Analysis and design of ship structure

Đây là tài liệu tính toán thiết kế tàu thủy. Phiên bản tiếng anh, giúp các bạn có thể học tập và nghiên cứu. tài liệu có hình ảnh minh họa rất hấp dẫn. chúc các bạn thành công trong con đường mình chọn. tài liệu hót

18.1 NOMENCLATURE

For specific symbols, refer to the definitions contained in

the various sections

IACS International Association of

Classifica-tion SocietiesISSC International Ship & Offshore Structures

CongressISOPE International Offshore and Polar Engi-

neering ConferenceISUM Idealized Structural Unit method

PRADS Practical Design of Ships and Mobile

Units,RINA Registro Italiano Navale

SNAME Society of naval Architects and marine

Engineers

C wave coefficient (Table 18.I)

CB hull block coefficient

m(x) longitudinal distribution of massI(x) geometric moment of inertia (beam sec-

tion x)

M(x) bending moment at section x of a beam

MT(x) torque moment at section x of a beam

q(x) resultant of sectional force acting on a

beam

V(x) shear at section x of a beams,w(low case) still water, wave induced componentv,h(low case) vertical, horizontal componentw(x) longitudinal distribution of weight

18.2 INTRODUCTION

The purpose of this chapter is to present the fundamentals

of 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 often

associated 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

Analysis and Design of Ship Structure

Philippe Rigo and Enrico Rizzuto

Analysis refers to stress and strength assessment of the

structure Analysis requires information on loads and needs

an initial structural scantling design Output of the structural

analysis is the structural response defined in terms of stresses,

deflections and strength Then, the estimated response is

compared to the design criteria Results of this comparison

as well as the objective functions (weight, cost, etc.) will

show 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 detailed

design stage (construction) To perform analysis, initial

de-sign is needed and analysis is required to dede-sign This

ex-plains why design and analysis are intimately linked, but

are absolutely different Of course design also relates to

topology and layout definition

The organization and framework of this chapter are based

on 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 duplicated

from these 2 books Other major references used to write this

chapter are Ship Structural Design (3) also published by

SNAME and the DNV 99-0394 Technical Report (4).

The present chapter is intimately linked with Chapter

11 – Parametric Design, Chapter 17 – Structural

Arrange-ment and Component Design and with Chapter 19 –

Reli-ability-Based Structural Design References to these

chapters will be made in order to avoid duplications In

ad-dition, as Chapter 8 deals with classification societies, the

present chapter will focus mainly on the direct analysis

methods available to perform a rationally based structural

design, even if mention is made to standard formulations

from Rules to quantify design loads

In the following sections of this chapter, steps of a global

analysis are presented Section 18.3 concerns the loads that

are necessary to perform a structure analysis Then, Sections

18.4, 18.5 and 18.6 concern, respectively, the stresses and

deflections (basic ship responses), the limit states, and the

fail-ures modes and associated structural capacity A review of

the available Numerical Analysis for Structural Design is

per-formed in Section 18.7 Finally Design Criteria (Section

18.8) and Design Procedures (Section 18.9) are discussed.

Structural modeling is discussed in Subsection 18.2.2 and

more 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 Hence

Hughes (3) states:

The complexities of modern ships and the demand for

greater reliability, efficiency, and economy require a

sci-entific, powerful, and versatile method for their structural design

But, even with the development of numerical techniques,design still remains based on the designer’s experience and

on 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 promises:

com-• 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 time

is not available For that reason, rule-based design orsimplified numerical analysis has to be performed

• to limit uncertainty and reduce conservatism in design, it

is 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 Design

There are basically two schools to perform analysis and sign of ship structure The first one, the oldest, is called

de-rule-based design It is mainly based on the rules defined

by the classification societies Hughes (3) states:

In the past, ship structural design has been largely ical, based on accumulated experience and ship perform- ance, and expressed in the form of structural design codes

empir-or rules published by the various ship classification eties These rules concern the loads, the strength and the design criteria and provide simplified and easy-to-use for- mulas for the structural dimensions, or “scantlings” of a ship This approach saves time in the design office and, since the ship must obtain the approval of a classification society, it also saves time in the approval process.

soci-The second school is the Rationally Based Structural Design; it is based on direct analysis Hughes, who could

be considered as a father of this methodology, (3) further

states:

There are several disadvantages to a completely “rulebook”

approach to design First, the modes of structural failure

are numerous, complex, and interdependent With such

simplified formulas the margin against failure remains

un-known; thus one cannot distinguish between structural

ad-equacy and over-adad-equacy Second, and most important,

these formulas involve a number of simplifying

assump-tions and can be used only within certain limits Outside

of this range they may be inaccurate

For these reasons there is a general trend toward direct

structural analysis.

Even if direct calculation has always been performed,

design based on direct analysis only became popular when

numerical analysis methods became available and were

cer-tified Direct analysis has become the standard procedure

in aerospace, civil engineering and partly in offshore

in-dustries In ship design, classification societies preferred to

offer updated rules resulting from numerical analysis

cali-bration For the designer, even if the rules were continuously

changing, the design remained rule-based There really were

two different methodologies

Hopefully, in 2002 this is no longer true The advantages

of 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 environment

of 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 overview

of 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 Analysis

General guidance on the modeling necessary for the tural analysis is that the structural model shall provide re-sults suitable for performing buckling, yield, fatigue and

struc-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

Buckling strength

Ultimate strength

Fatigue strength

Modeling technique Direct structural

vibration assessment of the relevant parts of the vessel This

is done by using a 3D model of the whole ship, supported

by one or more levels of sub models

Several approaches may be applied such as a detailed

3D 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 results

suited for yielding and buckling control but also to obtain

the displacements to apply as boundary conditions for sub

models with the purpose of determining the stress level in

more 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 (Subsection

18.6.6), collision and grounding (Subsection 18.6.7) and

vibration (Subsection 18.6.8) The hydrodynamic load

model must give a good representation of the wetted

sur-face of the ship, both with respect to geometry description

and with respect to hydrodynamic requirements The mass

model, which is part of the hydrodynamic load model, must

ensure a proper description of local and global moments of

inertia around the global ship axes

Ultimate hydrodynamic loads from the hydrodynamic

analysis should be combined with static loads in order to

form the basis for the yield, buckling and ultimate strength

checks 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 Design

For a ship structure, structural design consists of two

dis-tinct levels: the Preliminary Design and the Detailed sign about which Hughes (3) states:

De-The preliminary determines the location, spacing, and 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 the structure design and hence is the phase that offers very large potential savings This does not mean that detail de- sign is less important than preliminary design Each level

scant-is equally important for obtaining an efficient, safe and liable ship

re-During the detailed design there also are many fits to be gained by applying modern methods of engi- neering science, but the applications are different from preliminary design and the benefits are likewise different Since the items being designed are much smaller it is possible to perform full-scale testing, and since they are more repetitive it is possible to obtain the benefits of mass production, standardization and so on In fact, production aspects 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-service experience 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 design codes and standard designs that have been proven by ex- perience In other words, detail design is an area where a rule-based approach is very appropriate, and the rules that are published by the various ship classification societies contain a great deal of useful information on the design of local structure, structural connections, and other structural details.

bene-18.3 LOADS

Loads acting on a ship structure are quite varied and 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-

pecu-Figure 18.2 Strength Analysis Flow Chart (4)

