Analysis and design of ship structure

18.1 NOMENCLATURE
For specific symbols, refer to the definitions contained in
the various sections.
ABS American Bureau of Shipping
BEM Boundary Element Method
BV Bureau Veritas
DNV Det Norske Veritas
FEA Finite Element Analysis
FEM Finite Element Method
IACS International Association of Classifica-
tion Societies
ISSC International Ship & Offshore Structures
Congress
ISOPE International Offshore and Polar Engi-
neering Conference
ISUM Idealized Structural Unit method
NKK Nippon Kaiji Kyokai
PRADS Practical Design of Ships and Mobile
Units,
RINA Registro Italiano Navale
SNAME Society of naval Architects and marine
Engineers
SSC Ship Structure Committee.
a acceleration
A area
B breadth of the ship
C wave coefficient (Table 18.I)
CB hull block coefficient
D depth of the ship
g gravity acceleration
m(x) longitudinal distribution of mass
I(x) geometric moment of inertia (beam sec-
tion x)
L length of the ship
M(x) bending moment at section x of a beam
MT(x) torque moment at section x of a beam
p pressure
q(x) resultant of sectional force acting on a
beam
T draft of the ship
V(x) shear at section x of a beam
s,w (low case) still water, wave induced component
v,h (low case) vertical, horizontal component
w(x) longitudinal distribution of weight
θ roll angle
ρ density
ω angular frequency
18.2 INTRODUCTION
The purpose of this chapter is to present the fundamentals
of direct ship structure analysis based on mechanics and
strength of materials. Such analysis allows a rationally based
design that is practical, efficient, and versatile, and that has
already 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 for
the other even if there are some important differences be-
tween performing a design and completing an analysis.
18-1
Chapter 18
Analysis and Design of Ship Structure
Philippe Rigo and Enrico Rizzuto
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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 design. 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(basicshipresponses),thelimitstates,andthefail-
ures modes and associated structural capacity. A review of
theavailableNumericalAnalysisforStructuralDesign isper-
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
moreextensivelyinSubsection18.7.2forfiniteelementanaly-
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 the
strength criteria, but there is only one that is the optimum
solution (least cost, weight, etc.).
Ship structural analysis and design is a matter of com-
promises:
• compromise between accuracy and the available time to
perform the design. This is particularly challenging at
the preliminary design stage. A 3D Finite Element
Method (FEM) analysis would be welcome but the time
is not available. For that reason, rule-based design or
simplified numerical analysis has to be performed.
• to limit uncertainty and reduce conservatism in design, it
is important that the design methods are accurate. On the
other hand, simplicity is necessary to make repeated de-
sign analyses efficient. The results from complex analy-
sesshouldbeverifiedbysimplifiedmethodstoavoiderrors
and misinterpretation of results (checks and balances).
• compromise between weight and cost or compromise
between least construction cost, and global owner live
cycle cost (including operational cost, maintenance, etc.),
and
• builder optimum design may be different from the owner
optimum design.
18.2.1 Rationally Based Structural Design versus
Rules-Based Design
There are basically two schools to perform analysis and de-
sign of ship structure. The first one, the oldest, is called
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 empir-
ical, based on accumulated experience and ship perform-
ance, and expressed in the form of structural design codes
or rules published by the various ship classification soci-
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.
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:
18-2 Ship Design & Construction, Volume 1
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There are several disadvantages to a completely “rulebook”
approach to design. First, the modes of structural 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-adequacy. 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 societies
include, usually as an alternative, a direct analysis procedure
(numerical packages based on the finite element method,
seeTable 18.VIII, Subsection 18.7.5.2). In addition, for new
vessel 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 merging
procedure. Classification societies are now encouraging and
contributing greatly to the development of direct analysis
and 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 design
and construction is being made daily, at an unprecedented
pace. 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 to
analyze complex ship structures in a practical manner using
structural analysis techniques centering on FEM analysis.
The majority of ship designers strive to develop rational and
optimal designs based on direct strength analysis methods
using the latest technologies in order to realize the
shipowner’s requirements in the best possible way.
When carrying out direct strength analysis in order to
verify the equivalence of structural strength with rule re-
quirements, it is necessary for the classification society to
clarify the strength that a hull structure should have with
respect to each of the various steps taken in the analysis
process, 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 careful
consideration to more rational and accurate methods of di-
rect strength analysis.
Based on recognition of this need, extensive research
has 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 reports
regarding 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 requires
that all design decisions (objectives, criteria, priorities, con-
straints…) must be made before the design starts. This is a
major difficulty of this approach.
