that, once the ship’s main dimensions andgeneral arrangement are fixed, the remaining scantling de-velopment mainly deals with stiffened panels.

The panels are joined one to another by connecting lines(edges of the prismatic structures) and have longitudinaland transverse stiffening (Figures 18.23, 24 and 36).

• Longitudinal Stiffening includes

— longitudinals (equally distributed), used only for thedesign of longitudinally stiffened panels,

— girders (not equally distributed).

• Transverse Stiffening includes (Figure 18.23)

— transverse bulkheads (a),— the main transverse framing also called web-frames

(equally distributed; large spacing), used for longi-tudinally stiffened panels (b) and transversally stiff-ened panels (c).

18.4.3 Primary Response18.4.3.1 Beam Model and Hull Section ModulusThe structural members involved in the computation of pri-mary stress are, for the most part, the longitudinally contin-uous members such as deck, side, bottom shell, longitudinalbulkheads, and continuous or fully effective longitudinalprimary or secondary stiffening members.

Elementary beam theory (equation 29) is usually uti-lized in computing the component of primary stress, σ1, anddeflection due to vertical or lateral hull bending loads. Inassessing the applicability of this beam theory to ship struc-tures, it is useful to restate the underlying assumptions:

• the beam is prismatic, that is, all cross sections are thesame and there is no openings or discontinuities,

• plane cross sections remain plane after deformation, will


Figure 18.22 A Standard Stiffened Panel

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not deform in their own planes, and merely rotate as thebeam deflects.

• transverse (Poisson) effects on strain are neglected.• the material behaves elastically: the elasticity modulus

in tension and compression is equal.• Shear effects and bending (stresses, strains) are not cou-

pled. For torsional deformation, the effect of secondaryshear and axial stresses due to warping deformations areneglected.

Since stress concentrations (deck openings, side ports,etc.) cannot be avoided in a highly complex structure suchas a ship, their effects must be included in any comprehen-sive stress analysis. Methods dealing with stress concen-trations are presented in Subsection 18.6.6.3 as they arelinked to fatigue.

The elastic linear bending equations, equations 27 and28, are derived from basic mechanic principle presented atFigure 18.24.

EI (∂2w/∂x2) = M(x) [27]

or

EI (∂4w/∂x4) = q(x) [28]

where:

w = deflection (Figure 18.24), in mE = modulus of elasticity of the material, in N/m2

I = moment of inertia of beam cross section about ahorizontal axis through its centroid, in m4

M(x) = bending moment, in N.mq(x) = load per unit length in N/m

= ∂V(x)/∂x= ∂2M(x)/∂x2

= EI (∂4w/∂x4)

Hull Section Modulus: The plane section assumption to-gether with elastic material behavior results in a longitudi-nal stress, σ1, in the beam that varies linearly over the depthof the cross section.

The simple beam theory for longitudinal strength cal-culations of a ship is based on the hypothesis (usually at-tributed to Navier) that plane sections remain plane and inthe absence of shear, normal to the OXY plane (Figure18.24). This gives the well-known formula:

[29]

where:

M = bending moment (in N.m)σ = bending stress (in N/m2)

f pp

mpmP ( ) = −

0

2

02exp


Figure 18.23 Types of Stiffening (Longitudinal and Transverse)

Figure 18.24 Behavior of an Elastic Beam under Shear Force and Bending

Moment (2)

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I = Sectional moment of Inertia about the neutral axis(in m4)

c = distance from the neutral axis to the extreme mem-ber (in m)

SM = section modulus (I/c) (in m3)

For a given bending moment at a given cross section ofa ship, at any part of the cross section, the stress may be ob-tained (σ = M/SM = Mc/I) which is proportional to the dis-tance c of that part from the neutral axis. The neutral axiswill seldom be located exactly at half-depth of the section;hence two values of c and σ will be obtained for each sec-tion for any given bending moment, one for the top fiber(deck) and one for the bottom fiber (bottom shell).

A variation on the above beam equations may be of im-portance in ship structures. It concerns beams composed oftwo or more materials of different moduli of elasticity, forexample, steel and aluminum. In this case, the flexural rigid-ity, EI, is replaced by ∫A E(z) z2 dA, where A is cross sec-tional area and E(z) the modulus of elasticity of an elementof area dA located at distance z from the neutral axis. Theneutral axis is located at such height that ∫A E(z) z dA = 0.

Calculation of Section Modulus: An important step inroutine ship design is the calculation of the midship sectionmodulus. As defined in connection with equation 29, it in-dicates the bending strength properties of the primary hullstructure. The section modulus to the deck or bottom is ob-tained by dividing the moment of inertia by the distancefrom the neutral axis to the molded deck line at side or tothe base line, respectively.

In general, the following items may be included in thecalculation of the section modulus, provided they are con-tinuous or effectively developed:

• deck plating (strength deck and other effective decks).(See Subsection 18.4.3.9 for Hull/Superstructure Inter-action).

• shell and inner bottom plating,• deck and bottom girders,• plating and longitudinal stiffeners of longitudinal bulk-

heads,• all longitudinals of deck, sides, bottom and inner bot-

tom, and• continuous longitudinal hatch coamings.

In general, only members that are effective in both tensionand compression are assumed to act as part of the hull girder.

Theoretically, a thorough analysis of longitudinal strengthwould include the construction of a curve of section modulithroughout the length of the ship as shown in Figure 18.25.

Dividing the ordinates of the maximum bending-momentscurve (the envelope curve of maxima) by the corresponding

ordinates of the section-moduli curve yields stress values,and by using both the hogging and sagging moment curvesfour curves of stress can be obtained; that is, tension and com-pression values for both top and bottom extreme fibers.

It is customary, however, to assume the maximum bend-ing moment to extend over the midship portion of the ship.Minimum section modulus most often occurs at the loca-tion of a hatch or a deck opening. Accordingly, the classi-fication societies ordinarily require the maintenance of themidship scantlings throughout the midship four-tenthslength. This practice maintains the midship section area ofstructure practically at full value in the vicinity of maximumshear as well as providing for possible variation in the pre-cise location of the maximum bending moment.

Lateral Bending Combined with Vertical Bending: Up tothis point, attention has been focused principally upon the ver-tical longitudinal bending response of the hull. As the shipmoves through a seaway encountering waves from directionsother than directly ahead or astern, it will experience lateralbending loads and twisting moments in addition to the ver-tical loads. The former may be dealt with by methods thatare similar to those used for treating the vertical bendingloads, noting that there will be no component of still waterbending moment or shear in the lateral direction. The twist-ing or torsional loads will require some special consideration.Note that the response of the ship to the overall hull twistingloading should be considered a primary response.

The combination of vertical and horizontal bending mo-ment has as major effect to increase the stress at the ex-treme corners of the structure (equation 30).


Figure 18.25 Moment of Inertia and Section Modulus (1)

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[30]

where Mv, Iv, cv, and Mh, Ih, ch, correspond to the M, I, cdefined in equation 29, for the vertical bending and the hor-izontal bending respectively.

For a given vertical bending (Mv), the periodical waveinduced horizontal bending moment (Mh) increases stresses,alternatively, on the upper starboard and lower portside, andon the upper portside and lower starboard. This explainswhy these areas are usually reinforced.

Empirical interaction formulas between vertical bend-ing, horizontal bending and shear related to ultimate strengthof hull girder are given in Subsection 18.6.5.2.

Transverse Stresses: With regards to the validity of theNavier Equation (equation 29), a significant improvementmay be obtained by considering a longitudinal strengthmember composed of thin plate with transverse framing.This might, for example, represent a portion of the deckstructure of a ship that is subject to a longitudinal stress σx,from the primary bending of the hull girder. As a result ofthe longitudinal strain, εx, which is associated with σx, therewill exist a transverse strain, εs. For the case of a plate thatis free of constraint in the transverse direction, the twostrains will be of opposite sign and the ratio of their ab-solute values, given by | εs / εx | = ν, is a constant propertyof the material. The quantity ν is called Poisson’s Ratio and,for steel and aluminum, has a value of approximately 0.3.

Hooke’s Law, which expresses the relation between stressand strain in two dimensions, may be stated in terms of theplate strains (equation 31). This shows that the primary re-sponse induces both longitudinal (σx) and transversalstresses (σs) in plating.

εx = 1/E ( σx – v σS)[31]

εS = 1/E ( σS – ν σx)

As transverse plate boundaries are usually constrained(displacements not allowed), the transverse stress can betaken, in first approximation as:

σs = ν σx [32]

Equation 32 is only valid to assess the additional stressesin a given direction induced by the stresses in the perpen-dicular direction computed, for instance, with the Navierequation (equation 29).

18.4.3.2 Shear stress associated to shear forcesThe simple beam theory expressions given in the preced-ing section permit evaluation the longitudinal componentof the primary stress, σx. In Figure 18.26, it can be seen that

σ = ( ) + ( )M

I c

M

I cv

v v

h

h h

an element of side shell or deck plating may, in general besubject to two other components of stress, a direct stress inthe transverse direction and a shearing stress.

This figure illustrates these as the stress resultants, de-fined as the stress multiplied by plate thickness.

The stress resultants (N/m) are given by the followingexpressions:

Nx = t σx and Ns = t σs stress resultants, in N/m

N = t τ shear stress resultant or shear flow, in N/m

where:

σx, σs = stresses in the longitudinal and transverse direc-tions, in N/m2

τ = shear stress, in N/m2

t = plate thickness, in m

In many parts of the ship, the longitudinal stress, σx, isthe dominant component. There are, however, locations inwhich the shear component becomes important and underunusual circumstances the transverse component may, like-wise, become important. A suitable procedure for estimat-ing these other component stresses may be derived byconsidering the equations of static equilibrium of the ele-ment of plating (Figure 18.26). The static equilibrium con-ditions for a plate element subjected only to in-plane stress,that is, no plate bending, are:

∂Nx / ∂x + ∂N / ∂s = 0 [33-a]

∂Ns / ∂x + ∂N / ∂x = 0 [33-b]

In these equations, s, is the transverse coordinate meas-ured on the surface of the section from the x-axis as shownin Figure 18.26.

For vessels without continuous longitudinal bulkheads


Figure 18.26 Shear Forces (2)

ED: Correction on this equation is unclear.

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(single cell), having transverse symmetry and subject to abending moment in the vertical plane, the shear flow dis-tribution, N(s) is then given by:

[34]

and the shear stress, τ , at any point in the cross section is:

[35]

where:

V(x) = total shearing force (in N) in the hull for a givensection x

m(s) = in m3, is the first moment (or moment

= of area) about the neutral axis of the cross sectionalarea of the plating between the origin at the cen-terline and the variable location designated by s.This is the crosshatched area of the section shownin Figure 18.26

t(s) = thickness of material at the shear planeI(x) = moment of inertia of the entire section

The total vertical shearing force, V(x), at any point, x,in the ship’s length may be obtained by the integration ofthe load curve up to that point. Ordinarily the maximumvalue of the shearing force occurs at about one quarter ofthe vessel’s length from either end.

Since only the vertical, or nearly vertical, members ofthe hull girder are capable of resisting vertical shear, thisshear is taken almost entirely by the side shell, the contin-uous longitudinal bulkheads if present, and by the webs ofany deep longitudinal girders.

The maximum value of τ occurs in the vicinity of theneutral axis, where the value of t is usually twice the thick-ness of the side plating (Figure 18.27). For vessels with con-tinuous longitudinal bulkheads, the expression for shearstress is more complex.

Shear Flow in Multicell Sections: If the cross section ofthe ship shown in Figure 18.28 is subdivided into two ormore closed cells by longitudinal bulkheads, tank tops, ordecks, the problem of finding the shear flow in the bound-aries of these closed cells is statically indeterminate.

Equation 34 may be evaluated for the deck and bottomof the center tank space since the plane of symmetry atwhich the shear flow vanishes, lies within this space andforms a convenient origin for the integration. At thedeck/bulkhead intersection, the shear flow in the deck di-vides, but the relative proportions of the part in the bulk-head and the part in the deck are indeterminate. The sum

t s z dso

s( ) ,∫

t(s)V(x).m(s)t(s) I(x)

(in N / m )2=

N (s)V(x)I(x)

m (s)=

of the shear flows at two locations lying on a plane cuttingthe cell walls will still be given by equation 34, with m(s)equal to the moment of the shaded area (Figure 18.28).However, the distribution of this sum between the two com-ponents in bulkhead and side shell, requires additional in-formation for its determination.

This additional information may be obtained by con-sidering the torsional equilibrium and deflection of the cel-lular section. The way to proceed is extensively explainedin Lewis (2).

18.4.3.3 Shear stress associated with torsionIn order to develop the twisting equations, we consider aclosed, single cell, thin-walled prismatic section subjectonly to a twisting moment, MT, which is constant along thelength as shown in Figure 18.29. The resulting shear stressmay be assumed uniform through the plate thickness andis tangent to the mid-thickness of the material. Under thesecircumstances, the deflection of the tube will consist of atwisting of the section without distortion of its shape, andthe rate of twist, dθ/dx, will be constant along the length.


Figure 18.28 Shear Flow in Multicell Sections (2)

Figure 18.27 Shear Flow in Multicell Sections (1)

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Now consider equilibrium of forces in the x-direction forthe element dx.ds of the tube wall as shown in Figure 18.29.Since there is no longitudinal load, there will be no longi-tudinal stress, and only the shear stresses at the top and bot-tom edges need be considered in the expression for staticequilibrium. The shear flow, N = tτ, is therefore seen to beconstant around the section.

The magnitude of the moment, MT, may be computedby integrating the moment of the elementary force arisingfrom this shear flow about any convenient axis. If r is thedistance from the axis, 0, perpendicular to the resultant shearflow at location s:

[36]

Here the symbol indicates that the integral is taken en-tirely around the section and, therefore, Ω (m2) is the areaenclosed by the mid-thickness line of the tubular cross sec-tion. The constant shear flow, N (N/m), is then related tothe applied twisting moment by:

N = τ. t = MT /2Ω [37]

For uniform torsion of a closed prismatic section, theangle of torsion is:

(in radians) [38]

where:

MT = Twisting moment (torsion), in N.mL = Length of the girder, in mIp = Polar Inertia, in m4

G = E/2(1+ν), the shear Modulus, in N/m2

θ =M LG I

T

p

.

M r N ds N r ds NT = = =∫∫ 2 Ω

18.4.3.4 Twisting and warpingTorsional strength: Although torsion is not usually an im-portant factor in ship design for most ships, it does resultin significant additional stresses on ships, such as containerships, which have large hatch openings. These warpingstresses can be calculated by a beam analysis, which takesinto account the twisting and warping deflections. Therecan also be an interaction between horizontal bending andtorsion of the hull girder. Wave actions tending to bend thehull in a horizontal plane also induce torsion because of theopen cross section of the hull, which results in the shear cen-ter being below the bottom of the hull. Combined stressesdue to vertical bending, horizontal bending and torsion mustbe calculated.

In order to increase the torsional rigidity of the contain-ership cross sections, longitudinal and transverse closedbox girders are introduced in the upper side and deck struc-ture.

From previous studies, it has been established that spe-cial attention should be paid to the torsional rigidity distri-bution along the hull. Usually, toward the ship’s ends, thesection moduli are justifiably reduced base on bending. Onthe contrary the torsional rigidity, especially in the forwardhatches, should be gradually increased to keep the warpingstress as small as possible.

Twisting of opened section: A lateral seaway could in-duce severe twisting moment that is of the major importancefor ships having large deck openings. The equations for thetwist of a closed tube (equations 36 to 38) are applicableonly to the computation of the torsional response of closedthin-walled sections.

The relative torsional stiffness of closed and open sec-tions may be visualized by means of a very simple example.

Consider two circular tubes, one of which has a longi-tudinal slit over its full length as in Figure 18.30. The closedtube will be able to resist a much greater torque per unit an-gular deflection than the open tube because of the inabilityof the latter to sustain the shear stress across the slot. Thetwisting resistance of the thin material of which the tube iscomposed provides the only resistance to torsion in the case


Figure 18.29 Torsional Shear Flow (2). Figure 18.30 Twist of Open and Closed Tubes (2)

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of the open tube without longitudinal restraint. The resist-ance to twist of the entirely open section is given by the St.Venant torsion equation:

MT = G.J ∂θ/∂x (N.m) [39]

where:

∂θ/∂x = twist angle per unit length, in rad./m, which can beapproximated by θ/L for uniform torsion and uni-form section.

J = torsional constant of the section, in m4

= for a thin walled open section

= for a section composed of n different

= plates (bi= length, ti = thickness)

If warping resistance is present, that is, if the longitudi-nal displacement of the elemental strips shown in Figure18.30 is constrained, another component of torsional re-sistance is developed through the shear stresses that resultfrom this warping restraint. This is added to the torque givenby equation 39.

In ship structures, warping strength comes from foursources:

1. the closed sections of the structure between hatch open-ings,

2. the closed ends of the ship,3. double wall transverse bulkheads, and4. closed, torsionally stiff parts of the cross section (lon-

gitudinal torsion tubes or boxes, including double bot-tom, double side shell, etc.).

18.4.3.5 Racking and snakingRacking is the result of a transverse hull shape distortion andis caused by either dynamic loads due to rolling of the shipor by the transverse impact of seas against the topsides. Trans-verse bulkheads resist racking if the bulkhead spacing is closeenough to prevent deflection of the shell or deck plating inits own plane. Racking introduces primarily compressive andshearing forces in the plane of bulkhead plating.

With the usual spacing of transverse bulkheads the ef-fectiveness of side frames in resisting racking is negligible.However, when bulkheads are widely spaced or where thedeck width is small in way of very large hatch openings,side frames, in association with their top and bottom brack-ets, contribute significant resistance to racking. Racking incar-carriers is discussed in Chapters 17 and 34.

Racking stresses due to rolling reach a maximum in abeam sea each time the vessel completes an oscillation inone direction and is about to return.

13

3

1

b ti ii

n

=∑

1 3 30

/ t dss

∫

The angle between a deck beam and side frame tends toopen on one side and to close on the other side at the topand reverses its action at the bottom. The effect of the con-centration of stiff and soft sections results in a distortion pat-tern in the ship deck that is shown in Figure 18.31. The termsnaking is sometimes used in referring to this behavior andrelates to both twisting and racking.

18.4.3.6 Effective breadth and shear lagAn important effect of the edge shear loading of a platemember is a resulting nonlinear variation of the longitudi-nal stress distribution (Figure 18.32). In the real plate thelongitudinal stress decreases with increasing distance fromthe shear-loaded edge, and this is called shear lag. This isin contrast to the uniform stress distribution predicted inthe beam flanges by the elementary beam equation 29. Inmany practical cases, the difference from the value pre-dicted in equation 29 will be small. But in certain combi-nations of loading and structural geometry, the effect referredto by the term shear lag must be taken into considerationif an accurate estimate of the maximum stress in the mem-ber is to be made. This may be conveniently done by defin-ing an effective breadth of the flange member.

The ratio, be/b, of the effective breadth, be, to the realbreadth, b, is useful to the designer in determining the lon-gitudinal stress along the shear-loaded edge. It is a function


Figure 18.31 Snaking Behavior of a Container Vessel (2).