Structural model including necessary load definitions

Hydrodynamic/static loads

Load transfer to structural model Verified structural

model

Sub-models to be used in structural analysis Structural analysis

dures and practical methods for their evaluation will be

sum-marized

18.3.1 Classification of Loads

18.3.1.1 Time Duration

Static loads: These are the loads experienced by the ship in

still water They act with time duration well above the range

of sea wave periods Being related to a specific load

con-dition, they have little and very slow variations during a

voyage (mainly due to changes in the distribution of

con-sumables on board) and they vary significantly only during

loading and unloading operations

Quasi-static loads: A second class of loads includes

those with a period corresponding to wave actions (∼3 to

15 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) These

loads 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 resonance

modes, the dynamic properties of the structure have to be

considered 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) can

be classified in the same category

High frequency loads: Loads at frequencies higher than

the first resonance modes (> 10-20 Hz) also are present on

ships: this kind of excitation, however, involves more the

study of noise propagation on board than structural design

Other loads: All other loads that do not fall in the above

mentioned categories and need specific models can be

gen-erally grouped in this class Among them are thermal and

accidental loads

A large part of ship design is performed on the basis of

static and quasi-static loads, whose prediction procedures

are quite well established, having been investigated for a

long time However, specific and imposing requirements

can arise for particular ships due to the other load

cate-gories

18.3.1.2 Local and global loads

Another traditional classification of loads is based on the

structural scheme adopted to study the response

Loads acting on the ship as a whole, considered as a

beam (hull girder), are named global or primary loads and

the ship structural response is accordingly termed global or

primary response (see Subsection 18.4.3)

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

struc-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, represent

a contribution to the bending moment acting on the hullgirder

This terminology is typical of simplified structural 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

analy-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 to

be 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 tion of time, and

func-• the probability of exceeding the limit state surface in thedesign time (probability of crisis) is the element subject

to 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 uation corresponds to the highest load cycle and is charac-terized through the probability associated to the extremevalue in the reference (design) time

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

con-damage accumulation The analysis, therefore, does not

re-gard the magnitude of a single extreme load application, but

the 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 in

place of statistical distributions This usually is done, for

example, when calibrating a Partial Safety Factor format for

structural checks Such adoption implies the definition of a

single reference load value as representative of a whole

probability distribution This step is often performed by

as-signing an exceeding probability (or a return period) to each

variable and selecting the correspondent value from the

sta-tistical distribution

The exceeding probability for a stochastic variable has

the meaning of probability for the variable to overcome a

given value, while the return period indicates the mean time

to 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 108cycles, that is, with a return

period of the same order of the ship lifetime In first

yield-ing analyses, characteristic loads are associated to a higher

exceeding probability, usually in the range 10–4 to 10–6 In

fatigue analyses (see Subsection 18.6.6.2), reference loads

are often set with an exceeding probability in the range 10–3

to 10–5, corresponding to load cycles which, by effect of both

amplitude and frequency of occurrence, contribute more to

the 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 exceeding

probability

18.3.2 Definition of Global Hull Girder Loads

The global structural response of the ship is studied with

reference 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 this

scheme, only in terms of forces and moments acting in the

transverse 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 tained in the plane of symmetry), indicated as vertical re-sultant force qV; another force in the normal direction, (localhorizontal axis), termed horizontal resultant force qHand amoment mTabout the x axis All these actions are distrib-uted along the longitudinal axis x

(con-Five main load components are accordingly generatedalong the beam, related to sectional forces and momentthrough equation 1 to 5:

condi-These conditions impose constraints on the distributions

of qV, qHand mT

[6]

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

ob-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 Loads

Still water loads act on the ship floating in calm water, 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

usu-If the latter ones are not symmetric, a sectional torque

m (x) is generated (Figure 18.4), in addition to the

cal load qSV(x), obtained as a difference between buoyancy

b(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 bending

moment As regards equation 5, only mTg, if present, is to

be accounted for, to obtain the torque

18.3.3.1 Standard still water bending moments

While buoyancy distribution is known from an early stage

of the ship design, weight distribution is completely defined

only 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 of

weight items and their longitudinal distribution on board

The resulting approximated weight distribution, together

with the buoyancy distribution, allows computing shear and

bending moment

qSV(x)=b(x)−w(x) =gA (x)I −m(x)g

At an even earlier stage of design, parametric tions can be used to derive directly reference values for stillwater hull girder loads

formula-Common reference values for still water bending ment at mid-ship are provided by the major ClassificationSocieties (equation 8)

mo-[8]

where C = wave parameter (Table 18.I)

The formulations in equation 8 are sometimes explicitlyreported 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 analysis

of 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 buoyancy

forces, 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

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

the correspondence of area and barycenter of the trapezoid

respectively to the total weight and center of gravity of the

considered ship portion

The procedure is usually applied separately for

differ-ent types of weight items, grouping together the weights of

the 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 evaluation

A significant contribution to uncertainties in the evaluation

of still water loads comes from the inputs to the procedure,

in particular those related to quantification and location on

board of weight items

This lack of precision regards the weight distribution for

the ship in lightweight condition (hull structure, 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

machin-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 neglected

in 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 consider

an 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 loads

In 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, Tapply respectively tolongitudinal and transversal beams)

18.3.4 Wave Induced Global Loads

The prediction of the behaviour of the ship in waves 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

repre-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)

In particular, as regards global effects, the action of waves

modifies the pressure distribution along the wet hull

sur-face; the differential pressure between the situation in waves

and in still water generates, on the transverse section,

ver-tical and horizontal resultant forces (bWVand bWH) and a

moment component mTb

Analogous components come from the sectional

result-ants of inertial forces and moments induced on the section

by ship’s motions (Figure 18.9)

The total vertical and horizontal wave induced forces on

the section, as well as the total torsional component, are

found 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 moment

and torque can then be derived by integration (equations 1

to 5)

Such results are in principle obtained for each

instanta-neous wave pressure distribution, depending therefore, on

time, on type and direction of sea encountered and on the

ship geometrical and operational characteristics

In regular (sinusoidal) waves, vertical bending moments

tend to be maximized in head waves with length close to

the ship length, while horizontal bending and torque

com-ponents are larger for oblique wave systems

18.3.4.1 Statistical formulae for global wave loads

Simplified, first approximation, formulations are available

for the main wave load components, developed mainly on

the basis of past experience

Vertical wave-induced bending moment: IACS

[10]

Horizontal Wave-induced Bending Moment: Similar

for-mulations are available for reference values of horizontalwave induced bending moment, even though they are not

as uniform among different Societies as for the main cal component

verti-In Table 18.II, examples are reported of reference 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

val-Wave-induced Torque: A few reference formulations are

given also for reference wave torque at midship (see amples in Table 18.III) and for the inherent longitudinaldistributions

ex-18.3.4.2 Static Wave analysis of global wave loads

A traditional analysis adopted in the past for evaluation ofwave-induced loads was represented by a quasi-static wave

approach The ship is positioned on a freezed wave of given

characteristics 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 is

no 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 wave

loads Sometimes, however, the concept of equivalent static wave is adopted to associate a longitudinal distribution of

C L B C

(hog)(sag)

WV

B B

⋅

2 2

Figure 18.9 Sectional Forces and Moments in Waves

TABLE 18.II Reference Horizontal Bending Moments

Class Society M WH [N ⋅ m]

BV (9) RINA (10) 1600 L2.1TCBDNV (11) 220 L 9/4 (T + 0.3B)CBNKK (12) 320 L2C T L−35 / L

pressures to extreme wave loads, derived, for example, from

long term predictions based on other methods

18.3.4.3 Linear methods for wave loads

The most popular approach to the evaluation of wave loads

is represented by solutions of a linearized potential flow

problem based on the so-called strip theory in the frequency

domain (13)