18.2.2 Modeling and Analysis
General guidance on the modeling necessary for the struc-
tural analysis is that the structural model shall provide re-
sults suitable for performing buckling, yield, fatigue and
Chapter 18: Analysis and Design of Ship Structure 18-3
Figure 18.1 Direct Structural Analysis Flow Chart
Direct Load Analysis
Design Load
Study on Ocean Waves
Effect on
operationWave Load Response
Response function
of wave load
Structural analysis by
whole ship model
Stress response
function
Short term
estimation
Long term
estimation
Design
Sea State
Design wave Wave impact load
Structural response analysis
Strength Assessment
Yield
strength
Nonlinear influence
in large waves
Investigation on
corrosion
Buckling
strength
Ultimate
strength
Fatigue
strength
Modeling technique Direct structural
analysis
Stress Response
in Waves
Long term
estimation
Short term
estimation
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vibration assessment of the relevant parts of the vessel. 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 sub
models 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 De-
sign about which Hughes (3) states:
The preliminary determines the location, spacing, and scant-
lings of the principal structural members. The detailed de-
signdeterminesthegeometryandscantlingsoflocalstructure
(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
is equally important for obtaining an efficient, safe and re-
liable ship.
During the detailed design there also are many bene-
fits to be gained by applying modern methods of engi-
neering science, but the applications are different 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.
18.3 LOADS
Loads acting on a ship structure are quite varied and pecu-
liar, in comparison to those of static structures and also of
other vehicles. In the following an attempt will be made to
review the main typologies of loads: physical origins, gen-
eral interpretation schemes, available quantification proce-
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
Verification
of response
Verification
of model/
loads
Yes
No
Transfer of
displacements/forces
to sub-model?
Verification
of load
transfer
Structural drawings,
mass description and
loading conditions.
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dures and practical methods for their evaluation will be sum-
marized.
18.3.1 Classification of 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 struc-
tural models (stiffened panels, single beams, plate panels),
generally are termed local loads.
The distinction is purely formal, as the same external
forces can in fact be interpreted as global or local loads. For
instance, wave dynamic actions on a portion of the hull, if
described in terms of a bi-dimensional distribution of pres-
sures over the wet surface, represent a local load for the hull
panel, while, if integrated over the same surface, represent
a contribution to the bending moment acting on the hull
girder.
This terminology is typical of simplified structural analy-
ses, in which responses of the two classes of components
are evaluated separately and later summed up to provide
the total stress in selected positions of the structure.
In a complete 3D model of the whole ship, forces on the
structure are applied directly in their actual position and the
result is a total stress distribution, which does not need to
be decomposed.
18.3.1.3 Characteristic values for loads
Structural verifications are always based on a limit state
equation and on a design operational time.
Main aspects of reliability-based structural design and
analysis are (see Chapter 19):
• the state of the structure is identified by state variables
associated to loads and structural capacity,
• state variables are stochastically distributed as a func-
tion of time, and
• the probability of exceeding the limit state surface in the
design time (probability of crisis) is the element subject
to evaluation.
The situation to be considered is in principle the worst
combination of state variables that occurs within the design
time. The probability that such situation corresponds to an
out 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 a
time-invariant one. This is done by taking into account in
the analysis the worst situations as regards loads, and, sep-
arately, as regards capacity (reduced because of corrosion
and other degradation effects). The simplification lies in
considering these two situations as contemporary, which in
general is not the case.
When dealing with strength analysis, the worst load sit-
uation corresponds to the highest load cycle and is charac-
terized through the probability associated to the extreme
value in the reference (design) time.
In fatigue phenomena, in principle all stress cycles con-
tribute (to a different extent, depending on the range) to
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damage accumulation. The analysis, therefore, does not re-
gard the magnitude of a single extreme load application, 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 108
cycles, 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 (con-
tained in the plane of symmetry), indicated as vertical re-
sultant force qV; another force in the normal direction, (local
horizontal axis), termed horizontal resultant force qH and a
moment mT about the x axis. All these actions are distrib-
uted along the longitudinal axis x.
Five main load components are accordingly generated
along the beam, related to sectional forces and moment
through equation 1 to 5:
[1]
[2]
[3]
[4]
[5]
Due to total equilibrium, for a beam in free-free condi-
tions (no constraints at ends) all load characteristics have
zero values at ends (equations 6).
These conditions impose constraints on the distributions
of qV, qH and mT.
[6]
Global loads for the verification of the hull girder are ob-
tained with a linear superimposition of still water and wave-
induced global loads.
They are used, with different characteristic values, in
different 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, usu-
ally with the plane of symmetry normal to the still water
surface. In this condition, only a symmetric distribution of
hydrostatic pressure acts on each section, together with ver-
tical gravitational forces.