Figure 18.32 Shear Lag Effect in a Deck (2)

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of the external loading applied and the boundary conditionsalong the plate edges, but not its thickness. Figure 18.33gives the effective breadth ratio at mid-length for columnloading and harmonic-shaped beam loading, together witha common approximation for both cases:

[40]

The results are presented in a series of design charts,which are especially simple to use, and may be found inSchade (26).

A real situation in which such an alternating load dis-tribution may be encountered is a bulk carrier loaded witha dense ore cargo in alternate holds, the remainder beingempty.

An example of the computation of the effective breadthof bottom and deck plating for such a vessel is given inChapter VI of Taggart (1), using Figure 18.33.

It is important to distinguish the effective breadth (equa-tion 40) and the effective width (equations 54 and 55) pre-sented later in Subsection 18.6.3.2 for plate and stiffenedplate-buckling analysis.

18.4.3.7 Longitudinal deflectionThe longitudinal bending deflection of the ship girder is ob-tainable from the appropriate curvature equations (equa-tions 27 and 28) by integrating twice. A semi-empiricalapproximation for bending deflection amidships is:

bb

k Lb

e =6

w = k ( M L2/EI ) [41]

where the dimensionless coefficient k may be taken, for firstapproximation, as 0.09 (2).

Actual deflection in service is affected also by thermalinfluences, rigidity of structural components, and work-manship; furthermore, deflection due to shear is additive tothe bending deflection, though its amount is usually rela-tively small.

The same influences, which gradually increase nominaldesign stress levels, also increase flexibility. Additionally,draft limitations and stability requirements may force theL/D ratio up, as ships get larger. In general, therefore, mod-ern design requires that more attention be focused on flex-ibility than formerly.

No specific limits on hull girder deflections are given inthe classification rules. The required minimum scantlingshowever, as well as general design practices, are based ona limitation of the L/D ratio range.

18.4.3.8 Load diffusion into structureThe description of the computation of vertical shear andbending moment by integration of the longitudinal load dis-tribution implies that the external vertical load is resisteddirectly by the vertical shear carrying members of the hullgirder such as the side shell or longitudinal bulkheads. In alongitudinally framed ship, such as a tanker, the bottompressures are transferred principally to the widely spacedtransverse web frames or the transverse bulkheads where


Figure 18.33 Effective Breath Ratios at Midlength (1)

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they are transferred to the longitudinal bulkheads or sideshell, again as localized shear forces. Thus, in reality, theloading q(x), applied to the side shell or the longitudinalbulkhead will consist of a distributed part due to the directtransfer of load into the member from the bottom or deckstructure, plus a concentrated part at each bulkhead or webframe. This leads to a discontinuity in the shear curve at thebulkheads and webs.

18.4.3.9 Hull/superstructure interactionThe terms superstructure and deckhouse refer to a structureusually of shorter length than the entire ship and erectedabove the strength deck of the ship. If its sides are coplanarwith the ship’s sides it is referred to as a superstructure. Ifits width is less than that of the ship, it is called a deckhouse.

The prediction of the structural behavior of a super-structure constructed above the strength deck of the hullhas facets involving both the general bending response andimportant localized effects. Two opposing schools of thoughtexist concerning the philosophy of design of such erections.One attempts to make the superstructure effective in con-tributing to the overall bending strength of the hull, the otherpurposely isolates the superstructure from the hull so thatit carries only localized loads and does not experiencestresses and deflections associated with bending of the mainhull. This may be accomplished in long superstructures(>0.5Lpp) by cutting the deckhouse into short segments bymeans of expansion joints. Aluminum deckhouse con-struction is another alternative when the different materialproperties provide the required relief.

As the ship hull experiences a bending deflection in re-sponse to the wave bending moment, the superstructure isforced to bend also. However, the curvature of the super-structure may not necessarily be equal to that of the hull butdepends upon the length of superstructure in relation to thehull and the nature of the connection between the two, es-pecially upon the vertical stiffness or foundation modulusof the deck upon which the superstructure is constructed.The behavior of the superstructure is similar to that of abeam on an elastic foundation loaded by a system of nor-mal forces and shear forces at the bond to the hull.

The stress distributions at the midlength of the super-structure and the differential deflection between deckhouseand hull for three different degrees of superstructure effec-tiveness are shown on Figure 18.34.

The areas and inertias can be computed to account forshear lag in decks and bottoms. If the erection material dif-fers from that of the hull (aluminum on steel, for example)the geometric erection area Af and inertia If must be reducedaccording to the ratio of the respective material moduli; thatis, by multiplying by E (aluminum)/E (steel) (approximately

one-third). Further details on the design considerations fordeckhouses and superstructures may be found in Evans (27)and Taggart (1).

In addition to the overall bending, local stress concentra-tions may be expected at the ends of the house, since here thestructure is transformed abruptly from that of a beam consist-ing of the main hull alone to that of hull plus superstructure.

Recent works achieved in Norwegian University of Sci-ence & Technology have shown that the vertical stress dis-tribution in the side shell is not linear when there are largeopenings in the side shell as it is currently the case for upperdecks of passenger vessels. Approximated stress distribu-tions are presented at Figure 18.35. The reduced slope, θ,for the upper deck has been found equal to 0.50 for a cata-maran passenger vessel (28).

18.4.4 Secondary ResponseIn the case of secondary structural response, the principalobjective is to determine the distribution of both in-plane


Figure 18.34 Three Interaction Levels between Superstructure and Hull (1)

Figure 18.35 Vertical Stress Distribution in Passenger Vessels having Large

Openings above the Passenger Deck

Neutral axis

Passenger deck

x

z

( )zIMz =)(σ

)(.)( zzr σθσ =

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and normal loading, deflection and stress over the lengthand width dimensions of a stiffened panel. Remember thatthe primary response involves the determination of only thein-plane load, deflection, and stress as they vary over thelength of the ship. The secondary response, therefore, isseen to be a two-dimensional problem while the primaryresponse is essentially one-dimensional in character.

18.4.4.1 Stiffened panelsA stiffened panel of structure, as used in the present con-text, usually consists of a flat plate surface with its attachedstiffeners, transverse frames and/or girders (Figure 18.36).When the plating is absent the module is a grid or grillageof beam members only, rather than a stiffened panel.

In principle, the solution for the deflection and stress inthe stiffened panel may be thought of as a solution for theresponse of a system of orthogonal intersecting beams.

A second type of interaction arises from the two-di-mensional stress pattern in the plate, which may be thoughtof as forming a part of the flanges of the stiffeners. The platecontribution to the beam bending stiffness arises from thedirect longitudinal stress in the plate adjacent to the stiff-ener, modified by the transverse stress effects, and also fromthe shear stress in the plane of the plate. The maximum sec-ondary stress may be found in the plate itself, but more fre-quently it is found in the free flanges of the stiffeners, sincethese flanges are at a greater distance than the plate mem-ber from the neutral axis of the combined plate-stiffener.

At least four different procedures have been employed forobtaining the structural behavior of stiffened plate panelsunder normal loading, each embodying certain simplifyingassumptions: 1) orthotropic plate theory, 2) beam-on-elastic-

foundation theory, 3) grillage theory (intersecting beams), and4) the finite element method (FEM).

Orthotropic plate theory refers to the theory of bendingof plates having different flexural rigidities in the two or-thogonal directions. In applying this theory to panels hav-ing discrete stiffeners, the structure is idealized by assumingthat the structural properties of the stiffeners may be ap-proximated by their average values, which are assumed tobe distributed uniformly over the width or length of theplate. The deflections and stresses in the resulting contin-uum are then obtained from a solution of the orthotropicplate deflection differential equation:

[42]

where:

a1, a2, a3 = express the average flexural rigidity of the or-thotropic plate in the two directions

w(x,y) = is the deflection of the plate in the normal di-rection

p(x,y) = is the distributed normal pressure load per unitarea

Note that the behavior of the isotropic plate, that is, onehaving uniform flexural properties in all directions, is a spe-cial case of the orthotropic plate problem. The orthotropicplate method is best suited to a panel in which the stiffen-ers are uniform in size and spacing and closely spaced. Ithas been said that the application of this theory to cross-stiffened panels must be restricted to stiffened panels withmore than three stiffeners in each direction.

An advanced orthotropic procedure has been imple-mented by Rigo (29,30) into a computer-based scheme forthe optimum structural design of the midship section. It isbased on the differential equations of stiffened cylindricalshells (linear theory). Stiffened plates and cylindrical shellscan both be considered, as plates are particular cases of thecylindrical shells having a very large radius. A system ofthree differential equations, similar to equation 42, is es-tablished (8th order coupled differential equations). Fourierseries expansions are used to model the loads. Assumingthat the displacements (u,v,w) can also be expanded in sinand cosine, an analytical solution of u, v, and w(x,y) can beobtained for each stiffened panel.

This procedure can be applied globally to all the stiff-ened panels that compose a parallel section of a ship, typ-ically a cargo hold.

This approach has three main advantages. First the platebending behavior (w) and the inplane membrane behavior(u and v) are analyzed simultaneously. Then, in addition to

aw

xa

wx y

aw

yp1

4

4 2

4

2 2 3

4

4∂∂

+ ∂∂ ∂

+ ∂∂

= (x,y)


Figure 18.36 A Stiffened Panel with Uniformly Distributed Longitudinals, 4

Webframes, and 3 Girders.

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the flexural rigidity (bending), the inplane axial, torsional,transverse shear and inplane shear rigidities of the stiffen-ers in the both directions can also be considered. Finally,the approach is suited for stiffeners uniform in size andspacing, and closely spaced but also for individual mem-bers, randomly distributed such as deck and bottom gird-ers. These members considered through Heaviside functionsthat allow replacing each individual member by a set of 3forces and 2 bending moment load lines. Figure 18.36 showsa typical stiffened panel that can be considered. It includesuniformly distributed longitudinals and web frames, andthree prompt elements (girders).

The beam on elastic foundation solution is suitable for apanel in which the stiffeners are uniform and closely spacedin one direction and sparser in the other one. Each of thesemembers is treated individually as a beam on an elastic foun-dation, for which the differential equation of deflection is,

[43]

where:

w = is the deflectionI = is sectional moment of inertia of the longitudinal

stiffener, including adjacent platingk = is average spring constant per unit length of the

transverse stiffenersq(x) = is load per unit length on the longitudinal member

The grillage approach models the cross-stiffened panelas a system of discrete intersecting beams (in plane frame),each beam being composed of stiffener and associated ef-fective plating. The torsional rigidity of the stiffened paneland the Poisson ratio effect are neglected. The validity ofmodeling the stiffened panel by an intersecting beam (or gril-lage) may be critical when the flexural rigidities of stiffen-ers are small compared to the plate stiffness. It is knownthat the grillage approach may be suitable when the ratioof the stiffener flexural rigidity to the plate bending rigid-ity (EI/bD with I the moment of inertia of stiffener and Dthe plate bending rigidity) is greater than 60 (31) otherwiseif the bending rigidity of stiffener is smaller, an OrthotropicPlate Theory has to be selected.

The FEM approach is discussed in detail in section 18.7.2.

18.4.5 Tertiary Response18.4.5.1 Unstiffened plateTertiary response refers to the bending stresses and deflec-tions in the individual panels of plating that are bounded bythe stiffeners of a secondary panel. In most cases the loadthat induces this response is a fluid pressure from either the

EIw

xk w q

∂∂

+ =4

4(x)

water outside the ship or liquid or dry bulk cargo within.Such a loading is normal to and distributed over the surfaceof the panel. In many cases, the proportions, orientation, andlocation of the panel are such that the pressure may be as-sumed constant over its area.

As previously noted, the deflection response of anisotropic plate panel is obtained as the solution of a specialcase of the earlier orthotropic plate equation (equation 42),and is given by:

[44]

where:

D = plate flexural rigidity

= Et3 / 12(1 – ν)t = the uniform plate thickness

p(x,y) = distributed unit pressure load

Appropriate boundary conditions are to be selected torepresent the degree of fixity of the edges of the panel.Stresses and deflections are obtained by solving this equa-tion for rectangular plates under a uniform pressure distri-bution. Equation 44 is in fact a simplified case of the generalone (equation 42).

Information (including charts) on a plate subject to uni-form load and concentrated load (patch load) is availablein Hughes (3).

18.4.5.2 Local deflectionsLocal deflections must be kept at reasonable levels in orderfor the overall structure to have the proper strength andrigidity. Towards this end, the classification society rules maycontain requirements to ensure that local deflections are notexcessive.

Special requirements also apply to stiffeners. Trippingbrackets are provided to support the flanges, and they shouldbe in line with or as near as practicable to the flanges of struts.Special attention must be given to rigidity of members undercompressive loads to avoid buckling. This is done by pro-viding a minimum moment of inertia at the stiffener and as-sociated plating.

18.4.6 Transverse StrengthTransverse strength refers to the ability of the ship struc-ture to resist those loads that tend to cause distortion of thecross section. When it is distorted into a parallelogram shapethe effect is called racking. We recall that both the primarybending and torsional strength analyses are based upon theassumption of no distortion of the cross section. Thus, we

E t 3

12(1 )− ν

∂∂

+ ∂∂ ∂

+ ∂∂

=4

4

4

2 2

4

42

wx

wx y

wy

pD

(x,y)


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see that there is an inherent relationship between transversestrength and both longitudinal and torsional strength. Cer-tain structural members, including transverse bulkheads anddeep web frames, must be incorporated into the ship in orderto insure adequate transverse strength. These members pro-vide support to and interact with longitudinal members bytransferring loads from one part of a structure to another.For example, a portion of the bottom pressure loading onthe hull is transferred via the center girder and the longitu-dinals to the transverse bulkheads at the ends of theses lon-gitudinals. The bulkheads, in turn, transfer these loads asvertical shears into the side shell. Thus some of the loadsacting on the transverse strength members are also the loadsof concern in longitudinal strength considerations.

The general subject of transverse strength includes ele-ments taken from both the primary and secondary strengthcategories. The loads that cause effects requiring transversestrength analysis may be of several different types, de-pending upon the type of ship, its structural arrangement,mode of operation, and upon environmental effects.

Typical situations requiring attention to the transversestrength are:

• ship out of water: on building ways or on constructionor repair dry dock,

• tankers having empty wing tanks and full centerline tanksor vice versa,

• ore carriers having loaded centerline holds and largeempty wing tanks,

• all types of ships: torsional and racking effects causedby asymmetric motions of roll, sway and yaw, and

• ships with structural features having particular sensitiv-ity to transverse effects, as for instance, ships havinglargely open interior structure (minimum transverse bulk-heads) such as auto carriers, containers and RO-RO ships.

As previously noted, the transverse structural responseinvolves pronounced interaction between transverse andlongitudinal structural members. The principal loading con-sists of the water pressure distribution around the ship, andthe weights and inertias of the structure and hold contents.As a first approximation, the transverse response of such aframe may be analyzed by a two-dimensional frame re-sponse procedure that may or may not allow for support bylongitudinal structure. Such analysis can be easily performedusing 2D finite element analysis (FEA). Influence of lon-gitudinal girders on the frame would be represented by elas-tic attachments having finite spring constants (similar toequation 43). Unfortunately, such a procedure is very sen-sitive to the spring location and the boundary conditions.For this reason, a three-dimensional analysis is usually per-formed in order to obtain results that are useful for more

than comparative purposes. Ideally, the entire ship hull orat least a limited hold-model should be modeled. See Sub-section 18.7.2—Structural Finite Element Models (Figure18.57).

18.4.7 Superposition of StressesIn plating, each response induces longitudinal stresses, trans-verse stresses and shear stresses. These stresses can be cal-culated individually for each response. This is the traditionalway followed by the classification societies. With directanalysis such as finite element analysis (Subsection 18.7.2),it is not always possible to separate the different responses.

If calculated individually, all the longitudinal stresseshave to be added. Similar cumulative procedure must beachieved for the transverse stresses and the shear stresses.At the end they are combined through a criteria, which isusually for ship structure, the von-Mises criteria (equation45).

The standard procedure used by classification societiesconsiders that longitudinal stresses induced by primary re-sponse of the hull girder, can be assessed separately fromthe other stresses. Classification rules impose through al-lowable stress and minimal section modulus, a maximumlongitudinal stress induced by the hull girder bending mo-ment.

On the other hand, they recommend to combined stressesfrom secondary response and tertiary response, in platingand in members. These are combined through the von Misescriteria and compared to the classification requirements.

Such an uncoupled procedure is convenient to use butdoes not reflect reality. Direct analysis does not follow thisapproach. All the stresses, from the primary, secondary andtertiary responses are combined for yielding assessment.For buckling assessment, the tertiary response is discarded,as it does not induce in-plane stresses. Nevertheless the lat-eral load can be considered in the buckling formulation(Subsection 18.6.3). Tertiary stresses should be added forfatigue analysis.

Since all the methods of calculation of primary, sec-ondary, and tertiary stress presuppose linear elastic behav-ior of the structural material, the stress intensities computedfor the same member may be superimposed in order to ob-tain a maximum value for the combined stress. In performingand interpreting such a linear superposition, several con-siderations affecting the accuracy and significance of the re-sulting stress values must be borne in mind.

First, the loads and theoretical procedures used in com-puting the stress components may not be of the same ac-curacy or reliability. The primary loading, for example, maybe obtained using a theory that involves certain simplifica-


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tions in the hydrodynamics of ship and wave motion, andthe primary bending stress may be computed by simplebeam theory, which gives a reasonably good estimate of themean stress in deck or bottom but neglects certain localizedeffects such as shear lag or stress concentrations.

Second, the three stress components may not necessar-ily occur at the same instant in time as the ship movesthrough waves. The maximum bending moment amidships,which results in the maximum primary stress, does not nec-essarily occur in phase with the maximum local pressureon a midship panel of bottom structure (secondary stress)or panel of plating (tertiary stress).

Third, the maximum values of primary, secondary, andtertiary stress are not necessarily in the same direction oreven in the same part of the structure. In order to visualizethis, consider a panel of bottom structure with longitudinalframing. The forward and after boundaries of the panel willbe at transverse bulkheads. The primary stress (σ1) will actin the longitudinal direction, as given by equation 29. It willbe nearly equal in the plating and the stiffeners, and will beapproximately constant over the length of a midship panel.There will be a small transverse component in the plating,due to the Poison coefficient, and a shear stress given byequation 35. The secondary stress will probably be greaterin the free flanges of the stiffeners than in the plating, sincethe combined neutral axis of the stiffener/plate combina-tion is usually near the plate-stiffener joint. Secondarystresses, which vary over the length of the panel, are usu-ally subdivided into two parts in the case of single hull struc-ture. The first part (σ2) is associated with bending of a panelof structure bounded by transverse bulkheads and either theside shell or the longitudinal bulkheads. The principal stiff-eners, in this case, are the center and any side longitudinalgirders, and the transverse web frames. The second part,(σ2

*), is the stress resulting from the bending of the smallerpanel of plating plus longitudinal stiffeners that is boundedby the deep web frames. The first of these components (σ2),as a result of the proportions of the panels of structure, isusually larger in the transverse than in the longitudinal di-rection. The second (σ2

*) is predominantly longitudinal.The maximum tertiary stress (σ3) happens, of course, in theplate where biaxial stresses occur. In the case of longitudi-nal stiffeners, the maximum panel tertiary stress will act inthe transverse direction (normal to the framing system) atthe mid-length of a long side.