The theoretical background of this class of procedures

is 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

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 corresponding

to still water: fluid velocity normal to the free surface

equal 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 surface

is zero (impermeability condition), and

• linear decomposition into additive independent

compo-nents, separately solved for and later summed up

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

This subdivision also enables the de-coupling of the citation components from the response ones, thus avoiding

ex-a non-lineex-ar feedbex-ack 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 terms

of 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 sures on the hull and of the other derived responses, such

pres-as 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 Q w [N .m] (at mid-ship)

ABS (bulk carrier)

(e = vertical position of shear center)

valid only for small wave excitations, small motion

re-sponses and low speed of the ship

In practice, the field of successful applications extends

far beyond the limits suggested by the preservation of

re-alism in the base assumptions: the method is actually used

extensively to study even extreme loads and for fast

ves-sels

18.3.4.4 Limits of linear methods for wave loads

Due to the simplifications adopted on boundary conditions

to linearize the problem of ship response in waves, results

in terms of hydrodynamic pressures are given always up to

the still water level, while in reality the pressure

distribu-tion extends over the actual wetted surface This represents

a major problem when dealing with local loads in the side

region close to the waterline

Another effect of basic assumptions is that all responses

at 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 have

the same characteristics, while, for example, actual values

of vertical bending moment show marked differences

be-tween the hogging and sagging conditions Corrections to

account for this effect are often used, based on statistical

data (7) or on more advanced non-linear methods

A third implication of linearization regards the

super-imposition of static and dynamic loads Dynamic loads are

evaluated separately from the static ones and later summed

up: this results in an un-physical situation, in which weight

forces (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 forces

are directed along the true vertical direction, which depends

on roll and pitch angles, having therefore also components

in 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 input

force (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 characterization

The most widely adopted method to characterize the loads

in the probability domain is the so-called spectral method,

used in conjunction with linear frequency-domain methods

for the solution of the ship-wave interaction problem

From the frequency domain analysis response spectra

Sy(ω) are derived, which can be integrated to obtain

spec-tral moments m 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 to

the ship system is modeled as a stationary and ergodic Gaussian process with zero mean, the response of the sys-

tem (load) can be modeled as a process having the same acteristics

char-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 fluctuating

response 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 each

zero-crossing period (peaks: variable p),

• the probability distribution of the excursions betweenthe highest and the lowest value in each zero-crossing

period (range: variable r), and

• the probability distribution of the highest value in the

whole stationary period of the phenomenon (extreme value 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 sponse process and of independence between peaks The first

re-two probability distributions take the form of equations 15and 16 respectively, both Rayleigh density distributions (see14)

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

2exp

π

/

mny = ∫∞ωnS ( )dyω ω

0

The distribution for the extreme value in the stationary

period 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 derived

from 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 domain

solution of the ship motions problem

The results mentioned previously are derived for the

period Tsin which the input wave system can be

consid-ered as stationary (sea state: typically, a period of a few

hours) 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 advance

direction

To obtain a long-term prediction, relative to the ship life

(or any other design period Tdwhich can be described as a

series of stationary periods), the conditional hypothesis is

to be removed from short-term distributions In other words,

the probability of a certain response is to be weighed by the

probability of occurrence of the generating sea state

F(ySi) = probability for the response to be less than value

y, conditioned to occurrence of sea state Si(short

term 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 data

based on visual observations from commercial ships and/or

on 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

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 conditioning (that is removing the conditioning hypothesis).

de-The same procedure can be applied to any of the variablesstudied in the short term and it does not change the nature

of 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 Tsis achieved

In this case the result is further processed to get the bution of the extreme value in the design time Td This isdone with an additional application of the concept of sta-tistics of extremes

distri-In the hypothesis that the extremes of the various seastates are independent from each other, the extreme on time

Tdis 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 fected by numerous sources of uncertainties, such as:

af-F(extrTdy)=[F(extrTsy) ]Td/Ts

Figure 18.10 Short-term Distributions

• sea description: as above mentioned, scatter diagrams

are derived from direct observations on the field, which

are 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 and

mean 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 boundary

conditions

Numerical algorithms and specific procedures adopted

for the solution also influence results, creating differences

even between theoretically equivalent methods, and

• the de-conditioning procedure adopted to derive long

term predictions from short term ones can add further

uncertainties

18.3.5 Local Loads

As previously stated, local loads are applied to individual

structural members like panels and beams (stiffeners or

pri-mary supporting members)

They are once again traditionally divided into static and

dynamic loads, referred respectively to the situation in still

water 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 based

on 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 10–8value adopted in global loadstrength verifications

18.3.6 External Pressure Loads

Static and dynamic pressures generated on the wet surface

of the hull belong to external loads They act as local verse loads for the hull plating and supporting structures

trans-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)

In the case of the external pressure on the hull, hSresponds to the local draft of the load point (reference ismade to design waterline)

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

18.3.6.2 Dynamic pressures

The pressure distribution, as well as the wet portion of the

hull, is modified for a ship in a seaway with respect to the

still water (Figure 18.9) Pressures and areas of application

are in principle obtained solving the general problem of

ship 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 Tanks

Liquid cargoes generate normal pressures on the walls of

the 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 pressure

For a ship in still water, gravitation acceleration g

gener-ates a hydrostatic pressure, varying again according to

equa-tion 20 The static head hScorresponds here to the vertical

distance from the load point to the highest part of the tank,

increased to account for the vertical extension over that

point 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 example

to the setting pressure of outlet valves)

18.3.7.2 Dynamic internal pressure

When 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 on

the 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 the

fluid is considered as corresponding to the global ship

ac-celeration aw

The total pressure (equation 21) can be evaluated in terms

of the total acceleration aT, obtained summing awto

grav-ity g

The gravitational acceleration g is directed according to

the true vertical This means that its components in the ship

reference system depend on roll and pitch angles (in

Fig-ure 18.13 on roll angle θr)

In equation 21, hTis the distance between the load point

and the highest point of the tank in the direction of the total

acceleration vector aT(Figure 18.13)

If the tank is only partially filled, significant fluid

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

transver-If pitch or roll frequencies are close to the tank 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

reso-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 part

of the tank

Impact loads are very difficult to characterize, being 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

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

pres-Figure 18.13 Internal Fluid Pressure (full tank) Figure 18.12 Example of Simplified Distribution of External Pressure (10)

18.3.7.3 Dry bulk cargo

In the case of a dry bulk cargo, internal friction forces arise

within the cargo itself and between the cargo and the walls

of the hold As a result, the component normal to the wall

has a different distribution from the load corresponding to

a liquid cargo of the same density; also additional

tangen-tial components are present

18.3.8 Inertial Loads—Dry Cargo

To account for this effect, distributions for the components

of cargo load are approximated with empirical formulations

based on the material frictional characteristics, usually

ex-pressed by the angle of repose for the bulk cargo, and on

the slope of the wall Such formulations cover both the static

and the dynamic cases

18.3.8.1 Unit cargo

In the case of a unit cargo (container, pallet, vehicle or other)

the local translational accelerations at the centre of gravity

are applied to the mass to obtain a distribution of inertial

forces 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 or

supporting systems

Generally, this kind of load is modelled by one or more

concentrated 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 loads

transmitted by tyred vehicles when modelling the response

of the deck plate between stiffeners: in this case the load is

distributed uniformly on the tyre print

18.3.9 Dynamic Loads

18.3.9.1 Slamming and bow flare loads

When sailing in heavy seas, the ship can experience such

large heave motions that the forebody emerges completely

from the water In the following downward fall, the bottom

of the ship can hit the water surface, thus generating

con-siderable impact pressures

The phenomenon occurs in flat areas of the forward part

of the ship and it is strongly correlated to loading

condi-tions with a low forward draft

It affects both local structures (bottom panels) and the

global bending behaviour of the hull girder with generation

also of free vibrations at the first vertical flexural modes for

the hull (whipping).