If the latter ones are not symmetric, a sectional torque
mTg(x) is generated (Figure 18.4), in addition to the verti-
V (0) V (L) M (0) M (L) 0
V (0) V (L) M (0) M (L) 0
M (0) M (L) 0
V V V V
H H H H
T T
= = = =
= = = =
= =
M (x) m ( ) dT T
0
x
= ∫ ξ ξ
M (x) V ( ) dH H
0
x
= ∫ ξ ξ
V (x) q ) dH H
0
x
= ∫ (ξ ξ
M (x) V ( ) dV V
0
x
= ∫ ξ ξ
V (x) q ( ) dV V
0
x
= ∫ ξ ξ
Figure 18.3 Sectional Forces and Moment
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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.
q (x) b(x) w(x) gA (x) m(x)gSV I= − = −
At an even earlier stage of design, parametric formula-
tions can be used to derive directly reference values for still
water hull girder loads.
Common reference values for still water bending mo-
ment at mid-ship are provided by the major Classification
Societies (equation 8).
[8]
where C = wave parameter (Table 18.I).
The formulations in equation 8 are sometimes explicitly
reported in Rules, but they can anyway be indirectly de-
rived from prescriptions contained in (6, 7). The first re-
quirement (6) regards the minimum longitudinal strength
modulus and provides implicitly a value for the total bend-
ing moment; the second one (7), regards the wave induced
component 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 loads
Classification Societies require in general a direct analysis
of these types of load in the main loading conditions of the
ship, such as homogenous loading condition at maximum
draft, ballast conditions, docking conditions afloat, plus all
other conditions that are relevant to the specific ship (non-
homogeneous loading at maximum draft, light load at less
than maximum draft, short voyage or harbor condition, bal-
last exchange at sea, etc.).
The direct evaluation procedure requires, for a given
loading 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 is
subdivided into portions: for each of them, the total weight
andcenterofgravityisdeterminedsummingupcontributions
from all items present on board between the two bounding
sections. The distribution for w(x) is then usually approxi-
mated by a linear (trapezoidal) curve obtained by imposing
M N m
C L B 122.5 15 C (hogging)
C L B 45.5 65 C (sagging)s
2
B
2
B
⋅
−( )
+
[ ] =
( )
Figure 18.4 Sectional Resultant Forces in Still Water
Figure 18.5 Examples of Reference Still Water Bending Moment Distribution
(10). (a) oil tankers, bulk carriers, ore carriers, and (b) other ship types
TABLE 18.I Wave Coefficient Versus Length
Ship Length L Wave Coefficient C
90 ≤ L <300 m 10.75 – [(300 – L)/100]3/2
300 ≤ L <350 m 10.75
350 ≤ L 10.75 – [(300 – L)/150]3/2
(a)
(b)
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the correspondence of area and barycenter of the 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, machin-
ery, outfitting) but also the distribution of the various com-
ponents of the deadweight (cargo, ballast, consumables).
Ship types like bulk carriers are more exposed to uncer-
tainties on the actual distribution of cargo weight than, for
example, container ships, where actual weights of single
containers are kept under close control during operation.
In addition, model uncertainties arise from neglecting the
longitudinal components of the hydrostatic pressure (Fig-
ure 18.7), which generate an axial compressive force on the
hull girder.
As the resultant of such components is generally below
the neutral axis of the hull girder, it leads also to an addi-
tional hogging moment, which can reach up to 10% of the
total 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 sagging
moments.
All these compression and bending effects are neglected
in the hull beam model, which accounts only for forces and
moments acting in the transverse plane. This represents a
source of uncertainties.
Another approximation is represented by the fact that
buoyancy and weight are assumed in a direction normal to
the horizontal longitudinal axis, while they are actually ori-
ented along the true vertical.
Thisimpliesneglectingthestatictrimangleandtoconsider
an approximate equilibrium position, which often creates the
need for a few iterative corrections to the load curve qsv(x) in
order 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 to
conventional 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 for
shear, bending and torque; and L, T apply respectively to
longitudinal and transversal beams).
18.3.4 Wave Induced Global Loads
The prediction of the behaviour of the ship in waves repre-
sents a key point in the quantification of both global and
local loads acting on the ship. The solution of the seakeep-
ing problem yields the loads directly generated by external
pressures, 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 other
types of loads, like slamming and sloshing.
Figure 18.6 Weight Distribution Breakdown for Full Load Condition
Figure 18.7 Longitudinal Component of Pressure
Figure 18.8 Multi-hull Additional Still Water Loads (sketch)
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In particular, as regards global effects, the action of 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 (bWV and 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 classifi-
q (x) b (x) m(x)a (x)
q (x) b (x) m(x)a (x)
m (x) m (x) I (x)
WV WV V
WH WH H
TW Tb R
= −
= −
= − θ
cation societies provide a statistically based reference values
for the vertical component of wave-induced bending moment
MWV, expressed as a function of main ship dimensions.
