Analysis and Design of Ship Structure

MASTER SET SDC 18.qxd Page 18-1 4/28/03 1:30 PM

Chapter

18

Analysis and Design of Ship Structure Philippe Rigo and Enrico Rizzuto

18.1 NOMENCLATURE For specific symbols, refer to the definitions contained in the various sections. ABS BEM BV DNV FEA FEM IACS ISSC ISOPE ISUM NKK PRADS RINA SNAME SSC a A B C CB D g

American Bureau of Shipping Boundary Element Method Bureau Veritas Det Norske Veritas Finite Element Analysis Finite Element Method International Association of Classification Societies International Ship & Offshore Structures Congress International Offshore and Polar Engineering Conference Idealized Structural Unit method Nippon Kaiji Kyokai Practical Design of Ships and Mobile Units, Registro Italiano Navale Society of naval Architects and marine Engineers Ship Structure Committee. acceleration area breadth of the ship wave coefficient (Table 18.I) hull block coefficient depth of the ship gravity acceleration

m(x) I(x) L M(x) MT(x) p q(x) T V(x) s,w (low case) v,h (low case) w(x) θ ρ ω

18.2

longitudinal distribution of mass geometric moment of inertia (beam section x) length of the ship bending moment at section x of a beam torque moment at section x of a beam pressure resultant of sectional force acting on a beam draft of the ship shear at section x of a beam still water, wave induced component vertical, horizontal component longitudinal distribution of weight roll angle density angular frequency

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 between performing a design and completing an analysis.

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Ship Design & Construction, Volume 1

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 select the initial structural scantlings and to update these scantlings from the early design stage (bidding) to the detailed design stage (construction). To perform analysis, initial design is needed and analysis is required to design. This explains 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 Architecture (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 Arrangement and Component Design and with Chapter 19 – Reliability-Based Structural Design. References to these chapters will be made in order to avoid duplications. In addition, as Chapter 8 deals with classification societies, the present chapter will focus mainly on the direct analysis methods available to perform a rationally based structural design, even if mention is made to standard formulations from Rules to quantify design loads. In the following sections of this chapter, steps of a global analysis are presented. Section 18.3 concerns the loads that are necessary to perform a structure analysis. Then, Sections 18.4, 18.5 and 18.6 concern, respectively, the stresses and deflections (basic ship responses), the limit states, and the failures modes and associated structural capacity. A review of the available Numerical Analysis for Structural Design is performed in Section 18.7. Finally Design Criteria (Section 18.8) and Design Procedures (Section 18.9) are discussed. Structural modeling is discussed in Subsection 18.2.2 and more extensively in Subsection 18.7.2 for finite element analysis. 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 compromises: • 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 design analyses efficient. The results from complex analyses should be verified by simplified methods to avoid errors 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 design 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 empirical, based on accumulated experience and ship performance, and expressed in the form of structural design codes or rules published by the various ship classification societies. These rules concern the loads, the strength and the design criteria and provide simplified and easy-to-use formulas 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:


Chapter 18: Analysis and Design of Ship Structure

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 unknown; thus one cannot distinguish between structural adequacy and over-adequacy. Second, and most important, these formulas involve a number of simplifying assumptions 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 certified. Direct analysis has become the standard procedure in aerospace, civil engineering and partly in offshore industries. In ship design, classification societies preferred to offer updated rules resulting from numerical analysis calibration. For the designer, even if the rules were continuously changing, the design remained rule-based. There really were two different methodologies.

Design Load Direct Load Analysis Stress Response in Waves

Study on Ocean Waves

Effect on operation

Wave Load Response

Structural analysis by whole ship model Stress response function

Response function of wave load

Short term estimation

Design Sea State

Short term estimation

Long term estimation

Long term estimation

Nonlinear influence in large waves

Design wave

Wave impact load

Structural response analysis Modeling technique

Direct structural analysis

Investigation on corrosion

Strength Assessment Yield strength

Buckling strength

Ultimate strength

Fatigue strength

Figure 18.1 Direct Structural Analysis Flow Chart

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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, see Table 18.VIII, Subsection 18.7.5.2). In addition, for new vessel types or non-standard dimension, such direct procedure is the only way to assess the structural safety. Therefore 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 structures compared with other types of structures. They are subject 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 majority of specialists together with rapid advances in computer 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 requirements, 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 evaluation. In addition, in order to make this a practical and effective method of analysis, it is necessary to give careful consideration to more rational and accurate methods of direct strength analysis. Based on recognition of this need, extensive research has been conducted and a careful examination made, regarding the strength evaluation of hull structures. The results 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, constraints…) 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 structural analysis is that the structural model shall provide results suitable for performing buckling, yield, fatigue and


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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.

Structural drawings, mass description and loading conditions.

Verification of model/ loads

Structural model including necessary load definitions

Hydrodynamic/static loads

Verified structural model

Load transfer to structural model

Structural analysis

Sub-models to be used in structural analysis

Verification of load transfer

Verification of response Transfer of displacements/forces to sub-model?

Yes

No

Figure 18.2 Strength Analysis Flow Chart (4)

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 supported 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), buckling strength and ultimate strength checks of the ship. In addition, 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 surface 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

18.2.3 Preliminary Design versus Detailed Design For a ship structure, structural design consists of two distinct levels: the Preliminary Design and the Detailed Design about which Hughes (3) states: The preliminary determines the location, spacing, and scantlings of the principal structural members. The detailed design determines the geometry and scantlings of local structure (brackets, connections, cutouts, reinforcements, etc.). Preliminary design has the greatest influence on the structure design and hence is the phase that offers very large potential savings. This does not mean that detail design is less important than preliminary design. Each level is equally important for obtaining an efficient, safe and reliable ship. During the detailed design there also are many benefits to be gained by applying modern methods of engineering 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 detail 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 inefficient to attempt to design all of them from first principles. Instead it is generally more efficient to use design codes and standard designs that have been proven by experience. 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 peculiar, 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, general interpretation schemes, available quantification proce-



dures and practical methods for their evaluation will be summarized. 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 condition, they have little and very slow variations during a voyage (mainly due to changes in the distribution of consumables 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 induced by waves, but also those generated in the same frequency range by motions of the ship (inertial forces). These loads can be termed quasi-static because the structural response is studied with static models. Dynamic loads: When studying responses with frequency 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 (springing) or by mechanical excitation (main engine, propeller). Also transient impulsive loads that excite free structural vibrations (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 generally 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 categories. 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).

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Loads, defined in order to be applied to limited structural 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 pressures 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 analyses, 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 function 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, separately, 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 situation corresponds to the highest load cycle and is characterized through the probability associated to the extreme value in the reference (design) time. In fatigue phenomena, in principle all stress cycles contribute (to a different extent, depending on the range) to


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damage accumulation. The analysis, therefore, does not regard the magnitude of a single extreme load application, but the number of cycles and the shape of the probability distribution of all stress ranges in the design time. A further step towards the problem simplification is represented 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 assigning an exceeding probability (or a return period) to each variable and selecting the correspondent value from the statistical 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 typically represented by loads associated to an exceeding probability 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 yielding 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 analyses 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 monodimensional structural element with sectional characteristics 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

Figure 18.3 Sectional Forces and Moment

resultant force along the vertical axis of the section (contained in the plane of symmetry), indicated as vertical resultant 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 distributed 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: x

VV (x) =

∫ q V (ξ)

dξ

[1]

dξ

[2]

dξ

[3]

dξ

[4]

dξ

[5]

0

x

M V (x) =

∫ VV ( ξ ) 0

x

VH (x) =

∫ q H (ξ ) 0

x

M H (x) =

∫ VH ( ξ ) 0

x

M T (x) =

∫ m T (ξ) 0

Due to total equilibrium, for a beam in free-free conditions (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. VV (0) = VV (L) = M V (0) = M V (L) = 0 VH (0) = VH (L) = M H (0) = M H (L) = 0 M T (0) = M T (L) = 0

[6]

Global loads for the verification of the hull girder are obtained with a linear superimposition of still water and waveinduced global loads. They are used, with different characteristic values, in different types of analyses, such as ultimate state, first yielding, and fatigue. 18.3.3 Still Water Global Loads Still water loads act on the ship floating in calm water, usually 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 vertical gravitational forces. If the latter ones are not symmetric, a sectional torque mTg(x) is generated (Figure 18.4), in addition to the verti-



cal load qSV(x), obtained as a difference between buoyancy b(x) and weight w(x), as shown in equation 7 (2). q SV (x) = b(x) − w(x) = gA I (x) − m(x)g

[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 horizontal components of sectional forces in equation 3 and accordingly 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, calibrated on similar ships, are often used in the design development 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.

Figure 18.4 Sectional Resultant Forces in Still Water

(a)

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At an even earlier stage of design, parametric formulations can be used to derive directly reference values for still water hull girder loads. Common reference values for still water bending moment at mid-ship are provided by the major Classification Societies (equation 8). Ms [ N ⋅ m ] =

C L2 B (122.5 − 15 C B ) (hogging) [8] C L2 B ( 45.5 + 65 C B ) (sagging)

where C = wave parameter (Table 18.I). The formulations in equation 8 are sometimes explicitly reported in Rules, but they can anyway be indirectly derived from prescriptions contained in (6, 7). The first requirement (6) regards the minimum longitudinal strength modulus and provides implicitly a value for the total bending 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 Societies, (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 (nonhomogeneous loading at maximum draft, light load at less than maximum draft, short voyage or harbor condition, ballast exchange at sea, etc.). The direct evaluation procedure requires, for a given loading condition, a derivation, section by section, of vertical 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 and center of gravity is determined summing up contributions from all items present on board between the two bounding sections. The distribution for w(x) is then usually approximated by a linear (trapezoidal) curve obtained by imposing

TABLE 18.I Wave Coefficient Versus Length (b)

Figure 18.5 Examples of Reference Still Water Bending Moment Distribution (10). (a) oil tankers, bulk carriers, ore carriers, and (b) other ship types

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


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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)

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 different 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 loading 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, machinery, outfitting) but also the distribution of the various components of the deadweight (cargo, ballast, consumables). Ship types like bulk carriers are more exposed to uncertainties 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 (Figure 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 additional 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 counterbalanced 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 oriented along the true vertical. This implies neglecting the static trim angle and to consider 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 considered as a single longitudinal beam, also loads in the transversal direction can be significant, giving rise to shear, bending and torque in a transversal direction (see the simplified 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 represents a key point in the quantification of both global and local loads acting on the ship. The solution of the seakeeping 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 inertial loads and provide inputs for the evaluation of other types of loads, like slamming and sloshing.



In particular, as regards global effects, the action of waves modifies the pressure distribution along the wet hull surface; the differential pressure between the situation in waves and in still water generates, on the transverse section, vertical and horizontal resultant forces (bWV and bWH) and a moment component mTb. Analogous components come from the sectional resultants 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). q WV (x) = b WV (x) − m(x)a V (x) q WH (x) = b WH (x) − m(x)a H (x)

[9]

m TW (x) = m Tb (x) − I R (x) θ where IR(x) is the rotational inertia of section x. The longitudinal distributions along the hull girder of horizontal 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 instantaneous 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 components 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-

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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 probability of about 10–8 (once in the ship lifetime). M WV [ N ⋅ m ] =

(hog) 190 C L2 B C B [10] 2 −110 C L B ( C B + 0 . 7 ) (sag)

Horizontal Wave-induced Bending Moment: Similar formulations 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 vertical component. In Table 18.II, examples are reported of reference values of horizontal bending moment at mid-length for ships with unrestricted navigation. Simplified curves for the distribution in the longitudinal direction are also provided. Wave-induced Torque: A few reference formulations are given also for reference wave torque at midship (see examples 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 described for still water loads, with the difference that the waterline 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 effects. 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

TABLE 18.II Reference Horizontal Bending Moments

Figure 18.9 Sectional Forces and Moments in Waves

Class Society

MWH [N ⋅ m]

ABS (8)

180 C1L2DCB

BV (9) RINA (10)

1600 L2.1 TCB

DNV (11)

220 L9/4(T + 0.3B)CB

NKK (12)

320 L2C T L − 35 / L


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TABLE 18.III Examples of Reference Values for Wave Torque Class Society

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

ABS (bulk carrier)

2700 LB 2 T ( C W − 0 . 5 )

[

2

]

 e 0 .14  0 . 5  + 0 .1  0 .13 −    T  D  

(e = vertical position of shear center) BV RINA

 250 − 0 . 7 L  3  190 LB 2 C 2W  8.13 −     125  

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 presented: • inviscid, incompressible and homogeneous fluid in irrotational flow: Laplace equation 11 ∇2Φ = 0

[11]

where Φ = velocity potential • 2-dimensional solution of the problem • linearized boundary conditions: the quadratic component of velocity in the Bernoulli Equation is reformulated 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 condition); zero pressure, — on the hull: considered as a static surface, corresponding to the mean position of the hull: the component of the fluid velocity normal to the hull surface is zero (impermeability condition), and • linear decomposition into additive independent components, separately solved for and later summed up (equation 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): FroudeKrylov Φd = diffraction component, due to disturbance in the wave potential generated by the hull This subdivision also enables the de-coupling of the excitation 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 components 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 characteristics (spectrum and spatial distribution respect to the ship course). The output consists of response spectra of point pressures 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 probability 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 several contexts and are extensively used for predictions of wave loads. In principle, the base assumptions of the method are



valid only for small wave excitations, small motion responses and low speed of the ship. In practice, the field of successful applications extends far beyond the limits suggested by the preservation of realism in the base assumptions: the method is actually used extensively to study even extreme loads and for fast vessels. 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 distribution 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 fluctuations (symmetric with respect to a zero mean value). A consequence 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 between 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 superimposition 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 acting always along the vertical axis of the ship reference system (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-linearities 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 linear superposition of still water and wave loads is largely followed. 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 spectral moments m n of order n (equation 13).

18-11

∞

m ny = ∫ ω n S y (ω)dω

[13]

0

This information is the basis of the spectral method, whose theoretical framework (main hypotheses, assumptions 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 system (load) can be modeled as a process having the same characteristics. The Parseval theorem and the ergodicity property establish a correspondence between the area of the response spectrum (spectral moment of order 0: m0Y) and the variance of its Gaussian probability distribution (14). This allows expressing the density probability distribution of the Gaussian response y in terms of m0Y (equation 14). f Y (y) =

1 − y 2 / 2 m 20 Y ) e ( 2π m0Y

[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 extrTsy). The aforementioned distributions can be derived from the underlying Gaussian distribution of the response (equation 14) in the additional hypotheses of narrow band response 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 interesting for fatigue checks, as it can be adopted to describe stress ranges of fatigue cycles. fP ( p) =

 p p2  exp  −  m0  2m0 

[15]

fR ( r ) =

 r r2  exp  −  4m0  8m 0 

[16]


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The distribution for the extreme value in the stationary period Ts (short term extreme) can be modeled by a Poisson distribution (in equation 17: expression of the cumulative distribution) or other equivalent distributions derived from the statistics of extremes.  1 F extrTs p = exp  −  2 ∂

(

)

 m2 p2 exp  − m0 m0 2 

   Ts  [17]  

Figure 18.10 summarizes the various short-term distributions. It is interesting to note that all the mentioned distributions are expressed in terms of spectral moments of the response, 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 considered 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 distribution around the main wave direction (spreading function) 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 (equation18). F(y) =

n

∑ F ( y S i ) ⋅ P(S i )

[18]

i= 1

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 combinations. 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-dimensional histograms (scatter diagrams), containing classes

Figure 18.10 Short-term Distributions

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 deconditioning (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 extrTsy 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 distribution of the extreme value in the design time Td. This is done with an additional application of the concept of statistics 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:

(

) [ (

)]

Td/Ts F extrTd y = F extrTs y

[19]

where F(extrTdy) 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 methodologies 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 affected by numerous sources of uncertainties, such as:



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

• 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, biparametric formulations based on significant wave and mean wave period), • model for the ship’s response: as briefly outlined in Subsection 18.3.4.3, the model is greatly simplified, particularly 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 primary 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 example, 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 transverse 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 corresponds to the local draft of the load point (reference is made to design waterline).