In certain cases, there will be an appreciable shear stresscomponent present in the plate, and the proper interpreta-tion and assessment of the stress level will require the res-olution of the stress pattern into principal stress components.

From all these considerations, it is evident that, in manycases, the point in the structure having the highest stress level

will not always be immediately obvious, but must be foundby considering the combined stress effects at a number ofdifferent locations and times.

The nominal stresses produced from the analysis will bea combination of the stress components shown in Figures18.21 and 18.37.

18.4.7.1 von Mises equivalent stressThe yield strength of the material, σyield, is defined as themeasured stress at which appreciable nonlinear behavioraccompanied by permanent plastic deformation of the ma-terial occurs. The ultimate strength is the highest level ofstress achieved before the test specimen fractures. For mostshipbuilding steels, the yield and tensile strengths in ten-sion and compression are assumed equal.

The stress criterion that must be used is one in which itis possible to compare the actual multi-axial stress with thematerial strength expressed in terms of a single value forthe yield or ultimate stress.

For this purpose, there are several theories of materialfailure in use. The one usually considered the most suitablefor ductile materials such as ship steel is referred to as thevon Mises Theory:

[45]

Consider a plane stress field in which the componentstresses are σx, σy and τ. The distortion energy states that

σ σ σ σ σ τe x y x y= + − +( )2 2 21

23


Figure 18.37 Definition of Stress Components (4)

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failure through yielding will occur if the equivalent vonMises stress, σe, given by equation 45 exceeds the equiva-lent stress, σο, corresponding to yielding of the material testspecimen. The material yield strength may also be expressedthrough an equivalent stress at failure: σ0 = σyield (= σy).

18.4.7.2 Permissible stresses (Yielding)In actual service, a ship may be subjected to bending in theinclined position and to other forces, such as those, whichinduce torsion or side bending in the hull girder, not to men-tion the dynamic effects resulting from the motions of theship itself. Heretofore it has been difficult to arrive at theminimum scantlings for a large ship’s hull by first princi-ples alone, since the forces that the structure might be re-quired to withstand in service conditions are uncertain.Accordingly, it must be assumed that the allowable stressincludes an adequate factor of safety, or margin, for theseuncertain loading factors.

In practice, the margin against yield failure of the struc-ture is obtained by a comparison of the structure’s von Misesequivalent stress, σe, against the permissible stress (or al-lowable stress), σ0, giving the result:

σe ≤ σ0 = s1 × σy [46]

where:

s1 = partial safety factor defined by classification societies,which depends on the loading conditions and methodof analysis. For 20 years North Atlantic conditions(seagoing condition), the s1 factor is usually taken be-tween 0.85 and 0.95

σy = minimum yield point of the considered steel (mildsteel, high tensile steel, etc.)

For special ship types, different permissible stresses maybe specified for different parts of the hull structure. For ex-ample, for LNG carriers, there are special strain require-ments in way of the bonds for the containment system, whichin turn can be expressed as equivalent stress requirements.

For local areas subjected to many cycles of load rever-sal, fatigue life must be calculated and a reduced permissi-ble stress may be imposed to prevent fatigue failure (seeSubsection 18.6.6).

18.5 LIMIT STATES AND FAILURE MODES

Avoidance of structural failure is the goal of all structuraldesigners, and to achieve this goal it is necessary for the de-signer to be aware of the potential limit states, failure modesand methods of predicting their occurrence. This sectionpresents the basic types of failure modes and associated limit

states. A more elaborate description of the failure modes andmethods to assess the structural capabilities in relation tothese failure modes is available in Subsection 18.6.1.

Classically, the different limit states were divided in 2major categories: the service limit state and the ultimatelimit state. Today, from the viewpoint of structural design,it seems more relevant to use for the steel structures fourtypes of limit states, namely:

1. service or serviceability limit state,2. ultimate limit state,3. fatigue limit state, and4. accidental limit state.

This classification has recently been adopted by ISO.A service limit state corresponds to the situation where

the structure can no longer provide the service for which itwas conceived, for example: excessive deck deflection, elas-tic buckling in a plate, and local cracking due to fatigue.Typically they relate to aesthetic, functional or maintenanceproblem, but do not lead to collapse.

An ultimate limit state corresponds to collapse/failure,including collision and grounding. A classic example of ul-timate limit state is the ultimate hull bending moment (Fig-ure 18.46). The ultimate limit state is symbolized by thehigher point (C) of the moment-curvature curve (M-Φ).

Fatigue can be either considered as a third limit state or,classically, considered as a service limit state. Even if it isalso a matter of discussion, yielding should be consideredas a service limit state. First yield is sometimes used to as-sess the ultimate state, for instance for the ultimate hullbending moment, but basically, collapse occurs later. Mostof the time, vibration relates to service limit states.

In practice, it is important to differentiate service, ulti-mate, fatigue and accidental limit states because the partialsafety factors associated with these limit states are gener-ally different.

18.5.1 Basic Types of Failure ModesShip structural failure may occur as a result of a variety ofcauses, and the degree or severity of the failure may varyfrom a minor esthetic degradation to catastrophic failure re-sulting in loss of the ship. Three major failure modes aredefined:

1. tensile or compressive yield of the material (plasticity),2. compressive instability (buckling), and3. fracture that includes ductile tensile rupture, low-cycle

fatigue and brittle fracture.

Yield occurs when the stress in a structural member ex-ceeds a level that results in a permanent plastic deforma-


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tion of the material of which the member is constructed. Thisstress level is termed the material yield stress. At a some-what higher stress, termed the ultimate stress, fracture ofthe material occurs. While many structural design criteriaare based upon the prevention of any yield whatsoever, itshould be observed that localized yield in some portions ofa structure is acceptable. Yield must be considered as a serv-iceability limit state.

Instability and buckling failure of a structural memberloaded in compression may occur at a stress level that is sub-stantially lower than the material yield stress. The load atwhich instability or buckling occurs is a function of mem-ber geometry and material elasticity modulus, that is, slen-derness, rather than material strength. The most commonexample of an instability failure is the buckling of a simplecolumn under a compressive load that equals or exceedsthe Euler Critical Load. A plate in compression also willhave a critical buckling load whose value depends on theplate thickness, lateral dimensions, edge support conditionsand material elasticity modulus. In contrast to the column,however, exceeding this load by a small margin will notnecessarily result in complete collapse of the plate but onlyin an elastic deflection of the central portion of the plate awayfrom its initial plane. After removal of the load, the platemay return to its original un-deformed configuration (forelastic buckling). The ultimate load that may be carried bya buckled plate is determined by the onset of yielding at somepoint in the plate material or in the stiffeners, in the case ofa stiffened panel. Once begun, yield may propagate rapidlythroughout the entire plate or stiffened panel with furtherincrease in load.

Fatigue failure occurs as a result of a cumulative effectin a structural member that is exposed to a stress pattern al-ternating from tension to compression through many cy-cles. Conceptually, each cycle of stress causes some smallbut irreversible damage within the material and, after theaccumulation of enough such damage, the ability of themember to withstand loading is reduced below the level ofthe applied load. Two categories of fatigue damage are gen-erally recognized and they are termed high-cycle and low-cycle fatigue. In high-cycle fatigue, failure is initiated inthe form of small cracks, which grow slowly and whichmay often be detected and repaired before the structure isendangered. High-cycle fatigue involves several millionsof cycles of relatively low stress (less than yield) and is typ-ically encountered in machine parts rotating at high speedor in structural components exposed to severe and prolongedvibration. Low-cycle fatigue involves higher stress levels,up to and beyond yield, which may result in cracks beinginitiated after several thousand cycles.

The loading environment that is typical of ships and

ocean structures is of such a nature that the cyclical stressesmay be of a relatively low level during the greater part ofthe time, with occasional periods of very high stress levelscaused by storms. Exposure to such load conditions mayresult in the occurrence of low-cycle fatigue cracks after aninterval of a few years. These cracks may grow to serioussize if they are not detected and repaired.

Concerning brittle fracture, small cracks suddenly beginto grow and travel almost explosively through a major por-tion of the structure. The term brittle fracture refers to thefact that below a certain temperature, the ultimate tensilestrength of steel diminishes sharply (lower impact energy).The originating crack is usually found to have started as aresult of poor design or manufacturing practice. Fatigue(Subsection 18.6.6) is often found to play an important rolein the initiation and early growth of such originating cracks.The prevention of brittle fracture is largely a matter of ma-terial selection and proper attention to the design of struc-tural details in order to avoid stress concentrations. Thecontrol of brittle fracture involves a combination of designand inspection standards aimed toward the prevention ofstress concentrations, and the selection of steels having ahigh degree of notch toughness, especially at low tempera-tures. Quality control during construction and in-service in-spection form key elements in a program of fracture control.

In addition to these three failure modes, additional modesare:

• collision and grounding, and• vibration and noise.

Collision and Grounding is discussed in Subsection18.6.7 and Vibration in Subsection 18.6.8. Vibration as wellas noise is not a failure mode, while it could fall into theserviceability limit state.

18.6 ASSESSMENT OF THE STRUCTURALCAPACITY

18.6.1 Failure Modes ClassificationThe types of failure that may occur in ship structures aregenerally those that are characteristic of structures made upof stiffened panels assembled through welding. Figure 18.38presents the different structure levels: the global structure,usually a cargo hold (Level 1), the orthotropic stiffenedpanel or grillage (Level 2) and the interframe longitudi-nally stiffened panel (Level 3) or its simplified modeling:the beam-column (Level 3b). Level 4 (Figure 18.44a) is theunstiffened plate between two longitudinals and two trans-verse frames (also called bare plate).

The word grillage should be reserve to a structure com-


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posed of a grid of beams (without attached plating). Whenthe grid is fixed on a plate, orthotropic stiffened panel seemsto the authors more adequate to define a panel that is or-thogonally stiffened, and having thus orthotropic properties.

The relations between the different failure modes andstructure levels can be summarized as follows:

• Level 1: Ultimate bending moment, Mu, of the globalstructure (Figure 18.46).

• Level 2: Ultimate strength of compressed orthotropicstiffened panels (σu),

σu = min [σu (mode i)], i = I to VI,

the 6 considered failure modes.• Level 3:

Mode I: Overall buckling collapse (Figure 18.44d),Mode II: Plate/Stiffener YieldingMode III: Pult of interframe panels with a plate-stifener combination (Figure 18.44b) using a beam-col-umn model (Level 3b) or an orthotropic model (Level3), considering:

— plate induced failure (buckling)— stiffener induced failure (buckling or yielding)

Mode IV and V: Instability of stiffeners (local buck-ling, tripping—Figure 18.44c)Mode VI: Gross Yielding

• Level 4: Buckling collapse of unstiffened plate (bareplate, Figure 18.44a).

To avoid collapse related to the Mode I, a minimal rigid-ity is generally imposed for the transverse frames so that aninterframe panel collapse (Mode III) always occurs prior tooverall buckling (Mode I). It is a simple and easy constraintto implement, thus avoiding any complex calculation ofoverall buckling (mode I).

Note that the failure Mode III is influenced by the buck-ling of the bare plate (elementary unstiffened plate). Elas-tic buckling of theses unstiffened plates is usually notconsidered as an ultimate limit state (failure mode), butrather as a service limit state. Nevertheless, plate buckling(Level 4) may significantly affect the ultimate strength ofthe stiffened panel (Level 3).

Sources of the failures associated with the serviceabil-ity or ultimate limit states can be classified as follows:

18.6.1.1 Stiffened panel failure modesService limit state

• Upper and lower bounds (Xmin≤X≤Xmax): plate thick-ness, dimensions of longitudinals and transverse stiff-eners (web, flange and spacing).

• Maximum allowable stresses against first yield (Sub-section 18.4.7)

• Panel and plate deflections (Subsections 18.4.4.1 and18.4.5.2), and deflection of support members.

• Elastic buckling of unstiffened plates between two lon-gitudinals and two transverse stiffeners, frames or bulk-heads (Subsection 18.6.3),

• Local elastic buckling of longitudinal stiffeners (weband flange). Often the stiffener web/flange buckling doesnot induce immediate collapse of the stiffened panel astripping does. It could therefore be considered as a serv-iceability ultimate limit state. However, this failure modecould also be classified into the ultimate limit state sincethe plating may sometimes remain without stiffeningonce the stiffener web buckles.

• Vibration (Sub-ection 18.6.8)• Fatigue (Sub-ection 18.6.6)

Ultimate limit state (Subsection 18.6.4).

• Overall collapse of orthotropic panels (entire stiffenedplate structure),


Figure 18.38 Structural Modeling of the Structure and its Components

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• Collapse of interframe longitudinally stiffened panel,including torsional-flexural (lateral-torsional) bucklingof stiffeners (also called tripping).

18.6.1.2 Frame failure modesService limit state (Subsection 18.4.6).

• Upper and lower bounds (Xmin ≤ X ≤ Xmax),• Minimal rigidity to guarantee rigid supports to the in-

terframe panels (between two transverse frames).• Allowable stresses under the resultant forces (bending,

shear, torsion)

— Elastic analysis,— Elasto-plastic analysis.

• Fatigue (Subsection 18.6.6)

Ultimate limit state

• Frame bucklings: These failures modes are consideredas ultimate limit states rather than a service limit state.If one of them appears, the assumption of rigid supportsis no longer valid and the entire stiffened panel can reachthe ultimate limit state.

— Buckling of the compressed members,— Local buckling (web, flange).

18.6.1.3 Hull Girder Collapse modesService limit state

• Allowable stresses and first yield (Subsection 18.4.3.1),• Deflection of the global structure and relative deflec-

tions of components and panels (Subsection 18.4.3.7).

Ultimate limit state

• Global ultimate strength (of the hull girder/box girder).This can be done by considering an entire cargo hold oronly the part between two transverse web frames (Sub-section 18.6.5). Collapse of frames is assumed to onlyappear after the collapse of panels located between theseframes. This means that it is sufficient to verify the boxgirder ultimate strength between two frames to be pro-tected against a more general collapse including, for in-stance, one or more frame spans. This approach can beun-conservative if the frames are not stiff enough.

• Collision and grounding (Subsection 18.6.7), which isin fact an accidental limit state.

A relevant comparative list of the limit states was de-fined by the Ship Structure Committee Report No 375 (32).

18.6.2 YieldingAs explained in Subsection 18.5.1 yield occurs when thestress in a structural component exceeds the yield stress.

It is necessary to distinguish between first yield state andfully plastic state. In bending, first yield corresponds to thesituation when stress in the extreme fiber reaches the yieldstress. If the bending moment continues to increase the yieldarea is growing. The final stage corresponds to the PlasticMoment (Mp), where, both the compression and tensile sidesare fully yielded (as shown on Figure 18.47).

Yield can be assessed using basic bending theory, equa-tion 29, up to complex 3D nonlinear FE analysis. Designcriteria related to first yield is the von Mises equivalentstress (equation 45).

Yielding is discussed in detail in Section 18.4.

18.6.3 Buckling and Ultimate Strength of PlatesA ship stiffened plate structure can become unstable if ei-ther buckling or collapse occurs and may thus fail to per-form its function. Hence plate design needs to be such thatinstability under the normal operation is prevented (Figure18.44a). The phenomenon of buckling is normally dividedinto three categories, namely elastic buckling, elastic-plas-tic buckling and plastic buckling, the last two being calledinelastic buckling. Unlike columns, thin plating buckled inthe elastic regime may still be stable since it can normallysustain further loading until the ultimate strength is reached,even if the in-plane stiffness significantly decreases after theinception of buckling. In this regard, the elastic buckling ofplating between stiffeners may be allowed in the design,sometimes intentionally in order to save weight. Since sig-nificant residual strength of the plating is not expected afterbuckling occurs in the inelastic regime, however, inelasticbuckling is normally considered to be the ultimate strengthof the plate.

The buckling and ultimate strength of the structure de-pends on a variety of influential factors, namely geomet-ric/material properties, loading characteristics, fabricationrelated imperfections, boundary conditions and local dam-age related to corrosion, fatigue cracking and denting.

18.6.3.1 Direct AnalysisIn estimating the load-carrying capacity of plating betweenstiffeners, it is usually assumed that the stiffeners are sta-ble and fail only after the plating. This means that the stiff-eners should be designed with proper proportions that helpattain such behavior. Thus, webs, faceplates and flanges ofthe stiffeners or support members have to be proportionedso that local instability is prevented prior to the failure ofplating.


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Four load components, namely longitudinal compres-sion/tension, transverse compression/tension, edge shear andlateral pressure loads, are typically considered to act on shipplating between stiffeners, as shown in Figure 18.39, whilethe in-plane bending effects on plate buckling are also some-times accounted for. In actual ship structures, lateral pres-sure loading arises from water pressure and cargo weight.The still water magnitude of water pressure depends on thevessel draft, and the still water value of cargo pressure is de-termined by the amount and density of cargo loaded.

These still water pressure values may be augmented bywave action and vessel motion. Typically the larger in-planeloads are caused by longitudinal hull girder bending, bothin still water and in waves at sea, which is the source of theprimary stress as previously noted in Subsection 18.4.3.

The elastic plate buckling strength components undersingle types of loads, that is, σxE for σxav, σyE for σyav andτE for τav, can be calculated by taking into account the re-lated effects arising from in-plane bending, lateral pressure,cut-outs, edge conditions and welding induced residualstresses.

The critical (elastic-plastic) buckling strength compo-nents under single types of loads, that is, σxB for σxav, σyB

for σyav and τB for τav, are typically calculated by plasticitycorrection of the corresponding elastic buckling strengthusing the Johnson-Ostenfeld formula, namely:

[47]

where:

σE = elastic plate buckling strength

σ

σ σ σ

σσσ σ σB

E E F

FF

EE F

for

for=

≤

−

>

0 5

14

0 5

.

.

σB = critical buckling strength (that is, τB forshear stress)

σF = σY for normal stress= σY √

4

3 for shear stressσY = material yield stress

In 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.

For axial tensile loading, the critical strength may beconsidered to equal the material yield stress (σY).

Under single types of loads, the critical plate bucklingstrength must be greater than the corresponding appliedstress component with the relevant margin of safety. Forcombined biaxial compression/tension and edge shear, thefollowing type of critical buckling strength interaction cri-terion would need to be satisfied, for example:

[48]

where:

ηB = usage factor for buckling strength, which is typicallythe inverse of the conventional partial safety factor.

ηB = 1.0 is often taken for direct strength calculation, whileit is taken less than 1.0 for practical design in accor-dance with classification society rules.

Compressive stress is taken as negative while tensilestress is taken as positive and α = 0 if both σxav and σyav arecompressive, and α = 1 if either σxav or σyav or 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-dominantly longitudinal compressive loading before andafter 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 stressdistribution in the y direction may also become non-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 field is formed by theplate deflection since the plate edges remain straight.

With increase in the deflection of the plate keeping theedges straight, the upper and/or lower fibers inside the mid-dle of the plate element will initially yield by the action ofbending. However, as long as it is possible to redistribute

σσ α

σσ

σσ

σσ

ττ ηxav

xB

cxav

xB

yav

yB

yav

yB

cav

B

c

B

− +

+

≤


Figure 18.39 A Simply Supported Rectangular Plate Subject to Biaxial

Compression/tension, Edge Shear and Lateral Pressure Loads

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the applied loads to the straight plate boundaries by themembrane action, the plate element will not collapse. Col-lapse will then occur when the most stressed boundary lo-cations yield, since the plate element can not keep theboundaries straight any further, resulting in a rapid increaseof lateral plate deflection (33). Because of the nature of ap-plied axial compressive loading, the possible yield loca-

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-tions 49a and 49b can be expressed in terms of appliedstresses, lateral pressure loads and fabrication related ini-tial imperfections, by solving the nonlinear governing dif-ferential equations of plating, based on equilibrium andcompatibility equations. Note that equation 44 is the lineardifferential equation.