Such reference values for the midlength section of a ship
with unrestricted navigation are yielded by equation 10 for
hog and sag cases (7) and corresponds to an extreme value
with a return period of about 20 years or an exceeding prob-
ability of about 10–8
(once in the ship lifetime).
[10]
HorizontalWave-induced Bending Moment: Similar for-
mulations are available for reference values of horizontal
wave induced bending moment, even though they are not
as uniform among different Societies as for the main verti-
cal component.
In Table 18.II, examples are reported of reference val-
ues of horizontal bending moment at mid-length for ships
with unrestricted navigation. Simplified curves for the dis-
tribution in the longitudinal direction are also provided.
Wave-induced Torque: A few reference formulations are
given also for reference wave torque at midship (see ex-
amples in Table 18.III) and for the inherent longitudinal
distributions.
18.3.4.2 Static Wave analysis of global wave loads
A traditional analysis adopted in the past for evaluation of
wave-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 weight
and 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. By
definition, 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
M N m
C L B C
C L B C .
(hog)
(sag)WV
B
B
⋅[ ] =
− +( )
190
110 0 7
2
2
Figure 18.9 Sectional Forces and Moments in Waves
TABLE 18.II Reference Horizontal Bending Moments
Class Society MWH [N ⋅ m]
ABS (8) 180 C1L2
DCB
BV (9) RINA (10) 1600 L2.1
TCB
DNV (11) 220 L9/4
(T + 0.3B)CB
NKK (12) 320 L2C T L L− 35 /
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pressures to extreme wave loads, derived, for example, 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
∇2
Φ = 0 [11]
where Φ = velocity potential
• 2-dimensional solution of the problem
• linearized boundary conditions: the quadratic compo-
nent of velocity in the Bernoulli Equation is reformu-
lated in linear terms to express boundary conditions:
— on free surface: considered as a plane 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 (equa-
tion 12).
Φ = Φs + ΦFK + Φd + Φr [12]
where:
Φs = stationary component due to ship advancing in calm
water
Φr = radiation component due to the ship motions in calm
water
ΦFK = excitation component, due to the incident wave
(undisturbed by the presence of the ship): Froude-
Krylov
Φd = diffraction component, due to disturbance in the wave
potential generated by the hull
This subdivision also enables the de-coupling of the ex-
citation components from the response ones, thus avoiding
a non-linear feedback between the two.
Other key properties of linear systems that are used in
the analysis are:
• linear relation between the input and output amplitudes,
and
• superposition of effects (sum of inputs corresponds to
sum 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 spectral
representation 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 the
ship course).
The output consists of response spectra of point pres-
sures on the hull and of the other derived responses, such
as global loads and ship motions. Output spectra can be
used to derive short and long-term predictions for the prob-
ability distributions of the responses and of their extreme
values (see Subsection 18.3.4.5).
Despite the numerous and demanding simplifications at
the basis of the procedure, strip theory methods, developed
since the early 60s, have been validated over time in sev-
eral contexts and are extensively used for predictions of
wave loads.
In principle, the base assumptions of the method are
TABLE 18.III Examples of Reference Values for Wave Torque
Class Society Qw [N . m] (at mid-ship)
ABS (bulk carrier)
(e = vertical position of shear center)
BV RINA 190 8 13
250 0 7
125
2 2
3
LB C .
. L
W −
−









2700 0 5 0 1 0 13
0 142 2
0 5
LB T C . . .
e
D
.
TW
.
−( ) +[ ] − 









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valid only for small wave excitations, small motion re-
sponses and low speed of the ship.
In practice, the field of successful applications 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 mn of order n (equation 13).
[13]
This information is the basis of the spectral method,
whose theoretical framework (main hypotheses, assump-
tions and steps) is recalled in the following.
If the stochastic process representing the wave input 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 char-
acteristics.
The Parseval theorem and the ergodicity property es-
tablish a correspondence between the area of the response
spectrum (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 the
Gaussian 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 data
are represented by:
• the probability distribution of the response at selected
time instants, corresponding to the highest values in each
zero-crossing period (peaks: variable p),
• the probability distribution of the excursions between
the 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 extrTs
y).
The aforementioned distributions can be derived from
the underlying Gaussian distribution of the response (equa-
tion 14) in the additional hypotheses of narrow band re-
sponse process and of independence between peaks.The first
two probability distributions take the form of equations 15
and 16 respectively, both Rayleigh density distributions (see
14).
The distribution in equation 16 is particularly interest-
ing for fatigue checks, as it can be adopted to describe stress
ranges of fatigue cycles.