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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 Classification 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 transversal load for plate, stiffeners and primary supporting members of the tank walls. 18.3.7.1 Static internal pressure For a ship in still water, gravitation acceleration g generates a hydrostatic pressure, varying again according to equation 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 liquid) or, if applicable, for the ullage space pressure (the pressure 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 motions are generated in the liquid contained in a tank onboard, depending on the period of the ship motions and on the filling level: the internal pressure distribution varies accordingly. In a completely full tank, fluid internal velocities relative to the tank walls are small and the acceleration in the fluid is considered as corresponding to the global ship acceleration aw. The total pressure (equation 21) can be evaluated in terms of the total acceleration aT, obtained summing aw to gravity 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 Figure 18.13 on roll angle θr). pf = ρaThT

nal velocities can arise in the longitudinal and/or transversal directions, producing additional pressure loads (sloshing loads). If pitch or roll frequencies are close to the tank resonance frequency in the inherent direction (which can be evaluated on the basis of geometrical parameters and filling 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 quantification. Experimental techniques as well as 2D and 3D procedures 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 filling levels, in vertical or sub-vertical plates of the lower part of the tank. Impact loads are very difficult to characterize, being related to a number of effects, such as: local shape and velocity of the free surface, air trapping in the fluid and response of the structure. A complete model of the phenomenon would require a very detailed two-phase scheme for the fluid and a dynamic model for the structure including hydro-elasticity effects. Simplified distributions of sloshing and/or impact pressures are often provided by Classification Societies for structural verification (Figure 18.14).

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

[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-

Figure 18.13 Internal Fluid Pressure (full tank)



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 tangential 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 expressed 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 different ways, depending on the number and extension of contact 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 applied 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 considerable impact pressures. The phenomenon occurs in flat areas of the forward part of the ship and it is strongly correlated to loading conditions 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 motions relative to water, local angle formed at impact between

18-15

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 studied 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)


18-16


draft, local structural checks based on an additional external 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 additional term for the vertical wave-induced bending moment, 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 pressures 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 moment 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 excitation and the natural frequency associated to the first (two-node) flexural mode in the vertical plane, thus producing 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 comparatively 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 deflection and stresses is magnified by dynamic effects. For recent 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 during 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 transmitted 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 generated on the back of the profile (cavitation phenomenon). The vapor filled cavity collapses as soon as the angle of attack decreases in the propeller revolution and the local pressure rises again over the vapor saturation pressure. Cavitation further enhances pressure fluctuations, because 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 addition 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 simultaneously in the region of the stern counter, where disturbances 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 engines 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 natural 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 example, a vertical excitation in a nodal point of a vertical flexural mode has much less effect in exciting that mode than the same excitation placed on a point of maximum modal deflection.



18-17

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. FWiL,T = 1 / 2 C F L,T ( φ Wi ) φ A Wi VWi 2

[22]

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

[23]

where:

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

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 refrigerated cargo). What actually creates stresses is a non-uniform temperature distribution, which implies that the warmer part of the structure tends to expand while the rest opposes to this deformation. A peculiar aspect of this situation is that the portion 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 temperature distribution and in particular to the model for constraints. 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 supporting 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 component in the orthogonal direction can arise in particular situations. 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

φWi = the angle formed by the direction of the wind relative 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 parallel 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 sinusoidal time dependence (whose mean value is by definition zero). If ship motions in the wave direction are not constrained (for example, if the anchor chain is not in tension) the ship motion follows the excitation with similar time dependence 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-linear wave actions, with an average value different from zero, are also present, due to potential forces of higher order, formation 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 problem. For common ships, non-linear wave effects are usually neglected. A practical rule-of-thumb for taking into account wave actions for a ship at anchor in non protected waters is to increase of 75 to 100% the sum of the other force components. Once the total force on the ship is quantified, the tension in the mooring system (hawser, rope or chain) can be


18-18


derived by force decomposition, taking into account the angle formed with the external force in the horizontal and/or vertical plane.

integrated according to equations 1 and 2 to derive vertical shear and bending moment.

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 number 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 distributed 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 distributions 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 (equations 24 and 25, in which dynamic effects are neglected: quasi static approach).

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 poppet, if one is fitted) and it is at the maximum value. A considerable sagging moment is present in this situation, whose maximum value is usually lower than the design 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 temporary 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 percentage 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 components 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 dropping phases makes the ship oscillate around her final posi-

BT + FC – W = 0

[24]

xB BT + xF FC – xW W = 0

[25]

qVL(x) = w(x) – bL(x) – fC(x)

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 buoyancy 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 corresponds 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

Figure 18.17 End Launch: Sketch

Figure 18.18 Forces during Pivoting

[26]



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 studied 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 motions 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 excitation. 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 collisions, 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 number of forces acting during the event: forces induced by propeller, 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 response of the ship to waves. As regards collision forces, they can be described differently depending on the characteristics of the struck object (ship, platform, bridge pylon…) with different combinations of rigid, elastic or an elastic body models.

Figure 18.19 Side Launch (1, Chapter XVII)

18-19

Governing equations for the problem are given by conservation of momentum and of energy. Within this framework, time domain simulations can evaluate the magnitude of contact forces and the energy, which is absorbed by structure 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 energy 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 potential 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 magnitude 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 18.20 Sagging Moments for a Grounded Ship: Simulation Results (22)


18-20


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 probability (usually the same of the single components). This cannot be obtained by simple superposition of the characteristic values of single contributing loads, as the probability 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 expressed 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 approximation would consist in modeling the joint distribution through its first and second order moments, that is mean values 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 statistical 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 combinations 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 proposed 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 complete 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 available; 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 component of the velocity potential (while the other ones are treated linearly). In particular, they account for the actual geometry 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 surface or (d) the incident wave profile (neglecting the interaction 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 models based on the solution of Reynolds Averaged Navier Stokes (RANS) equations in the time domain. These methods represent the most recent trend in the field of ship motions and loads prediction and their use is limited to a few research groups.



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 associated analyses involving stresses are referred to under the general term of strength. The strength of a structural component would be inadequate if it experiences a loss of load-carrying 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 rendering 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 associated 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 bending strength, with torsional strength requiring particular attention 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 associated 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 stiffened panels (for single hull ship) or to the behavior of double bottom, double sides, etc., for double hull ships:

18-21

• 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 between 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 hydrostatic pressure of the longitudinals between transverse 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 longitudinally, transversally or both. This global bending induces 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)


18-22


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

tect deals principally with beam theory, plate theory, and combinations of both.

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 membrane 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 deflections are large compared to the plate thickness. In many instances, there is little or no interaction between the three (primary, secondary, tertiary) component stresses or deflections, and each component may be computed 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 membrane stresses. In stiffeners, only primary and secondary responses induce 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 response are shown with their associated stresses (σ1, σ2, σ2* and σ3). These considerations point to the inherent simplicity of the underlying theory. The structural naval archi-

18.4.2 Basic Structural Components Structural components are extensively discussed in Chapter 17 – Structure Arrangement Component Design. In this section, only the basic structural component used extensively 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 modeling the hull girder with a series of stiffened panels. Stiffened panels are the main components of a ship. Almost any part of the ship can be modeled as stiffened panels (plane or cylindrical). This means that, once the ship’s main dimensions and general arrangement are fixed, the remaining scantling development 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 longitudinally stiffened panels (b) and transversally stiffened 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 primary stress are, for the most part, the longitudinally continuous 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 utilized 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 structures, it is useful to restate the underlying assumptions:

Figure 18.22 A Standard Stiffened Panel

• 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



18-23

Figure 18.23 Types of Stiffening (Longitudinal and Transverse)

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 coupled. 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 comprehensive stress analysis. Methods dealing with stress concentrations 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 (∂2w/∂x2) = M(x)

[27]

EI (∂4w/∂x4) = q(x)

[28]

or

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 = ∂2M(x)/∂x2 = EI (∂4w/∂x4)

Figure 18.24 Behavior of an Elastic Beam under Shear Force and Bending Moment (2)

Hull Section Modulus: The plane section assumption together with elastic material behavior results in a longitudinal stress, σ1, in the beam that varies linearly over the depth of the cross section. The simple beam theory for longitudinal strength calculations of a ship is based on the hypothesis (usually attributed 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: fP ( p) =

 p p2  exp  −  m0  2m0 

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

[29]


18-24


I = Sectional moment of Inertia about the neutral axis (in m4) c = distance from the neutral axis to the extreme member (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 obtained (σ = M/SM = Mc/I) which is proportional to the distance 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 section 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 importance 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 rigidity, EI, is replaced by ∫A E(z) z2 dA, where A is cross sectional 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 indicates the bending strength properties of the primary hull structure. The section modulus to the deck or bottom is obtained 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 continuous or effectively developed:

ordinates of the section-moduli curve yields stress values, and by using both the hogging and sagging moment curves four curves of stress can be obtained; that is, tension and compression values for both top and bottom extreme fibers. It is customary, however, to assume the maximum bending moment to extend over the midship portion of the ship. Minimum section modulus most often occurs at the location of a hatch or a deck opening. Accordingly, the classification 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 precise location of the maximum bending moment. Lateral Bending Combined with Vertical Bending: Up to this point, attention has been focused principally upon the vertical 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 vertical 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 twisting 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 moment has as major effect to increase the stress at the extreme corners of the structure (equation 30).

• deck plating (strength deck and other effective decks). (See Subsection 18.4.3.9 for Hull/Superstructure Interaction). • shell and inner bottom plating, • deck and bottom girders, • plating and longitudinal stiffeners of longitudinal bulkheads, • all longitudinals of deck, sides, bottom and inner bottom, and • continuous longitudinal hatch coamings. In general, only members that are effective in both tension and compression are assumed to act as part of the hull girder. Theoretically, a thorough analysis of longitudinal strength would include the construction of a curve of section moduli throughout the length of the ship as shown in Figure 18.25. Dividing the ordinates of the maximum bending-moments curve (the envelope curve of maxima) by the corresponding

Figure 18.25 Moment of Inertia and Section Modulus (1)



18-25

ED: Correction on this equation is unclear. σ=

Mv Mh + I I c ( v v ) ( h ch )

[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 horizontal 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 bending, 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 absolute 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 response induces both longitudinal (σx) and transversal stresses (σs) in plating. εx = 1/E ( σx – v σS) εS = 1/E ( σS – ν σx)

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, defined 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 directions, 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, likewise, become important. A suitable procedure for estimating these other component stresses may be derived by considering the equations of static equilibrium of the element of plating (Figure 18.26). The static equilibrium conditions 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 measured on the surface of the section from the x-axis as shown in Figure 18.26. For vessels without continuous longitudinal bulkheads

[31]

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 perpendicular 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 preceding section permit evaluation the longitudinal component of the primary stress, σx. In Figure 18.26, it can be seen that

Figure 18.26 Shear Forces (2)


18-26


(single cell), having transverse symmetry and subject to a bending moment in the vertical plane, the shear flow distribution, N(s) is then given by: V(x)  N (s) =  m (s)  I(x) 

[34]

and the shear stress, τ , at any point in the cross section is: t(s) =

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

(in N / m 2 )

[35]

where: V(x) = total shearing force (in N) in the hull for a given section x m(s) =

s

∫o t ( s ) z ds, in m , is the first moment (or moment 3

= of area) about the neutral axis of the cross sectional area of the plating between the origin at the centerline and the variable location designated by s. This is the crosshatched area of the section shown in Figure 18.26 t(s) = thickness of material at the shear plane I(x) = moment of inertia of the entire section The total vertical shearing force, V(x), at any point, x, in the ship’s length may be obtained by the integration of the load curve up to that point. Ordinarily the maximum value of the shearing force occurs at about one quarter of the vessel’s length from either end. Since only the vertical, or nearly vertical, members of the hull girder are capable of resisting vertical shear, this shear is taken almost entirely by the side shell, the continuous longitudinal bulkheads if present, and by the webs of any deep longitudinal girders. The maximum value of τ occurs in the vicinity of the neutral axis, where the value of t is usually twice the thickness of the side plating (Figure 18.27). For vessels with continuous longitudinal bulkheads, the expression for shear stress is more complex. Shear Flow in Multicell Sections: If the cross section of the ship shown in Figure 18.28 is subdivided into two or more closed cells by longitudinal bulkheads, tank tops, or decks, the problem of finding the shear flow in the boundaries of these closed cells is statically indeterminate. Equation 34 may be evaluated for the deck and bottom of the center tank space since the plane of symmetry at which the shear flow vanishes, lies within this space and forms a convenient origin for the integration. At the deck/bulkhead intersection, the shear flow in the deck divides, but the relative proportions of the part in the bulkhead and the part in the deck are indeterminate. The sum

of the shear flows at two locations lying on a plane cutting the cell walls will still be given by equation 34, with m(s) equal to the moment of the shaded area (Figure 18.28). However, the distribution of this sum between the two components in bulkhead and side shell, requires additional information for its determination. This additional information may be obtained by considering the torsional equilibrium and deflection of the cellular section. The way to proceed is extensively explained in Lewis (2). 18.4.3.3 Shear stress associated with torsion In order to develop the twisting equations, we consider a closed, single cell, thin-walled prismatic section subject only to a twisting moment, MT, which is constant along the length as shown in Figure 18.29. The resulting shear stress may be assumed uniform through the plate thickness and is tangent to the mid-thickness of the material. Under these circumstances, the deflection of the tube will consist of a twisting of the section without distortion of its shape, and the rate of twist, dθ/dx, will be constant along the length.

Figure 18.27 Shear Flow in Multicell Sections (1)

Figure 18.28 Shear Flow in Multicell Sections (2)



Now consider equilibrium of forces in the x-direction for the element dx.ds of the tube wall as shown in Figure 18.29. Since there is no longitudinal load, there will be no longitudinal stress, and only the shear stresses at the top and bottom edges need be considered in the expression for static equilibrium. The shear flow, N = tτ, is therefore seen to be constant around the section. The magnitude of the moment, MT, may be computed by integrating the moment of the elementary force arising from this shear flow about any convenient axis. If r is the distance from the axis, 0, perpendicular to the resultant shear flow at location s: MT =

∫ r N ds = N ∫ r ds = 2 NΩ

[36]

Here the symbol indicates that the integral is taken entirely around the section and, therefore, Ω (m2) is the area enclosed by the mid-thickness line of the tubular cross section. The constant shear flow, N (N/m), is then related to the applied twisting moment by: N = τ. t = MT /2Ω

[37]

For uniform torsion of a closed prismatic section, the angle of torsion is: θ=

MT .L (in radians) G Ip

where: MT = Twisting moment (torsion), in N.m L = Length of the girder, in m Ip = Polar Inertia, in m4 G = E/2(1+ν), the shear Modulus, in N/m2

Figure 18.29 Torsional Shear Flow (2).

[38]

18-27

18.4.3.4 Twisting and warping Torsional strength: Although torsion is not usually an important factor in ship design for most ships, it does result in significant additional stresses on ships, such as container ships, which have large hatch openings. These warping stresses can be calculated by a beam analysis, which takes into account the twisting and warping deflections. There can also be an interaction between horizontal bending and torsion of the hull girder. Wave actions tending to bend the hull in a horizontal plane also induce torsion because of the open cross section of the hull, which results in the shear center being below the bottom of the hull. Combined stresses due to vertical bending, horizontal bending and torsion must be calculated. In order to increase the torsional rigidity of the containership cross sections, longitudinal and transverse closed box girders are introduced in the upper side and deck structure. From previous studies, it has been established that special attention should be paid to the torsional rigidity distribution along the hull. Usually, toward the ship’s ends, the section moduli are justifiably reduced base on bending. On the contrary the torsional rigidity, especially in the forward hatches, should be gradually increased to keep the warping stress as small as possible. Twisting of opened section: A lateral seaway could induce severe twisting moment that is of the major importance for ships having large deck openings. The equations for the twist of a closed tube (equations 36 to 38) are applicable only to the computation of the torsional response of closed thin-walled sections. The relative torsional stiffness of closed and open sections may be visualized by means of a very simple example. Consider two circular tubes, one of which has a longitudinal slit over its full length as in Figure 18.30. The closed tube will be able to resist a much greater torque per unit angular deflection than the open tube because of the inability of the latter to sustain the shear stress across the slot. The twisting resistance of the thin material of which the tube is composed provides the only resistance to torsion in the case

Figure 18.30 Twist of Open and Closed Tubes (2)


18-28


of the open tube without longitudinal restraint. The resistance to twist of the entirely open section is given by the St. Venant torsion equation: MT = G.J ∂θ/∂x (N.m)

[39]

where: ∂θ/∂x = twist angle per unit length, in rad./m, which can be approximated by θ/L for uniform torsion and uniform section. J = torsional constant of the section, in m4 = 1/3 =

1 3

s

∫0 t 3 ds for a thin walled open section

n

∑ b i t 3i for a section composed of n different i =1

= plates (bi= length, ti = thickness) If warping resistance is present, that is, if the longitudinal displacement of the elemental strips shown in Figure 18.30 is constrained, another component of torsional resistance is developed through the shear stresses that result from this warping restraint. This is added to the torque given by equation 39. In ship structures, warping strength comes from four sources: 1. the closed sections of the structure between hatch openings, 2. the closed ends of the ship, 3. double wall transverse bulkheads, and 4. closed, torsionally stiff parts of the cross section (longitudinal torsion tubes or boxes, including double bottom, double side shell, etc.). 18.4.3.5 Racking and snaking Racking is the result of a transverse hull shape distortion and is caused by either dynamic loads due to rolling of the ship or by the transverse impact of seas against the topsides. Transverse bulkheads resist racking if the bulkhead spacing is close enough to prevent deflection of the shell or deck plating in its own plane. Racking introduces primarily compressive and shearing forces in the plane of bulkhead plating. With the usual spacing of transverse bulkheads the effectiveness of side frames in resisting racking is negligible. However, when bulkheads are widely spaced or where the deck width is small in way of very large hatch openings, side frames, in association with their top and bottom brackets, contribute significant resistance to racking. Racking in car-carriers is discussed in Chapters 17 and 34. Racking stresses due to rolling reach a maximum in a beam sea each time the vessel completes an oscillation in one direction and is about to return.