On the other hand, the plate ultimate edge shear strength,τu , is often taken τu =τB (equation 47, with τB instead ofσB).Also, an empirical formula obtained by curve fitting basedon nonlinear finite 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.

σ σ σ σ σx min x min y max y max2 2 2− + = Y

σ σ σ σ σx max x max y min y min2 2 2− + = Y


Figure 18.40 Membrane Stress Distribution Inside the Plate Element under

Predomianntly Longitudinal Compressive Loads; (a) Before buckling, (b) After

buckling, unloaded edges move freely in plane, (c) After buckling, unloaded

edges kept straight

Figure 18.41 Possible Locations for the Initial Plastic Yield at the Plate Edges

(Expected yield locations, T: Tension, C: Compression); (a) Yield at longitudinal

mid-edges under longitudinal uniaxial compression, (b) Yield at transverse

mid-edges under transverse uniaxial compression)

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For combined biaxial compression/tension, edge shearand lateral pressure loads, the last being usually regardedas a given constant secondary load, the plate ultimatestrength interaction criterion may also be given by an ex-pression similar to equation 48, but replacing the criticalbuckling strength components by the corresponding ulti-mate strength components, as follows:

[50]

where:

α and c = variables defined in equation 48ηu = usage factors for the ultimate limit state

σxu and σyu = solutions of equation 49a with regard to σxav

and equation 49b with regard to σyav, respec-tively

18.6.3.2 Simplified 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, σB ,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 kc is a function of the plate aspect ratio, α = 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 deflected platein the longitudinal direction, which is taken as an integersatisfying the condition For long plate inα = m (m + 1).

km

mc = +

α

α 2

σ πνB ck

E tb

=−

2

2

2

12 1( )

σσ α

σσ

σσ

σσ

ττ ηxav

xu

cxav

xu

yav

yu

yav

yu

cav

u

c

u

− +

+

≤

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, τB, canalso be obtain by equation 51 and the buckling coefficientfor simply supported edges is:

kc = 5.34 + 4(b/a)2 [53]

Figure 18.42 presents, kc, versus the aspect ratio, a/b, fordifferent configurations of rectangular plates in compression.

For the simplified 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 σx max = σY for σxav or σy max = σY for σyav.

This assumption is relevant when the unloaded edgesmove 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 components


Figure 18.42 Compressive Buckling Coefficient for Plates in Compression; for

5 Configurations (2) (A, B, C, D and E) where Boundary Conditions of Unloaded

Edges are: SS: Simply Supported, C: Clamped, and F: Free

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under uniaxial compressive stresses (σxu and σyu), as fol-low:

[54]

where aeu and beu are 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 exemplified 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:

β = is the plate slenderness

E = the Young’s modulust = the plate thickness

c1 , c2 = typically taken as c1 = 2 and c2 = 1

The plate ultimate strength components under uniaxialcompressive loads are therefore predicted by substitutingthe plate effective width formulae (equation 55a) into equa-tion 54.

More charts and formulations are available in manybooks, for example, 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-plied stress component with an appropriate target partialsafety factor. In a multiple load component case, the struc-tural safety check is made with equation 48 against buck-ling and equation 50 against ultimate limit state beingsatisfied.

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 transverse

σbt E

Y

aa

ba

eu = + −

0 9 1 91

0 92 2. . .

β β β

bb

for

c cfor

eu =<

− ≥

1 1

11 22

β

β ββ

σσ

σσ

xu

Y

eu yu

Y

eubb

aa

= =and

compression do not occur simultaneously. For instance,DNV (4) recommends:

• maximum compression, σx, in a plate field and phaseangle associated with σy, τ (buckling control),

• maximum compression, σy, in a plate field and phaseangle associated with σx, τ (buckling control),

• absolute maximum shear stress, τ, in a plate field andphase angle associated with σx, σy (buckling control),and

• maximum equivalent von Mises stress, σe, at given po-sitions (yield control).

In order to get σx and σy, the following stress compo-nents may normally be considered for the buckling control:

σ1 = stress from primary response, andσ2 = stress from secondary response (that is, double

bottom bending).

As the lateral bending effects should be normally in-cluded in the buckling strength formulation, stresses fromlocal bending of stiffeners (secondary response), σ2

*, andlocal bending of plate (tertiary response), σ3, must there-fore not to be included in the buckling control. If FE-analy-sis is performed the local plate bending stress, σ3, 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-tudinal girders, transverse webs and deep beams are de-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(Figure 18.43a). The stiffeners and webs/girders are at-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 influence of in-plane bending effects may be negligible.

However, for a large stiffened panel such as that in sideshell of ships, the effect of in-plane bending may not benegligible, since the panel may collapse by failure of stiff-


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eners which are loaded by largest added portion of axialcompression due to in-plane bending moments.

When the stiffeners are relatively small so that theybuckle 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 influence of in-plane bending. When the stiff-eners are relatively stiff so that the plating between stiffen-ers buckles before failure of the stiffeners, the ultimatestrength 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 cases

different 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-verse webs and deep beams, it is often assumed that thepanel edges are simply supported, with zero deflection andzero rotational restraints along four edges, with all edgeskept straight.

This idealization may provide somewhat pessimistic,but adequate predictions of the ultimate strength of stiffenedpanels supported by heavy longitudinal girders, transversewebs and deep beams (or bulkheads).

Today, direct non-linear strength assessment methodsusing recognized programs is usual (38). The model should


Figure 18.43 A Stiffened Steel Panel Under Biaxial Compression/Tension,

Biaxial In-plane Bending, Edge Shear and Lateral Pressure Loads. (a) Stiffened

Panel—Longitudinals 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 (tripping—mode V).

(d) Overall stiffened panel buckling (grillage or gross panel buckling—mode I).

(a)

(b)

(c)

(d)

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be capable of capturing all relevant buckling modes anddetrimental interactions between them. The fabrication re-lated initial imperfections in the form of initial deflections(plates, stiffeners) and residual stresses can in some casessignificantly affect (usually reduce) the ultimate strength ofthe panel so that they should be taken into account in thestrength computations as parameters of influence.

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 failure—yielding of the plate-

stiffener combination at panel edges,• Mode III: Plate induced failure—flexural buckling fol-

lowed by yielding of the plate-stiffener combination atmid-span,

• Mode IV: Stiffener induced failure—local buckling ofstiffener web,

• Mode V: Stiffener induced failure—tripping 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-cation related initial imperfections (initial deflection andwelding 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 find 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. Many simplified design methods have of coursebeen previously developed to estimate the panel ultimatestrength, considering one or more of the failure modesamong those mentioned above. Some of those methods havebeen reviewed by the ISSC’2000 (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).

Assessment of different formulations by comparison

with 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 Simplified modelsExisting simplified methods for predicting the ultimatestrength of stiffened panels typically use one or more of thefollowing approaches:

• orthotropic plate approach,• plate-stiffener combination approach (or beam-column

approach), 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 strengthfor the overall grillage collapse mode (Mode I, Figure18.44d), (31,46,48).

The plate-stiffener combination approach (also calledbeam-column approach) models the stiffened panel behav-ior by that of a single “beam” consisting of a stiffener to-gether with the attached plating, as representative of thestiffened 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-fect and the effect of the intersecting beams are all neg-lected. The beam-column approach is useful for thecomputation 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 tothe rigidity of stiffeners and/or under significant biaxialloading.

Stiffened panels are asymmetric in geometry about theplate-plane. This necessitates strength control for both plateinduced failure and stiffener-induced failure.

Plate induced failure: Deflection away from the plate as-sociated with yielding in compression at the connection be-tween plate and stiffener. The characteristic bucklingstrength for the plate is to be used.

Stiffener induced failure: Deflection 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 as


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the 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 fitting 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 mostdesign rules of classification societies, the so-called John-son-Ostenfeld formulation is used to account for this behav-ior (equation 47). On the other hand, in the so-calledPerry-Robertson formulation, the strength expression as-sumes that the stiffener with associated plating will collapseas a beam-column when the maximum compressive stress inthe extreme fiber reaches the yield strength of the material.

In empirical approaches, the ultimate strength formula-tions are developed by curve fitting based on mechanicalcollapse test results or numerical solutions. Even if limitedto a range of applicability (load types, slenderness ranges,assumed level of initial imperfections, etc.) they are veryuseful for preliminary design stage, uncertainty assessmentand as constraint in optimization package. While a vast num-ber of empirical formulations (sometimes called columncurves) 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-sion and with initial imperfections (initial deflections andresidual stresses) at an average level. Since the ultimatestrength of columns (σu) must be less than the elastic col-umn buckling strength (σE), the Paik-Thayamballi empiri-cal formula for a plate-stiffener combination is given by:

[56]

and

with

β σ= bt

YE

σσ λ

σσ

u

Y

E

Y≤ =1

2

σσ λ β λ β λ

u

Y=

+ + + −

1

0 995 0 936 2 0 17 2 0 188 2 2 0 067 4. . . . .

and

where:

r = radius of gyration= √

4

I / 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-stiffener combination, that is, without considering an ef-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-umn ultimate strength for a plate-stiffener combinationvarying the column slenderness ratios, with selected initialeccentricity and plate slenderness ratios. In usage of thePerry-Roberson formula, the lower strength as obtainedfrom either plate induced failure or stiffener-induced fail-ure is adopted herein. Interaction between bending axial

λ π σ σσ= =a

rYE

YE


Figure 18.45 A Comparison of the Ultimate Strength Formulations for

Plate-stiffener Combinations under Axial Compression (η relates to the

initial deflection)

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compression and lateral pressure can, within the same fail-ure mode (Flexural Buckling—Mode 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 defined by equation 50, but using the corre-sponding stiffened panel ultimate strength and stressparameters. 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 satisfied. 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 II–VI 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 combinedvertical and horizontal bending moments (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, Mu is 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 present

comprehensive 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.

Computation of Mu depends closely on the ultimatestrength of the structure’s constituent panels, and particularlyon the ultimate strength in compressed panels or components.Figure 18.46 shows that in sagging, the deck is compressed(σdeck) and reaches the ultimate limit state when σdeck = σu.On the other hand, the bottom is in tensile and reaches its ul-timate limit state after complete yielding, σbottom = σ0 (σ0

being the yield stress).Basically, there exist two main approaches to evaluate

the hull girder ultimate strength of a ship’s hull under lon-gitudinal bending moments. One, the approximate analy-sis, is to calculate the ultimate bending moment directly(Mu, point C on Figure 18.46), and the other is to performprogressive collapse analysis on a hull girder and obtain,both, Mu and the curves M-φ.

The first approach, approximate analysis, requires anassumption 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 (σu), and• the average stress-average strain relationship (σ−ε), to

perform a progressive collapse analysis.

For an approximate assessment, such as the Caldwellmethod, only the ultimate strength of each compressed panel(σu) is required.

18.6.5.1 Direct analysisThe direct analysis corresponds to the Progressive collapseanalysis. The methods include the typical numerical analy-


Figure 18.46 The Moment-Curvature Curve (M-Φ)

Figure 18.47 Typical Stress Distributions Used by Approximate Methods. (a)

First Yield. (b) Sagging Bending Moment (c) Evans (d) Paik—Mansour (e)

Caldwell Modified (f) Plastic Bending Moment.

(a) (b) (c) (d) (e) (f)

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sis such as Finite Element Method (FEM) and the Idealizedstructural Element method (ISUM) and Smith’s method,which is a simplified 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 ship’s hull girder. Both material and geometrical non-linearities can be considered.

A 3D analysis of a hold or a ship’s section is funda-mentally possible but very difficult to perform. This is be-cause a ship’s 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 fine mesh using ordinary elements. For instance, theinvestigation committee on the causes of the Nakhodka ca-sualty performed elastoplastic large deflection 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 sufficiently extends enough in the longitudinal di-rection to model the characteristic behavior. Thus, a typi-cal analysis may concern one frame spacing in a wholecompartment (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 difficulty to model the residual stresses.

Idealized Structural Unit Method (ISUM): presented inSubsection 18.7.3.1, can also be used to perform progres-sive collapse analysis. It allows calculating the ultimatebending 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.

Smith’s Method (Figure 18.48): A convenient alterna-tive to FEM is the Smith’s progressive collapse analysis(54), which consists of the following three steps (55).

Step 1: Modeling (mesh modeling of the cross-sectioninto elements),

Step 2: Derivation of average stress-average strain rela-tionship of each element (σ−ε curve), Figure18.49a.

Step 3: To perform progressive collapse analysis, Figure18.49b.


Figure 18.49 Influence of Element Average Stress-Average Strain Curves

(σ−ε) on Progressive Collapse Behavior. (a) Average stress-average strain

relationships of element, and (b) moment-curvature relationship of cross-

section.

(a)

(b)

Figure 18.48 The Smith’s Progressive Collapse Method

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In 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 (σ−ε) of this stiffener element is derived underthe axial load considering the influences of buckling andyielding. Step 3 can be explained as follows:

• axial rigidities of individual elements are calculated usingthe average stress-average strain relationships (σ−ε),

• flexural 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,

• the corresponding incremental bending moments areevaluated and so the strain and stress increments in in-dividual elements, and

• incremental curvatures and bending moments of thecross-section as well as incremental strains and stressesof elements are summed up to provide their cumulativevalues.

Figure 18.48 shows that the σ−ε 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 specifically 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 ε = –y φ, where φ is the hull curvatureand y is the distance from the neutral axis (simple beam as-sumption). The average stress-average strain curve (σ-ε)will then provide an estimate of the longitudinal stress (σi)acting on the section. Individual moments about the neu-tral axis are then summed to give the total bending momentfor a particular curvature φi.

The accuracy of the calculated ultimate bending mo-ment depends on the accuracy of the average stress-aver-age strain relationships of individual elements. Maindifficulties concern the modeling of initial imperfections(deflection and welding residual stress) and the boundaryconditions (multi-span model, interaction between adjacentelements, etc.).

Many formulations and methods to calculate these av-erage stress-average strain relationships are available: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, and the associated bending moment-curva-ture relationships (M-φ). Four typical σ−ε curves are con-sidered, which are:

Case A: Linear relationship (elastic). The M-φrelationshipis free from the influences of yielding and buck-ling, and is linear.

Case B: Bi-linear relationship (elastic-perfectly plastic,without buckling).

Case C: With buckling but without strength reduction be-yond the ultimate strength.

Case D: With buckling and a strength reduction beyondthe 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 flexural buckling, tripping, etc. ForCase C, it is assumed that the structural components can con-tinue to carry load after attaining their ultimate strength.The collapse behavior (M-φcurve) 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 defined as the ultimatelongitudinal bending moment of the hull girder (Mu).

Shortcomings and limitations of the Smith’s 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 suchas shear lag, warping and racking are thus ignored. 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 deflect/fail and significantly affect the stiff-ened plate structure and hull girder bending capacity.


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18.6.5.2 Simplified modelsCaldwell (64) was the first 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 influence of buckling and yielding ofstructural members composing a ship’s hull (Figure 18.47).

He idealised a stiffened cross-section of a ship’s hull toan unstiffened cross-section with equivalent thickness. Ifbuckling takes place at the compression side of bending,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.

Several authors have proposed improvements for theCaldwell 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-tic neutral axis is estimated using considerations oflongitudinal equilibrium. The ultimate bending moment isthen the sum of individual moments of all elements aboutthe plastic neutral axis.

In Caldwell’s Method, and Caldwell Modified Methods,reduction in the capacity of structural members beyond theirultimate strength is not explicitly taken into account. Thismay cause the overestimation of the ultimate strength ingeneral (Case C, Figure 18.49).

Empirical Formulations: In contrast to all the previousrational methods, there are some empirical formulationsusually calibrated for a type of specific 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 first estimate of the ultimate hull girder moment.

Interactions: In order to raise the problem of combinedloads (vertical and horizontal bending moments and shearforces), several authors have proposed empirical interac-

tion equations to predict the ultimate strength. Each loadcomponent is supposed to act separately. These methodswere reviewed by ISSC (68) and are often formulated asequation 57.

[57]

where:

Mv and Mh = vertical and horizontal bending momentsMvu and Mhu = ultimate vertical and horizontal bending mo-

mentsa, b and α = empirical constants

For instance, Mansour et al (47) proposes a=1, b=2 andα= 0.8 based on analysis on one container, one tanker and2 cruisers, and Gordo and Soares (60) 1.5<a=b<1.66 andα= 1.0 for tankers. Hu et al (69) has proposed similar for-mulations for bulk carriers. Paik et al (70) proposes an em-pirical formulation that includes the shear forces in additionto the bending moments.

18.6.5.3 Design CriteriaFor design purpose, the value of the ultimate longitudinalbending moment (capability) has to be compared with theextreme bending moment (load) that may act on a ship’s hullgirder. To estimate the extreme bending moment, the mostsevere loading condition has to be selected to provide themaximum still water bending moment. Regarding the wavebending moment, the IACS unified requirement is a majorreference (71,72), but more precise discussions can be foundin the ISSC 2000 report (24).

To evaluate the ultimate longitudinal strength, variousmethods can be applied ranging from simple to complicatedmethods. In 2000, many of the available methods were ex-amined and assessed by an ISSC’2000 Committee (50). Thegrading of each method with respect to each capability isquantitatively performed by scoring 1 through 5. The com-mittee concluded that the appropriate methods should be se-lected according to the designer’s needs and the designstage. That is, at early design stage, a simple method basedon an Assumed Stress Distribution can be used to obtain arough estimate of the ultimate bending moment. At laterstages, a more accurate method such as Progressive Col-lapse Analysis with calculated σ−εcurves (Smith’s Method)or ISUM has to be applied.

Main sensitive model capability with regards to the as-sessment of ultimate strength can be ranked in 3 classes, re-spectively, high (H), medium (M) and low (L) consequenceof omitting capability (Table 18.IV).

Based on the different sources of uncertainties (model-

MM

MM

v

vu

ah

hu

b

+

=α 1


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ing, σ−ε curves, curvature incrementation), the global un-certainty on the ultimate bending moment is usually large(55). A bias of 10 to 15% must be considered as acceptable.

For intact hull the design criteria for Mu, defined by clas-sification societies, is given by:

MS + s1 Mw ≤ s2 MU [58]

where:

s1 = the partial safety factor for load (typically 1.10)s2 = the material partial safety factor (typically 0.85)

MS = still water momentMw = design wave moment (20 year return period)

18.6.6 Fatigue and Fracture18.6.6.1 GeneralDesign criteria stated expressly in terms of fatigue damageresistance were in the past seldom employed in ship struc-tural design although cumulative fatigue criteria have beenused in offshore structure design. It was assumed that fa-tigue resistance is implicitly included in the conventionalsafety factors or acceptable stress margins based on pastexperience.