[15]
[16]f r
r
m
r
mR ( ) = −





4 80
2
0
exp
f p
p
m
p
mP ( ) = −






0
2
02
exp
f (y)
m
eY
Y
y m Y=
−( )1
2 0
22
0
2
π
/
m S ( )dny
n
y= ∫
∞
ω ω ω
0
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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 Ts in 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 Td which 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 (equa-
tion18).
[18]
where:
F(y) = probability for the response to be less than value
y (unconditioned).
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
F y F y S P(S )i i
i
n
( ) = ( )⋅
=
∑
1
F p
m
m
p
m
TextrT
s
s
( ) = − −














exp exp
1
2 2
2
0
2
0∂
of signiﬁcant wave heights and mean periods. Such scatter
diagrams are catalogued according to sea zones, such as
shown in Figure 18.11 (the subdivision of the world atlas),
and main wave direction. Seasonal characteristics are also
available.
The process described in equation 18 can be termed de-
conditioning (that is removing the conditioning hypothesis).
The same procedure can be applied to any of the variables
studied in the short term and it does not change the nature
of the variable itself. If a range distribution is processed, a
long-term distribution for ranges of single oscillations is
obtained (useful data for a fatigue analysis).
If the distribution of variable extrTs
y is de-conditioned, a
weighed average of the highest peak in time Ts is achieved.
In this case the result is further processed to get the distri-
bution of the extreme value in the design time Td. This is
done with an additional application of the concept of sta-
tistics of extremes.
In the hypothesis that the extremes of the various sea
states are independent from each other, the extreme on time
Td is given by equation 19:
[19]
where F(extrTd
y) is the cumulative probability distribution
for the highest response peak in time Td (long-term extreme
distribution in time Td).
18.3.4.6 Uncertainties in long-term predictions
The theoretical framework of the above presented spectral
method, coupled to linear frequency domain methodolo-
gies like those summarized in Subsection 18.3.4.3, allows
the characterization, in the probability domain, of all the
wave induced load variables of interest both for strength
and fatigue checks.
The results of this linear prediction procedure are af-
fected by numerous sources of uncertainties, such as:
F y F yextrTd extrTs
Td/Ts
( ) = ( )[ ]
Figure 18.10 Short-term Distributions
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• sea description: as above mentioned, scatter 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, which
are 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, as
well as the motions and accelerations on which they are
based, are therefore implicitly calibrated at an exceeding
probability higher than the 10–8
value adopted in global load
strength 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 trans-
verse loads for the hull plating and supporting structures.
18.3.6.1 Static external pressures
Hydrostatic pressure is related through equation 20 to the
vertical distance between the free surface and the load point
(static head hS).
pS = ρghS [20]
In the case of the external pressure on the hull, hS cor-
responds to the local draft of the load point (reference is
made to design waterline).
Figure 18.11 Map of Sea Zones of the World (15)
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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 hS corresponds 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 aw to 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).
pf = ρaThT [21]
In equation 21, hT is 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 longitudinal and/or transver-
sal directions, producing additional pressure loads (slosh-
ing loads).
If pitch or roll frequencies are close to the tank reso-
nance frequency in the inherent direction (which can be
evaluated on the basis of geometrical parameters and fill-
ing ratio), kinetic energy tends to concentrate in the fluid
and sloshing phenomena are enhanced.
The resulting pressure field can be quite complicated
and 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 more
details see references 16 and 17.
A further type of excitation is represented by impacts that
can occur on horizontal or sub-horizontal plates of the upper
part 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 re-
lated to a number of effects, such as: local shape and ve-
locity of the free surface, air trapping in the fluid and
response of the structure. A complete model of the phe-
nomenon would require a very detailed two-phase scheme
for the fluid and a dynamic model for the structure includ-
ing hydro-elasticity effects.
Simplified distributions of sloshing and/or impact pres-
sures are often provided by Classification Societies for struc-
tural verification (Figure 18.14).
Figure 18.13 Internal Fluid Pressure (full tank)
Figure 18.12 Example of Simplified Distribution of External Pressure (10)
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18.3.7.3 Dry bulk 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, presence
and extension of air trapped between fluid and ship bottom
and structural dynamic behavior (18,19).
While slamming probability of occurrence can be stud-
ied on the basis only of predictions of ship relative motions
(which should in principle include non-linear effects due to
extreme motions), a quantification of slamming pressure
involves necessarily all the other mentioned phenomena
and is very difficult to attain, both from a theoretical and
experimental point of view (18,19).
From a practical point of view, Class Societies prescribe,
for ships with loading conditions corresponding to a low fore
Figure 18.14 Example of Simplified Distributions of Sloshing and Impact
Pressures (11)
Figure 18.15 Scheme of Local Forces Transmitted by a Container to the
Support System (8)
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draft, local structural checks based on an additional exter-
nal pressure.