The angle between a deck beam and side frame tends to open on one side and to close on the other side at the top and reverses its action at the bottom. The effect of the concentration of stiff and soft sections results in a distortion pattern in the ship deck that is shown in Figure 18.31. The term snaking is sometimes used in referring to this behavior and relates to both twisting and racking. 18.4.3.6 Effective breadth and shear lag An important effect of the edge shear loading of a plate member is a resulting nonlinear variation of the longitudinal stress distribution (Figure 18.32). In the real plate the longitudinal stress decreases with increasing distance from the shear-loaded edge, and this is called shear lag. This is in contrast to the uniform stress distribution predicted in the beam flanges by the elementary beam equation 29. In many practical cases, the difference from the value predicted in equation 29 will be small. But in certain combinations of loading and structural geometry, the effect referred to by the term shear lag must be taken into consideration if an accurate estimate of the maximum stress in the member is to be made. This may be conveniently done by defining an effective breadth of the flange member. The ratio, be/b, of the effective breadth, be, to the real breadth, b, is useful to the designer in determining the longitudinal stress along the shear-loaded edge. It is a function

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

Figure 18.32 Shear Lag Effect in a Deck (2)



of the external loading applied and the boundary conditions along the plate edges, but not its thickness. Figure 18.33 gives the effective breadth ratio at mid-length for column loading and harmonic-shaped beam loading, together with a common approximation for both cases: be k L = 6 b b

[40]

The results are presented in a series of design charts, which are especially simple to use, and may be found in Schade (26). A real situation in which such an alternating load distribution may be encountered is a bulk carrier loaded with a dense ore cargo in alternate holds, the remainder being empty. An example of the computation of the effective breadth of bottom and deck plating for such a vessel is given in Chapter VI of Taggart (1), using Figure 18.33. It is important to distinguish the effective breadth (equation 40) and the effective width (equations 54 and 55) presented later in Subsection 18.6.3.2 for plate and stiffened plate-buckling analysis. 18.4.3.7 Longitudinal deflection The longitudinal bending deflection of the ship girder is obtainable from the appropriate curvature equations (equations 27 and 28) by integrating twice. A semi-empirical approximation for bending deflection amidships is:

w = k ( M L2/EI )

18-29

[41]

where the dimensionless coefficient k may be taken, for first approximation, as 0.09 (2). Actual deflection in service is affected also by thermal influences, rigidity of structural components, and workmanship; furthermore, deflection due to shear is additive to the bending deflection, though its amount is usually relatively small. The same influences, which gradually increase nominal design stress levels, also increase flexibility. Additionally, draft limitations and stability requirements may force the L/D ratio up, as ships get larger. In general, therefore, modern design requires that more attention be focused on flexibility than formerly. No specific limits on hull girder deflections are given in the classification rules. The required minimum scantlings however, as well as general design practices, are based on a limitation of the L/D ratio range. 18.4.3.8 Load diffusion into structure The description of the computation of vertical shear and bending moment by integration of the longitudinal load distribution implies that the external vertical load is resisted directly by the vertical shear carrying members of the hull girder such as the side shell or longitudinal bulkheads. In a longitudinally framed ship, such as a tanker, the bottom pressures are transferred principally to the widely spaced transverse web frames or the transverse bulkheads where

Figure 18.33 Effective Breath Ratios at Midlength (1)


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they are transferred to the longitudinal bulkheads or side shell, again as localized shear forces. Thus, in reality, the loading q(x), applied to the side shell or the longitudinal bulkhead will consist of a distributed part due to the direct transfer of load into the member from the bottom or deck structure, plus a concentrated part at each bulkhead or web frame. This leads to a discontinuity in the shear curve at the bulkheads and webs. 18.4.3.9 Hull/superstructure interaction The terms superstructure and deckhouse refer to a structure usually of shorter length than the entire ship and erected above the strength deck of the ship. If its sides are coplanar with the ship’s sides it is referred to as a superstructure. If its width is less than that of the ship, it is called a deckhouse. The prediction of the structural behavior of a superstructure constructed above the strength deck of the hull has facets involving both the general bending response and important localized effects. Two opposing schools of thought exist concerning the philosophy of design of such erections. One attempts to make the superstructure effective in contributing to the overall bending strength of the hull, the other purposely isolates the superstructure from the hull so that it carries only localized loads and does not experience stresses and deflections associated with bending of the main hull. This may be accomplished in long superstructures (>0.5Lpp) by cutting the deckhouse into short segments by means of expansion joints. Aluminum deckhouse construction is another alternative when the different material properties provide the required relief. As the ship hull experiences a bending deflection in response to the wave bending moment, the superstructure is forced to bend also. However, the curvature of the superstructure may not necessarily be equal to that of the hull but depends upon the length of superstructure in relation to the hull and the nature of the connection between the two, especially upon the vertical stiffness or foundation modulus of the deck upon which the superstructure is constructed. The behavior of the superstructure is similar to that of a beam on an elastic foundation loaded by a system of normal forces and shear forces at the bond to the hull. The stress distributions at the midlength of the superstructure and the differential deflection between deckhouse and hull for three different degrees of superstructure effectiveness are shown on Figure 18.34. The areas and inertias can be computed to account for shear lag in decks and bottoms. If the erection material differs from that of the hull (aluminum on steel, for example) the geometric erection area Af and inertia If must be reduced according to the ratio of the respective material moduli; that is, by multiplying by E (aluminum)/E (steel) (approximately

one-third). Further details on the design considerations for deckhouses and superstructures may be found in Evans (27) and Taggart (1). In addition to the overall bending, local stress concentrations may be expected at the ends of the house, since here the structure is transformed abruptly from that of a beam consisting of the main hull alone to that of hull plus superstructure. Recent works achieved in Norwegian University of Science & Technology have shown that the vertical stress distribution in the side shell is not linear when there are large openings in the side shell as it is currently the case for upper decks of passenger vessels. Approximated stress distributions are presented at Figure 18.35. The reduced slope, θ, for the upper deck has been found equal to 0.50 for a catamaran passenger vessel (28). 18.4.4 Secondary Response In the case of secondary structural response, the principal objective is to determine the distribution of both in-plane

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

z Passenger deck Neutral axis

σr (z) =θ .σ(z)

( I )z

σ (z) = M x

Figure 18.35 Vertical Stress Distribution in Passenger Vessels having Large Openings above the Passenger Deck



and normal loading, deflection and stress over the length and width dimensions of a stiffened panel. Remember that the primary response involves the determination of only the in-plane load, deflection, and stress as they vary over the length of the ship. The secondary response, therefore, is seen to be a two-dimensional problem while the primary response is essentially one-dimensional in character. 18.4.4.1 Stiffened panels A stiffened panel of structure, as used in the present context, usually consists of a flat plate surface with its attached stiffeners, transverse frames and/or girders (Figure 18.36). When the plating is absent the module is a grid or grillage of beam members only, rather than a stiffened panel. In principle, the solution for the deflection and stress in the stiffened panel may be thought of as a solution for the response of a system of orthogonal intersecting beams. A second type of interaction arises from the two-dimensional stress pattern in the plate, which may be thought of as forming a part of the flanges of the stiffeners. The plate contribution to the beam bending stiffness arises from the direct longitudinal stress in the plate adjacent to the stiffener, modified by the transverse stress effects, and also from the shear stress in the plane of the plate. The maximum secondary stress may be found in the plate itself, but more frequently it is found in the free flanges of the stiffeners, since these flanges are at a greater distance than the plate member from the neutral axis of the combined plate-stiffener. At least four different procedures have been employed for obtaining the structural behavior of stiffened plate panels under normal loading, each embodying certain simplifying assumptions: 1) orthotropic plate theory, 2) beam-on-elastic-

Figure 18.36 A Stiffened Panel with Uniformly Distributed Longitudinals, 4 Webframes, and 3 Girders.

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foundation theory, 3) grillage theory (intersecting beams), and 4) the finite element method (FEM). Orthotropic plate theory refers to the theory of bending of plates having different flexural rigidities in the two orthogonal directions. In applying this theory to panels having discrete stiffeners, the structure is idealized by assuming that the structural properties of the stiffeners may be approximated by their average values, which are assumed to be distributed uniformly over the width or length of the plate. The deflections and stresses in the resulting continuum are then obtained from a solution of the orthotropic plate deflection differential equation: a1

∂4w ∂4w ∂4w + a2 + a3 = p (x,y) 4 2 2 ∂x ∂x ∂y ∂y 4

[42]

where: a1, a2, a3 = express the average flexural rigidity of the orthotropic plate in the two directions w(x,y) = is the deflection of the plate in the normal direction p(x,y) = is the distributed normal pressure load per unit area Note that the behavior of the isotropic plate, that is, one having uniform flexural properties in all directions, is a special case of the orthotropic plate problem. The orthotropic plate method is best suited to a panel in which the stiffeners are uniform in size and spacing and closely spaced. It has been said that the application of this theory to crossstiffened panels must be restricted to stiffened panels with more than three stiffeners in each direction. An advanced orthotropic procedure has been implemented by Rigo (29,30) into a computer-based scheme for the optimum structural design of the midship section. It is based on the differential equations of stiffened cylindrical shells (linear theory). Stiffened plates and cylindrical shells can both be considered, as plates are particular cases of the cylindrical shells having a very large radius. A system of three differential equations, similar to equation 42, is established (8th order coupled differential equations). Fourier series expansions are used to model the loads. Assuming that the displacements (u,v,w) can also be expanded in sin and cosine, an analytical solution of u, v, and w(x,y) can be obtained for each stiffened panel. This procedure can be applied globally to all the stiffened panels that compose a parallel section of a ship, typically a cargo hold. This approach has three main advantages. First the plate bending behavior (w) and the inplane membrane behavior (u and v) are analyzed simultaneously. Then, in addition to


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the flexural rigidity (bending), the inplane axial, torsional, transverse shear and inplane shear rigidities of the stiffeners in the both directions can also be considered. Finally, the approach is suited for stiffeners uniform in size and spacing, and closely spaced but also for individual members, randomly distributed such as deck and bottom girders. These members considered through Heaviside functions that allow replacing each individual member by a set of 3 forces and 2 bending moment load lines. Figure 18.36 shows a typical stiffened panel that can be considered. It includes uniformly distributed longitudinals and web frames, and three prompt elements (girders). The beam on elastic foundation solution is suitable for a panel in which the stiffeners are uniform and closely spaced in one direction and sparser in the other one. Each of these members is treated individually as a beam on an elastic foundation, for which the differential equation of deflection is, EI

∂4w + k w = q (x) ∂x 4

[43]

where: w = is the deflection I = is sectional moment of inertia of the longitudinal stiffener, including adjacent plating k = is average spring constant per unit length of the transverse stiffeners q(x) = is load per unit length on the longitudinal member The grillage approach models the cross-stiffened panel as a system of discrete intersecting beams (in plane frame), each beam being composed of stiffener and associated effective plating. The torsional rigidity of the stiffened panel and the Poisson ratio effect are neglected. The validity of modeling the stiffened panel by an intersecting beam (or grillage) may be critical when the flexural rigidities of stiffeners are small compared to the plate stiffness. It is known that the grillage approach may be suitable when the ratio of the stiffener flexural rigidity to the plate bending rigidity (EI/bD with I the moment of inertia of stiffener and D the plate bending rigidity) is greater than 60 (31) otherwise if the bending rigidity of stiffener is smaller, an Orthotropic Plate Theory has to be selected. The FEM approach is discussed in detail in section 18.7.2. 18.4.5 Tertiary Response 18.4.5.1 Unstiffened plate Tertiary response refers to the bending stresses and deflections in the individual panels of plating that are bounded by the stiffeners of a secondary panel. In most cases the load that induces this response is a fluid pressure from either the

water outside the ship or liquid or dry bulk cargo within. Such a loading is normal to and distributed over the surface of the panel. In many cases, the proportions, orientation, and location of the panel are such that the pressure may be assumed constant over its area. As previously noted, the deflection response of an isotropic plate panel is obtained as the solution of a special case of the earlier orthotropic plate equation (equation 42), and is given by: ∂4w ∂4w ∂ 4 w p (x,y) +2 2 2 + = 4 D ∂x ∂x ∂y ∂y 4

[44]

where: D = plate flexural rigidity

E t3 12(1 − ν )

= Et3 / 12(1 – ν) t = the uniform plate thickness p(x,y) = distributed unit pressure load

Appropriate boundary conditions are to be selected to represent the degree of fixity of the edges of the panel. Stresses and deflections are obtained by solving this equation for rectangular plates under a uniform pressure distribution. Equation 44 is in fact a simplified case of the general one (equation 42). Information (including charts) on a plate subject to uniform load and concentrated load (patch load) is available in Hughes (3). 18.4.5.2 Local deflections Local deflections must be kept at reasonable levels in order for the overall structure to have the proper strength and rigidity. Towards this end, the classification society rules may contain requirements to ensure that local deflections are not excessive. Special requirements also apply to stiffeners. Tripping brackets are provided to support the flanges, and they should be in line with or as near as practicable to the flanges of struts. Special attention must be given to rigidity of members under compressive loads to avoid buckling. This is done by providing a minimum moment of inertia at the stiffener and associated plating. 18.4.6 Transverse Strength Transverse strength refers to the ability of the ship structure to resist those loads that tend to cause distortion of the cross section. When it is distorted into a parallelogram shape the effect is called racking. We recall that both the primary bending and torsional strength analyses are based upon the assumption of no distortion of the cross section. Thus, we



see that there is an inherent relationship between transverse strength and both longitudinal and torsional strength. Certain structural members, including transverse bulkheads and deep web frames, must be incorporated into the ship in order to insure adequate transverse strength. These members provide support to and interact with longitudinal members by transferring loads from one part of a structure to another. For example, a portion of the bottom pressure loading on the hull is transferred via the center girder and the longitudinals to the transverse bulkheads at the ends of theses longitudinals. The bulkheads, in turn, transfer these loads as vertical shears into the side shell. Thus some of the loads acting on the transverse strength members are also the loads of concern in longitudinal strength considerations. The general subject of transverse strength includes elements taken from both the primary and secondary strength categories. The loads that cause effects requiring transverse strength analysis may be of several different types, depending upon the type of ship, its structural arrangement, mode of operation, and upon environmental effects. Typical situations requiring attention to the transverse strength are: • ship out of water: on building ways or on construction or repair dry dock, • tankers having empty wing tanks and full centerline tanks or vice versa, • ore carriers having loaded centerline holds and large empty wing tanks, • all types of ships: torsional and racking effects caused by asymmetric motions of roll, sway and yaw, and • ships with structural features having particular sensitivity to transverse effects, as for instance, ships having largely open interior structure (minimum transverse bulkheads) such as auto carriers, containers and RO-RO ships. As previously noted, the transverse structural response involves pronounced interaction between transverse and longitudinal structural members. The principal loading consists of the water pressure distribution around the ship, and the weights and inertias of the structure and hold contents. As a first approximation, the transverse response of such a frame may be analyzed by a two-dimensional frame response procedure that may or may not allow for support by longitudinal structure. Such analysis can be easily performed using 2D finite element analysis (FEA). Influence of longitudinal girders on the frame would be represented by elastic attachments having finite spring constants (similar to equation 43). Unfortunately, such a procedure is very sensitive to the spring location and the boundary conditions. For this reason, a three-dimensional analysis is usually performed in order to obtain results that are useful for more