Today, fatigue considerations become more and moreimportant in the design of details such as hatch corners, re-inforcements for openings in structural members and so on.Since the ship-loading environment consists in large partof alternating loads, ship structures are highly sensitive tofatigue failures. Since 1990, fatigue is maybe the most sen-sitive point at the detailed design stage. Tools are available

but they are time consuming and there is large uncertaintyof using simplified methods.

With the introduction of higher tensile steels in hull struc-tures, at first in deck and bottom to increase hull girderstrength, and later in local structures, the fatigue problembecame more imminent. The fatigue strength does not in-crease according to the yield strength of the steel. In fact,fatigue is found to be independent of the yield strength. Thehigher stress levels in modern hull structures using highertensile steel have therefore led to a growing number of fa-tigue crack problems.

To ensure that the structure will fulfill its intended func-tion, fatigue assessment should be carried out for each in-dividual type of structural detail that is subjected to extensivedynamic loading. It should be noted that every welded jointand attachment or other form of stress concentration is po-tentially a source of fatigue cracking and should be indi-vidually considered.

This section gives an overview of feasible analysis to beperformed. A more complete description of the different fa-tigue procedures, S-N curves, stress concentration factors,and so on, are given in: Almar-Naess (73), DNV (4), Frickeet al (74), Maddox (75), Niemi (76), NRC (77) and Peter-shagen et al (78). Reliability-based fatigue procedure is pre-sented by Ayyub and Assakkaf in Chapter 19. These authorsalso have contributed to this section.

18.6.6.2 Basic fatigue theoriesFatigue analyses can be performed based on:

• simplified analytical expressions,• more refined analysis where loadings/load effects are

calculated by numerical analysis, and• a combination of simplified and refined techniques.`

There are generally two major technical approaches forfatigue life assessment of welded joints the Fracture Me-chanics Approach and the Characteristic S-N Curves Ap-proach.

The Fracture Mechanics Approach is based on crackgrowth data assuming that the crack initiation already ex-ists. The initiation phase is not modeled as it is assumed thatthe lifetime can be predicted only using fracture mechan-ics method of the growing cracks (after initiation). The frac-ture mechanics approach is obviously more detailed thanthe S-N curves approach. It involves examining crack growthand determining the number of load cycles that are neededfor small initial defects to grow into cracks large enough tocause fractures. The growth rate is proportional to the stressrange, S (or ∆σ) that is expressed in terms of a stress in-tensity factor, K, which accounts for the magnitude of thestress, current crack size, and weld and joint details. The


TABLE 18.IV Sensitivity Factors for Ultimate StrengthAssessment of Hull Girder.

Model Capability Impact

Plate buckling H

Stiffened plate buckling H

Post buckling behavior H

Plate welding residual stress H

M-φcurve (post collapse prediction) H

Plate initial deflection M

Stiffener initial deflection M

Stiffener welding residual stress M

Multi-span model (instead of single span) H(see Figure 19.12 – Chapter 19)

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basic equation that governs crack growth (79) is known asthe Paris Law is:

[59]

where:

a = crack size,N = number of fatigue cycles (fatigue life),

∆K = S.Y(a) range of stress intensity factor, (Kmax

– Kmin)C, m = crack propagation parameters,

S = constant amplitude stress range,= ∆σ = σmax – σmin

Y(a) = function of crack geometry.

Fatigue life prediction based on the fracture mechanicsapproach shall be computed according to the followingequation:

[60]

Equation 60 involves a variety of sources of uncertaintyand practical difficulties to define, for instance, the a and ao

crack size. The crack propagation parameter C in this equa-tion is treated as random variable (80). However, in moresophisticated models, equation 60 is treated as a stochasticdifferential equation and C is allowed to vary during thecrack growth process. State of art on the Fracture Mechan-ics Approach is available in Niemi (76) and Harris (81).

The characteristic S-N curves approach is based on fa-tigue test data (S-N curves—Figure 18.50) and on the as-sumption that fatigue damage accumulation is a linearphenomenon (Miner’s rule). According to Miner (82) thetotal fatigue life under a variety of stress ranges is theweighted sum of the individual lives at constant stress rangeS as given by the S-N curves (Figure 18.50), with each beingweighted according to fractional exposure to that level ofstress range.

The S-N curve approach related mainly to the crack ini-tiation and a maximum allowable crack size. After, crackspropagate based on the fracture mechanics concept as shownin Figure 18.51. The propagation is not explicitly consid-ered by the S-N curve approach.

Fatigue life strength prediction based on both the S-Napproach and Miner’s cumulative damage shall be evalu-ated with equation 61 or, in logarithmic form, with equa-tion 62 (Figure 18.50).

[61]NA

k SSm

em

= ∆

NC S

da

Ym ma

a= ∫1

0.

π. a ,

dadN

C m= . ( K)∆

logN = log (∆A) – m log (∆σ) [62]

where:

∆ = fatigue damage ratio (≤ 1)log(∆A) = intercept of the S-N curve of the Log N axis

–1 / m = slope of the S-N curve, (≅ 3 ≤ m ≤ ≅ 7)S–e= mean of the Miner’s equivalent stress range Se, de-

fined at Table 18.VkS = fatigue stress uncertainty factor

∆σ = kS. S–e (or the constant amplitude stress range for fail-ure at N cycles)

N = fatigue life, or number of loading cycles expected dur-ing the life of a detail

The Miner’s equivalent stress range, Se, can be evalu-ated based on the models provided in Table 18.V (83). Themost refined model would start with a scatter diagram ofsea-states, information on ship’s routes and operating char-


Figure 18.50 A Typical S-N Curve

Figure 18.51 Comparison between the Characteristic S-N Curve and Fracture

Mechanics Approach

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acteristics, and use of a ship response computer program toprovide a detailed history of stress ranges over the servicelife of the ship. For such model, the wave exceedance dia-gram (deterministic method) and the spectral method (prob-abilistic method) can be employed (Table 18.V).

S-N curves are obtained from fatigue tests and are avail-able in different design codes for various structural detailsin bridges, ships, and offshore structures. The design S-Ncurves are based on the mean-minus-two-standard-devia-tion curves for relevant experimental data (Figure 18.50).They are thus associated with a 97.6% probability of sur-vival. Some classification societies use 90%.

In practice, the actual probabilities of failure associatedwith fatigue design lives is usually higher due to uncer-tainties associated with the calculated stresses, the variousS-N curve correction factors, and the critical value of thecumulative fatigue damage ratio, ∆.

Cumulative damage: The damage may either be calculatedon basis of the long-term stress range distribution usingWeibull parameters (simplified method), or on summation ofdamage from each short-term distribution in the scatter dia-gram (probabilistic and deterministic methods, Table 18.V).

The stress range (S or ∆σ): The procedure for the fa-tigue analysis is based on the assumption that it is only nec-essary to consider the ranges of cyclic principal stresses indetermining the fatigue endurance. However, some reduc-tion in the fatigue damage accumulation can be creditedwhen parts of the stress cycle range are in compression.

Fatigue areas: The potential for fatigue damage is de-pendent on weather conditions, ship type, corrosion level,location on ship, structural detail and weld geometry andworkmanship. The potential danger of fatigue damage willalso vary according to crack location and number of po-tential damage points. Fatigue strength assessment shallnormally be carried out for:

• longitudinal and transverse element in:

— bottom/inner bottom (side),— longitudinal and transverse bulkheads.

• strength deck in the midship region and forebody, and• other highly stressed structural details in the midship re-

gion and forebody, like panel knuckles.

Time at sea: Vessel response may differ significantly fordifferent loading conditions. It is therefore of major im-portance to include response for actual loading conditions.Since fatigue is a result of numerous cyclic loads, only themost frequent loading conditions are included in the fatigueanalysis. These will normally be ballast and full load con-dition. Under certain circumstances, other loading condi-tions may be used.

Environmental conditions: The long-term distributionof load responses for fatigue analyses may be estimatedusing the wave climate, represented by the distribution ofHs and Ts, representing the sea operation conditions. Asguidance to the choice between these data sets, one shouldconsider the average wave environment the vessel is ex-pected to encounter during its design life. The world widesailing routes will therefore normally apply. For shuttletankers and vessels that will sail frequently on the North At-lantic, or in other harsh environments, the wave data givenin accordance with this should be applied. For vessels thatwill sail in more smooth sailing routes, less harsh environ-mental data may be applied. This should be decided uponfor each case.

Geometrical imperfections: The fatigue life of a weldedjoint is much dependent on the local stress concentrationsfactors arising from surface imperfections during the fab-rication process, consisting of weld discontinuities and geo-metrical deviations. Surface weld discontinuities are weldtoe undercuts, cracks, overlaps, incomplete penetration, etc.Geometrical imperfections are defined as misalignment, an-gular distortion, excessive weld reinforcement and other-wise poor weld shapes.

Effect of grinding of welds: For welded joints involvingpotential fatigue cracking from the weld toe an improve-ment in strength by a factor of at least 2 on fatigue life canbe obtained by controlled local machining or grinding ofthe weld toe. Note that grinding of welds should not be usedas a “design tool”, but rather as a mean to lower the fatiguedamage when special circumstances have made it necessary.This should be used as a reserve if the stress in special areasturns out to be larger than estimated at an earlier stage ofthe design.

18.6.6.3 Stress concentration and hot spot stressThe stress level obtained from a structural analysis, such asFEA, will depend on the fineness of the model. The differ-ent analysis models described in Subsection 18.7.2 willtherefore lead to different levels of result processing in orderto complete the fatigue calculations.

In order to correctly determine the stresses to be used infatigue analyses, it is important to note the definition of thedifferent stress categories (Figure 18.52).

Nominal stresses are those, typically, derived from coarsemesh FE models. Stress concentrations resulting from thegross shape of the structure, for example, shear lag effects,have to be included in the nominal stresses derived fromstress analysis.

Geometric stresses include nominal stresses and stressesdue to structural discontinuities and presence of attach-ments, but excluding stresses due to presence of welds.


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Stresses derived from fine mesh FE models are geometricstresses. Effects caused by fabrication imperfections as mis-alignment of structural parts, are normally not included inFEA, and must be separately accounted for, using, for in-stance (equation 65).

Hot spot stress is the greatest value of the extrapolationto the weld toe of the geometric stress distribution imme-diately outside the region affected by the geometry of theweld (Figure 18.52).

Notch stress is the total stress at the weld toe (hot spotlocation) and includes the geometric stress and the stressdue to the presence of the weld. The notch stress may becalculated by multiplying the hot spot stress by a stress con-centration factor, or more precisely the theoretical notchfactor, K2 (equation 65).

FE may be used to directly determine the notch stress.However, because of the small notch radius and the steepstress gradient at a weld, a very fine mesh is needed.

In practice, the stress concentration factors (K-factors)may be determined based on fine mesh FE analyses, or, al-ternatively, from the selection of factors for typical details.

The notch stress range governs the fatigue life of a de-tail. For components other than smooth specimens the notchstress is obtained by multiplication of the nominal stress byK-factors (equation 63). The K-factors in this document arethus defined as

[63]

The relation between the notch stress range to be usedtogether with the S-N-curve and the nominal stress rangeis

[64]

All stress risers have to be considered when evaluating

S Knotch nominal= = =∆ ∆ ∆σ σ σ.

K notch

nominal=

σσ

the notch stress. This can be done by multiplication of K-factors arising from different causes. The resulting K-fac-tor to be used for calculation of notch stress is:

K = K1 . K2 . K3 . K4 . K5 [65]

where:

K1 = stress concentration factor due to the gross geometryof the detail considered

K2 = stress concentration factor due to the weld geometry(notch factor); K2 = 1.5 if not stated otherwise

K3 = additional stress concentration factor due to eccen-tricity tolerance

K4 = additionally stress concentration factor due to angu-lar mismatch

K5 = additional stress concentration factor for un-symmet-rical stiffeners on laterally loaded panels, applicablewhen the nominal stress is derived from simple beamanalyses

Fatigue cracks are assumed to be independent of princi-pal stress direction within 45° of the normal to the weld toe.

Hot spot stress extrapolation procedure: The hot spotstress extrapolation procedure (Figure 18.52) is only to beused for stresses that are derived from stress concentrationmodels (fine mesh). Nominal stresses found from othermodels should be multiplied with appropriate stress con-centration factors (equation 65). The stress extrapolationprocedure is specific to each classification societies (74).Today, there is unfortunately no standard procedure.

18.6.6.4 Direct analysisSeveral S-N fatigue approaches exists, they all have ad-vantages and disadvantages. The different approaches aretherefore suitable for different areas. Load effects, accu-racy of the analysis, computer demands, etc. should be eval-uated before one of the approaches is chosen.

Full stochastic fatigue analysis: The full stochastic analy-sis, for example the Spectral Model of Table 18.V, is ananalysis where all load effects from global and local loads,are included. This is ensured by use of stress concentrationmodels and direct load transfer to the structural model.Hence, all stress components are combined using the cor-rect phasing and without simplifications or omissions ofany stress component.

This method usually will be the most exact for determi-nation of fatigue damage and will normally be used togetherwith fine meshed stress concentration models. The methodmay, however, not be suitable when non-linearities in theloading are of importance (side longitudinals). This is es-pecially the case for areas where wave or tank pressures inthe surface region are of major importance. This is due to


Figure 18.52 Definition of Stress Categories (4)

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the fact that all load effects result in one set of combinedstresses, making it difficult to modify the stress caused byone of the load effects.

The approach is suitable for areas where the stress con-centration factors are unknown (knuckles, bracket and flangeterminations of main girder, stiffeners subjected to largerelative deformations).

18.6.6.5 Simplified modelsThe stress component based stochastic fatigue analysis:The idea of the stress component based fatigue analysis isto change the direct load transfer functions calculated fromthe hydrodynamic load program into stress transfer func-

tions by use of load/stress ratios, Hi (equation 66). The loadtransfer functions, Hi, normally include the global hull girderbending sectional forces and moments, the pressures for allpanels of the 3-D diffraction model, the internal tank pres-sures.

The stress transfer functions, Hi, are combined to a totalstress transfer function, Hσ, by a linear complex summationof the different transfer functions (4), as:

[66]

where:

Ai = stress per unit axial force defined as the local stressresponse in the considered detail due to a unit sec-tional load for load component i.

Ησ = total transfer function for the combined local stress,Hi = transfer function for the load component i, that is, axial

force, bending moments, twisting and lateral load.

This approach enables the use of separate load factors oneach load component and thus includes loads non-linearities.Few load cases have to be analyzed and it is possible to usesimplified formulas for the area of interest but errors are eas-ily made in the combination of stresses, manual definition ofextra load cases may cause errors and simplifications are usu-ally made in loading. Suitable areas are components wheregeometric stress concentration factors, K1, are available (lon-gitudinals, plating, cut-outs and standard hopper knuckles)and areas where side pressure is of importance.

The simplified design wave approach (Weibull Model,Table 18.V) is a simplification to the previous componentbased stochastic fatigue analyses. In this simplified ap-proach, the extreme load response effect over a specifiednumber of load cycles, for example, 104 cycles, is deter-mined. The resulting stress range, ∆σ, is then representa-tive for the stress at a probability level of exceedance of10-4 per cycle. The derived extreme stress response is com-bined with a calculated Weibull shape parameter, k, to de-fine the long-term stress range distribution (Table 18.V).The Weibull shape parameter, k, for the stress responseshould be determined from the long-term distribution of thedominating load calculated in the hydrodynamic analysis.

This simplified approach only requires the considera-tion of one load case. It is easy and fast to perform but itcan only be used if one load dominates the response andthe results are very sensitive to selection of design wave.Suitable areas concern components where one load is dom-inating the response, that is, deck areas and other areas with-out local loading.

H A Hi ii

σ = ∑


TABLE 18.V Commonly Used Expressions for EvaluatingMiner’s Equivalent Stress Range (Se), (83)

1. Wave Exceedance Diagram (Deterministic Method)

Si = stress range

Fi = fraction of cycles in the ith stress block

nb = number of stress block

2. Spectral Method (Probabilistic Method)

λ(m) = rainflow correction

Γ(.) = gamma function

γι = fraction of time in ith sea-state

fi = frequency of wave loading in ith sea-state

σι = RMS of stress process in ith sea-state

3. Weibull Model for Stress Ranges (Simplified Method)

Sd = stress range that is exceeded on the average once out ofNd stress cycles

Γ(.) = gamma function

k = Weibull shape parameter

Nd = total number of stress ranges in design life

S f S S f Sem

i im

i

n

e i im

i

n

mb b

= → =∑ ∑

S mf

mfe

m

m

i i im

i

= ( )( )

+

∑λ γ σ

2 2

21

0Γ

S f S S f Sem

i im

i

n

e i im

i

n

mb b

= → =∑ ∑

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18.6.6.6 Design criteriaThe standard fatigue design criterion is basically the ex-pected lifetime before that significant damage appears(cracks). It usually is taken as being 20 years. Then, the de-signer’s target is to design structural details for which thefatigue failure happens after, for instance, 20 years. If ithappens before, the fixing cost is very high and inducesowner losses. If the first failure only happens after 30 yearsor later, the structural detail scantlings were globally over-estimated, the hull weight too high and, therefore, that theowner had lost payload during 20 years.

Partial safety factors, additional stress concentration fac-tors and the stress extrapolation procedure are typically de-fined by the classifications societies.

18.6.7 Collision and Grounding18.6.7.1 Present design approachesThe OPA 90 and equivalent IMO requirements must be sat-isfied in structural design of ships carrying dangerous or pol-lutant cargoes, for example, chemicals, bulk oil, liquefiedgas. The primary requirements are to arrange a double bot-tom of a required minimum height, and double sides of arequired minimum width. In this context, to reduce the out-flow of pollutant cargoes in ship collision or grounding ac-cident, OPA 90 and IMO both require that the minimumvertical height, h, of each double bottom ballast tank or voidspace is not to be less than 2.0 m or B/15 (B = ship’s beam),whichever is the lesser, but in no case is the height to beless than 1.0 m. OPA and IMO also require that the mini-mum width, w, of each wing ballast tank or void space isnot to be less than 0.5+DWT/20 000 (m) or w =2.0 (m),whichever is the lesser, where DWT is the deadweight ofthe ship in tonnes. In no case is w to be less than 1.0 (m).More detailed information is available in Chapter 29 on OilTanker.

18.6.7.2 Direct analysisTo reduce the probability of outflow of hazardous cargo inship collisions and grounding, the kinetic energy loss dur-ing the accident should be entirely absorbed by damage ofouter structures, that is, before the inner shell in contactwith the cargo can rupture. Of crucial importance, then, ishow to arrange or make the scantlings of strength membersin the implicated ship structures such that the initial kineticenergy is effectively consumed and the structural perform-ance against an accident will be maximized. For this pur-pose, the structural crashworthiness of ships in collisionsand grounding must be analyzed using accurate and efficientprocedures (84).

Figure 18.53 shows direct design procedures of ship

structures against collision and grounding (85). For the ac-cidental limit state design, the integrity of a structure canbe checked in two steps. In the first step, the structural per-formance against design accident events will be assessed,while post-accident effects such as likely oil outflow areevaluated in the second step.

The primary concern of the accidental limit state designin such cases is to maintain the water tightness of ship com-partments, the containment of dangerous or pollutant car-goes, and the integrity of critical spaces (reactor compart-ments of nuclear powered ships or tanks in LNG ships) atthe greatest possible levels, and to minimize the release/out-flow of cargo. To facilitate a rescue mission, it is also nec-essary keep the residual strength of damaged structures ata certain level, so that the ship can be towed to safe harboror a repair yard as may be required.