Such additional pressure is formulated as a function 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 the
structure.
A second effect is due to axial and non axial forces and
moments 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 the
increased angle of attack, the local pressure on the back of
blade profiles can, for any rotation angle, fall below the
vapor 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 water
volume during the growing phase of the vapor bubble and
because of the following implosion when conditions for its
existence are removed.
All of the three mentioned types of excitation have their
main components at the propeller rotational frequency, at
the blade frequency, and at their first harmonics. In addi-
tion to the above frequencies, the cavitation pressure field
contains also other components at higher frequency, related
to the dynamics of the vapor cavity.
Propellers with skewed blades perform better as regards
induced 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, pressure
fluctuations are distributed over a longer time period and
peak values are lower.
Bearing forces and pressures induced on the stern counter
by cavitating and non cavitating propellers can be calculated
with dedicated numerical simulations (18).
18.3.9.4 Main engine excitation
Another major source of dynamic excitation for the hull
girder is represented by the main engine. Depending on
general arrangement and on number of cylinders, diesel en-
gines generate internally unbalanced forces and moments,
mainly at the engine revolution frequency, at the cylinders
firing frequency and inherent harmonics (Figure 18.16).
The excitation due to the first harmonics of low speed
diesel 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 and
location 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 than
the same excitation placed on a point of maximum modal
deflection.
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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 elongation 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 MWiz about the vertical
axis (equation 23), all applied at the center of gravity.
[22]
[23]
where:
φWi = the angle formed by the direction of the wind rela-
tive to the ship
CMz(φWi), CFL(φWi), CFT(φWi) are all coefficients 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 speed
The empirical formulas in equations 22 and 23 account
also 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 and
variables 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 definition
zero). 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 on
the mooring system is very small (a few percent of the other
actions).
If the ship is constrained, significant loads arise on the
mooring 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, non-lin-
ear wave actions, with an average value different from zero,
are also present, due to potential forces of higher order, for-
mation of vortices, and viscous effects. These components
can be significant on off-shore floating structures, which
often feature also complicated mooring systems: in those
cases the dynamic behavior of the mooring system is to be
included in the analysis, to solve a specific motion prob-
lem. For common ships, non-linear wave effects are usu-
ally neglected.
A practical rule-of-thumb for taking into account wave
actions for a ship at anchor in non protected waters is to in-
crease of 75 to 100% the sum of the other force components.
Once the total force on the ship is quantified, the ten-
sion in the mooring system (hawser, rope or chain) can be
M C A L VWiz Mz Wi Wi Wi= ( )1 2 2/ φ φ
F C A VWiL,T F L,T Wi Wi Wi= ( )1 2 2/ φ φ
Figure 18.16 Propeller, Shaft and Engine Induced Actions (20)
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derived by force decomposition, taking into account 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).
BT + FC – W = 0 [24]
xB BT + xF FC – xW W = 0 [25]
where:
W, BT, FC = (respectively) weight, buoyancy and cradle
force resultants
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 FC and 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 vertical
shear and bending moment.
qVL(x) = w(x) – bL(x) – fC(x) [26]
This computation is performed for various intermediate
positions of the cradle during the launching in order to check
all phases. However, the most demanding situation for the
hull 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 pop-
pet, if one is fitted) and it is at the maximum value.
A considerable sagging moment is present in this situ-
ation, whose maximum value is usually lower than the de-
sign one, but tends to be located in the fore part of the ship,
where bending strength is not as high as at midship.
Furthermore, the ship at launching could still have tem-
porary openings or incomplete structures (lower strength)
in the area of maximum bending moment.
Another matter of concern is the concentrated force at
the 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 temporary
internal 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 (see
Figure 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 ship
tilts and drops into water.
The kinetic energy gained during the tilting and drop-
ping phases makes the ship oscillate around her final posi-
Figure 18.17 End Launch: Sketch
Figure 18.18 Forces during Pivoting
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tion at rest. The amplitude of heave and roll motions 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 con-
servation of momentum and of energy. Within this frame-
work, time domain simulations can evaluate the magnitude
of contact forces and the energy, which is absorbed by struc-
ture deformation: these quantities, together with the response
characteristics of the structure (energy absorption capacity),
allow an evaluation of the damage penetration (21).
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 en-
ergy is (a) dissipated in the deformation of the lower part
of the bow (b) dissipated in friction of the same area against
the ground, (c) spent in deformation work of the ground (if
soft: sand, gravel) and (d) converted into gravitational po-
tential energy (work done against the weight force, which
resists to the vertical raising of the ship barycenter).