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than comparative purposes. Ideally, the entire ship hull or at least a limited hold-model should be modeled. See Subsection 18.7.2—Structural Finite Element Models (Figure 18.57). 18.4.7 Superposition of Stresses In plating, each response induces longitudinal stresses, transverse stresses and shear stresses. These stresses can be calculated individually for each response. This is the traditional way followed by the classification societies. With direct analysis such as finite element analysis (Subsection 18.7.2), it is not always possible to separate the different responses. If calculated individually, all the longitudinal stresses have to be added. Similar cumulative procedure must be achieved for the transverse stresses and the shear stresses. At the end they are combined through a criteria, which is usually for ship structure, the von-Mises criteria (equation 45). The standard procedure used by classification societies considers that longitudinal stresses induced by primary response of the hull girder, can be assessed separately from the other stresses. Classification rules impose through allowable stress and minimal section modulus, a maximum longitudinal stress induced by the hull girder bending moment. On the other hand, they recommend to combined stresses from secondary response and tertiary response, in plating and in members. These are combined through the von Mises criteria and compared to the classification requirements. Such an uncoupled procedure is convenient to use but does not reflect reality. Direct analysis does not follow this approach. All the stresses, from the primary, secondary and tertiary responses are combined for yielding assessment. For buckling assessment, the tertiary response is discarded, as it does not induce in-plane stresses. Nevertheless the lateral load can be considered in the buckling formulation (Subsection 18.6.3). Tertiary stresses should be added for fatigue analysis. Since all the methods of calculation of primary, secondary, and tertiary stress presuppose linear elastic behavior of the structural material, the stress intensities computed for the same member may be superimposed in order to obtain a maximum value for the combined stress. In performing and interpreting such a linear superposition, several considerations affecting the accuracy and significance of the resulting stress values must be borne in mind. First, the loads and theoretical procedures used in computing the stress components may not be of the same accuracy or reliability. The primary loading, for example, may be obtained using a theory that involves certain simplifica-


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tions in the hydrodynamics of ship and wave motion, and the primary bending stress may be computed by simple beam theory, which gives a reasonably good estimate of the mean stress in deck or bottom but neglects certain localized effects such as shear lag or stress concentrations. Second, the three stress components may not necessarily occur at the same instant in time as the ship moves through waves. The maximum bending moment amidships, which results in the maximum primary stress, does not necessarily occur in phase with the maximum local pressure on a midship panel of bottom structure (secondary stress) or panel of plating (tertiary stress). Third, the maximum values of primary, secondary, and tertiary stress are not necessarily in the same direction or even in the same part of the structure. In order to visualize this, consider a panel of bottom structure with longitudinal framing. The forward and after boundaries of the panel will be at transverse bulkheads. The primary stress (σ1) will act in the longitudinal direction, as given by equation 29. It will be nearly equal in the plating and the stiffeners, and will be approximately constant over the length of a midship panel. There will be a small transverse component in the plating, due to the Poison coefficient, and a shear stress given by equation 35. The secondary stress will probably be greater in the free flanges of the stiffeners than in the plating, since the combined neutral axis of the stiffener/plate combination is usually near the plate-stiffener joint. Secondary stresses, which vary over the length of the panel, are usually subdivided into two parts in the case of single hull structure. The first part (σ2) is associated with bending of a panel of structure bounded by transverse bulkheads and either the side shell or the longitudinal bulkheads. The principal stiffeners, in this case, are the center and any side longitudinal girders, and the transverse web frames. The second part, (σ2*), is the stress resulting from the bending of the smaller panel of plating plus longitudinal stiffeners that is bounded by the deep web frames. The first of these components (σ2), as a result of the proportions of the panels of structure, is usually larger in the transverse than in the longitudinal direction. The second (σ2*) is predominantly longitudinal. The maximum tertiary stress (σ3) happens, of course, in the plate where biaxial stresses occur. In the case of longitudinal stiffeners, the maximum panel tertiary stress will act in the transverse direction (normal to the framing system) at the mid-length of a long side. In certain cases, there will be an appreciable shear stress component present in the plate, and the proper interpretation and assessment of the stress level will require the resolution of the stress pattern into principal stress components. From all these considerations, it is evident that, in many cases, the point in the structure having the highest stress level

will not always be immediately obvious, but must be found by considering the combined stress effects at a number of different locations and times. The nominal stresses produced from the analysis will be a combination of the stress components shown in Figures 18.21 and 18.37. 18.4.7.1 von Mises equivalent stress The yield strength of the material, σyield, is defined as the measured stress at which appreciable nonlinear behavior accompanied by permanent plastic deformation of the material occurs. The ultimate strength is the highest level of stress achieved before the test specimen fractures. For most shipbuilding steels, the yield and tensile strengths in tension and compression are assumed equal. The stress criterion that must be used is one in which it is possible to compare the actual multi-axial stress with the material strength expressed in terms of a single value for the yield or ultimate stress. For this purpose, there are several theories of material failure in use. The one usually considered the most suitable for ductile materials such as ship steel is referred to as the von Mises Theory:

(

σ e = σ 2x + σ 2y − σ x σ y + 3 τ 2

)

1

2

[45]

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

Figure 18.37 Definition of Stress Components (4)



failure through yielding will occur if the equivalent von Mises stress, σe, given by equation 45 exceeds the equivalent stress, σο, corresponding to yielding of the material test specimen. The material yield strength may also be expressed through an equivalent stress at failure: σ0 = σyield (= σy). 18.4.7.2 Permissible stresses (Yielding) In actual service, a ship may be subjected to bending in the inclined position and to other forces, such as those, which induce torsion or side bending in the hull girder, not to mention the dynamic effects resulting from the motions of the ship itself. Heretofore it has been difficult to arrive at the minimum scantlings for a large ship’s hull by first principles alone, since the forces that the structure might be required to withstand in service conditions are uncertain. Accordingly, it must be assumed that the allowable stress includes an adequate factor of safety, or margin, for these uncertain loading factors. In practice, the margin against yield failure of the structure is obtained by a comparison of the structure’s von Mises equivalent stress, σe, against the permissible stress (or allowable stress), σ0, giving the result: σe ≤ σ0 = s1 × σy

[46]

where: s1 = partial safety factor defined by classification societies, which depends on the loading conditions and method of analysis. For 20 years North Atlantic conditions (seagoing condition), the s1 factor is usually taken between 0.85 and 0.95 σy = minimum yield point of the considered steel (mild steel, high tensile steel, etc.) For special ship types, different permissible stresses may be specified for different parts of the hull structure. For example, for LNG carriers, there are special strain requirements in way of the bonds for the containment system, which in turn can be expressed as equivalent stress requirements. For local areas subjected to many cycles of load reversal, fatigue life must be calculated and a reduced permissible stress may be imposed to prevent fatigue failure (see Subsection 18.6.6).

18.5 LIMIT STATES AND FAILURE MODES Avoidance of structural failure is the goal of all structural designers, and to achieve this goal it is necessary for the designer to be aware of the potential limit states, failure modes and methods of predicting their occurrence. This section presents the basic types of failure modes and associated limit

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states. A more elaborate description of the failure modes and methods to assess the structural capabilities in relation to these failure modes is available in Subsection 18.6.1. Classically, the different limit states were divided in 2 major categories: the service limit state and the ultimate limit state. Today, from the viewpoint of structural design, it seems more relevant to use for the steel structures four types of limit states, namely: 1. 2. 3. 4.

service or serviceability limit state, ultimate limit state, fatigue limit state, and accidental limit state.

This classification has recently been adopted by ISO. A service limit state corresponds to the situation where the structure can no longer provide the service for which it was conceived, for example: excessive deck deflection, elastic buckling in a plate, and local cracking due to fatigue. Typically they relate to aesthetic, functional or maintenance problem, but do not lead to collapse. An ultimate limit state corresponds to collapse/failure, including collision and grounding. A classic example of ultimate limit state is the ultimate hull bending moment (Figure 18.46). The ultimate limit state is symbolized by the higher point (C) of the moment-curvature curve (M-Φ). Fatigue can be either considered as a third limit state or, classically, considered as a service limit state. Even if it is also a matter of discussion, yielding should be considered as a service limit state. First yield is sometimes used to assess the ultimate state, for instance for the ultimate hull bending moment, but basically, collapse occurs later. Most of the time, vibration relates to service limit states. In practice, it is important to differentiate service, ultimate, fatigue and accidental limit states because the partial safety factors associated with these limit states are generally different. 18.5.1 Basic Types of Failure Modes Ship structural failure may occur as a result of a variety of causes, and the degree or severity of the failure may vary from a minor esthetic degradation to catastrophic failure resulting in loss of the ship. Three major failure modes are defined: 1. tensile or compressive yield of the material (plasticity), 2. compressive instability (buckling), and 3. fracture that includes ductile tensile rupture, low-cycle fatigue and brittle fracture. Yield occurs when the stress in a structural member exceeds a level that results in a permanent plastic deforma-


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tion of the material of which the member is constructed. This stress level is termed the material yield stress. At a somewhat higher stress, termed the ultimate stress, fracture of the material occurs. While many structural design criteria are based upon the prevention of any yield whatsoever, it should be observed that localized yield in some portions of a structure is acceptable. Yield must be considered as a serviceability limit state. Instability and buckling failure of a structural member loaded in compression may occur at a stress level that is substantially lower than the material yield stress. The load at which instability or buckling occurs is a function of member geometry and material elasticity modulus, that is, slenderness, rather than material strength. The most common example of an instability failure is the buckling of a simple column under a compressive load that equals or exceeds the Euler Critical Load. A plate in compression also will have a critical buckling load whose value depends on the plate thickness, lateral dimensions, edge support conditions and material elasticity modulus. In contrast to the column, however, exceeding this load by a small margin will not necessarily result in complete collapse of the plate but only in an elastic deflection of the central portion of the plate away from its initial plane. After removal of the load, the plate may return to its original un-deformed configuration (for elastic buckling). The ultimate load that may be carried by a buckled plate is determined by the onset of yielding at some point in the plate material or in the stiffeners, in the case of a stiffened panel. Once begun, yield may propagate rapidly throughout the entire plate or stiffened panel with further increase in load. Fatigue failure occurs as a result of a cumulative effect in a structural member that is exposed to a stress pattern alternating from tension to compression through many cycles. Conceptually, each cycle of stress causes some small but irreversible damage within the material and, after the accumulation of enough such damage, the ability of the member to withstand loading is reduced below the level of the applied load. Two categories of fatigue damage are generally recognized and they are termed high-cycle and lowcycle fatigue. In high-cycle fatigue, failure is initiated in the form of small cracks, which grow slowly and which may often be detected and repaired before the structure is endangered. High-cycle fatigue involves several millions of cycles of relatively low stress (less than yield) and is typically encountered in machine parts rotating at high speed or in structural components exposed to severe and prolonged vibration. Low-cycle fatigue involves higher stress levels, up to and beyond yield, which may result in cracks being initiated after several thousand cycles. The loading environment that is typical of ships and

ocean structures is of such a nature that the cyclical stresses may be of a relatively low level during the greater part of the time, with occasional periods of very high stress levels caused by storms. Exposure to such load conditions may result in the occurrence of low-cycle fatigue cracks after an interval of a few years. These cracks may grow to serious size if they are not detected and repaired. Concerning brittle fracture, small cracks suddenly begin to grow and travel almost explosively through a major portion of the structure. The term brittle fracture refers to the fact that below a certain temperature, the ultimate tensile strength of steel diminishes sharply (lower impact energy). The originating crack is usually found to have started as a result of poor design or manufacturing practice. Fatigue (Subsection 18.6.6) is often found to play an important role in the initiation and early growth of such originating cracks. The prevention of brittle fracture is largely a matter of material selection and proper attention to the design of structural details in order to avoid stress concentrations. The control of brittle fracture involves a combination of design and inspection standards aimed toward the prevention of stress concentrations, and the selection of steels having a high degree of notch toughness, especially at low temperatures. Quality control during construction and in-service inspection form key elements in a program of fracture control. In addition to these three failure modes, additional modes are: • collision and grounding, and • vibration and noise. Collision and Grounding is discussed in Subsection 18.6.7 and Vibration in Subsection 18.6.8. Vibration as well as noise is not a failure mode, while it could fall into the serviceability limit state.

18.6 ASSESSMENT OF THE STRUCTURAL CAPACITY 18.6.1 Failure Modes Classification The types of failure that may occur in ship structures are generally those that are characteristic of structures made up of stiffened panels assembled through welding. Figure 18.38 presents the different structure levels: the global structure, usually a cargo hold (Level 1), the orthotropic stiffened panel or grillage (Level 2) and the interframe longitudinally stiffened panel (Level 3) or its simplified modeling: the beam-column (Level 3b). Level 4 (Figure 18.44a) is the unstiffened plate between two longitudinals and two transverse frames (also called bare plate). The word grillage should be reserve to a structure com-



posed of a grid of beams (without attached plating). When the grid is fixed on a plate, orthotropic stiffened panel seems to the authors more adequate to define a panel that is orthogonally stiffened, and having thus orthotropic properties. The relations between the different failure modes and structure levels can be summarized as follows: • Level 1: Ultimate bending moment, Mu, of the global structure (Figure 18.46). • Level 2: Ultimate strength of compressed orthotropic stiffened panels (σu), σu = min [σu (mode i)], i = I to VI, the 6 considered failure modes. • Level 3: Mode I: Overall buckling collapse (Figure 18.44d), Mode II: Plate/Stiffener Yielding Mode III: Pult of interframe panels with a plate-stif ener combination (Figure 18.44b) using a beam-column model (Level 3b) or an orthotropic model (Level 3), considering:

18-37

— plate induced failure (buckling) — stiffener induced failure (buckling or yielding) Mode IV and V: Instability of stiffeners (local buckling, tripping—Figure 18.44c) Mode VI: Gross Yielding • Level 4: Buckling collapse of unstiffened plate (bare plate, Figure 18.44a). To avoid collapse related to the Mode I, a minimal rigidity is generally imposed for the transverse frames so that an interframe panel collapse (Mode III) always occurs prior to overall buckling (Mode I). It is a simple and easy constraint to implement, thus avoiding any complex calculation of overall buckling (mode I). Note that the failure Mode III is influenced by the buckling of the bare plate (elementary unstiffened plate). Elastic buckling of theses unstiffened plates is usually not considered as an ultimate limit state (failure mode), but rather as a service limit state. Nevertheless, plate buckling (Level 4) may significantly affect the ultimate strength of the stiffened panel (Level 3). Sources of the failures associated with the serviceability or ultimate limit states can be classified as follows: 18.6.1.1 Stiffened panel failure modes Service limit state • Upper and lower bounds (Xmin≤X≤Xmax): plate thickness, dimensions of longitudinals and transverse stiffeners (web, flange and spacing). • Maximum allowable stresses against first yield (Subsection 18.4.7) • Panel and plate deflections (Subsections 18.4.4.1 and 18.4.5.2), and deflection of support members. • Elastic buckling of unstiffened plates between two longitudinals and two transverse stiffeners, frames or bulkheads (Subsection 18.6.3), • Local elastic buckling of longitudinal stiffeners (web and flange). Often the stiffener web/flange buckling does not induce immediate collapse of the stiffened panel as tripping does. It could therefore be considered as a serviceability ultimate limit state. However, this failure mode could also be classified into the ultimate limit state since the plating may sometimes remain without stiffening once the stiffener web buckles. • Vibration (Sub-ection 18.6.8) • Fatigue (Sub-ection 18.6.6) Ultimate limit state (Subsection 18.6.4).