18.6.7.3 Simplified modelsSince the response of ships in collision or grounding acci-dent includes relatively complicated behavior such as crush-ing, tearing and yielding, existing simplified methods arenot always adequate. However, many simplified modelsuseful for predicting accident induced structural damagesand residual strength of damaged ship structures have beendeveloped and continue to be successfully used. Simplifiedmodels for collision are rather different from those ofgrounding since both are different in the nature of the me-chanics involved. As it is impossible to describe them in alimited space, valuable references are Ohtsubo et al (86),and Kaminski et al (39).

18.6.7.4 Design criteriaThe structural design criteria for ship collisions and ground-ing are based on limiting accidental consequences such asstructural damage, fire and explosion, and environmentalpollution, and to make sure that the main safety functionsof ship structures are not impaired to a significant extent dur-ing any accidental event or within a certain time periodthereafter.

Structural performance of a ship against collision orgrounding can be measured by:

• energy absorption capability,• maximum penetration in an accident,• spillage amount of hazardous cargo, for example, crude

oil, and• hull girder ultimate strength of damaged ships (Section

18.6.5).

Design acceptance criteria may be based on the follow-ing parameters (87):


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• minimum distance of cargo containment from the outershell,

• ship speed above which a critical event (breaching ofcargo containment) happens,

• allowable quantity of oil outflow, and• minimum values of section modulus or ultimate hull

girder strength.

And the design results must satisfy:

• cargo tanks/holds are not breached in an accident so thatthere will be no danger of pollution, or

• if the cargo tanks are breached, the oil outflow follow-ing an accident is limited, and/or

• the ship has adequate residual hull girder strength so thatit will survive an accident and will not break apart, min-imizing a second chance of pollution.

18.6.8 Vibration18.6.8.1 Present Vibration Design ApproachesThe traditional design methodology for vibration is based onrules, defined by classification societies. Vibrations are notexplicitly covered by class rules but their prediction is neededto achieve a good design. Ship structures are excited by nu-merous dynamic oscillating forces. Excitation may originatewithin the ship or outside the ship by external forces. Reci-procating machinery such as large main propulsion dieselproduce important forces at low frequency. Pressure fluctu-ations due to propeller at blade rate frequency induce pres-sure variation on the ship’s hull. Varying hull pressuresassociated with waves belong also to external excitations. Allthese forces can be approximated by a combination of har-monic forces. If their frequencies coincide with the structureeigen frequencies, resonant behavior will happen.


Figure 18.53 Structural Design Procedures of Ships for Collision and Grounding (85)

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It is of prime importance to avoid global main hull vi-brations. If they do occur, the remedial action will proba-bly be very costly. So, during early design, the hull girderfrequencies must be compared to wave excitation (spring-ing risk), and to propeller and engine excitation. Table 18.VIgives some typical values of the first hull girder frequen-cies in Hz of some ship types.

Hull girder frequencies and modes should be computedusing approximate empirical formulae (88), simple beammodels for long prismatic structures (VLCC, container ships,etc.) associated with lumped added mass models, or using3D finite element models for complex ships (RO-RO, cruiseship), LNG, and short and non-prismatic structures (tug,catamaran, etc.).

18.6.8.2 Fluid structure interactionFluid structure interaction is evidenced in the dynamic be-havior of ships. As a first approximation, the ship is con-sidered as a rigid body, for the sea keeping analyses (waveinduced motions and loads).

Wave vibration induced: An early determination of hullgirder vibration modes and frequencies is important to avoidserious problems that would be difficult to solve at a laterstage of the project.

Risk of springing (occurring when first hull girder fre-quency equals wave encounter frequency) has to be detectedvery early. Springing may occur for long and/or flexibleships and for high speed craft and it increases the numberof cyclic loads contributing to human fatigue. Various meth-ods to assess the first hull girder frequency can be used atpreliminary design stage.

Engine/propeller vibration induced: Resonance prob-lems may also appear on small ships like tugs, where hullgirder frequency can be close to the propulsion excitation(around 7Hz). High vibration levels contribute to humanfatigue and dysfunction, besides the discomfort aspect.

Fluid added mass: Hull girder vibrations induce dis-

placement of the surrounding fluid. Therefore imparting ki-netic energy in the fluid. This phenomenon can be takeninto account for the hull girder modes and frequencies cal-culation as added mass terms. Various methods can be usedfor the determination of added mass term. Lumped mass ap-proach is the simplest one (89) but is only valid for simpleprismatic slender shapes, and for a single mode. Fluid fi-nite and semi-infinite elements or boundary integral for-mulation lead to the calculation of more accurate addedmass matrices (90), especially for complex hull forms andappendices study (rudder). Added mass matrices associatedwith 3D finite element model of the structure, allow for anaccurate determination of hull girder modes and frequen-cies. Added mass terms may also be needed for the vibra-tions of tank walls. The corresponding methods andassociated software are available for industrial usage (Fig-ure 18.54) and numerical simulations are today predictablewith good accuracy (91). Figure 18.54 shows a fluid-struc-ture coupled FE-model of a 230 m long passenger vesselusing 150 000 degrees of freedom.

A difficult coupled problem is the fluid impact occur-ring in slamming or due to sloshing in tanks. The local de-formation of the impacted shells and plating influences the


TABLE 18.VI Typical Values of the First Hull GirderFrequencies (in Hertz)

LargeOrder Cruise Fast (mode) ship monohull LNG VLCC Frigate Tug

1 1.0 Hz 1.8 0.9 0.8 1.9 7.0

2 1.5 Hz 2.9 2.0 1.7 3.8 13

3 2.6 Hz — — — 5.8 —

4 3.2 Hz — — — 7.8 —Figure 18.54 Fluid/Structure FE-Model of a Passenger Vessel (Principia

Marine, France)

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pressures and fluid velocities. Moreover, air trapped in suchan impact may have a cushioning effect, softening its sever-ity. The numerical simulation of those heavily coupled prob-lems still belongs to the research domain, though itsindustrial importance for the design of ship structures (92).

18.6.8.3 Direct analysisVibration problems are critical for passenger ships with typ-ically a 12-Hertz blade excitation. Ship owners demand verylow vertical velocity levels incabins and public areas (lessthan 1.2 mm/s in the 5-25 Hz frequency band).

Numerical simulation using 3D finite element models isthe only method to predict ship response (including the var-ious frequency modes) to pressure fluctuation on the shiphull. Such simulation is now used as a design tool to selectappropriate scantlings of decks, location of pillars, detectpossible resonance, and select the number of propellerblades. The main difficulty is to perform this analysis earlyenough in a very short design cycle.

Local analyses also have to be performed, based on fi-nite element models to check the potential risk of vibrationof local areas, when local modes can be considered as de-coupled from global hull girder modes. Decks, superstruc-ture, appendices (rudder, radar mast, etc.) can be analyzedto check scantling and avoid the risk of resonance.

Slamming impacts generate impulsive response of thehull girder (whipping), which affects comfort and fatigue.Prediction of stress fluctuations and vibration levels in var-

ious parts of the ship can only be performed by simulationin the time domain based on 3D detailed finite element mod-els (Figure 18.55). The main difficulty is the determinationof the time and space dependent slamming forces.

18.6.8.4 Simplified modelsUnfortunately, they are of little use for simplified vibrationpredictions. Beam models associated to database can beused for an approximate determination of hull girder modesand frequencies at early stage of the project. Decks zonesand equipment frequencies may also be estimated by for-mulas given by reference books (94).

Dedicated software has also been written for the studyof shafting, including journal and bearing stiffness andwhirling effect (95).

18.6.8.5 Design criteriaThe most effective way to control vibration resides in thereduction of the excitation. This can be achieved by bal-ancing all forces in reciprocating and rotary machinery andusing special mounts. Hydrodynamic forces can be reducedby improving the flow around the propeller and siting itclear of the hull. Propulsion using pods can dramatically re-duce pressure fluctuations. Excitation frequencies can alsobe modified by changing the number of propeller blades.

A good design, ensuring continuity of vertical bulkheads,avoiding cantilevered and stiff or mass discontinuities, con-tributes to improving the dynamic behavior of the ship. The


Figure 18.55 Hull Girder Vibration—Mode #3 (Principia Marine-France)

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second action consists in avoiding resonance by modifica-tion of the hull scantlings, and addition of pillars, in orderto increase or lower the eigen frequencies.

Reduction of unavoidable vibration levels can beachieved for local vibrations by dynamic isolation for equip-ments, passive damping solutions (floating floors on ab-sorbing material), and dynamic energy absorbers. All thesecurative actions are usually difficult, costly, only applica-ble for local vibrations and nearly impossible for vibrationsdue to global modes. Local modes determination is diffi-cult at early stage of the design mainly due to the uncer-tainty on mass distribution, non-structural mass (outfittingand equipments) being of the some order of magnitude asthe steelwork part.

18.6.9 Special ConsiderationsIn addition to the considerations for LNG tank, containership, bulk carrier and passenger vessel, special considera-tions are available in Volume II of this book. Moreover,ISSC committees 1997 and 2000 also provide valuable in-formation on specific ship types, that is, high-speed vesselsand ships sailing in ice conditions.

18.6.9.1 LNG TanksGeneral information on such ships is available in Chapter32 – Liquefied Gas Carriers. These ships contain usually adouble hull (sides and bottom). Major structural concernsdeal with the tanks themselves and with their support legs.Dilatation, tightness and thermal isolation are important as-pects. There are several patented concepts: independenttanks, membrane tanks, semi-membranes tanks and inte-gral tanks. Excepted for the integral tanks, the tanks are self-supporting and are not essential to the hull strength. Whensupported by legs, these legs require a particular attention.Integral tanks form a structural part of the ship’s hull andare influenced in the same manner by wave loads.

18.6.9.2 Container shipsThe design of container ships of 5000 and 6000 TEU hav-ing a beam of 40m has increased the standard torsional prob-lem of ships having a large open deck. Torsional strengthand limitation of the equivalent stress (equation 45) at thehatch corners are the major issues in the evaluation of thestrength of main hull structure. Use of multicell structuresin side shell and double bottom is recommended. More-over, the torsional moment distribution must be assessedwith care.

As hatch covers are not considered as hull strength mem-bers, omission of hatch covers does not impose any partic-

ular effects in the structural design of a main hull structure.The general characteristics of container ships are detailedin Chapter 36 – Container Ships.

18.6.9.3 Bulk carriersCasualty of bulk carriers was very high in the early 1990s.The main reasons were a lack of maintenance, excessive cor-rosion and fatigue (77). Weak point of these ships is thelower part of the side plate at the junction with the bilgehopper. Now, classification societies are aware about thisproblem and had updated their rules and associated struc-tural details. The general design practice on bulk carriers isdetailed in Chapter 33 – Bulk Carriers.

18.6.9.4 Passenger vesselsShip strength analysis is based on a beam model. The com-plexity of large passenger ships, with a low resistant deckand wide openings, windows and openings in the side in-duces a much more complex behavior. Rational approachis necessary to get a realistic understanding of the flux offorces and capture the complex behavior of such ships.Due to the large openings and discontinuities, racking andstress concentration are two major concerns. For archi-tectural reason, pillars are often omitted in large publicareas (theater, lounge, etc.). Today, 3D FEA is usually car-ried out to design large passenger vessels (Figures 18.54and 18.55). Due to large opening in the side shells, the ver-tical stress distribution is not linear (Figure 18.35). Thismeans that the basic beam bending formulation is no valid(equation 29). More general information related to pas-senger vessels is available in Chapter 37 – Passenger Shipsand in reference 68.

18.6.9.5 Composite materialFiberglass boat building started in the 1960s. Today, de-signers are trying to plan composite construction of shipsup to 100 meters in length. A comprehensive guide for thedesign of ship structures in composites is the Ship Struc-ture Committee Report SSC-403 of Greene (96). Designmethodology, materiel properties, micro and macro me-chanic of composites and failures modes are deeply dis-cussed.

In addition to the classic failure modes of steel and alu-minum structures presented in Subsection 18.6.1, compos-ites are subject to specific failure modes.

In compression, there are the crimping, skin wrinklingand dimpling of the honeycomb cores (Figure 18.56). Inbending, instead of the traditional first yield bending mo-ment, for composites, the design limit load corresponds tothe first ply failure.

The creep behavior and the long-term damage from


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water, UV and temperature, and their performance in firesare other specific structural problems of composites. A re-view of the performance of composite structures is pro-posed by Jensen et al (98).

18.6.9.6 Aluminum structuresCompared to steel, the reduced specific weight of aluminum(2.70 kN/m3 for aluminum and7.70 kN/m3 for steel) is a veryinteresting property for a ship designer. The yield stress ofunwelded aluminum alloys can be comparable to mild steel(235 MPa) but changes drastically from one alloy to an-other (125 MPa for ALU 5083-O and 215 MPa for ALU5083-H321). The modulus of elasticity of aluminum alloysis one-third of steel.

The main difficulty for the use of aluminum use dealswith its mechanical properties after welding. The yield stressof aluminum alloys may decrease significantly after weld-ing (remains at 125 MPa for

ALU 5083-O but drop to 140 MPa for ALU 5083-H321).The area close to a weld is called Heat Affected Zone (HAZ).It is characterized by reduced strength properties. HAZ isparticularly important to assess the buckling and ultimatestrength of welded components such as beam-column ele-ments, stiffened panels, etc.

For marine applications ALU 5083, 5086 and 6061 canbe used. Nevertheless, the mechanical and strength prop-erties of aluminum change a lot with the alloy compositionand the production processing. Thus, the alloy selectionmust be done with care with regard to the yield strength be-fore and after welding, the welding and extruding capabil-ities, the marine behavior, etc.

Fire strength is another concerns when using aluminumalloys as it quickly loses its strength when the temperaturerises.

Despite the aforementioned shortcomings aluminum al-loys will be more extensively use in the future for the de-

sign of fast vessels, for which the structural weight is veryimportant to reach higher speed (for high speed mono hull,catamaran and trimaran vessels). The good extruding ca-pability of aluminum alloys has to be enhanced throughscantling standardization. That helps to lower to produc-tion cost ($/man-hour) and compensate the initial highermaterial cost of aluminum, which is approximately 3 timeshigher that mild steel ($/kg).

18.6.9.7 CorrosionCorrosion does not present a structural design problem, asalmost all the classification societies base their rules on anet scantling. This means that the thickness to consider inanalysis (for empirical formulations up to complex FEA)is the reduced thickness (without corrosion allowance) andnot the actual thickness. The difference between the reducedthickness and the actual one is usually fixed by the classi-fication but can also change according to the owner re-quirements. This is an economic choice and not a structuralproblem.

For bulk carriers, thickness reduction due to corrosionis generally assumed to be 5 mm for hold frames and 3 mmfor side shell plating.

18.7 NUMERICAL ANALYSIS FOR STRUCTURALDESIGN

18.7.1 Motivation for Numerical AnalysisIn most of the cases, a ship is a one of a kind product, evenif limited series may exist in some cases. The design, studyand production cycle is very short and major decision haveto be taken very early in the project. It is well known thatthe cost of a late modification is very high and such a situ-ation has to be avoided. Also experience-based design canbe an obstacle to the introduction of innovation. Numericalanalysis clearly is needed to improve the design (innova-tion) but also to control safety margins. Moreover, it givesaccess to local and detailed analysis, which is not possiblewith simplified methods. The concept of numerical mock up,used in aerospace and car industry has proven its efficiency.Shipbuilding is clearly moving in the same direction.

18.7.1.1 Static and quasi-static analysisStatic and quasi-static analysis represents the traditionalway to perform stress and strength analysis of a ship struc-ture. Loads are assessed separately of the strength structureand, even if their origins are dynamic (flow induced), theyare assumed to be static (do not change with the time). Thisassumption may be correct for the hydrostatic pressure but


Figure 18.56 Potential Failure Modes of Sandwich Panels (100), (a) Face

yielding/fracture, (b) Core shear failure, (c-d) Face wrinkling, (e) Buckling, (f)

Shear crimping, (g) Face dimpling, (h) Local indentation.

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not when the dynamic wave loads are changed to static loadsapplied on the side plates of the hull.

In the future, even if the assumption of static loads is notverified, static analysis will continue to be performed, as itis easier and faster to perform. In addition, tens of experi-ence years have shown that they provide accurate resultswhen stresses and deflections assessment are the main tar-get (as defined in Section 18.4).

Such analysis is also the standard procedure for fatigueassessment to determine the hot spot stress through finemesh FEA.

18.7.1.2 Dynamic analysisWhen problems occur on a ship due to dynamic effects, itis very often late in the design and building stage and evenin service, and corrective actions are costly. Simplified meth-ods can only predict the first hull girder modes frequencies.Numerical finite element based simulation is mature enoughto predict up to second propeller harmonic, the vibrationlevel, giving a design tool to comply with ISO or ship ownerrequirements. Moreover, possible dynamic problems canbe detected early enough in the design to allow for correc-tive actions.

18.7.1.3 Nonlinearities analysisNonlinear structural analysis is mainly used to analyze buck-ling, ultimate strength and accidental or extreme situations(explosions, collisions, grounding, blast). The results ofsuch costly and difficult analysis are often used to calibratesimplified methods or rules. But they are also very usefulto understand possible failure modes and mechanical be-havior under severe loads.

18.7.1.4 Emerging trendsLike the automotive and aerospace industry, there is a cleartrend towards the reduction of design cycle time. Numeri-cal mock up or virtual ship approach (97), especially for oneof a kind product, is clearly a way to achieve this. Requiredcomputing power is available and will no longer be a con-straint. The first difficulty is to establish an efficient modelof complex physical problems, associated with increasingdemand for accuracy. The second difficulty is the manpowerneeded to prepare and check the models, which will besolved by the development of integrated solutions for shipdescription and modeling (99).

Advances are expected in the field of FE-modeling. Thetrend is toward one structure description, one model and sev-eral applications. This is the field for multiphysics and cou-pling analysis. The base modeling will be re-used andadapted to perform successively,

• static, fatigue and fracture analysis,• buckling and ultimate strength analysis,• vibration and acoustics analysis, and• vulnerability assessment.

Progress is expected by the utilization of reliability meth-ods already used in offshore industry, where uncertaintiesand dispersions of the loads, geometrical defaults, initialstresses and strains, material properties are defined as sto-chastic (non deterministic) data, leading to the calculationof a probability of failure. This philosophy can be appliedto fatigue and ultimate strength, but also to dynamic re-sponse, leading to a more robust design, less sensitive todefaults, imperfections, uncertainties and stochastic natureof loads. Reliability-based analyses using probabilistic con-cept are presented in Chapter 19.

In the future, safety aspects related to structural prob-lems will also be tackled such as ultimate strength using non-linear methods. Collision and grounding damages andimproved design to increase ship safety will be studied bynumerical simulation, whereas experimental approach isnearly impossible and/or too costly. Explicit codes, used incar crash simulation (101), will be adapted to specific as-pects of ship structure (size and presence of fluid). In tra-ditional sea keeping analysis, the ship is considered as arigid body. In coupled problems such as slamming situa-tions, this hypothesis is no more valid and a part of the en-ergy is absorbed by ship deformation. Hydro-elasticitymethods (102) aim taking into account the interaction of theflexible ship structure with the surrounding water. Nonlin-ear effects due to bow and aft part of the ship, ship veloc-ity, diffraction radiation effects contribute to the complexityof the problem. The simulation of catamaran, trimaran andfast monohulls behavior need the development of new meth-ods to take into account the high velocities and the com-plex 3D phenomena.