A full description of the slamming phenomenon involves

a 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 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)

stud-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)

draft, local structural checks based on an additional

exter-nal pressure

Such additional pressure is formulated as a function of

ship main characteristics, of local geometry of the ship

(width of flat bottom, local draft) and, in some cases, of the

first 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% and

more) in the design value

A phenomenon quite similar to bottom slamming can

occur 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 on

ships prone to it, but with lower pressures than in bottom

slamming The incremental effect on vertical bending

mo-ment can however be significant

A quantification of bow flare effects implies taking into

account the variation of the local breadth of the section as

a 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 Springing

Another phenomenon which involves the dynamic response

of 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 Subsection

18.6.8.2)

The phenomenon has been observed in particular on Great

Lakes 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 of

quasi-static wave bending moment and can be studied with

the 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 references

16 and 17)

18.3.9.3 Propeller induced pressures and forces

Due to the wake generated by the presence of the after part

of the hull, the propeller operates in a non-uniform incident

velocity field

Blade profiles experience a varying angle of attack

dur-ing the revolution and the pressure field generated around

the 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, related

to 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 counter

by 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 cause

of 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

In addition to low frequency hull vibrations, components

at higher frequencies from the same sources can give rise

to resonance in local structures, which can be predicted by

suitable dynamic structural models (18,19)

18.3.10 Other Loads

18.3.10.1 Thermal loads

A ship experiences loads as a result of thermal effects, which

can be produced by external agents (the sun heating the

deck), 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 the

structure 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 elongapor-tion is compressed and

vice-versa, which is contrary to the normal experience

It is very difficult to quantify thermal loads, the main

problems 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 a

qualitative way (1, Chapter VI)

18.3.10.2 Mooring loads

For a moored vessel, loads are exerted from external actions

on 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 in

the 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 on

extension and geometry of the exposed part of the ship The

action due to wind can be described in terms of two force

components; a longitudinal one FWiL, and a transverse one

FWiT(equation 22), and a moment MWizabout the verticalaxis (equation 23), all applied at the center of gravity

[22]

[23]where:

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

rela-CMz(φWi), CFL(φWi), CFT(φWi) are all coefficients depending

on the shape of exposed part of the ship and on angle φWi

AWi= the reference area for the surface of the ship exposed

to wind, (usually the area of the cross section)

VWi= the wind speedThe 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 the

hull 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 order

of magnitude of the stationary forces due to the other actions

In addition to the linear effects discussed above, 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

non-lin-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

MWiz =1 2/ CMz(φWi)φAWiL VWi2

FWiL,T =1 2/ CF L,T(φWi)φAWiVWi2

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

derived by force decomposition, taking into account the

angle formed with the external force in the horizontal and/or

vertical plane

18.3.10.3 Launching loads

The launch is a unique moment in the life of the ship For

a successful completion of this complex operation, a

num-ber of practical, organizational and technical elements are

to be kept under control (as general reference see Reference

1, Chapter XVII)

Here only the aspect of loads acting on the ship will be

discussed, so, among the various types of launch, only those

which present peculiarities as regards ship loads will be

considered: end launch and side launch

End Launch: In end launch, resultant forces and motions

are contained in the longitudinal plane of the ship (Figure

18.17)

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 to

the cradle and from the latter to the ship’s bottom (in the

following: 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 of

launching loads is based on the computation of these two

distributions

Such computation, repeated for various positions of the

cradle, is based on the global static equilibrium s

(equa-tions 24 and 25, in which dynamic effects are neglected:

quasi static approach)

xW, xB, xF= their longitudinal positions

In a first phase of launching, when the cradle is still in

contact for a certain length with the ground way, the

buoy-ancy distribution is known and the cradle force resultant

and position is derived

In a second phase, beginning when the cradle starts to

rotate (pivoting phase: Figure 18.18), the position xF

cor-responds steadily to the fore end of the cradle and what is

unknown is the magnitude of FCand the actual aft draft of

the ship (and consequently, the buoyancy distribution)

The total sectional vertical force distribution is found as

the 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 to

the bow, at the fore end of the cradle itself (on the fore pet, if one is fitted) and it is at the maximum value.

pop-A considerable sagging moment is present in this 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)

situ-in the area of maximum bendsitu-ing 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 represents

a strong local load and often requires additional temporaryinternal strengthening structures, to distribute the force on

a 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 in

a comparatively short time by buoyancy forces when the shiptilts and drops into water

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

drop-Figure 18.17 End Launch: Sketch

Figure 18.18 Forces during Pivoting

tion at rest The amplitude of heave and roll motions and

accelerations 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, but

need 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 that

here motions are due to a free oscillation of the system due

to an unbalanced initial condition and not to an external

ex-citation

Another difference with respect to end launch is that

both ground reaction (first) and buoyancy forces (later) are

always distributed along the whole length of the ship and

are not concentrated in a portion of it

18.3.10.4 Accidental loads

Accidental loads (collision and grounding) are discussed

in more detail by ISSC (21).

Collision: When defining structural loads due to

colli-sions, the general approach is to model the dynamics of the

accident itself, in order to define trajectories of the unit(s)

involved

In general terms, the dynamics of collision should be

formulated 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 the

units, hydrodynamic pressure due to motions

Normally, theoretical models confine the analysis to

components in the horizontal plane (3 degrees of freedom)

and to collision forces and motion-induced hydrodynamic

pressures The latter are evaluated with potential methods

of 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 different

combinations of rigid, elastic or an elastic body models

Governing equations for the problem are given by servation of momentum and of energy Within this frame-work, time domain simulations can evaluate the magnitude

con-of contact forces and the energy, which is absorbed by ture deformation: these quantities, together with the responsecharacteristics of the structure (energy absorption capacity),allow an evaluation of the damage penetration (21)

struc-Grounding: In grounding, dominant effects are forces and

motions 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 ergy is (a) dissipated in the deformation of the lower part

en-of the bow (b) dissipated in friction en-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 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)

magni-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)

18.3.11 Combination of Loads

When dealing with the characterization of a set of loads

acting simultaneously, the interest lies in the definition of

a total loading condition with the required exceeding

prob-ability (usually the same of the single components) This

cannot 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 lower

than 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 which

characteristic values occur

In the probability domain, the complete formulation of

the problem would imply, in principle, the definition of a

joint probability distribution of the various loads, in order

to quantify the distribution for the total load An

approxi-mation would consist in modeling the joint distribution

through its first and second order moments, that is mean

val-ues and covariance matrix (composed by the variances of

the single variables and by the covariance calculated for

each couple of variables) However, also this level of

sta-tistical characterization is difficult to obtain

As a practical solution to the problem, empirically based

load cases are defined in Rules by means of combination

coefficients (with values generally ≤1) applied to single

loads 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 The

result of the verification is governed by the one, which turns

out to be the most conservative for the specific structure

This procedure needs a higher number of checks (which, on

the other hand, can be easily automated today), but allows

considering various load situations (defined with different

combinations of the same base loads), without choosing a

priori the worst one.