Figure 18.38 Structural Modeling of the Structure and its Components

• Overall collapse of orthotropic panels (entire stiffened plate structure),


18-38


• Collapse of interframe longitudinally stiffened panel, including torsional-flexural (lateral-torsional) buckling of stiffeners (also called tripping). 18.6.1.2 Frame failure modes Service limit state (Subsection 18.4.6). • Upper and lower bounds (Xmin ≤ X ≤ Xmax), • Minimal rigidity to guarantee rigid supports to the interframe panels (between two transverse frames). • Allowable stresses under the resultant forces (bending, shear, torsion) — Elastic analysis, — Elasto-plastic analysis. • Fatigue (Subsection 18.6.6) Ultimate limit state • Frame bucklings: These failures modes are considered as ultimate limit states rather than a service limit state. If one of them appears, the assumption of rigid supports is no longer valid and the entire stiffened panel can reach the ultimate limit state. — Buckling of the compressed members, — Local buckling (web, flange). 18.6.1.3 Hull Girder Collapse modes Service limit state • Allowable stresses and first yield (Subsection 18.4.3.1), • Deflection of the global structure and relative deflections of components and panels (Subsection 18.4.3.7). Ultimate limit state • Global ultimate strength (of the hull girder/box girder). This can be done by considering an entire cargo hold or only the part between two transverse web frames (Subsection 18.6.5). Collapse of frames is assumed to only appear after the collapse of panels located between these frames. This means that it is sufficient to verify the box girder ultimate strength between two frames to be protected against a more general collapse including, for instance, one or more frame spans. This approach can be un-conservative if the frames are not stiff enough. • Collision and grounding (Subsection 18.6.7), which is in fact an accidental limit state. A relevant comparative list of the limit states was defined by the Ship Structure Committee Report No 375 (32).

18.6.2 Yielding As explained in Subsection 18.5.1 yield occurs when the stress in a structural component exceeds the yield stress. It is necessary to distinguish between first yield state and fully plastic state. In bending, first yield corresponds to the situation when stress in the extreme fiber reaches the yield stress. If the bending moment continues to increase the yield area is growing. The final stage corresponds to the Plastic Moment (Mp), where, both the compression and tensile sides are fully yielded (as shown on Figure 18.47). Yield can be assessed using basic bending theory, equation 29, up to complex 3D nonlinear FE analysis. Design criteria related to first yield is the von Mises equivalent stress (equation 45). Yielding is discussed in detail in Section 18.4. 18.6.3 Buckling and Ultimate Strength of Plates A ship stiffened plate structure can become unstable if either buckling or collapse occurs and may thus fail to perform its function. Hence plate design needs to be such that instability under the normal operation is prevented (Figure 18.44a). The phenomenon of buckling is normally divided into three categories, namely elastic buckling, elastic-plastic buckling and plastic buckling, the last two being called inelastic buckling. Unlike columns, thin plating buckled in the elastic regime may still be stable since it can normally sustain further loading until the ultimate strength is reached, even if the in-plane stiffness significantly decreases after the inception of buckling. In this regard, the elastic buckling of plating between stiffeners may be allowed in the design, sometimes intentionally in order to save weight. Since significant residual strength of the plating is not expected after buckling occurs in the inelastic regime, however, inelastic buckling is normally considered to be the ultimate strength of the plate. The buckling and ultimate strength of the structure depends on a variety of influential factors, namely geometric/material properties, loading characteristics, fabrication related imperfections, boundary conditions and local damage related to corrosion, fatigue cracking and denting. 18.6.3.1 Direct Analysis In estimating the load-carrying capacity of plating between stiffeners, it is usually assumed that the stiffeners are stable and fail only after the plating. This means that the stiffeners should be designed with proper proportions that help attain such behavior. Thus, webs, faceplates and flanges of the stiffeners or support members have to be proportioned so that local instability is prevented prior to the failure of plating.



Four load components, namely longitudinal compression/tension, transverse compression/tension, edge shear and lateral pressure loads, are typically considered to act on ship plating between stiffeners, as shown in Figure 18.39, while the in-plane bending effects on plate buckling are also sometimes accounted for. In actual ship structures, lateral pressure loading arises from water pressure and cargo weight. The still water magnitude of water pressure depends on the vessel draft, and the still water value of cargo pressure is determined by the amount and density of cargo loaded. These still water pressure values may be augmented by wave action and vessel motion. Typically the larger in-plane loads are caused by longitudinal hull girder bending, both in still water and in waves at sea, which is the source of the primary stress as previously noted in Subsection 18.4.3. The elastic plate buckling strength components under single types of loads, that is, σxE for σxav, σyE for σyav and τE for τav, can be calculated by taking into account the related effects arising from in-plane bending, lateral pressure, cut-outs, edge conditions and welding induced residual stresses. The critical (elastic-plastic) buckling strength components under single types of loads, that is, σxB for σxav, σyB for σyav and τB for τav, are typically calculated by plasticity correction of the corresponding elastic buckling strength using the Johnson-Ostenfeld formula, namely:  σ E for σ E  σB =  σF   σ F  1 − 4 σ E 

≤ 0.5 σ F   for σ E > 0.5 σ F 

where: σE = elastic plate buckling strength

Figure 18.39 A Simply Supported Rectangular Plate Subject to Biaxial Compression/tension, Edge Shear and Lateral Pressure Loads

[47]

18-39

σB = critical buckling strength (that is, τB for shear stress) σF = σY for 4 normal stress = σY √3 for shear stress σY = material yield stress In ship rules and books, equation 47 may appear with somewhat different constants depending on the structural proportional limit assumed. The above form assumes a structural proportional limit of a half the applicable yield value. For axial tensile loading, the critical strength may be considered to equal the material yield stress (σY). Under single types of loads, the critical plate buckling strength must be greater than the corresponding applied stress component with the relevant margin of safety. For combined biaxial compression/tension and edge shear, the following type of critical buckling strength interaction criterion would need to be satisfied, for example: c σ xav σ yav  σ yav  σ xav  +   −α σ xB σ yB  σ yB  σ xB 

c

  τ av  +  τ  B

c

  ≤ η B [48] 

where: ηB = usage factor for buckling strength, which is typically the inverse of the conventional partial safety factor. ηB = 1.0 is often taken for direct strength calculation, while it is taken less than 1.0 for practical design in accordance with classification society rules. Compressive stress is taken as negative while tensile stress is taken as positive and α = 0 if both σxav and σyav are compressive, and α = 1 if either σxav or σyav or both are tensile. The constant c is often taken as c = 2. Figure 18.40 shows a typical example of the axial membrane stress distribution inside a plate element under predominantly longitudinal compressive loading before and after buckling occurs. It is noted that the membrane stress distribution in the loading (x) direction can become nonuniform as the plate element deforms. The membrane stress distribution in the y direction may also become non-uniform with the unloaded plate edges remaining straight, while no membrane stresses will develop in the y direction if the unloaded plate edges are free to move in plane. As evident, the maximum compressive membrane stresses are developed around the plate edges that remain straight, while the minimum membrane stresses occur in the middle of the plate element where a membrane tension field is formed by the plate deflection since the plate edges remain straight. With increase in the deflection of the plate keeping the edges straight, the upper and/or lower fibers inside the middle of the plate element will initially yield by the action of bending. However, as long as it is possible to redistribute


18-40


Figure 18.41 Possible Locations for the Initial Plastic Yield at the Plate Edges (Expected yield locations, T: Tension, C: Compression); (a) Yield at longitudinal mid-edges under longitudinal uniaxial compression, (b) Yield at transverse mid-edges under transverse uniaxial compression)

tions are longitudinal mid-edges for longitudinal uniaxial compressive loads and transverse mid-edges for transverse uniaxial compressive loads, as shown in Figure 18.41. The occurrence of yielding can be assessed by using the von Mises yield criterion (equation 45). The following conditions for the most probable yield locations will then be found. (a) Yielding at longitudinal edges: σ 2x max − σ x max σ y min + σ 2y min = σ 2Y

[49a]

(b) Yielding at transverse edges: Figure 18.40 Membrane Stress Distribution Inside the Plate Element under Predomianntly Longitudinal Compressive Loads; (a) Before buckling, (b) After buckling, unloaded edges move freely in plane, (c) After buckling, unloaded edges kept straight

the applied loads to the straight plate boundaries by the membrane action, the plate element will not collapse. Collapse will then occur when the most stressed boundary locations yield, since the plate element can not keep the boundaries straight any further, resulting in a rapid increase of lateral plate deflection (33). Because of the nature of applied axial compressive loading, the possible yield loca-

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

[49b]

The maximum and minimum membrane stresses of equations 49a and 49b can be expressed in terms of applied stresses, lateral pressure loads and fabrication related initial imperfections, by solving the nonlinear governing differential equations of plating, based on equilibrium and compatibility equations. Note that equation 44 is the linear differential equation. On the other hand, the plate ultimate edge shear strength, τu , is often taken τu =τB (equation 47, with τB instead of σB). Also, an empirical formula obtained by curve fitting based on nonlinear finite element solutions may be utilized (33). The effect of lateral pressure loads on the plate ultimate edge shear strength may in some cases need to be accounted for.



18-41

For combined biaxial compression/tension, edge shear and lateral pressure loads, the last being usually regarded as a given constant secondary load, the plate ultimate strength interaction criterion may also be given by an expression similar to equation 48, but replacing the critical buckling strength components by the corresponding ultimate strength components, as follows: c σ xav σ yav  σ yav  σ xav  +   −α σ xu σ yu  σ xu   σ yu

c

  τ av  +  τ  u

c

  ≤ η u [50] 

where: α and c = variables defined in equation 48 ηu = usage factors for the ultimate limit state σxu and σyu = solutions of equation 49a with regard to σxav and equation 49b with regard to σyav, respectively 18.6.3.2 Simplified models In the interest of simplicity, the elastic plate buckling strength components under single types of loads may sometimes be calculated by neglecting the effects of in-plane bending or lateral pressure loads. Without considering the effect of lateral pressure, the resulting elastic buckling strength prediction would be pessimistic. While the plate edges are often supposed to be simply supported, that is, without rotational restraints along the plate/stiffener junctions, the real elastic buckling strength with rotational restraints would of course be increased by a certain percentages, particularly for heavy stiffeners. This arises from the increased torsional restraint provided at the plate edges in such cases. The theoretical solution for critical buckling stress, σB , in the elastic range has been found for a number of cases of interest. For rectangular plate subject to compressive inplane stress in one direction: σB = kc

2 π2E  t 12 (1 − ν 2 )  b 

[51]

Here kc is a function of the plate aspect ratio, α = a/ b, the boundary conditions on the plate edges and the type of loading. If the load is applied uniformly to a pair of opposite edges only, and if all four edges are simply supported, then kc is given by: m α 2 kc =  +   α m

[52]

where m is the number of half-waves of the deflected plate in the longitudinal direction, which is taken as an integer satisfying the condition α = m (m + 1). For long plate in

Figure 18.42 Compressive Buckling Coefficient for Plates in Compression; for 5 Configurations (2) (A, B, C, D and E) where Boundary Conditions of Unloaded Edges are: SS: Simply Supported, C: Clamped, and F: Free

compression (a > b), kc = 4, and for wide plate (a ≤ b) in compression, kc = (1 + b2 / a2)2, for simply supported edges. For shear force, the critical buckling shear stress, τB, can also be obtain by equation 51 and the buckling coefficient for simply supported edges is: kc = 5.34 + 4(b/a)2

[53]

Figure 18.42 presents, kc, versus the aspect ratio, a/b, for different configurations of rectangular plates in compression. For the simplified prediction of the plate ultimate strength under uniaxial compressive loads, one of the most common approaches is to assume that the plate will collapse if the maximum compressive stress at the plate corner reaches the material yield stress, namely σx max = σY for σxav or σy max = σY for σyav. This assumption is relevant when the unloaded edges move freely in plane as that shown in Figure 40(b). Another approximate method is to use the plate effective width concept, which provides the plate ultimate strength components


18-42


under uniaxial compressive stresses (σxu and σyu), as follow: σ yu σ xu b a = eu and = eu σY b σY a

[54]

where aeu and beu are the plate effective length and width at the ultimate limit state, respectively. While a number of the plate effective width expressions have been developed, a typical approach is exemplified by Faulkner, who suggests an empirical effective width (beu /b) formula for simply supported steel plates, as follows, • for longitudinal axial compression (34), 1 for β < 1 b eu  c2 =  c1 b  β − β 2 for β ≥ 1 

[55a]

• for transverse axial compression (35), a eu 0.9 b 1.9  0.9  = + 1−  a a β  β2 β2 

[55b]

where: σY β= b is the plate slenderness t E E = the Young’s modulus t = the plate thickness c1 , c2 = typically taken as c1 = 2 and c2 = 1 The plate ultimate strength components under uniaxial compressive loads are therefore predicted by substituting the plate effective width formulae (equation 55a) into equation 54. More charts and formulations are available in many books, for example, Bleich (36), ECCS-56 (37), Hughes (3) and Lewis (2). In addition, the design strength of plate (unstiffened panels) is detailed in Chapter 19, Subsection 19.5.4.1, including an example of reliability-based design and alternative equations to equations 56 and 57. 18.6.3.3 Design criteria When a single load component is involved, the buckling or ultimate strength must be greater than the corresponding applied stress component with an appropriate target partial safety factor. In a multiple load component case, the structural safety check is made with equation 48 against buckling and equation 50 against ultimate limit state being satisfied. To ensure that the possible worst condition is met (buckling and yield) for the ship, several stress combination must be considered, as the maximum longitudinal and transverse

compression do not occur simultaneously. For instance, DNV (4) recommends: • maximum compression, σx, in a plate field and phase angle associated with σy, τ (buckling control), • maximum compression, σy, in a plate field and phase angle associated with σx, τ (buckling control), • absolute maximum shear stress, τ, in a plate field and phase angle associated with σx, σy (buckling control), and • maximum equivalent von Mises stress, σe, at given positions (yield control). In order to get σx and σy, the following stress components may normally be considered for the buckling control: σ1 = stress from primary response, and σ2 = stress from secondary response (that is, double bottom bending). As the lateral bending effects should be normally included in the buckling strength formulation, stresses from local bending of stiffeners (secondary response), σ2*, and local bending of plate (tertiary response), σ3, must therefore not to be included in the buckling control. If FE-analysis is performed the local plate bending stress, σ3, can easily be excluded using membrane stresses. 18.6.4 Buckling and Ultimate Strength of Stiffened Panels For the structural capacity analysis of stiffened panels, it is presumed that the main support members including longitudinal girders, transverse webs and deep beams are designed with proper proportions and stiffening systems so that their instability is prevented prior to the failure of the stiffened panels they support. In many ship stiffened panels, the stiffeners are usually attached in one direction alone, but for generality, the design criteria often consider that the panel can have stiffeners in one direction and webs or girders in the other, this arrangement corresponds to a typical ship stiffened panels (Figure 18.43a). The stiffeners and webs/girders are attached to only one side of the panel. The number of load components acting on stiffened steel panels are generally of four types, namely biaxial loads, that is compression or tension, edge shear, biaxial in-plane bending and lateral pressure, as shown in Figure 18.43. When the panel size is relatively small compared to the entire structure, the influence of in-plane bending effects may be negligible. However, for a large stiffened panel such as that in side shell of ships, the effect of in-plane bending may not be negligible, since the panel may collapse by failure of stiff-



eners which are loaded by largest added portion of axial compression due to in-plane bending moments. When the stiffeners are relatively small so that they buckle together with the plating, the stiffened panel typically behaves as an orthotropic plate. In this case, the average values of the applied axial stresses may be used by neglecting the influence of in-plane bending. When the stiffeners are relatively stiff so that the plating between stiffeners buckles before failure of the stiffeners, the ultimate strength is eventually reached by failure of the most highly stressed stiffeners. In this case, the largest values of the axial compressive or tensile stresses applied at the location of the stiffeners are used for the failure analysis of the stiffeners. In stiffened panels of ship structures, material properties of the stiffeners including the yield stress are in some cases

18-43

different from that of the plate. It is therefore necessary to take into account this effect in the structural capacity formulations, at least approximately. For analysis of the ultimate strength capacity of stiffened panels which are supported by longitudinal girders, transverse webs and deep beams, it is often assumed that the panel edges are simply supported, with zero deflection and zero rotational restraints along four edges, with all edges kept straight. This idealization may provide somewhat pessimistic, but adequate predictions of the ultimate strength of stiffened panels supported by heavy longitudinal girders, transverse webs and deep beams (or bulkheads). Today, direct non-linear strength assessment methods using recognized programs is usual (38). The model should

(a) (a)

(b)

(b)

(c)

(d)

Figure 18.44 Modes of Failures by Buckling of a Stiffened Panel (2). (a) Elastic buckling of plating between stiffeners (serviceability limit state). (b) Flexural buckling of stiffeners including plating (plate-stiffener combination, Figure 18.43 A Stiffened Steel Panel Under Biaxial Compression/Tension,

mode III).