18.7.2 Finite Element AnalysisThe main aim of using the finite element method (FEM) instructural analysis is to obtain an accurate calculation of thestress response in the hull structure. Several types or levelsof FE-models may be used in the analyses:

• global stiffness model,• cargo hold model,• frame and girder models,• local structure models, and• stress concentration models.

The model or sets of models applied is to give a properrepresentation of the following structure:


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• longitudinal plating,• transverse bulkheads/frames,• stringers/girders, and• longitudinals or other structural stiffeners.

The finer mesh models are usually referred to as sub-models. These models may be solved separately by trans-fer of boundary deformations/ boundary forces from thecoarser model. This requires that the various mesh modelsare compatible, meaning that the coarser models havemeshes producing deformations and/or forces applicable asboundary conditions for the finer mesh models.

18.7.2.1 Structural finite element modelsGlobal stiffness model:A relatively coarse mesh that is usedto represent the overall stiffness and global stress distribu-tion of the primary members of the total hull length. Typi-cal models are shown in Figure 18.57. The mesh density ofthe model has to be sufficient to describe deformations andnominal stresses from the following effects:

• vertical hull girder bending including shear lag effects,• vertical shear distribution between ship side and bulk-

heads,• horizontal hull girder bending including shear lag ef-

fects, torsion of the hull girder, and• transverse shear and bending.

Stiffened panels may be modeled by means of layeredelements, anisotropic elements or frequently by a combi-nation of plate and beam elements. It is important to havea good representation of the overall membrane panel stiff-ness in the longitudinal/transverse directions. Structure notcontributing to the global strength of the vessel may be dis-regarded; the mass of these elements shall nevertheless beincluded (for vibration). The scantling is to be modeled withreduced scantling, that is, corrosion addition is to be de-ducted from the actual scantling.

All girder webs should be modeled with shell elements.Flanges may be modeled using beam and truss elements.Web and flange properties are to be according to the realgeometry.

The performance of the model is closely linked to thetype of elements and the mesh topology that is used. As astandard practice, it is recommended to use 4-node shell ormembrane elements in combination with 2-node beam ortruss elements are used. The shape of 4-node elementsshould be as rectangular as possible as skew elements willlead to inaccurate element stiffness properties. The elementformulation of the 4-node elements requires all four nodesto be in the same plane. Double curved surfaces shouldtherefore not be modeled with 4-node elements. 3-node el-ements should be used instead.

The minimum element sizes to be used in a global struc-tural model (coarse mesh) for 4–node elements (finer meshdivisions may of course be used and is welcomed, speciallywith regard to sub-models):

• main model: 1 element between transverse frames/gird-ers; 1element between structural deck levels and mini-mum three elements between longitudinal bulkheads,

• girders: 3 elements over the height, and• plating: 1 element between 2 longitudinals.


Figure 18.58 Cargo Hold Model (Based on the Fine Mesh of the Frame

Model), (4)

Figure 18.59 Frame and Girder Model (Web Frame), (4)

Figure 18.57 Global Finite Element Model of Container Vessel Including a 4

Cargo Holds Sub-model (4).

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Cargo hold model: The model is used to analyze the de-formation response and nominal stresses of the primarymembers of the midship area. The model will normallycover 1/2+1+1/2 cargo hold/tank length in the midship re-gion. Typical models are shown in Figure 18.58.

Frame and girder models: These models are used to an-alyze nominal stresses in the main framing/girder system(Figure 18.59). The element mesh is to be fine enough todescribe stress increase in critical areas (such as bracketwith continuous flange). This model may be included in thecargo hold model, or run separately with prescribed bound-ary deformations/forces. However, if sufficient computercapacity is available, it will normally be convenient to com-bine the two analyses into one model.

Local structure analyses are used to analyze stresses inlocal areas. Stresses in laterally loaded local plates and stiff-eners subjected to large relative deformations between gird-ers/frames and bulkheads may be necessary to investigatealong with stress increase in critical areas, such as brack-ets with continuous flanges.

As an example, the areas to model are normally the fol-lowing for a tanker:

• longitudinals in double bottom and adjoining verticalbulkhead members,

• deck longitudinals and adjoining vertical bulkhead mem-bers,

• double side longitudinals and adjoining horizontal bulk-head members,

• hatch corner openings, and• corrugations and supporting structure.

The magnitude of the stiffener bending stress includedin the stress results depends on the mesh division and theelement type that is used. Figure 18.60 shows that the stiff-ener bending stress, using FEM, is dependent on the meshsize for 4-node shell elements. One element between floorsresults in zero stiffener bending. Two elements betweenfloors result in a linear distribution with approximately zerobending in the middle of the elements.

Stress concentration models are used for fatigue analy-ses of details were the geometrical stress concentration isunknown. A typical detail is presented Figure 18.61.

Local FE analyses may be used for calculation of localgeometric stresses at the hot spots and for determination ofassociated K-factors to be used in subsequent fatigue analy-ses (equation 63). The aim of the FE analysis is normallynot to calculate directly the notch stress at a detail, but tocalculate the geometric stress distribution in the region ofthe hot spot. These stresses can then be used either directlyin the fatigue assessment of given details or as a basis forderivation of stress concentration factors. FE stress con-

centration models are generally very sensitive to elementtype and mesh size.

Several FEA benchmarks of such structural details wereperformed by ISSC technical committees (68,103). They as-sess the uncertainties of different FE packages associatedwith coarse and fine mesh models. Variation is usuallyaround 10% but is sometime much larger.

This implies that element sizes in the order of the platethickness are to be used for the modeling. If solid model-ing is used, the element size in way of the hot spot mayhave to be reduced to half the plate thickness in case theoverall geometry of the weld is included in the model rep-resentation.

18.7.2.2. Uncertainties related to FEAAn important issue in structural analysis is the verificationof the analysis. The FEM is basically reliable but manysources of errors can appear, mainly induced by inappro-priate modeling and wrong data. For this reason, different


Figure 18.61 Stress Concentration Model of Hopper Tank Knuckle (4)

Figure 18.60 Stiffener Bending Stress with FEM (from left to right: using 1, 2

or 8 elements), (4)

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levels of verification of the analysis should be performedin order to ensure trustworthiness of the analysis results. Ver-ification must be achieved at the following steps:

• basic input,• assumptions and simplifications made in modeling/

analysis,• models,• loads and load transfer,• analysis,• results, and• strength calculations.

One important step in the verification is the understandingof the physics and check of deformations and stress flowagainst expected patterns/levels. However, all levels of ver-ification are important in order to verify the results.

Verifications of structural models:Assumptions and sim-plifications will have to be made for most structural mod-els. These should be listed such that an evaluation of theirinfluence on the results can be made.

The boundary conditions for the global structural modelshould reflect simple supporting to avoid built in stresses. Thefixation points should be located away from areas wherestresses are of interest. Fixation points are often applied in thecenterline close to the aft and the forward ends of the vessel.

Verification of loads: Inaccuracy in the load transfer fromthe hydrodynamic analysis to the structural model is amongthe main error sources in this type of analysis. The loadtransfer can be checked on basis of the structural responseor on basis on the load transfer itself.

Verification of response: The response should be veri-fied at several levels to ensure correctness of the analysis:

• global displacement patterns/magnitude,• local displacement patterns/magnitude,• global sectional forces,• stress levels and distribution,• sub-model boundary displacement/forces, and• reaction forces and moments.

18.7.2.3 FEM backgroundToday the finite element method is studied worldwide in uni-versities, in mechanical engineering, civil engineering, navalarchitecture, etc. Hundreds of papers are published yearly.Many commercial packages are available including pre andpost processors and many books are published each year onthe subject. Classification Societies also present technicalreports and guidelines associated with their own directanalysis package (Table 18.VIII).

It is not the purpose of this chapter to present the FE the-ory and a state of art. This topic is reviewed periodically by

ISSC. For instance, Sumi et al (68) presents finite elementguidelines and a comprehensive review of the available soft-ware. Mesh modeling is discussed in ISSC’2000 by Por-cari et al (103). Hughes (3) proposes in Chapter VI and VIIof his book published by SNAME an easy way to learnFEM that does not require knowledge of variational calcu-lus or of FEM. The Ship Structure Committee Reports (SSC387 and 399) contains also Guideline for FEM (43,104).

18.7.3 Other Numerical ApproachesAs an alternative to FEA, two other approaches are pre-sented, namely: the idealized Structural Unit Method (ISUM)and the Boundary Element Method (BEM). Both are gen-eral purpose oriented. Many others exist but they are usu-ally dedicated to a special purpose. For instance, at thepreliminary design stage, the LBR-5 package founded on theanalytical solution of the governing differential equations ofstiffened plates is a convenient alternative to standard FEA.Such an approach (30,105) allows structural design opti-mization to be performed at the earliest design stage but doesnot have the capability to perform detailed analysis includ-ing stress concentration and non-linear analysis.

18.7.3.1 Idealized structural unit method (ISUM)When subjected to extreme or accidental loading, ship struc-tures can be involved in highly non-linear response associ-ated with yielding, buckling, crushing and sometimesrupture of individual structural components. Quite accuratesolutions of the non-linear structural response can be ob-tained by application of the conventional FEM. However,a weak feature of the conventional FEM is that it requiresenormous modeling effort and computing time for non-lin-ear analysis of large sized structures. Therefore, most ef-forts in the development of new non-linear finite elementmethods have focused on reducing modeling and comput-ing times.

The most obvious way to reduce modeling effort andcomputing time is to reduce the number of degrees of free-dom so that the number of unknowns in the finite elementstiffness equation decreases. Modeling the object structurewith very large sized structural units is perhaps the best wayto do that. Properly formulated structural units or super el-ements in such an approach can then be used to efficientlymodel the actual non-linear behavior of large structuralunits. The idealized structural unit method (ISUM), whichis a type of simplified non-linear FEM, is one of such meth-ods (106). Since ship structures are composed of severaldifferent types of structural members such as beams,columns, rectangular plates and stiffened panels, it is nec-essary in the ISUM approach to develop various ISUM units


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for each type of structural member in advance. The non-lin-ear behavior of each type of structural member is idealizedand expressed in the form of a set of failure functions defin-ing the necessary conditions for different failures whichmay take place in the corresponding ISUM unit, and setsof stiffness matrices representing the non-linear relationshipbetween the nodal force vector and the nodal displacementvector until the limit state is reached. The ISUM super el-ements so developed are typically used within the frame-work of a non-linear matrix displacement procedureapplying the incremental method.

Figure 18.62 shows a cantilevers box girder and Figures18.63 and 18.64 show typical FEM and ISUM models forthe non-linear analysis. For a recent state-of-the-art reviewon ISUM theory and applications to ship structures, thereader is referred to Paik and Hughes (107).

With the existing standard ISUM elements, the main dif-ficulty is that computation of the post-collapse behavior inthe structural elements beyond their ultimate strength aswell as the flexural-torsional collapse behavior of stiffen-ers is not very successful.

In fact, ISUM elements accommodating post-collapsebehavior have previously been already developed but im-provements are under development to better accommodatesuch behavior (107, 108).

Usage of ISUM is limited to some specific problems andis not a general-purpose methodology. In contrast to FEM,for instance, it is necessary to formulate/develop ISUM el-ements specifically; by including buckling and collapse be-havior for ultimate strength analysis or by including tearingand crushing for collision strength analysis. The former typeelement cannot be used for the purpose of latter type analy-sis and vice versa. ISUM is also not adequate for linearstress analysis.

ISUM is very flexible, new closed form expressions ofthe ultimate strength can be directly utilized by replacingin the existing ISUM element the previous ultimate strengthformulations with the new ones.

18.7.3.2 Boundary Element Method (BEM)In contrast to FEM, the boundary element method (BEM)is a type of semi-numerical method involving integral equa-tions along the boundary of the integral domain (or vol-ume). To solve a problem that involves the boundary integralequations, BEM typically uses an appropriate numerical in-tegration technique so that the problem is discretized by di-viding only the boundary of the integral domain into anumber of segments or boundary elements, while the con-ventional FEM uses a mesh (finite elements) over the en-tire domain (or volume), that is, inside as well as itsboundary. For a specific problem with a relatively simple

boundary domain, linear or flat boundary elements may beemployed so that analytical solutions for the integral equa-tions can be adopted, while higher degree boundary ele-ments must be used for modeling an integral domain withmore complex characteristics with the integration gener-ally needing to be carried out numerically. Figure 18.65shows typical FEM and BEM models for analysis of a pres-sure vessel (109).

Since the publication of an early book on BEM, manyengineering applications using BEM have been achieved.More recent developments of BEM together with the basic


Figure 18.63 A Typical FEM Model for NonLinear Analysis of the Cantilever

Box Girder

Figure 18.64 A Typical ISUM Model for Nonlinear Analysis of the

Cantilever Box Girder

Figure 18.62 Cantilever Box Girder

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idea may be found in Brebbia and Dominguez (109). Whilethere are some problem areas to overcome in use of BEMfor non-linear analysis, it has been recognized that BEM isa powerful alternative to FEM particularly for problems in-volving stress concentration or fracture mechanics, and forcases in which the integral domain extends to infinity. Forexample, to design the cathodic corrosion protection sys-tems for ships, offshore structures and pipelines, it has beensuggested that BEM should be employed, with the regionof interest extending to infinity. BEM can also be appliedto problems other than stress or temperature analysis, in-cluding fluid flow and diffusion (for example, for fluid-structure interaction, Subsection 18.6.8.2).

Main advantages of BEM are due that very complex ex-pressions of integral equations can be adopted, resulting inhigher accuracy of the results.

In this regard, BEM can be involved in the usage of morerefined mathematical treatment than FEM. However, to cal-culate the integral equations using BEM, appropriate nu-merical techniques should be used, otherwise the integrationresults may not be accurate. For most linear problems, lin-ear or flat boundary elements along the boundary of the in-tegral domain can be used so that we don’t have to carryout numerical integration. If analytical solutions are avail-able the required computing times will be very small and

the accuracy high. Nevertheless as the required computa-tional times with the BEM is in general significant, BEMmay be more appropriate for linear analysis of solids andfor fluid mechanics problems.

18.7.4 Presentation of the Stress ResultAfter performing an analysis, the presentation of the stressand deformation is very important. It should be based onstresses acting at the middle of element thickness, exclud-ing plate-bending stress, in the form of ISO-stress contoursin general. Numerical values should also be presented forhighly stressed areas or locations where openings are notincluded in the model.

The following results should be presented for parts ofthe vessel covered by the global model, such as, cargo holdmodel and frame and girder models:

• deformed shape for each loading condition,• In-plane maximum normal stresses (σx and σy) in the

global axis system, shear stresses (_) and equivalent vonMises stress (σe) of the following elements:

— bottom,— inner bottom,— deck,— side shell,— inner side including hopper tank top,— longitudinal and transverse bulkheads, and— longitudinal and transverse girders.

• Axial stress of free flanges,• Deformations of supporting brackets for main frames

including longitudinals connected to these when appli-cable,

• Deformation of supports for longitudinals subject tolarge relative deformation when applicable.

For parts of the vessel covered by the local model, thefollowing stresses are to be presented:

• Equivalent stress of plate/membrane elements,• Axial stress of truss elements,• Axial forces, bending moments and shear forces for beam

elements.

18.7.5 Relevant Structural Analysis Methods forSpecific Design StagesShipbuilding design offices face very challenging situations(especially for passenger and other complex ships). Theproducts are one-of-a-kind or at least on short series andthe resulting ships are designed and built within two years


Figure 18.65 A Typical FEM/BEM Model for Analysis of the

Pressure Vessel (109). (a) Typical BEM model, and (b) Typical FEM model.

(b)

Author:Pleaseadvisewhatsymboldisneeded.

(a)

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for 20 to 30 years of operation. Another impact on designactivities that is also challenging is that the design overlapsthe production. To clarify the actual situation, a commonview of the design workflow for a commercial ship in theshipyard is shown in Table 18.VII.

18.7.5.1 Basic designThe Basic Design is the design activities performed beforeorder. This phase does not overlap with the production butis very short and will become the technical basis for thecontract. The shipyard must be sure that no technical prob-lem will appear later on, to avoid extra costs not includedin the contract. The structural analysis carried out in thisphase must be as fast as possible because the allocated timeis short. The most time consuming task for analysis is thedata input. The more detailed are the data more accurate theresults. There are three kinds of early analysis:

1. First principles methods:Very simplified geometric rep-resentation of the structure. These methods are dedicatedto an assessment of the global behavior of the ship. Theymainly use empirical or semi-empirical formulas.

2. Two-dimensional (or almost 2D) geometry-based meth-ods: These methods are based on one or more 2D viewsof the ship sections. The expected results may be:

• Verification of main section scantlings,• Global strength assessment,• Global vibration levels prediction,• Ultimate strength determination, and• Early assessment of fatigue

Two main approaches exist:

— The main section of the ship is modeled a 2D way(including geometry and scantlings) then global, andpossibly local, loadings are applied (bending mo-ments, pressures, etc.). All major Classification So-cieties provide today the designer with such tools(Table 18.VIII).

— Various significant sections are described as beamcross section properties (areas, inertias, etc.) and thenthe ship is represented by a beam with variable prop-erties on which global loading is applied.

3. Simple three-dimensional models: These models are use-ful when a more detailed response is needed. The ideais to include main surfaces and actual scantlings (or fromthe main section when not available) in a 3D model thatcan be achieved in one or two weeks. This approach ismainly dedicated to novel ship designs for which thefeedback is rather small.

18.7.5.2 Production designThe most popular method for structural analysis at the pro-duction design stage remains the Finite Elements Analysis(FEA). This method is commonly used by Shipyards, Classi-fication Societies, Research Institutes and Universities. It isvery versatile and may be applied to various types of analysis:

• global and local strength,• global and local vibration analysis (natural frequencies

with or without external water, forced response to thepropeller excitation, etc.),

• ultimate strength, and• detailed stress for local fatigue assessment,• fatigue life cycle assessment,• analysis of various non-linearities (material, geometry,

contact, etc.), and• collision and grounding studies.

The two main approaches for solving the physical prob-lem are:

1. implicit method is used to solve large problems (both lin-ear and non linear) with a matrix-based method. This is


TABLE 18.VII Timing of a Design Project

Basic Design

Concept Design 1 or 2 days

Preliminary Design About 1 week

Contract Design Months

Receive Order

Production Design

Complete Functional Design 1 or 2 months

Production Design 6–10 months

TABLE 18.VIII Classification Society Tools Overview (110)

Classification Society Product

American Bureau of Shipping (ABS) ABS Safe Hull

Bureau Veritas (BV) VeriSTAR

Det Norske Veritas (DNV) Electronic Rulebook &Nauticus HULL

Germanisher Lloyd (GL) GL-Rules & POSEIDON

Korean Register of shipping (KRS) KR-RULES, KR-TRAS

Lloyd’s Register of Shipping (LR) Rulefinder, ShipRight

Nippon Kaiji Kyokai (NK) PrimeShip BOSUN

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the favored method for solving global and local linearstrength and vibration problems. But it can also be ap-plied to non linear calculations when the time step re-mains rather large (about 1/10 to 1 second), and

2. explicit method is mainly used for fast dynamics (as col-lision and grounding or explosion) where time step isquite smaller. This method allows using different for-mulations for structural elements (Lagrangian) and fluidelements (Eulerian).