18.3.12 New Trends and Load Non-linearities

A large part of research efforts is still devoted to a better

definition of wave loads New procedures have been

pro-posed in the last decades to improve traditional 2D linear

methods, overcoming some of the simplifications adopted

to 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 of

the main features of the procedures reported in literature is

here 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, while

in other cases an intermediate approach is followed, withboundary conditions formulated part in 2D, part in 3D

18.3.12.2 Body boundary conditions

In 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 terms

of 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 methods

modified to include non-linear effects in a single 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)

compo-18.3.12.3 Free surface boundary conditionsBoundary conditions on free surface can be set, depending

on the various methods, with reference to: (a) a free stream

at constant velocity, corresponding to ship advance, (b) a

double body flow, accounting for the disturbance induced

by 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 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

mod-18.4 STRESSES AND DEFLECTIONS

The reactions of structural components of the ship hull to

external loads are usually measured by either stresses or

deflections 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 component

would be inadequate if it experiences a loss of

load-carry-ing ability through material fracture, yield, bucklload-carry-ing, or

some other failure mechanism in response to the applied

loading Excessive deflection may also limit the structural

effectiveness of a member, even though material failure

does not occur, if that deflection results in a misalignment

or other geometric displacement of vital components of the

ship’s machinery, navigational equipment, etc., thus

ren-dering the system ineffective

The present section deals with the determination of the

responses, in the form of stress and deflection, of structural

members to the applied loads Once these responses are

known it is necessary to determine whether the structure is

adequate to withstand the demands placed upon it, and this

requires consideration of the different failure modes

asso-ciated to the limit states, as discussed in Sections 18.5 and

18.6

Although longitudinal strength under vertical bending

moment and vertical shear forces is the first important

strength consideration in almost all ships, a number of other

strength considerations must be considered Prominent

amongst these are transverse, torsional and horizontal

bend-ing strength, with torsional strength requirbend-ing particular

at-tention on open ships with large hatches arranged close

together All these are briefly presented in this Section More

detailed information is available in Lewis (2) and Hughes

(3), both published by SNAME, and Rawson (25) Note

that the content of Section 18.4 is influenced mainly from

Lewis (2)

18.4.1 Stress and Deflection Components

The 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, when

the ship bends as a beam under the longitudinal distribution

of load The associated primary stresses (σ1) are those, which

are usually called the longitudinal bending stresses, but the

general 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, doudou-ble sides, etc., for doudou-ble hull ships:

• Stresses in the plating of stiffened panel under lateral

pressure may have different origins (σ2and 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 duces stress (σ2) and deflection In addition, there is also

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

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

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 deflection

and associated stress of an individual unstiffened plate panel

included between 2 longitudinals and 2 transverse web

frames 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 plate

thickness Tertiary stresses, which result from the bending

of the plate member itself vary through the thickness, but

may 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) component

stresses or deflections, and each component may be

com-puted by methods and considerations entirely independent

of the other two The resultant stress, in such a case, is then

obtained by a simple superposition of the three component

stresses (Subsection 18.4.7) An exception is the case of

plate (tertiary) deflections, which are large compared to the

thickness of plate

In plating, each response induces longitudinal stresses,

transverse stresses and shear stresses This is due to the

Poisson’s Ratio Both primary and secondary stresses are

bending 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 shear

stresses 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 Components

Structural components are extensively discussed in ter 17 – Structure Arrangement Component Design In thissection, only the basic structural component used exten-

Chap-sively is presented It is basically a stiffened panel.

The global ship structure is usually referred to as being

a box girder or hull girder Modeling of this hull girder is

the first task of the designer It is usually done by 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)

model-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 longitudinal and 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 Response

18.4.3.1 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 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:

uti-• 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

not deform in their own planes, and merely rotate as the

beam 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 secondary

shear and axial stresses due to warping deformations are

neglected

Since stress concentrations (deck openings, side ports,

etc.) cannot be avoided in a highly complex structure such

as 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 are

linked to fatigue

The elastic linear bending equations, equations 27 and

28, are derived from basic mechanic principle presented at

E = modulus of elasticity of the material, in N/m2

I = moment of inertia of beam cross section about a

horizontal axis through its centroid, in m4

Hull Section Modulus: The plane section assumption

to-gether with elastic material behavior results in a nal stress,σ1, in the beam that varies linearly over the depth

longitudi-of the cross section

The simple beam theory for longitudinal strength 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:

cal-[29]

where:

M = bending moment (in N.m)

σ= bending stress (in N/m2)

m

pm

2exp

Figure 18.23 Types of Stiffening (Longitudinal and Transverse)

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

Moment (2)

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 of

a 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 axis

will seldom be located exactly at half-depth of the section;

hence two values of c and σwill 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 of

two or more materials of different moduli of elasticity, for

example, steel and aluminum In this case, the flexural

rigid-ity, EI, is replaced by ∫AE(z) z2dA, where A is cross

sec-tional area and E(z) the modulus of elasticity of an element

of area dA located at distance z from the neutral axis The

neutral axis is located at such height that ∫A E(z) z dA = 0

Calculation of Section Modulus: An important step in

routine ship design is the calculation of the midship section

modulus As defined in connection with equation 29, it

in-dicates the bending strength properties of the primary hull

structure The section modulus to the deck or bottom is

ob-tained by dividing the moment of inertia by the distance

from the neutral axis to the molded deck line at side or to

the base line, respectively

In general, the following items may be included in the

calculation 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 tension

and compression are assumed to act as part of the hull girder

Theoretically, a thorough analysis of longitudinal strength

would include the construction of a curve of section moduli

throughout the length of the ship as shown in Figure 18.25

Dividing the ordinates of the maximum bending-moments

curve (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 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

bend-Lateral Bending Combined with Vertical Bending: Up to

this point, attention has been focused principally upon the 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

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

mo-Figure 18.25 Moment of Inertia and Section Modulus (1)

where Mv, Iv, cv, and Mh, Ih, ch, correspond to the M, I, c

defined in equation 29, for the vertical bending and the

hor-izontal bending respectively

For a given vertical bending (Mv), the periodical wave

induced horizontal bending moment (Mh) increases stresses,

alternatively, on the upper starboard and lower portside, and

on the upper portside and lower starboard This explains

why these areas are usually reinforced

Empirical interaction formulas between vertical

bend-ing, horizontal bending and shear related to ultimate strength

of hull girder are given in Subsection 18.6.5.2

Transverse Stresses: With regards to the validity of the

Navier Equation (equation 29), a significant improvement

may be obtained by considering a longitudinal strength

member composed of thin plate with transverse framing

This might, for example, represent a portion of the deck

structure of a ship that is subject to a longitudinal stress σx,

from the primary bending of the hull girder As a result of

the longitudinal strain,εx, which is associated with σx, there

will exist a transverse strain,εs For the case of a plate that

is free of constraint in the transverse direction, the two

strains will be of opposite sign and the ratio of their

ab-solute values, given by | εs / εx | = ν, is a constant property

of the material The quantity νis called Poisson’s Ratio and,

for steel and aluminum, has a value of approximately 0.3

Hooke’s Law, which expresses the relation between stress

and strain in two dimensions, may be stated in terms of the

plate strains (equation 31) This shows that the primary

re-sponse induces both longitudinal (σx) and transversal

stresses (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 be

taken, in first approximation as:

Equation 32 is only valid to assess the additional stresses

in a given direction induced by the stresses in the

perpen-dicular direction computed, for instance, with the Navier

equation (equation 29)