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

(c) Lateral-torsional buckling of stiffeners (tripping—mode V).

Panel—Longitudinals and Frames (4), and (b) A Generic Stiffened Panel (38).

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


18-44


be capable of capturing all relevant buckling modes and detrimental interactions between them. The fabrication related initial imperfections in the form of initial deflections (plates, stiffeners) and residual stresses can in some cases significantly affect (usually reduce) the ultimate strength of the panel so that they should be taken into account in the strength computations as parameters of influence. 18.6.4.1 Direct analysis The primary modes for the ultimate limit state of a stiffened panel subject to predominantly axial compressive loads may be categorized as follows (Figure 18.44): • Mode I: Overall collapse after overall buckling, • Mode II: Plate induced failure—yielding of the platestiffener combination at panel edges, • Mode III: Plate induced failure—flexural buckling followed by yielding of the plate-stiffener combination at mid-span, • Mode IV: Stiffener induced failure—local buckling of stiffener web, • Mode V: Stiffener induced failure—tripping of stiffener, and • Mode VI: Gross yielding. Calculation of the ultimate strength of the stiffened panel under combined loads taking into account all of the possible failure modes noted above is not straightforward, because of the interplay of the various factors previously noted such as geometric and material properties, loading, fabrication related initial imperfections (initial deflection and welding induced residual stresses) and boundary conditions. As an approximation, the collapse of stiffened panels is then usually postulated to occur at the lowest value among the various ultimate loads calculated for each of the above collapse patterns. This leads to the easier alternative wherein one calculates the ultimate strengths for all collapse modes mentioned above separately and then compares them to find the minimum value which is then taken to correspond to the real panel ultimate strength. The failure mode of stiffened panels is a broad topic that cannot be covered totally within this chapter. Many simplified design methods have of course been previously developed to estimate the panel ultimate strength, considering one or more of the failure modes among those mentioned above. Some of those methods have been reviewed by the ISSC’2000 (39). On the other hand, a few authors provide a complete set of formulations that cover all the feasible failure modes noted previously, namely, Dowling et al (40), Hughes (3), Mansour et al (41,42), and more recently Paik (38). Assessment of different formulations by comparison

with experimental and/or FE analysis are available (43-45). An example of reliability-based assessment of the stiffened panel strength is presented in Chapter 19. Formulations of Herzog, Hughes and Adamchack are also discussed. 18.6.4.2 Simplified models Existing simplified methods for predicting the ultimate strength of stiffened panels typically use one or more of the following approaches: • orthotropic plate approach, • plate-stiffener combination approach (or beam-column approach), and • grillage approach. These approaches are similar to those presented in Subsection 18.4.4.1 for linear analysis. All have the same background but, here, the buckling and the ultimate strength is considered. In the orthotropic plate approach, the stiffened panel is idealized as an equivalent orthotropic plate by smearing the stiffeners into the plating. The orthotropic plate theory will then be useful for computation of the panel ultimate strength for the overall grillage collapse mode (Mode I, Figure 18.44d), (31,46,48). The plate-stiffener combination approach (also called beam-column approach) models the stiffened panel behavior by that of a single “beam” consisting of a stiffener together with the attached plating, as representative of the stiffened panel (Figure 18.38, level 3b). The beam is considered to be subjected to axial and lateral line loads. The torsional rigidity of the stiffened panel, the Poisson ratio effect and the effect of the intersecting beams are all neglected. The beam-column approach is useful for the computation of the panel ultimate strength based on Mode III, which is usually an important failure mode that must be considered in design. The degree of accuracy of the beamcolumn idealization may become an important consideration when the plate stiffness is relatively large compared to the rigidity of stiffeners and/or under significant biaxial loading. Stiffened panels are asymmetric in geometry about the plate-plane. This necessitates strength control for both plate induced failure and stiffener-induced failure. Plate induced failure: Deflection away from the plate associated with yielding in compression at the connection between plate and stiffener. The characteristic buckling strength for the plate is to be used. Stiffener induced failure: Deflection towards the plate associated with yielding in compression in top of the stiffener or torsional buckling of the stiffener. Various column strength formulations have been used as



the basis of the beam-column approach, three of the more common types being the following: • Johnson-Ostenfeld (or Bleich-Ostenfeld) formulation, • Perry-Robertson formulation, and • empirical formulations obtained by curve fitting experimental or numerical data. A stocky panel that has a high elastic buckling strength will not buckle in the elastic regime and will reach the ultimate limit state with a certain degree of plasticity. In most design rules of classification societies, the so-called Johnson-Ostenfeld formulation is used to account for this behavior (equation 47). On the other hand, in the so-called Perry-Robertson formulation, the strength expression assumes that the stiffener with associated plating will collapse as a beam-column when the maximum compressive stress in the extreme fiber reaches the yield strength of the material. In empirical approaches, the ultimate strength formulations are developed by curve fitting based on mechanical collapse test results or numerical solutions. Even if limited to a range of applicability (load types, slenderness ranges, assumed level of initial imperfections, etc.) they are very useful for preliminary design stage, uncertainty assessment and as constraint in optimization package. While a vast number of empirical formulations (sometimes called column curves) for ultimate strength of simple beams in steel framed structures have been developed, relevant empirical formulae for plate-stiffener combination models are also available. As an example of the latter type, Paik and Thayamballi (49) developed an empirical formula for predicting the ultimate strength of a plate-stiffener combination under axial compression in terms of both column and plate slenderness ratios, based on existing mechanical collapse test data for the ultimate strength of stiffened panels under axial compression and with initial imperfections (initial deflections and residual stresses) at an average level. Since the ultimate strength of columns (σu) must be less than the elastic column buckling strength (σE), the Paik-Thayamballi empirical formula for a plate-stiffener combination is given by: σu = σY

1 0.995

+ 0.936 λ 2 + 0.17 β 2 + 0.188 λ 2 β 2 − 0.067 λ 4

18-45

and λ=

a πr

σ

Y = E

σY σE

where: r = radius 4 of gyration = √I / A, (m) I = inertia, (m4) A = cross section of the plate-stiffener combination with full attached plating, (m2) t = plate thickness, (m) a = span of the stiffeners, (m) b = spacing between 2 longitudinals, (m) Note that A, I, r, ... refer to the full section of the platestiffener combination, that is, without considering an effective plating. Figure 18.45 compares the Johnson-Ostenfeld formula (equation 47), the Perry-Robertson formula and the PaikThayamballi empirical formula (equation 56) for on the column ultimate strength for a plate-stiffener combination varying the column slenderness ratios, with selected initial eccentricity and plate slenderness ratios. In usage of the Perry-Roberson formula, the lower strength as obtained from either plate induced failure or stiffener-induced failure is adopted herein. Interaction between bending axial

[56]

and σu σ 1 ≤ 2 = E σY σY λ with

Figure 18.45 A Comparison of the Ultimate Strength Formulations for

b β= t

Y σ E

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


18-46


compression and lateral pressure can, within the same failure mode (Flexural Buckling—Mode III), leads to three-failure scenario: plate induced failure, stiffener induced failure or a combined failure of stiffener and plating (see Chapter 19 – Figure 19.11 ). 18.6.4.3 Design criteria The ultimate strength based design criteria of stiffened panels can also be defined by equation 50, but using the corresponding stiffened panel ultimate strength and stress parameters. Either all of the six design criteria, that is, against individual collapse modes I to VI noted above, or a single design criterion in terms of the real (minimum) ultimate strength components must be satisfied. For stiffened panels following Mode I behavior, the safety check is similar to a plate, using average applied stress components. The applied axial stress components for safety evaluation of the stiffened panel following Modes II–VI behavior will use the maximum axial stresses at the most highly stressed stiffeners. 18.6.5 Ultimate Bending Moment of Hull Girder Ultimate hull girder strength relates to the maximum load that the hull girder can support before collapse. These loads induce vertical and horizontal bending moment, torsional moment, vertical and horizontal shear forces and axial force. For usual seagoing vessels axial force can be neglected. As the maximun shear forces and maximum bending moment do not occur at the same place, ultimate hull girder strength should be evaluated at different locations and for a range of bending moments and shear forces. The ultimate bending moment (Mu) refers to a combined vertical and horizontal bending moments (Mv, Mh); the transverse shear forces (Vv,Vh) not being considered. Then, the ultimate bending moment only corresponds to one of the feasible loading cases that induce hull girder collapse. Today, Mu is considered as being a relevant design case. Two major references related to the ultimate strength of hull girder are, respectively, for extreme load and ultimate strength, Jensen et al (24) and Yao et al (50). Both present

comprehensive works performed by the Special Task Committees of ISSC 2000. Yao (51) contains an historical review and a state of art on this matter. Computation of Mu depends closely on the ultimate strength of the structure’s constituent panels, and particularly on the ultimate strength in compressed panels or components. Figure 18.46 shows that in sagging, the deck is compressed (σdeck) and reaches the ultimate limit state when σdeck = σu. On the other hand, the bottom is in tensile and reaches its ultimate limit state after complete yielding, σbottom = σ0 (σ0 being the yield stress). Basically, there exist two main approaches to evaluate the hull girder ultimate strength of a ship’s hull under longitudinal bending moments. One, the approximate analysis, is to calculate the ultimate bending moment directly (Mu, point C on Figure 18.46), and the other is to perform progressive collapse analysis on a hull girder and obtain, both, Mu and the curves M-φ. The first approach, approximate analysis, requires an assumption on the longitudinal stress distribution. Figure 18.47 shows several distributions corresponding to different methods. On the other hand, the progressive collapse analysis does not need to know in advance this distribution. Accordingly, to determine the global ultimate bending moment (Mu), one must know in advance • the ultimate strength of each compressed panel (σu), and • the average stress-average strain relationship (σ−ε), to perform a progressive collapse analysis. For an approximate assessment, such as the Caldwell method, only the ultimate strength of each compressed panel (σu) is required. 18.6.5.1 Direct analysis The direct analysis corresponds to the Progressive collapse analysis. The methods include the typical numerical analy-

(a)

(b)

(c)

(d)

(e)

(f)

Figure 18.47 Typical Stress Distributions Used by Approximate Methods. (a) First Yield. (b) Sagging Bending Moment (c) Evans (d) Paik—Mansour (e) Figure 18.46 The Moment-Curvature Curve (M-Φ)

Caldwell Modified (f) Plastic Bending Moment.



sis such as Finite Element Method (FEM) and the Idealized structural Element method (ISUM) and Smith’s method, which is a simplified procedure to perform progressive collapse analysis. FEM: is the most rational way to evaluate the ultimate hull girder strength through a progressive collapse analysis on a ship’s hull girder. Both material and geometrical nonlinearities can be considered. A 3D analysis of a hold or a ship’s section is fundamentally possible but very difficult to perform. This is because a ship’s hull is too large and complicated for such kind of analysis. Nevertheless, since 1983 results of FEM analyses have been reported (52). Today, with the development of computers, it is feasible to perform progressive collapse analysis on a hull girder subjected to longitudinal bending with fine mesh using ordinary elements. For instance, the investigation committee on the causes of the Nakhodka casualty performed elastoplastic large deflection analysis with nearly 200 000 elements (53). However, the modeling and analysis of a complete hull girder using FEM is an enormous task. For this reason the analysis is more conveniently performed on a section of the hull that sufficiently extends enough in the longitudinal direction to model the characteristic behavior. Thus, a typical analysis may concern one frame spacing in a whole compartment (cargo tank). These analyses have to be supplemented by information on the bending and shear loads that act at the fore and aft transverse loaded sections. Such Finite Element Analysis (FEA) has shown that accuracy is limited because of the boundary conditions along the transverse sections where the loading is applied, the position of the neutral axis along the length of the analyzed section and the difficulty to model the residual stresses. Idealized Structural Unit Method (ISUM): presented in Subsection 18.7.3.1, can also be used to perform progressive collapse analysis. It allows calculating the ultimate bending moment through a 3D progressive collapse analysis of an entire cargo hold. For that purpose, new elements to simulate the actual collapse of deck and bottom plating are actually underdevelopment. Smith’s Method (Figure 18.48): A convenient alternative to FEM is the Smith’s progressive collapse analysis (54), which consists of the following three steps (55). Step 1: Modeling (mesh modeling of the cross-section into elements), Step 2: Derivation of average stress-average strain relationship of each element (σ−ε curve), Figure 18.49a. Step 3: To perform progressive collapse analysis, Figure 18.49b.

18-47

Figure 18.48 The Smith’s Progressive Collapse Method

(a)

(b)

Figure 18.49 Influence of Element Average Stress-Average Strain Curves (σ−ε) on Progressive Collapse Behavior. (a) Average stress-average strain relationships of element, and (b) moment-curvature relationship of crosssection.


18-48


In Step 1, the cross-section of a hull girder is divided into elements composed of a longitudinal stiffener and attached plating. In Step 2, the average stress-average strain relationship (σ−ε) of this stiffener element is derived under the axial load considering the influences of buckling and yielding. Step 3 can be explained as follows: • axial rigidities of individual elements are calculated using the average stress-average strain relationships (σ−ε), • flexural rigidity of the cross-section is evaluated using the axial rigidities of elements, • vertical and horizontal curvatures of the hull girder are applied incrementally with the assumption that the plane cross-section remains plane and that the bending occurs about the instantaneous neutral axis of the cross-section, • the corresponding incremental bending moments are evaluated and so the strain and stress increments in individual elements, and • incremental curvatures and bending moments of the cross-section as well as incremental strains and stresses of elements are summed up to provide their cumulative values. Figure 18.48 shows that the σ−ε curves are used to estimate the bending moment carried by the complete transverse section (Mi). The contribution of each element (dM) depends on its location in the section, and specifically on its distance from the current position of the neutral axis (Yi). The contribution will then also depend on the strain that is applied to it, since ε = –y φ, where φ is the hull curvature and y is the distance from the neutral axis (simple beam assumption). The average stress-average strain curve (σ-ε) will then provide an estimate of the longitudinal stress (σi) acting on the section. Individual moments about the neutral axis are then summed to give the total bending moment for a particular curvature φi. The accuracy of the calculated ultimate bending moment depends on the accuracy of the average stress-average strain relationships of individual elements. Main difficulties concern the modeling of initial imperfections (deflection and welding residual stress) and the boundary conditions (multi-span model, interaction between adjacent elements, etc.). Many formulations and methods to calculate these average stress-average strain relationships are available: Adamchack (56), Beghin et al (57), Dow et al (58), Gordo and Guedes Soares (59,60) and, Yao and Nikolov (61,62). The FEM can even be used to get these curves (Smith 54). For most of the methods, typical element types are: plate element, beam-column element (stiffener and attached plate) and hard corner.

An interesting well-studied ship that reached its ultimate bending moment is the Energy Concentration (63). It frequently is used as a reference case (benchmark) by authors to validate methods. Figure 18.49 shows typical average stress-average strain relationships, and the associated bending moment-curvature relationships (M-φ). Four typical σ−ε curves are considered, which are: Case A: Linear relationship (elastic). The M-φ relationship is free from the influences of yielding and buckling, and is linear. Case B: Bi-linear relationship (elastic-perfectly plastic, without buckling). Case C: With buckling but without strength reduction beyond the ultimate strength. Case D: With buckling and a strength reduction beyond the ultimate strength (actual behavior). In Case B, where yielding takes place but no buckling, the deck initially undergoes yielding and then the bottom. With the increase in curvature, yielded regions spread in the side shell plating and the longitudinal bulkheads towards the plastic neutral axis. In this case, the maximum bending moment is the fully plastic bending moment (Mp) of the cross-section and its absolute value is the same both in the sagging and the hogging conditions. For Cases C and D, the element strength is limited by plate buckling, stiffener flexural buckling, tripping, etc. For Case C, it is assumed that the structural components can continue to carry load after attaining their ultimate strength. The collapse behavior (M-φ curve) is similar to that of Case B, but the ultimate strength is different in the sagging and the hogging conditions, since the buckling collapse strength is different in the deck and the bottom. Case D is the actual case; the capacity of each structural member decreases beyond its ultimate strength. In this case, the bending moment shows a peak value for a certain value of the curvature. This peak value is defined as the ultimate longitudinal bending moment of the hull girder (Mu). Shortcomings and limitations of the Smith’s method relates to the fact that a typical analysis concerns one frame spacing of a whole cargo hold and not a complete 3D hold. As simple linear beam theory is used, deviations such as shear lag, warping and racking are thus ignored. This method may be a little un-conservative if the structure is predominantly subjected to lateral pressure loads as well as axial compression, and if it is not realized that the transverse frames can deflect/fail and significantly affect the stiffened plate structure and hull girder bending capacity.