One interesting result from research that is being intro-duced today is the reliability approach (see Chapter 19).This approach introduces uncertainties within the model(non planar plates, residual stresses from welding, dis-crepancies in the thickness…) to provide the designer witha level of reliability for a given result instead of a deter-ministic value.

For FEA models, the modeling time is usually assumedto be 70% of the overall calculation time and results ex-ploitation 30%. The computation itself is regarded as neg-ligible (excepted for explicit analysis). So the main effortstoday are focused on reducing the modeling time.

18.7.6 OptimizationOptimization is a field in which much research has been car-ried out over a long time. It is included today in many soft-ware tools and many designers are using it. The aim ofoptimization is to give the designers the opportunity tochange design variables (such as thickness, number andcross section of stiffeners, shape or topology) to design abetter structure for a given objective (lower weight or cost).

Optimization can be performed both at basic and pro-duction design stages:

• Basic Design: Even with simplified models, the designercan optimize the scantlings. It can be used for instanceto find out the minimal scantlings for a novel ship forwhich the yard have a lack of feedback,

• Production Design: Optimization can be used for threemain purposes:

— Scantlings optimization, which gives the user theminimum scantlings for a given structure. The num-ber of longitudinals and the frame spacing for a givencargo hold/tank can also be optimized (105).

— Shape optimization (111), which uses a given topol-ogy and scantlings to provide the user the minimum,required area of material (reducing holes in a platefor instance), and to improve the hull shape consid-ering the fluid-structure interaction.

— Topology optimization (112) which uses a givenscantlings and allows the user to find out where to

put material. An academic example of topology op-timization is given on Figure 18.66.

Weight is the most usual objective function for structureoptimization. Minimizing weight is of particular impor-tance in deadweight carriers, in ships required to have alimited draft, and in fast fine lined ships, for example, pas-senger vessels. However, it is well know that the lowestweight solution is not usually the lowest acquisition cost.Today, cost is becoming the usual objective function for op-timization (124).

For the other ship types it is still desirable to minimizesteel weight to reduce material cost but only when this canbe done without increasing labor costs to an extent that ex-ceeds the saving in material costs. On the other hand, a re-duction in structural labor cost achieved by simplifyingconstruction methods may still be worthwhile even if thisis obtained at the expense of increasing the steel weight.

Rigo (105) presents extensive review of ship structureoptimization focusing on scantling optimization. Vander-plaats (113), and Sen and Yang (114) are standard referencebooks about optimization techniques. Catley et al (115),Hughes (3) and Chapter 11 of this book also contain valu-able information on structure optimization.

18.7.6.1 Scantling optimization procedureA standard optimization problem is defined as follows:

• Xi (i = 1, N), the N design variables,• F(Xi), the objective function to minimize,• Cj(Xi) ≤ CMj (j = 1, M), the M structural and geomet-

rical constraints,• Xi min ≤ Xi ≤ Xi max upper and lower bounds of the Xi de-

sign variables: technological bounds (also called sideconstraints).


Figure 18.66 Topology Optimization

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Constraints are linear or nonlinear functions, either ex-plicit or implicit of the design variables (XI). These con-straints are analytical translations of the limitations that theuser wants to impose on the design variables themselves orto parameters like displacement, stress, ultimate strength,etc. Note that these parameters must be functions of the de-sign variables.

So it is possible to distinguish:

Technological constraints (or side constraints) that providethe upper and lower bounds of the design variables. For ex-ample:

Xi min = 4mm ≤ Xi ≤ Xi max = 40 mm,

with:

Xi min = a thickness limit dues to corrosion,Xi max = a technological limit of manufacturing or assembly.

Geometrical constraints that impose relationships betweendesign variables in order to guarantee a functional, feasi-ble, reliable structure. They are generally based on goodpractice rules to avoid local strength failures (web or flangebuckling, stiffener tripping, etc.), or to guarantee weldingquality and easy access to the welds. For instance, weldinga plate of 30 mm thick with one that is 5 mm thick is notrecommended. Hence, the constraints can be 0.5 ≤ X2 / X1≤ 2 with X1, the web thickness of a stiffener and X2, theflange thickness.

Structural constraints represent limit states in order to avoidyielding, buckling, cracks, etc. and to limit deflection, stress,etc. These constraints are based on solid-mechanics phe-nomena and modeled with rational equations. Rational equa-tions mean a coherent and homogeneous group of analysismethods based on physics, solid mechanics, strength andstability treatises, etc. and that differ from empirical andparametric formulations. Such standard rational structuralconstraints can limit:

• the deflection level (absolute or relative) in a point of thestructure,

• the stress level in an element: σx , σy, and σc = σvon Mises,• the safety level related to buckling, ultimate resistance,

tripping, etc. For example: σ /σult ≤ 0.5.

For each constraint, or solid-mechanics phenomenon,the selected behavior model is especially important sincethis model fixes the quality of the constraint modeling. Thesebehavior models can be so complex that it is no longer pos-sible to explicitly express the relation between the param-eters being studied (stress, displacement, etc.) and the designvariables (XI). This happens when one uses mathematicalmodels (FEM, ISUM, BEM, etc.). In this case, one gener-

ally uses a numeric procedure that consists of replacing theimplicit function by an explicit approximated function ad-justed in the vicinity of the initial values of the design vari-ables (for instance using the first or second order Taylorseries expansions). This way, the optimization process be-comes an iterative analysis based on a succession of localapproximations of the behavior models.

At least one constraint should be defined for each fail-ure mode and limit state considered in the Subsection 18.6.1.When going from the local to the general (Figure 18.38),there are three types of constraints: 1) constraints on stiff-ened panels and its components, 2) constraints on trans-verse frames and transversal stiffening, and 3) constraintson the global structure.

Constraints on stiffened panels (Figure 18.22): Panelsare limited by their lateral edges (junctions with other pan-els, AA’ and BB’) either by transverse bulkheads or trans-verse frames. These panels are orthotropic plates and shellssupported on their four sides, laterally loaded (bending) andsubmitted, at their extremities, to in-plane loads (compres-sion/tensile and shearing).

Global buckling of panels (including the local transverseframes) must also be considered. Panel supports, in partic-ular those corresponding to the reinforced frames, are as-sumed infinitely rigid. This means that they can distortthemselves significantly only after the stiffened panel col-lapse.

Constraints on the transverse frames (Figure 18.23): Theframes take the lateral loads (pressure, dead weight, etc.)and are therefore submitted to combined loads (large bend-ing and compression). The rigidity of these frames must beassured in order to respect the hypotheses on panel bound-ary conditions (undeformable supports).

Constraints on the global structure (box girder/hullgirder) (Figure 18.46): The ultimate strength of the globalstructure or a section (block) located between two rigidframes (or bulkheads) must be considered as well as theelastic bending moment of the hull girder (against yielding).

18.8 DESIGN CRITERIA

In ship design, the structural analysis phase is concernedwith the prediction of the magnitude of the stresses and de-flections that are developed in the structural members as aresult of the action of the sea and other external and inter-nal causes. Many of the failure mechanisms, particularlythose that determine the ultimate strength and collapse ofthe structure, involve non-linear material and structural be-havior that are beyond the range of applicability of the lin-ear structural analysis procedures in Section 18.4, which are


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commonly used in design practice. Most of the availablemethods of non-linear structural analysis are briefly intro-duced in Sections 18.6 and 18.7. Sometimes, these meth-ods are limited in their applicability to a narrow class ofproblems.

One of the difficulties facing the structural designer is thatlinear analysis tools must often be used in predicting the be-havior of a structure in which the ultimate capability is gov-erned by non-linear phenomena. This is one of the importantsources of uncertainty related to strength assessment.

After performing an analysis, the adequacy or inade-quacy of the member and/or the entire ship structure mustthen be judged through comparison with some kind of cri-terion of performance (Design Criteria). The conventionalcriteria that are commonly used today in ship structural de-sign are usually stated in terms of acceptable levels of stressin comparison to the yield or ultimate strength of the ma-terial, or as acceptable stress levels compared to the criti-cal buckling strength and ultimate strength of the structuralmember. Such criteria are, therefore, intended specificallyfor the prevention of yielding (hull girder, frames, longitu-dinals, etc), plate and stiffened plate buckling, plate andstiffened plate ultimate strength, ultimate strength of hullgirder, fatigue, collision, grounding, vibration and manyother failure modes specific to particular vessel types. In-formation related to the design criteria is given in Section18.6 for each specific failure mode (see also Beghin et al(116)).

18.8.1 Structural Reliability as a Design BasisThree categories of design methodology are basically avail-able. They are usually classified as:

1. deterministic method,2. semiprobabilistic method, and3. full probabilistic method.

The deterministic method uses a global safety factor. Itassumes that loads and strength are fully determined. Thismeans that no aspect of randomness is considered. Every-thing is assumed to be deterministic. The global safety fac-tor is compared to the ratio between the actual strength andthe required strength.

The full probabilistic method is an ideal approach as-suming that all the randomness can be exactly consideredwithin a global probabilistic approach. All the actual devel-opment in structural reliability and reliability analysis showthe huge effort actually done to reach that aims. Chapter 19presents in detail the reliability concept with examples of thereliability-based strength analysis of plates, stiffened pan-els, hull girder and fatigue. See also Mansour et al (42).

The semiprobabilistic method corresponds to the cur-rent practice used by codes and the major classifications so-cieties. Load, strength, dimensions are random parametersbut their distribution is basically not known. To overcomethis, partial safety factor are used. Each safety factor cor-responds to a load type, failure mode, etc. This is an inter-mediate step between the deterministic and the fullprobabilistic methods.

18.9 DESIGN PROCEDURE

It does not seem possible to unify all of the design proce-dures (117-122). They differ from country to country, fromshipyard to shipyard and differ between naval ships, com-mercial ships and advanced high-speed catamaran passen-ger vessels. So, as an example of one feasible methodology,the design procedure for commercial vessel such as tanker,container, and VLCC is selected. It corresponds to the ac-tual current shipyard procedure.

This structural design procedure can be defined as fol-lows:

• receive general arrangement from the basic design group,• define structural arrangement based on the general

arrangement,• determine initial scantling of structural members within

design criteria (rule-based).,• check longitudinal and transverse strength,• change the structural arrangement or scantling, and• transfer the structural arrangement and scantling to the

production design group.

The structural design can also be classified according toavailable design tool:

• use data of existing ship or past experience—expert sys-tem, (1st level)

• use of a structural analysis software like FEM (2nd level)• use optimization software (3rd level)

The adequacy of the relevant analysis method to use fora specific design stage is discussed in Subsection 18.7.5.Here the discussion concerns the procedure from a designpoint of view and not from the analysis point of view.

18.9.1 Initial ScantlingAt the basic design stage, principal dimensions, hull form,double bottom height, location of longitudinal bulkheads andtransverse bulkheads, maximum still-water bending mo-ment, etc. have already been determined to meet the owner’srequirements such as deadweight and ship’s speed. Such a


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parametric design procedure presented in Chapter 11 is rel-evant for this stage.

For the structural design stage, the structural arrangementis carried out to define the material property, plate breadth,stiffener spacing, stiffener type, slot type, shape of open-ings, and frame spacing. The initial scantling of longitudi-nal members such as plate thickness and section area ofstiffener can be determined by applying the classificationrules which give minimum required value to meet the bend-ing, shear and buckling strength. As there are usually no suit-able rules for the transverse members, the initial scantlingof transverse members such as height and thickness of web,breadth and thickness of flange are determined by referenceto similar ships or using empirical shipyard database.

18.9.2 Strength AssessmentThe purpose of the strength assessment is to validate the ini-tial design, that is, to evaluate quantitatively the strength ca-pability of the initial design. This problem was extensivelypresented in previous Sections 18.4, 18.5 and 18.6.

In general, the longitudinal members are subjected toseveral kinds of stresses in the sea-going condition: pri-mary, secondary and tertiary stresses (Subsection 18.4.1).As all these stresses act simultaneously, the superpositionof these stresses should not exceed the allowable equiva-lent stress given by the classification rules (equations 45and 46).

There are two kinds of strength to design the longitudi-nal members. One is the local strength to avoid collapse,and the other is the longitudinal strength to consider thecollapse of the ships’ hull girder. The local strength is au-tomatically satisfied if the design is based on the classifi-cation rules. The hull girder longitudinal strength can beassessed with the hull section modulus (SM) at bottom anddeck where the extreme stresses are taken place (equation29). The hull section modulus is calculated easily by usingavailable software.

If the hull section modulus at bottom or deck part is big-ger than the required value, this design can be consideredas finished but this design might be too expensive. If thesection modulus at the deck or at the bottom is less than therequired value, the designer should change the initial scant-lings.

If the calculated hull section modulus at deck part is lessthan required, he can increase, step by step, the deck scant-ling (for example, 0.5 mm for the plate thickness) until therequirement is satisfied.

The designer also has to modify the scantling (usuallyplate thickness) of transverse members, for which the stressexceeds the allowable value. The designer estimates the in-

creased thickness according to the difference between theactual stress and allowable stress. If the difference is small,it is not necessary to perform a new strength assessmentand the design may be completed with only small changes.If the difference is large, the design should be drasticallychanged and it will be necessary to analyze the structureagain (see previous step in this Subsection).

Then, the designer has to check the transverse strengthby comparing the actual stresses in the transverse frameswith the allowable stresses given by the classification rules.The actual stresses such as equivalent stress and shear stresscan be obtained using commercial FEA packages. If thestress in some of elements exceeds the allowable stress, thedesigner should increase the initial scantling. These changesare performed at the third step Structural Design using theresults of the Strength Assessment and by comparison withthe design criteria.

18.9.3 Structural DesignIf all of local scantlings are determined by the rule mini-mum values, and if the longitudinal strength satisfies the rulestrength requirement, the design is completed. But, even ifthis design is strong enough, it might be too heavy and/ortoo expensive and it should be refined. In practice, refiningan already feasible design is a difficult task and requires ex-perience. The designer can change the structural arrange-ment, especially the dimensions such as frame spacing, andmaterial properties to better fit with the longitudinal strengthrequirements. This work has to be done in agreement withthe basic design team.

Instead of the trial and error procedure discussed above,an automatic optimization technique can be used to obtainthe minimum weight and/or cost for the longitudinal andtransverse structural member. The object function(s) can bestructural weight and/or fabrication cost, using either a sin-gle object function approach or a multiple objective func-tion method. The design variables can be longitudinal andtransverse spacing, deck/bottom scantlings for the longitu-dinal and transverse members (web height and thickness,flange width and thickness). The constraints and limitationsof the optimization process can be the range of each designvariable as well as the required hull section modulus andminimum deck/bottom scantlings for the longitudinal mem-bers, and allowable bending and shear stresses for the trans-verse members (see Optimization in Subsection 18.7.6).

18.9.4 A Generic Design FrameworkBy comparison with the previous standard procedure, Fig-ure 18.67 shows a new generic and advanced design method-


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ology where the performance of the system, the manufac-turing process of the system and the associated life cyclecosts are considered in an integrated fashion (120). De-signing ship structures systems involves achieving simul-taneous, though sometimes competing, objectives. Thestructure must perform its function while conforming tostructural, economic and production constraints. The pres-ent design framework consists of establishing the structuralsystem and composite subsystems, which optimally satisfythe topology, shape, loading and performance constraintswhile simultaneously considering the manufacturing or fab-rication processes in a cost effective manner.

The framework is used within a computerized virtualenvironment in which CAD product models, physics-basedmodels, production process models and cost models areused simultaneously by a designer or design team. The per-formance of the product or process is in general judged bysome time independent parameter, which is referred to asa response metric (R). Specifications for the system mustbe established in terms of these Response Metrics. The for-mulation of the design problem is thus the same whetherthe product or process systems (or both) are considered.

The general framework consists of a system definitionmodule, a simulation module and a design module.

The system definition module [Y(U,V,W)] is used tobuild an environmental model [U], a product model [V] anda process model [W]. The system definition module receivesoperational requirements [Z] such as owner’s requirements.These operational parameters are presumed fixed through-out the design.

They of course can eventually be changed if no accept-able design is established, but presumably any design wouldhave operational parameters, which would not be sacrificed.The environmental model [U] includes the still water andwave loading conditions and the product model [V] con-tains the production information, for example. The processmodel [W] is built to consider or define the fabrication se-quence. A translator (simulation based design translator)assigns some [Y] model parameters to the simulation pa-rameters [T] and design variables [X].

These parameters are selected based on the availablesimulation tools [S] that require specific data ([T],[X] andtime).

The simulation module [S(T, X, time)] is used to pro-duce simulation responses such as Response Metrics [R[S(T,X)]]. The time is needed to consider the dynamic effects andactual dynamic load conditions [U].

The optimum design module includes the Design Cri-teria, the Design Assessment and the Optimization compo-nents. The design criteria module provides constraints [G(T,X, Y, Z)] and objective functions [F(R, T, X, Y, Z)]. Theseare used to assess the design through the Design Assess-ment component of the module (for example R≤G). Theconstraints are obtained by considering not only the simu-lation parameters [T] and the design variables [X] but alsothe operational requirements [Z] and the system definitionparameter [Y]. Also, the objective function [F] is calculatedusing the response metrics [R], the operational requirements[Z], the system definition parameter [Y] as well as the de-sign variables [X] and simulation parameters [T].

Based on the results of the Design Assessment (Min(F)and R≤G) several strategies for the design procedure (iter-ations) can be followed:

• if the object function does not reach its minimum valueor the response metrics do not satisfy the constraints, anoptimization algorithm (steepest descent, dual approachand convex linearization, evolutionary strategies, etc.) isadopted to find a new set of design variables. Standardalgorithms are presented in (113,114,123):

— if the optimizer fails to find an improved solution (un-feasible design space), it is required to change thesimulation parameter values [T] and/or design vari-ables selection [X] or even to modify the Model Pa-rameters [Y].


Figure 18.67 A Generic Design Framework (120)

Operational Requirements

Parameters Z

System Definition Model Parameters Y

Environmental Model Product Model Process ModelParameters U Parameters V Parameters W

Simulation Based Design Translator

Simulation Parameters T

Design Variables X

Simulations

Simulation Response S(T ,X ,time) Design Criteria

Constraints G(T,X,Y,Z)

Objective Function F(R,T,X,Y,Z)Response Metrics R [S(T ,X )]

Design AssessmentMin (F) ?R < G ?

Conditions Satisfied ?

Is Design SpaceFeasible?

Redesign? Stop

OptimizationSteepest Descent

Convex Linearization Yes

Yes

NoYes

No

No

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— otherwise, the design space is feasible, and a changeof design variable values [X] is performed based onthe optimizer solution (in other words a new itera-tion).

• if the object function reaches its minimum value and theresponse metrics satisfy the constraints, two alternativesare examined:

— change the operational requirements parameters [Z],repeat the previous procedure and to compare withother alternative designs, or

— end the design procedure.

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