18.4.3.2 Shear stress associated to shear forces

The simple beam theory expressions given in the

preced-ing section permit evaluation the longitudinal component

of the primary stress,σ In Figure 18.26, it can be seen that

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 σxand Ns = t σs stress resultants, in N/m

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

For vessels without continuous longitudinal bulkheads

Figure 18.26 Shear Forces (2)

(single cell), having transverse symmetry and subject to a

bending moment in the vertical plane, the shear flow

dis-tribution, N(s) is then given by:

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

= of area) about the neutral axis of the cross sectional

area 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 shown

in Figure 18.26

t(s) = thickness of material at the shear plane

I(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 of

the load curve up to that point Ordinarily the maximum

value of the shearing force occurs at about one quarter of

the vessel’s length from either end

Since only the vertical, or nearly vertical, members of

the hull girder are capable of resisting vertical shear, this

shear is taken almost entirely by the side shell, the

contin-uous longitudinal bulkheads if present, and by the webs of

any deep longitudinal girders

The maximum value of τoccurs in the vicinity of the

neutral 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 shear

stress is more complex

Shear Flow in Multicell Sections: If the cross section of

the ship shown in Figure 18.28 is subdivided into two or

more closed cells by longitudinal bulkheads, tank tops, or

decks, 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 bottom

of the center tank space since the plane of symmetry at

which the shear flow vanishes, lies within this space and

forms a convenient origin for the integration At the

deck/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

This additional information may be obtained by sidering the torsional equilibrium and deflection of the cel-lular section The way to proceed is extensively explained

con-in Lewis (2)

18.4.3.3 Shear stress associated with torsion

In 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 and

is 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)

Now consider equilibrium of forces in the x-direction for

the 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 static

equilibrium The shear flow, N = tτ, is therefore seen to be

constant around the section

The magnitude of the moment, M T , may be computed

by integrating the moment of the elementary force arising

from this shear flow about any convenient axis If r is the

distance from the axis, 0, perpendicular to the resultant shear

flow at location s:

[36]

Here the symbol indicates that the integral is taken

en-tirely around the section and, therefore,Ω(m2) is the area

enclosed by the mid-thickness line of the tubular cross

sec-tion The constant shear flow, N (N/m), is then related to

the applied twisting moment by:

For uniform torsion of a closed prismatic section, the

angle of torsion is:

where:

MT= Twisting moment (torsion), in N.m

L = Length of the girder, in m

.

MT =∫r N ds=N r ds∫ =2NΩ

18.4.3.4 Twisting and warping

Torsional strength: Although torsion is not usually an

im-portant factor in ship design for most ships, it does result

in 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 the

open 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 must

be calculated

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

contain-From previous studies, it has been established that 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

spe-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 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 inability

sec-of 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)

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 be

approximated 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 Figure

18.30 is constrained, another component of torsional

re-sistance is developed through the shear stresses that result

from this warping restraint This is added to the torque given

2 the closed ends of the ship,

3 double wall transverse bulkheads, and

4 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 snaking

Racking is the result of a transverse hull shape distortion and

is caused by either dynamic loads due to rolling of the ship

or by the transverse impact of seas against the topsides

Trans-verse bulkheads resist racking if the bulkhead spacing is close

enough to prevent deflection of the shell or deck plating in

its own plane Racking introduces primarily compressive and

shearing 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 the

deck 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 in

car-carriers is discussed in Chapters 17 and 34

Racking stresses due to rolling reach a maximum in a

beam sea each time the vessel completes an oscillation in

one direction and is about to return

1

3

3 1

snaking is sometimes used in referring to this behavior and

relates to both twisting and racking

18.4.3.6 Effective breadth and shear lag

An 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 from

the shear-loaded edge, and this is called shear lag This is

in 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 referred

to by the term shear lag must be taken into consideration

if an accurate estimate of the maximum stress in the ber is to be made This may be conveniently done by defin-

mem-ing an effective breadth of the flange member.

The ratio, be/b, of the effective breadth, b e, 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)

of the external loading applied and the boundary conditions

along the plate edges, but not its thickness Figure 18.33

gives the effective breadth ratio at mid-length for column

loading and harmonic-shaped beam loading, together with

a 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 in

Schade (26)

A real situation in which such an alternating load

dis-tribution may be encountered is a bulk carrier loaded with

a dense ore cargo in alternate holds, the remainder being

empty

An example of the computation of the effective breadth

of bottom and deck plating for such a vessel is given in

Chapter VI of Taggart (1), using Figure 18.33

It is important to distinguish the effective breadth

(equa-tion 40) and the effective width (equa(equa-tions 54 and 55)

pre-sented later in Subsection 18.6.3.2 for plate and stiffened

plate-buckling analysis

18.4.3.7 Longitudinal deflection

The 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-empirical

approximation for bending deflection amidships is:

bb

k Lb

e =6

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 on

a 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)

they are transferred to the longitudinal bulkheads or side

shell, again as localized shear forces Thus, in reality, the

loading q(x), applied to the side shell or the longitudinal

bulkhead will consist of a distributed part due to the direct

transfer of load into the member from the bottom or deck

structure, plus a concentrated part at each bulkhead or web

frame This leads to a discontinuity in the shear curve at the

bulkheads and webs

18.4.3.9 Hull/superstructure interaction

The terms superstructure and deckhouse refer to a structure

usually of shorter length than the entire ship and erected

above the strength deck of the ship If its sides are coplanar

with the ship’s sides it is referred to as a superstructure If

its 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 hull

has facets involving both the general bending response and

important localized effects Two opposing schools of thought

exist 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 other

purposely isolates the superstructure from the hull so that

it carries only localized loads and does not experience

stresses and deflections associated with bending of the main

hull This may be accomplished in long superstructures

(>0.5Lpp) by cutting the deckhouse into short segments by

means of expansion joints Aluminum deckhouse

con-struction is another alternative when the different material

properties provide the required relief

As the ship hull experiences a bending deflection in

re-sponse to the wave bending moment, the superstructure is

forced to bend also However, the curvature of the

super-structure may not necessarily be equal to that of the hull but

depends upon the length of superstructure in relation to the

hull and the nature of the connection between the two,

es-pecially upon the vertical stiffness or foundation modulus

of the deck upon which the superstructure is constructed

The behavior of the superstructure is similar to that of a

beam 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 deckhouse

and hull for three different degrees of superstructure

effec-tiveness are shown on Figure 18.34

The areas and inertias can be computed to account for

shear 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 Afand inertia Ifmust be reduced

according to the ratio of the respective material moduli; that

is, 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 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)

concentra-18.4.4 Secondary Response

In 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 axisPassenger deck

x

z

I M

z ) =

(

σ

) ( )

r θ σ

and normal loading, deflection and stress over the length

and width dimensions of a stiffened panel Remember that

the primary response involves the determination of only the

in-plane load, deflection, and stress as they vary over the

length of the ship The secondary response, therefore, is

seen to be a two-dimensional problem while the primary

response is essentially one-dimensional in character

18.4.4.1 Stiffened panels

A stiffened panel of structure, as used in the present

con-text, usually consists of a flat plate surface with its attached

stiffeners, transverse frames and/or girders (Figure 18.36)

When the plating is absent the module is a grid or grillage

of beam members only, rather than a stiffened panel.