In addition to soil characteristics, key parameters for the
description are: slope and geometry of the ground, initial
speed and direction of the ship relative to ground, shape of
the bow (with/without bulb).
The final position (grounded ship) governs the magni-
tude of the vertical reaction force and the distribution of
shear and sagging moment that are generated in the hull
girder. Figure 18.20 gives an idea of the magnitude of
grounding loads for different combinations of ground slopes
and coefficients of friction for a 150 000 tanker (results of
simulations from reference 22).
In addition to numerical simulations, full and model
scale tests are performed to study grounding events (21).
Figure 18.19 Side Launch (1, Chapter XVII) Figure 18.20 Sagging Moments for a Grounded Ship: Simulation Results (22)
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18.3.11 Combination of 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 models
Three-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, with
boundary conditions formulated part in 2D, part in 3D.
18.3.12.2 Body boundary conditions
In linear methods, body boundary conditions are set with
reference to the mean position of the hull (in still water).
Perturbation terms take into account, in the frequency or in
the time domain, first order variations of hydrodynamic and
hydrostatic coefficients around the still water line.
Other non-linear methods account for perturbation terms
of a higher order. In this case, body boundary conditions
are still linear (mean position of the hull), but second order
variations of the coefficients are accounted for.
Mixed or blending procedures consist in linear methods
modified to include non-linear effects in a single compo-
nent of the velocity potential (while the other ones are treated
linearly). In particular, they account for the actual geome-
try of wetted hull (non-linear body boundary condition) in
the Froude-Krylov potential only. This effect is believed to
have a major role in the definition of global loads.
More evolved (and complex) methods are able to take
properly into account the exact body boundary condition
(actual wetted surface of the hull).
18.3.12.3 Free surface boundary conditions
Boundary 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 on
the uniform flow, (c) the flow corresponding to the steady
advance 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 free
surface conditions have also been published.
18.3.12.4 Fluid characteristics
All the methods above recalled are based on an inviscid
fluid potential scheme.
Some results have been published of viscous flow mod-
els based on the solution of Reynolds Averaged Navier
Stokes (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 few
research groups.
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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, buckling, 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 requiring 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, double sides, etc., for double hull ships:
• Stresses in the plating of stiffened panel under lateral
pressure may have different origins (σ2 and σ2*). For a
stiffened panel, there is the stress (σ2) and deflection of
the global bending of the orthotropic stiffened panels,
for example, the panel of bottom structure contained be-
tween two adjacent transverse bulkheads. The stiffener
and the attached plating bend under the lateral load and
the plate develops additional plane stresses since the
plate acts as a flange with the stiffeners. In longitudinally
framed ships there is also a second type of secondary
stresses: σ2* corresponds to the bending under the hy-
drostatic pressure of the longitudinals between trans-
verse frames (web frames). For transversally framed
panels, σ2* may also exist and would correspond to the
bending of the equally spaced frames between two stiff
longitudinal girders.
• A double bottom behaves as box girder but can bend lon-
gitudinally, transversally or both.This global bending in-
duces stress (σ2) and deflection. In addition, there is also
Figure 18.21 Primary (Hull), Secondary (Double Bottom and Stiffened Panels)
and Tertiary (Plate) Structural Responses (1, 2)
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the σ2* stress that corresponds to the bending of the lon-
gitudinals (for example, in the inner and outer bottom)
between two transverse elements (floors).
Tertiary response describes the out-of-plane 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, and
combinations of both.
18.4.2 Basic Structural Components
Structural components are extensively discussed in Chap-
ter 17 – Structure Arrangement Component Design. In this
section, only the basic structural component used exten-
sively is presented. It is basically a stiffened panel.
The global ship structure is usually referred to as being
a box girder or hull girder. Modeling of this hull girder is
the first task of the designer. It is usually done by model-
ing the hull girder with a series of stiffened panels.
Stiffened panels are the main components of a ship. Al-
most any part of the ship can be modeled as stiffened pan-
els (plane or cylindrical).
This means that, once the ship’s main dimensions and
general 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 the
design 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 Modulus
The 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, longitudinal
bulkheads, and continuous or fully effective longitudinal
primary or secondary stiffening members.
Elementary beam theory (equation 29) is usually uti-
lized in computing the component of primary stress, σ1, and
deflection due to vertical or lateral hull bending loads. In
assessing the applicability of this beam theory to ship struc-
tures, it is useful to restate the underlying assumptions:
• the beam is prismatic, that is, all cross sections are the
same and there is no openings or discontinuities,
• plane cross sections remain plane after deformation, will
Figure 18.22 A Standard Stiffened Panel
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not deform in their own planes, and merely rotate as 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
Figure 18.24.