18.6.5.2 Simplified models Caldwell (64) was the first who tried to theoretically evaluate the ultimate hull girder strength of a ship subjected to longitudinal bending. He introduced a so-called Plastic Design considering the influence of buckling and yielding of structural members composing a ship’s hull (Figure 18.47). He idealised a stiffened cross-section of a ship’s hull to an unstiffened cross-section with equivalent thickness. If buckling takes place at the compression side of bending, compressive stress cannot reach the yield stress, and the fully plastic bending moment (Mp) cannot be attained. Caldwell introduced a stress reduction factor in the compression side of bending, and the bending moment produced by the reduced stress was considered as the ultimate hull girder strength. Several authors have proposed improvements for the Caldwell formulation (65). Each of them is characterized by an assumed stress distribution (Figure 18.47). Such methods aim at providing an estimate of the ultimate bending moment without attempting to provide an insight into the behaviour before, and more importantly, after, collapse of the section. The tracing out of a progressive collapse curve is replaced by the calculation of the ultimate bending moment for a particular distribution of stresses. The quality of the direct approximate method is directly dependent on the quality of the stress distribution at collapse. It is assumed that at collapse the stresses acting on the members that are in tension are equal to yield throughout whereas the stresses in the members that are in compression are equal to the individual inelastic buckling stresses. On this basis, the plastic neutral axis is estimated using considerations of longitudinal equilibrium. The ultimate bending moment is then the sum of individual moments of all elements about the plastic neutral axis. In Caldwell’s Method, and Caldwell Modified Methods, reduction in the capacity of structural members beyond their ultimate strength is not explicitly taken into account. This may cause the overestimation of the ultimate strength in general (Case C, Figure 18.49). Empirical Formulations: In contrast to all the previous rational methods, there are some empirical formulations usually calibrated for a type of specific vessels (66,67). Yao et al (50), found that initial yielding strength of the deck can provide in general a little higher but reasonably accurate estimate of the ultimate sagging bending moment. On the other hand, the initial buckling strength of the bottom plate gives a little lower but accurate estimate of the ultimate hogging bending moment. These in effect can provide a first estimate of the ultimate hull girder moment. Interactions: In order to raise the problem of combined loads (vertical and horizontal bending moments and shear forces), several authors have proposed empirical interac-

18-49

tion equations to predict the ultimate strength. Each load component is supposed to act separately. These methods were reviewed by ISSC (68) and are often formulated as equation 57.  Mv   M vu

a  Mh   + α  M hu 

b

  =1 

[57]

where: Mv and Mh = vertical and horizontal bending moments Mvu and Mhu = ultimate vertical and horizontal bending moments a, b and α = empirical constants For instance, Mansour et al (47) proposes a=1, b=2 and α= 0.8 based on analysis on one container, one tanker and 2 cruisers, and Gordo and Soares (60) 1.5

18-50


TABLE 18.IV Sensitivity Factors for Ultimate Strength Assessment of Hull Girder. Model Capability Plate buckling

Impact H

Stiffened plate buckling

H

Post buckling behavior

H

Plate welding residual stress

H

M-φ curve (post collapse prediction)

H

Plate initial deflection

M

Stiffener initial deflection

M

Stiffener welding residual stress

M

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

H

ing, σ−ε curves, curvature incrementation), the global uncertainty on the ultimate bending moment is usually large (55). A bias of 10 to 15% must be considered as acceptable. For intact hull the design criteria for Mu, defined by classification societies, is given by: MS + s1 Mw ≤ s2 MU

but they are time consuming and there is large uncertainty of using simplified methods. With the introduction of higher tensile steels in hull structures, at first in deck and bottom to increase hull girder strength, and later in local structures, the fatigue problem became more imminent. The fatigue strength does not increase according to the yield strength of the steel. In fact, fatigue is found to be independent of the yield strength. The higher stress levels in modern hull structures using higher tensile steel have therefore led to a growing number of fatigue crack problems. To ensure that the structure will fulfill its intended function, fatigue assessment should be carried out for each individual type of structural detail that is subjected to extensive dynamic loading. It should be noted that every welded joint and attachment or other form of stress concentration is potentially a source of fatigue cracking and should be individually considered. This section gives an overview of feasible analysis to be performed. A more complete description of the different fatigue procedures, S-N curves, stress concentration factors, and so on, are given in: Almar-Naess (73), DNV (4), Fricke et al (74), Maddox (75), Niemi (76), NRC (77) and Petershagen et al (78). Reliability-based fatigue procedure is presented by Ayyub and Assakkaf in Chapter 19. These authors also have contributed to this section.

[58]

where:

18.6.6.2 Basic fatigue theories Fatigue analyses can be performed based on:

s1 = the partial safety factor for load (typically 1.10) s2 = the material partial safety factor (typically 0.85) MS = still water moment Mw = design wave moment (20 year return period)

• simplified analytical expressions, • more refined analysis where loadings/load effects are calculated by numerical analysis, and • a combination of simplified and refined techniques.`

18.6.6 Fatigue and Fracture 18.6.6.1 General Design criteria stated expressly in terms of fatigue damage resistance were in the past seldom employed in ship structural design although cumulative fatigue criteria have been used in offshore structure design. It was assumed that fatigue resistance is implicitly included in the conventional safety factors or acceptable stress margins based on past experience. Today, fatigue considerations become more and more important in the design of details such as hatch corners, reinforcements for openings in structural members and so on. Since the ship-loading environment consists in large part of alternating loads, ship structures are highly sensitive to fatigue failures. Since 1990, fatigue is maybe the most sensitive point at the detailed design stage. Tools are available

There are generally two major technical approaches for fatigue life assessment of welded joints the Fracture Mechanics Approach and the Characteristic S-N Curves Approach. The Fracture Mechanics Approach is based on crack growth data assuming that the crack initiation already exists. The initiation phase is not modeled as it is assumed that the lifetime can be predicted only using fracture mechanics method of the growing cracks (after initiation). The fracture mechanics approach is obviously more detailed than the S-N curves approach. It involves examining crack growth and determining the number of load cycles that are needed for small initial defects to grow into cracks large enough to cause fractures. The growth rate is proportional to the stress range, S (or ∆σ) that is expressed in terms of a stress intensity factor, K, which accounts for the magnitude of the stress, current crack size, and weld and joint details. The



basic equation that governs crack growth (79) is known as the Paris Law is: da = C . ( ∆K) m dN

[59]

where: a = crack size, N = number of fatigue cycles (fatigue life), ∆K = S.Y(a) . π a , range of stress intensity factor, (Kmax – Kmin) C, m = crack propagation parameters, S = constant amplitude stress range, = ∆σ = σmax – σmin Y(a) = function of crack geometry. Fatigue life prediction based on the fracture mechanics approach shall be computed according to the following equation: N=

1 C . Sm

a

∫a

0

da Ym

logN = log (∆A) – m log (∆σ)

∆A m

k Sm S e

[62]

where: ∆ = fatigue damage ratio (≤ 1) log(∆A) = intercept of the S-N curve of the Log N axis –1 / m = slope of the S-N curve, (≅3 ≤ m ≤ ≅7) – Se= mean of the Miner’s equivalent stress range Se, defined at Table 18.V kS = fatigue stress uncertainty factor – ∆σ = kS. Se (or the constant amplitude stress range for failure at N cycles) N = fatigue life, or number of loading cycles expected during the life of a detail The Miner’s equivalent stress range, Se, can be evaluated based on the models provided in Table 18.V (83). The most refined model would start with a scatter diagram of sea-states, information on ship’s routes and operating char-

[60]

Equation 60 involves a variety of sources of uncertainty and practical difficulties to define, for instance, the a and ao crack size. The crack propagation parameter C in this equation is treated as random variable (80). However, in more sophisticated models, equation 60 is treated as a stochastic differential equation and C is allowed to vary during the crack growth process. State of art on the Fracture Mechanics Approach is available in Niemi (76) and Harris (81). The characteristic S-N curves approach is based on fatigue test data (S-N curves—Figure 18.50) and on the assumption that fatigue damage accumulation is a linear phenomenon (Miner’s rule). According to Miner (82) the total fatigue life under a variety of stress ranges is the weighted sum of the individual lives at constant stress range S as given by the S-N curves (Figure 18.50), with each being weighted according to fractional exposure to that level of stress range. The S-N curve approach related mainly to the crack initiation and a maximum allowable crack size. After, cracks propagate based on the fracture mechanics concept as shown in Figure 18.51. The propagation is not explicitly considered by the S-N curve approach. Fatigue life strength prediction based on both the S-N approach and Miner’s cumulative damage shall be evaluated with equation 61 or, in logarithmic form, with equation 62 (Figure 18.50). N=

18-51

[61]

Figure 18.50 A Typical S-N Curve

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


18-52


acteristics, and use of a ship response computer program to provide a detailed history of stress ranges over the service life of the ship. For such model, the wave exceedance diagram (deterministic method) and the spectral method (probabilistic method) can be employed (Table 18.V). S-N curves are obtained from fatigue tests and are available in different design codes for various structural details in bridges, ships, and offshore structures. The design S-N curves are based on the mean-minus-two-standard-deviation curves for relevant experimental data (Figure 18.50). They are thus associated with a 97.6% probability of survival. Some classification societies use 90%. In practice, the actual probabilities of failure associated with fatigue design lives is usually higher due to uncertainties associated with the calculated stresses, the various S-N curve correction factors, and the critical value of the cumulative fatigue damage ratio, ∆. Cumulative damage: The damage may either be calculated on basis of the long-term stress range distribution using Weibull parameters (simplified method), or on summation of damage from each short-term distribution in the scatter diagram (probabilistic and deterministic methods, Table 18.V). The stress range (S or ∆σ): The procedure for the fatigue analysis is based on the assumption that it is only necessary to consider the ranges of cyclic principal stresses in determining the fatigue endurance. However, some reduction in the fatigue damage accumulation can be credited when parts of the stress cycle range are in compression. Fatigue areas: The potential for fatigue damage is dependent on weather conditions, ship type, corrosion level, location on ship, structural detail and weld geometry and workmanship. The potential danger of fatigue damage will also vary according to crack location and number of potential damage points. Fatigue strength assessment shall normally be carried out for: • longitudinal and transverse element in: — bottom/inner bottom (side), — longitudinal and transverse bulkheads. • strength deck in the midship region and forebody, and • other highly stressed structural details in the midship region and forebody, like panel knuckles. Time at sea: Vessel response may differ significantly for different loading conditions. It is therefore of major importance to include response for actual loading conditions. Since fatigue is a result of numerous cyclic loads, only the most frequent loading conditions are included in the fatigue analysis. These will normally be ballast and full load condition. Under certain circumstances, other loading conditions may be used.

Environmental conditions: The long-term distribution of load responses for fatigue analyses may be estimated using the wave climate, represented by the distribution of Hs and Ts, representing the sea operation conditions. As guidance to the choice between these data sets, one should consider the average wave environment the vessel is expected to encounter during its design life. The world wide sailing routes will therefore normally apply. For shuttle tankers and vessels that will sail frequently on the North Atlantic, or in other harsh environments, the wave data given in accordance with this should be applied. For vessels that will sail in more smooth sailing routes, less harsh environmental data may be applied. This should be decided upon for each case. Geometrical imperfections: The fatigue life of a welded joint is much dependent on the local stress concentrations factors arising from surface imperfections during the fabrication process, consisting of weld discontinuities and geometrical deviations. Surface weld discontinuities are weld toe undercuts, cracks, overlaps, incomplete penetration, etc. Geometrical imperfections are defined as misalignment, angular distortion, excessive weld reinforcement and otherwise poor weld shapes. Effect of grinding of welds: For welded joints involving potential fatigue cracking from the weld toe an improvement in strength by a factor of at least 2 on fatigue life can be obtained by controlled local machining or grinding of the weld toe. Note that grinding of welds should not be used as a “design tool”, but rather as a mean to lower the fatigue damage when special circumstances have made it necessary. This should be used as a reserve if the stress in special areas turns out to be larger than estimated at an earlier stage of the design. 18.6.6.3 Stress concentration and hot spot stress The stress level obtained from a structural analysis, such as FEA, will depend on the fineness of the model. The different analysis models described in Subsection 18.7.2 will therefore lead to different levels of result processing in order to complete the fatigue calculations. In order to correctly determine the stresses to be used in fatigue analyses, it is important to note the definition of the different stress categories (Figure 18.52). Nominal stresses are those, typically, derived from coarse mesh FE models. Stress concentrations resulting from the gross shape of the structure, for example, shear lag effects, have to be included in the nominal stresses derived from stress analysis. Geometric stresses include nominal stresses and stresses due to structural discontinuities and presence of attachments, but excluding stresses due to presence of welds.



Stresses derived from fine mesh FE models are geometric stresses. Effects caused by fabrication imperfections as misalignment of structural parts, are normally not included in FEA, and must be separately accounted for, using, for instance (equation 65). Hot spot stress is the greatest value of the extrapolation to the weld toe of the geometric stress distribution immediately outside the region affected by the geometry of the weld (Figure 18.52). Notch stress is the total stress at the weld toe (hot spot location) and includes the geometric stress and the stress due to the presence of the weld. The notch stress may be calculated by multiplying the hot spot stress by a stress concentration factor, or more precisely the theoretical notch factor, K2 (equation 65). FE may be used to directly determine the notch stress. However, because of the small notch radius and the steep stress gradient at a weld, a very fine mesh is needed. In practice, the stress concentration factors (K-factors) may be determined based on fine mesh FE analyses, or, alternatively, from the selection of factors for typical details. The notch stress range governs the fatigue life of a detail. For components other than smooth specimens the notch stress is obtained by multiplication of the nominal stress by K-factors (equation 63). The K-factors in this document are thus defined as K=

σ notch σ nominal

[63]

The relation between the notch stress range to be used together with the S-N-curve and the nominal stress range is S = ∆σ = ∆σ notch = K . ∆σ nominal

[64]

All stress risers have to be considered when evaluating

Figure 18.52 Definition of Stress Categories (4)

18-53

the notch stress. This can be done by multiplication of Kfactors arising from different causes. The resulting K-factor to be used for calculation of notch stress is: K = K1 . K2 . K3 . K4 . K5

[65]

where: K1 = stress concentration factor due to the gross geometry of the detail considered K2 = stress concentration factor due to the weld geometry (notch factor); K2 = 1.5 if not stated otherwise K3 = additional stress concentration factor due to eccentricity tolerance K4 = additionally stress concentration factor due to angular mismatch K5 = additional stress concentration factor for un-symmetrical stiffeners on laterally loaded panels, applicable when the nominal stress is derived from simple beam analyses Fatigue cracks are assumed to be independent of principal stress direction within 45° of the normal to the weld toe. Hot spot stress extrapolation procedure: The hot spot stress extrapolation procedure (Figure 18.52) is only to be used for stresses that are derived from stress concentration models (fine mesh). Nominal stresses found from other models should be multiplied with appropriate stress concentration factors (equation 65). The stress extrapolation procedure is specific to each classification societies (74). Today, there is unfortunately no standard procedure. 18.6.6.4 Direct analysis Several S-N fatigue approaches exists, they all have advantages and disadvantages. The different approaches are therefore suitable for different areas. Load effects, accuracy of the analysis, computer demands, etc. should be evaluated before one of the approaches is chosen. Full stochastic fatigue analysis: The full stochastic analysis, for example the Spectral Model of Table 18.V, is an analysis where all load effects from global and local loads, are included. This is ensured by use of stress concentration models and direct load transfer to the structural model. Hence, all stress components are combined using the correct phasing and without simplifications or omissions of any stress component. This method usually will be the most exact for determination of fatigue damage and will normally be used together with fine meshed stress concentration models. The method may, however, not be suitable when non-linearities in the loading are of importance (side longitudinals). This is especially the case for areas where wave or tank pressures in the surface region are of major importance. This is due to