In principle, the solution for the deflection and stress in

the stiffened panel may be thought of as a solution for the

response 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 thought

of as forming a part of the flanges of the stiffeners The plate

contribution to the beam bending stiffness arises from the

direct longitudinal stress in the plate adjacent to the

stiff-ener, modified by the transverse stress effects, and also from

the 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, since

these 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 for

obtaining the structural behavior of stiffened plate panels

under normal loading, each embodying certain simplifying

assumptions: 1) orthotropic plate theory, 2)

beam-on-elastic-foundation theory, 3) grillage theory (intersecting beams), and 4) the finite element method (FEM).

Orthotropic plate theory refers to the theory of bending

of plates having different flexural rigidities in the two 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 to

or-be 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 directionsw(x,y) = is the deflection of the plate in the normal di-

rectionp(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 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

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

stiff-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

4

4 4

Figure 18.36 A Stiffened Panel with Uniformly Distributed Longitudinals, 4

Webframes, and 3 Girders.

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 and

spacing, and closely spaced but also for individual

mem-bers, randomly distributed such as deck and bottom

gird-ers These members considered through Heaviside functions

that allow replacing each individual member by a set of 3

forces and 2 bending moment load lines Figure 18.36 shows

a typical stiffened panel that can be considered It includes

uniformly distributed longitudinals and web frames, and

three prompt elements (girders)

The beam on elastic foundation solution is suitable for a

panel in which the stiffeners are uniform and closely spaced

in one direction and sparser in the other one Each of these

members is treated individually as a beam on an elastic

foun-dation, for which the differential equation of deflection is,

[43]

where:

w = is the deflection

I = is sectional moment of inertia of the longitudinal

stiffener, including adjacent plating

k = is average spring constant per unit length of the

transverse stiffeners

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

The grillage approach models the cross-stiffened panel

as 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 panel

and the Poisson ratio effect are neglected The validity of

modeling 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 known

that the grillage approach may be suitable when the ratio

of the stiffener flexural rigidity to the plate bending

rigid-ity (EI/bD with I the moment of inertia of stiffener and D

the plate bending rigidity) is greater than 60 (31) otherwise

if the bending rigidity of stiffener is smaller, an Orthotropic

Plate Theory has to be selected.

The FEM approach is discussed in detail in section 18.7.2.

18.4.5 Tertiary Response

18.4.5.1 Unstiffened plate

Tertiary response refers to the bending stresses and

deflec-tions in the individual panels of plating that are bounded by

the stiffeners of a secondary panel In most cases the load

that induces this response is a fluid pressure from either the

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:

Information (including charts) on a plate subject to

uni-form load and concentrated load (patch load) is available

in 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 should

be 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 Strength

Transverse strength refers to the ability of the ship ture to resist those loads that tend to cause distortion of thecross section When it is distorted into a parallelogram shape

struc-the effect is called racking We recall that both struc-the primary

bending and torsional strength analyses are based upon theassumption of no distortion of the cross section Thus, we

E t312(1− ν)

4

2 2

4 4

2wx

w

wy

pD(x,y)

see that there is an inherent relationship between transverse

strength and both longitudinal and torsional strength

Cer-tain structural members, including transverse bulkheads and

deep web frames, must be incorporated into the ship in order

to insure adequate transverse strength These members

pro-vide support to and interact with longitudinal members by

transferring loads from one part of a structure to another

For example, a portion of the bottom pressure loading on

the 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 as

vertical shears into the side shell Thus some of the loads

acting on the transverse strength members are also the loads

of concern in longitudinal strength considerations

The general subject of transverse strength includes

ele-ments taken from both the primary and secondary strength

categories The loads that cause effects requiring transverse

strength 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 transverse

strength are:

• ship out of water: on building ways or on construction

or repair dry dock,

• tankers having empty wing tanks and full centerline tanks

or vice versa,

• ore carriers having loaded centerline holds and large

empty wing tanks,

• all types of ships: torsional and racking effects caused

by asymmetric motions of roll, sway and yaw, and

• ships with structural features having particular

sensitiv-ity to transverse effects, as for instance, ships having

largely open interior structure (minimum transverse

bulk-heads) such as auto carriers, containers and RO-RO ships

As previously noted, the transverse structural response

involves pronounced interaction between transverse and

longitudinal structural members The principal loading

con-sists of the water pressure distribution around the ship, and

the weights and inertias of the structure and hold contents

As a first approximation, the transverse response of such a

frame may be analyzed by a two-dimensional frame

re-sponse procedure that may or may not allow for support by

longitudinal structure Such analysis can be easily performed

using 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 to

equation 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 or

at least a limited hold-model should be modeled See section 18.7.2—Structural Finite Element Models (Figure18.57)

Sub-18.4.7 Superposition of Stresses

In plating, each response induces longitudinal stresses, 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),

trans-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 eral load can be considered in the buckling formulation(Subsection 18.6.3) Tertiary stresses should be added forfatigue analysis

lat-Since all the methods of calculation of primary, 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

sec-First, the loads and theoretical procedures used in puting the stress components may not be of the same ac-curacy or reliability The primary loading, for example, may

com-be obtained using a theory that involves certain

simplifica-tions in the hydrodynamics of ship and wave motion, and

the primary bending stress may be computed by simple

beam theory, which gives a reasonably good estimate of the

mean stress in deck or bottom but neglects certain localized

effects 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 moves

through waves The maximum bending moment amidships,

which results in the maximum primary stress, does not

nec-essarily occur in phase with the maximum local pressure

on a midship panel of bottom structure (secondary stress)

or panel of plating (tertiary stress)

Third, the maximum values of primary, secondary, and

tertiary stress are not necessarily in the same direction or

even in the same part of the structure In order to visualize

this, consider a panel of bottom structure with longitudinal

framing The forward and after boundaries of the panel will

be at transverse bulkheads The primary stress (σ1) will act

in the longitudinal direction, as given by equation 29 It will

be nearly equal in the plating and the stiffeners, and will be

approximately 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 by

equation 35 The secondary stress will probably be greater

in the free flanges of the stiffeners than in the plating, since

the combined neutral axis of the stiffener/plate

combina-tion is usually near the plate-stiffener joint Secondary

stresses, 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 panel

of structure bounded by transverse bulkheads and either the

side shell or the longitudinal bulkheads The principal

stiff-eners, in this case, are the center and any side longitudinal

girders, and the transverse web frames The second part,

(σ2), is the stress resulting from the bending of the smaller

panel of plating plus longitudinal stiffeners that is bounded

by the deep web frames The first of these components (σ2),

as a result of the proportions of the panels of structure, is

usually 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 the

plate where biaxial stresses occur In the case of

longitudi-nal stiffeners, the maximum panel tertiary stress will act in

the transverse direction (normal to the framing system) at

the mid-length of a long side

In certain cases, there will be an appreciable shear stress

component 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 many

cases, the point in the structure having the highest stress level

will not always be immediately obvious, but must be found

by considering the combined stress effects at a number ofdifferent locations and times

The nominal stresses produced from the analysis will be

a 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 it

is 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 the

von Mises Theory:

[45]Consider a plane stress field in which the componentstresses are σx,σyand τ The distortion energy states that

σe =(σx2 +σ2y −σ σx y +3τ2)1

Figure 18.37 Definition of Stress Components (4)

failure through yielding will occur if the equivalent von

Mises stress,σe, given by equation 45 exceeds the

equiva-lent stress,σο, corresponding to yielding of the material test

specimen The material yield strength may also be expressed

through 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 the

inclined position and to other forces, such as those, which

induce torsion or side bending in the hull girder, not to

men-tion the dynamic effects resulting from the momen-tions of the

ship itself Heretofore it has been difficult to arrive at the

minimum 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 stress

includes an adequate factor of safety, or margin, for these

uncertain loading factors

In practice, the margin against yield failure of the

struc-ture is obtained by a comparison of the strucstruc-ture’s von Mises

equivalent stress,σe, against the permissible stress (or

al-lowable stress),σ0, giving the result:

where:

s1= partial safety factor defined by classification societies,

which depends on the loading conditions and method

of analysis For 20 years North Atlantic conditions

(seagoing condition), the s 1factor is usually taken

be-tween 0.85 and 0.95

σy= minimum yield point of the considered steel (mild

steel, high tensile steel, etc.)