EI (∂2
w/∂x2
) = M(x) [27]
or
EI (∂4
w/∂x4
) = q(x) [28]
where:
w = deﬂection (Figure 18.24), in m
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
M(x) = bending moment, in N.m
q(x) = load per unit length in N/m
= ∂V(x)/∂x
= ∂2
M(x)/∂x2
= EI (∂4
w/∂x4
)
Hull Section Modulus: The plane section assumption to-
gether with elastic material behavior results in a longitudi-
nal stress, σ1, in the beam that varies linearly over the depth
of the cross section.
The simple beam theory for longitudinal strength cal-
culations of a ship is based on the hypothesis (usually at-
tributed to Navier) that plane sections remain plane and in
the absence of shear, normal to the OXY plane (Figure
18.24). This gives the well-known formula:
[29]
where:
M = bending moment (in N.m)
σ = bending stress (in N/m2
)
f p
p
m
p
mP ( ) = −






0
2
02
exp
Figure 18.23 Types of Stiffening (Longitudinal and Transverse)
Figure 18.24 Behavior of an Elastic Beam under Shear Force and Bending
Moment (2)
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I = Sectional moment of Inertia about the neutral axis
(in m4
)
c = distance from the neutral axis to the extreme mem-
ber (in m)
SM = section modulus (I/c) (in m3
)
For a given bending moment at a given cross section 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 σ will be obtained for each sec-
tion for any given bending moment, one for the top fiber
(deck) and one for the bottom fiber (bottom shell).
A variation on the above beam equations may be of im-
portance in ship structures. It concerns beams composed 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 ∫A E(z) z2
dA, 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.
Dividingtheordinatesofthemaximumbending-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 curves
fourcurvesofstresscanbeobtained;thatis,tensionandcom-
pression values for both top and bottom extreme fibers.
It is customary, however, to assume the maximum bend-
ing moment to extend over the midship portion of the ship.
Minimum section modulus most often occurs at the loca-
tion of a hatch or a deck opening. Accordingly, the classi-
fication societies ordinarily require the maintenance of the
midship scantlings throughout the midship four-tenths
length. This practice maintains the midship section area of
structure practically at full value in the vicinity of maximum
shear as well as providing for possible variation in the pre-
cise location of the maximum bending moment.
Lateral Bending Combined with Vertical Bending: Up to
thispoint,attentionhasbeenfocusedprincipallyuponthever-
tical longitudinal bending response of the hull. As the ship
moves through a seaway encountering waves from directions
other than directly ahead or astern, it will experience lateral
bending loads and twisting moments in addition to the ver-
tical loads. The former may be dealt with by methods that
are similar to those used for treating the vertical bending
loads, noting that there will be no component of still water
bending 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 twisting
loading should be considered a primary response.
The combination of vertical and horizontal bending mo-
ment has as major effect to increase the stress at the ex-
treme corners of the structure (equation 30).
Figure 18.25 Moment of Inertia and Section Modulus (1)
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[30]
where Mv, Iv, cv, and Mh, Ih, ch, correspond to the M, I, 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 ν is called Poisson’s Ratio and,
for steel and aluminum, has a value of approximately 0.3.
Hooke’s Law, which expresses the relation between 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:
σs = ν σx [32]
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, σx. In Figure 18.26, it can be seen that
σ =
( )
+
( )
M
I c
M
I c
v
v v
h
h h
an element of side shell or deck plating may, in general be
subject to two other components of stress, a direct stress in
the transverse direction and a shearing stress.
This figure illustrates these as the stress resultants, de-
fined as the stress multiplied by plate thickness.
The stress resultants (N/m) are given by the following
expressions:
Nx = t σx and Ns = t σs stress resultants, in N/m
N = t τ shear stress resultant or shear flow, in N/m
where:
σx, σs = stresses in the longitudinal and transverse direc-
tions, in N/m2
τ = shear stress, in N/m2
t = plate thickness, in m
In many parts of the ship, the longitudinal stress, σx, is
the dominant component. There are, however, locations in
which the shear component becomes important and under
unusual circumstances the transverse component may, like-
wise, become important. A suitable procedure for estimat-
ing these other component stresses may be derived by
considering the equations of static equilibrium of the ele-
ment of plating (Figure 18.26). The static equilibrium con-
ditions for a plate element subjected only to in-plane stress,
that is, no plate bending, are:
∂Nx / ∂x + ∂N / ∂s = 0 [33-a]
∂Ns / ∂x + ∂N / ∂x = 0 [33-b]
In these equations, s, is the transverse coordinate meas-
ured on the surface of the section from the x-axis as shown
in Figure 18.26.
For vessels without continuous longitudinal bulkheads
Figure 18.26 Shear Forces (2)
ED: Correction on this equation is unclear.
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