For special ship types, different permissible stresses may

be 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, which

in 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 (see

Subsection 18.6.6)

18.5 LIMIT STATES AND FAILURE MODES

Avoidance of structural failure is the goal of all structural

designers, and to achieve this goal it is necessary for the

de-signer to be aware of the potential limit states, failure modes

and methods of predicting their occurrence This section

presents 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 2

major categories: the service limit state and the ultimate limit 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, and

4 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 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)

ul-Fatigue can be either considered as a third limit state or,

classically, considered as a service limit state Even if it is

also a matter of discussion, yielding should be considered

as a service limit state First yield is sometimes used to sess the ultimate state, for instance for the ultimate hullbending moment, but basically, collapse occurs later Most

as-of the time, vibration relates to service limit states.

In practice, it is important to differentiate service, mate, fatigue and accidental limit states because the partial

ulti-safety factors associated with these limit states are ally different

gener-18.5.1 Basic Types of Failure Modes

Ship 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), and

3 fracture that includes ductile tensile rupture, low-cyclefatigue and brittle fracture

Yield occurs when the stress in a structural member

ex-ceeds a level that results in a permanent plastic

deforma-tion of the material of which the member is constructed This

stress level is termed the material yield stress At a

some-what higher stress, termed the ultimate stress, fracture of

the material occurs While many structural design criteria

are based upon the prevention of any yield whatsoever, it

should be observed that localized yield in some portions of

a structure is acceptable Yield must be considered as a

serv-iceability limit state

Instability and buckling failure of a structural member

loaded in compression may occur at a stress level that is

sub-stantially lower than the material yield stress The load at

which 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 common

example of an instability failure is the buckling of a simple

column under a compressive load that equals or exceeds

the Euler Critical Load A plate in compression also will

have a critical buckling load whose value depends on the

plate thickness, lateral dimensions, edge support conditions

and material elasticity modulus In contrast to the column,

however, exceeding this load by a small margin will not

necessarily result in complete collapse of the plate but only

in an elastic deflection of the central portion of the plate away

from its initial plane After removal of the load, the plate

may return to its original un-deformed configuration (for

elastic buckling) The ultimate load that may be carried by

a buckled plate is determined by the onset of yielding at some

point in the plate material or in the stiffeners, in the case of

a stiffened panel Once begun, yield may propagate rapidly

throughout the entire plate or stiffened panel with further

increase in load

Fatigue failure occurs as a result of a cumulative effect

in 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 small

but irreversible damage within the material and, after the

accumulation of enough such damage, the ability of the

member to withstand loading is reduced below the level of

the applied load Two categories of fatigue damage are

gen-erally recognized and they are termed high-cycle and

low-cycle fatigue In high-low-cycle fatigue, failure is initiated in

the form of small cracks, which grow slowly and which

may often be detected and repaired before the structure is

endangered High-cycle fatigue involves several millions

of cycles of relatively low stress (less than yield) and is

typ-ically encountered in machine parts rotating at high speed

or in structural components exposed to severe and prolonged

vibration Low-cycle fatigue involves higher stress levels,

up to and beyond yield, which may result in cracks being

initiated 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 begin

to grow and travel almost explosively through a major

por-tion of the structure The term brittle fracture refers to the

fact 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 role

in 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 Subsection 18.6.7 and Vibration in Subsection 18.6.8 Vibration as well

as noise is not a failure mode, while it could fall into theserviceability limit state

18.6 ASSESSMENT OF THE STRUCTURAL CAPACITY

18.6.1 Failure Modes Classification

The types of failure that may occur in ship structures aregenerally those that are characteristic of structures made up

of stiffened panels assembled through welding Figure 18.38

presents the different structure levels: the global structure, usually a cargo hold (Level 1), the orthotropic stiffened panel 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 the unstiffened plate between two longitudinals and two trans-

verse frames (also called bare plate)

The word grillage should be reserve to a structure

com-posed of a grid of beams (without attached plating) When

the grid is fixed on a plate, orthotropic stiffened panel seems

to 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 and

structure levels can be summarized as follows:

• Level 1: Ultimate bending moment, M u, of the global

structure (Figure 18.46)

• Level 2: Ultimate strength of compressed orthotropic

stiffened 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 Yielding

Mode III: Pultof interframe panels with a plate-stif

ener combination (Figure 18.44b) using a

beam-col-umn model (Level 3b) or an orthotropic model (Level

3), 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 (bare

plate, Figure 18.44a)

To avoid collapse related to the Mode I, a minimal

rigid-ity is generally imposed for the transverse frames so that an

interframe panel collapse (Mode III) always occurs prior to overall buckling (Mode I) It is a simple and easy constraint

to implement, thus avoiding any complex calculation of

overall buckling (mode I).

Note that the failure Mode III is influenced by the

buck-ling of the bare plate (elementary unstiffened plate) tic buckling of theses unstiffened plates is usually not

Elas-considered as an ultimate limit state (failure mode), but rather 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 ity or ultimate limit states can be classified as follows:

serviceabil-18.6.1.1 Stiffened panel failure modes

Service limit state

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

thick-• Maximum allowable stresses against first yield section 18.4.7)

(Sub-• 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 gitudinals and two transverse stiffeners, frames or bulk-heads (Subsection 18.6.3),

lon-• Local elastic buckling of longitudinal stiffeners (weband flange) Often the stiffener web/flange buckling doesnot induce immediate collapse of the stiffened panel as

tripping does It could therefore be considered as a iceability ultimate limit state However, this failure mode could also be classified into the ultimate limit state since

serv-the 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

• Collapse of interframe longitudinally stiffened panel,

including torsional-flexural (lateral-torsional) buckling

of stiffeners (also called tripping)

18.6.1.2 Frame failure modes

Service 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 considered

as ultimate limit states rather than a service limit state

If one of them appears, the assumption of rigid supports

is no longer valid and the entire stiffened panel can reach

the ultimate limit state

— Buckling of the compressed members,

— Local buckling (web, flange)

18.6.1.3 Hull Girder Collapse modes

Service 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 or

only the part between two transverse web frames

(Sub-section 18.6.5) Collapse of frames is assumed to only

appear after the collapse of panels located between these

frames This means that it is sufficient to verify the box

girder 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 be

un-conservative if the frames are not stiff enough

• Collision and grounding (Subsection 18.6.7), which is

in 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 Yielding

As explained in Subsection 18.5.1 yield occurs when the

stress 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, tion 29, up to complex 3D nonlinear FE analysis Designcriteria related to first yield is the von Mises equivalentstress (equation 45)

equa-Yielding is discussed in detail in Section 18.4

18.6.3 Buckling and Ultimate Strength of Plates

A ship stiffened plate structure can become unstable if 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 strength

ei-of the plate

The buckling and ultimate strength of the structure 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

de-18.6.3.1 Direct Analysis

In 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 proportioned

so that local instability is prevented prior to the failure ofplating