18-54


TABLE 18.V Commonly Used Expressions for Evaluating Miner’s Equivalent Stress Range (Se), (83) 1. Wave Exceedance Diagram (Deterministic Method)

S em =

nb

∑ f i S im

→ Se = m

nb

∑ f i S im i

i

Si = stress range

tions by use of load/stress ratios, Hi (equation 66). The load transfer functions, Hi, normally include the global hull girder bending sectional forces and moments, the pressures for all panels of the 3-D diffraction model, the internal tank pressures. The stress transfer functions, Hi, are combined to a total stress transfer function, Hσ, by a linear complex summation of the different transfer functions (4), as: Hσ =

Fi = fraction of cycles in the ith stress block

where:

2. Spectral Method (Probabilistic Method)

(2 2 )

m

f0

m Γ  + 1  2 

∑ γ i f i σ im i

λ(m) = rainflow correction Γ(.) = gamma function γι = fraction of time in ith sea-state fi = frequency of wave loading in ith sea-state σι = RMS of stress process in ith sea-state 3. Weibull Model for Stress Ranges (Simplified Method)

S em =

nb

∑

f i S im → S e = m

i

[66]

i

nb = number of stress block

S em = λ ( m )

∑ AiHi

nb

∑ f i S im i

Sd = stress range that is exceeded on the average once out of Nd stress cycles Γ(.) = gamma function k = Weibull shape parameter Nd = total number of stress ranges in design life

the fact that all load effects result in one set of combined stresses, making it difficult to modify the stress caused by one of the load effects. The approach is suitable for areas where the stress concentration factors are unknown (knuckles, bracket and flange terminations of main girder, stiffeners subjected to large relative deformations). 18.6.6.5 Simplified models The stress component based stochastic fatigue analysis: The idea of the stress component based fatigue analysis is to change the direct load transfer functions calculated from the hydrodynamic load program into stress transfer func-

Ai = stress per unit axial force defined as the local stress response in the considered detail due to a unit sectional load for load component i. Ησ = total transfer function for the combined local stress, Hi = transfer function for the load component i, that is, axial force, bending moments, twisting and lateral load. This approach enables the use of separate load factors on each load component and thus includes loads non-linearities. Few load cases have to be analyzed and it is possible to use simplified formulas for the area of interest but errors are easily made in the combination of stresses, manual definition of extra load cases may cause errors and simplifications are usually made in loading. Suitable areas are components where geometric stress concentration factors, K1, are available (longitudinals, plating, cut-outs and standard hopper knuckles) and areas where side pressure is of importance. The simplified design wave approach (Weibull Model, Table 18.V) is a simplification to the previous component based stochastic fatigue analyses. In this simplified approach, the extreme load response effect over a specified number of load cycles, for example, 104 cycles, is determined. The resulting stress range, ∆σ, is then representative for the stress at a probability level of exceedance of 10-4 per cycle. The derived extreme stress response is combined with a calculated Weibull shape parameter, k, to define the long-term stress range distribution (Table 18.V). The Weibull shape parameter, k, for the stress response should be determined from the long-term distribution of the dominating load calculated in the hydrodynamic analysis. This simplified approach only requires the consideration of one load case. It is easy and fast to perform but it can only be used if one load dominates the response and the results are very sensitive to selection of design wave. Suitable areas concern components where one load is dominating the response, that is, deck areas and other areas without local loading.



18.6.6.6 Design criteria The standard fatigue design criterion is basically the expected lifetime before that significant damage appears (cracks). It usually is taken as being 20 years. Then, the designer’s target is to design structural details for which the fatigue failure happens after, for instance, 20 years. If it happens before, the fixing cost is very high and induces owner losses. If the first failure only happens after 30 years or later, the structural detail scantlings were globally overestimated, the hull weight too high and, therefore, that the owner had lost payload during 20 years. Partial safety factors, additional stress concentration factors and the stress extrapolation procedure are typically defined by the classifications societies. 18.6.7 Collision and Grounding 18.6.7.1 Present design approaches The OPA 90 and equivalent IMO requirements must be satisfied in structural design of ships carrying dangerous or pollutant cargoes, for example, chemicals, bulk oil, liquefied gas. The primary requirements are to arrange a double bottom of a required minimum height, and double sides of a required minimum width. In this context, to reduce the outflow of pollutant cargoes in ship collision or grounding accident, OPA 90 and IMO both require that the minimum vertical height, h, of each double bottom ballast tank or void space is not to be less than 2.0 m or B/15 (B = ship’s beam), whichever is the lesser, but in no case is the height to be less than 1.0 m. OPA and IMO also require that the minimum width, w, of each wing ballast tank or void space is not to be less than 0.5+DWT/20 000 (m) or w =2.0 (m), whichever is the lesser, where DWT is the deadweight of the ship in tonnes. In no case is w to be less than 1.0 (m). More detailed information is available in Chapter 29 on Oil Tanker. 18.6.7.2 Direct analysis To reduce the probability of outflow of hazardous cargo in ship collisions and grounding, the kinetic energy loss during the accident should be entirely absorbed by damage of outer structures, that is, before the inner shell in contact with the cargo can rupture. Of crucial importance, then, is how to arrange or make the scantlings of strength members in the implicated ship structures such that the initial kinetic energy is effectively consumed and the structural performance against an accident will be maximized. For this purpose, the structural crashworthiness of ships in collisions and grounding must be analyzed using accurate and efficient procedures (84). Figure 18.53 shows direct design procedures of ship

18-55

structures against collision and grounding (85). For the accidental limit state design, the integrity of a structure can be checked in two steps. In the first step, the structural performance against design accident events will be assessed, while post-accident effects such as likely oil outflow are evaluated in the second step. The primary concern of the accidental limit state design in such cases is to maintain the water tightness of ship compartments, the containment of dangerous or pollutant cargoes, and the integrity of critical spaces (reactor compartments of nuclear powered ships or tanks in LNG ships) at the greatest possible levels, and to minimize the release/outflow of cargo. To facilitate a rescue mission, it is also necessary keep the residual strength of damaged structures at a certain level, so that the ship can be towed to safe harbor or a repair yard as may be required. 18.6.7.3 Simplified models Since the response of ships in collision or grounding accident includes relatively complicated behavior such as crushing, tearing and yielding, existing simplified methods are not always adequate. However, many simplified models useful for predicting accident induced structural damages and residual strength of damaged ship structures have been developed and continue to be successfully used. Simplified models for collision are rather different from those of grounding since both are different in the nature of the mechanics involved. As it is impossible to describe them in a limited space, valuable references are Ohtsubo et al (86), and Kaminski et al (39). 18.6.7.4 Design criteria The structural design criteria for ship collisions and grounding are based on limiting accidental consequences such as structural damage, fire and explosion, and environmental pollution, and to make sure that the main safety functions of ship structures are not impaired to a significant extent during any accidental event or within a certain time period thereafter. Structural performance of a ship against collision or grounding can be measured by: • energy absorption capability, • maximum penetration in an accident, • spillage amount of hazardous cargo, for example, crude oil, and • hull girder ultimate strength of damaged ships (Section 18.6.5). Design acceptance criteria may be based on the following parameters (87):


18-56


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

• minimum distance of cargo containment from the outer shell, • ship speed above which a critical event (breaching of cargo containment) happens, • allowable quantity of oil outflow, and • minimum values of section modulus or ultimate hull girder strength. And the design results must satisfy: • cargo tanks/holds are not breached in an accident so that there will be no danger of pollution, or • if the cargo tanks are breached, the oil outflow following an accident is limited, and/or • the ship has adequate residual hull girder strength so that it will survive an accident and will not break apart, minimizing a second chance of pollution.

18.6.8 Vibration 18.6.8.1 Present Vibration Design Approaches The traditional design methodology for vibration is based on rules, defined by classification societies. Vibrations are not explicitly covered by class rules but their prediction is needed to achieve a good design. Ship structures are excited by numerous dynamic oscillating forces. Excitation may originate within the ship or outside the ship by external forces. Reciprocating machinery such as large main propulsion diesel produce important forces at low frequency. Pressure fluctuations due to propeller at blade rate frequency induce pressure variation on the ship’s hull. Varying hull pressures associated with waves belong also to external excitations. All these forces can be approximated by a combination of harmonic forces. If their frequencies coincide with the structure eigen frequencies, resonant behavior will happen.



It is of prime importance to avoid global main hull vibrations. If they do occur, the remedial action will probably be very costly. So, during early design, the hull girder frequencies must be compared to wave excitation (springing risk), and to propeller and engine excitation. Table 18.VI gives some typical values of the first hull girder frequencies in Hz of some ship types. Hull girder frequencies and modes should be computed using approximate empirical formulae (88), simple beam models for long prismatic structures (VLCC, container ships, etc.) associated with lumped added mass models, or using 3D finite element models for complex ships (RO-RO, cruise ship), LNG, and short and non-prismatic structures (tug, catamaran, etc.). 18.6.8.2 Fluid structure interaction Fluid structure interaction is evidenced in the dynamic behavior of ships. As a first approximation, the ship is considered as a rigid body, for the sea keeping analyses (wave induced motions and loads). Wave vibration induced: An early determination of hull girder vibration modes and frequencies is important to avoid serious problems that would be difficult to solve at a later stage of the project. Risk of springing (occurring when first hull girder frequency equals wave encounter frequency) has to be detected very early. Springing may occur for long and/or flexible ships and for high speed craft and it increases the number of cyclic loads contributing to human fatigue. Various methods to assess the first hull girder frequency can be used at preliminary design stage. Engine/propeller vibration induced: Resonance problems may also appear on small ships like tugs, where hull girder frequency can be close to the propulsion excitation (around 7Hz). High vibration levels contribute to human fatigue and dysfunction, besides the discomfort aspect. Fluid added mass: Hull girder vibrations induce dis-

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placement of the surrounding fluid. Therefore imparting kinetic energy in the fluid. This phenomenon can be taken into account for the hull girder modes and frequencies calculation as added mass terms. Various methods can be used for the determination of added mass term. Lumped mass approach is the simplest one (89) but is only valid for simple prismatic slender shapes, and for a single mode. Fluid finite and semi-infinite elements or boundary integral formulation lead to the calculation of more accurate added mass matrices (90), especially for complex hull forms and appendices study (rudder). Added mass matrices associated with 3D finite element model of the structure, allow for an accurate determination of hull girder modes and frequencies. Added mass terms may also be needed for the vibrations of tank walls. The corresponding methods and associated software are available for industrial usage (Figure 18.54) and numerical simulations are today predictable with good accuracy (91). Figure 18.54 shows a fluid-structure coupled FE-model of a 230 m long passenger vessel using 150 000 degrees of freedom. A difficult coupled problem is the fluid impact occurring in slamming or due to sloshing in tanks. The local deformation of the impacted shells and plating influences the

TABLE 18.VI Typical Values of the First Hull Girder Frequencies (in Hertz) Order (mode)

Large Cruise ship

1

1.0 Hz

1.8

2

1.5 Hz

2.9

3

2.6 Hz

4

3.2 Hz

Fast monohull LNG

VLCC

Frigate

Tug

0.9

0.8

1.9

2.0

1.7

3.8

—

—

—

5.8

—

—

—

—

7.8

—

7.0 13 Figure 18.54 Fluid/Structure FE-Model of a Passenger Vessel (Principia Marine, France)


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pressures and fluid velocities. Moreover, air trapped in such an impact may have a cushioning effect, softening its severity. The numerical simulation of those heavily coupled problems still belongs to the research domain, though its industrial importance for the design of ship structures (92). 18.6.8.3 Direct analysis Vibration problems are critical for passenger ships with typically a 12-Hertz blade excitation. Ship owners demand very low vertical velocity levels incabins and public areas (less than 1.2 mm/s in the 5-25 Hz frequency band). Numerical simulation using 3D finite element models is the only method to predict ship response (including the various frequency modes) to pressure fluctuation on the ship hull. Such simulation is now used as a design tool to select appropriate scantlings of decks, location of pillars, detect possible resonance, and select the number of propeller blades. The main difficulty is to perform this analysis early enough in a very short design cycle. Local analyses also have to be performed, based on finite element models to check the potential risk of vibration of local areas, when local modes can be considered as decoupled from global hull girder modes. Decks, superstructure, appendices (rudder, radar mast, etc.) can be analyzed to check scantling and avoid the risk of resonance. Slamming impacts generate impulsive response of the hull girder (whipping), which affects comfort and fatigue. Prediction of stress fluctuations and vibration levels in var-

ious parts of the ship can only be performed by simulation in the time domain based on 3D detailed finite element models (Figure 18.55). The main difficulty is the determination of the time and space dependent slamming forces. 18.6.8.4 Simplified models Unfortunately, they are of little use for simplified vibration predictions. Beam models associated to database can be used for an approximate determination of hull girder modes and frequencies at early stage of the project. Decks zones and equipment frequencies may also be estimated by formulas given by reference books (94). Dedicated software has also been written for the study of shafting, including journal and bearing stiffness and whirling effect (95). 18.6.8.5 Design criteria The most effective way to control vibration resides in the reduction of the excitation. This can be achieved by balancing all forces in reciprocating and rotary machinery and using special mounts. Hydrodynamic forces can be reduced by improving the flow around the propeller and siting it clear of the hull. Propulsion using pods can dramatically reduce pressure fluctuations. Excitation frequencies can also be modified by changing the number of propeller blades. A good design, ensuring continuity of vertical bulkheads, avoiding cantilevered and stiff or mass discontinuities, contributes to improving the dynamic behavior of the ship. The

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



second action consists in avoiding resonance by modification of the hull scantlings, and addition of pillars, in order to increase or lower the eigen frequencies. Reduction of unavoidable vibration levels can be achieved for local vibrations by dynamic isolation for equipments, passive damping solutions (floating floors on absorbing material), and dynamic energy absorbers. All these curative actions are usually difficult, costly, only applicable for local vibrations and nearly impossible for vibrations due to global modes. Local modes determination is difficult at early stage of the design mainly due to the uncertainty on mass distribution, non-structural mass (outfitting and equipments) being of the some order of magnitude as the steelwork part.

18.6.9 Special Considerations In addition to the considerations for LNG tank, container ship, bulk carrier and passenger vessel, special considerations are available in Volume II of this book. Moreover, ISSC committees 1997 and 2000 also provide valuable information on specific ship types, that is, high-speed vessels and ships sailing in ice conditions. 18.6.9.1 LNG Tanks General information on such ships is available in Chapter 32 – Liquefied Gas Carriers. These ships contain usually a double hull (sides and bottom). Major structural concerns deal with the tanks themselves and with their support legs. Dilatation, tightness and thermal isolation are important aspects. There are several patented concepts: independent tanks, membrane tanks, semi-membranes tanks and integral tanks. Excepted for the integral tanks, the tanks are selfsupporting and are not essential to the hull strength. When supported by legs, these legs require a particular attention. Integral tanks form a structural part of the ship’s hull and are influenced in the same manner by wave loads. 18.6.9.2 Container ships The design of container ships of 5000 and 6000 TEU having a beam of 40m has increased the standard torsional problem of ships having a large open deck. Torsional strength and limitation of the equivalent stress (equation 45) at the hatch corners are the major issues in the evaluation of the strength of main hull structure. Use of multicell structures in side shell and double bottom is recommended. Moreover, the torsional moment distribution must be assessed with care. As hatch covers are not considered as hull strength members, omission of hatch covers does not impose any partic-

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ular effects in the structural design of a main hull structure. The general characteristics of container ships are detailed in Chapter 36 – Container Ships. 18.6.9.3 Bulk carriers Casualty of bulk carriers was very high in the early 1990s. The main reasons were a lack of maintenance, excessive corrosion and fatigue (77). Weak point of these ships is the lower part of the side plate at the junction with the bilge hopper. Now, classification societies are aware about this problem and had updated their rules and associated structural details. The general design practice on bulk carriers is detailed in Chapter 33 – Bulk Carriers. 18.6.9.4 Passenger vessels Ship strength analysis is based on a beam model. The complexity of large passenger ships, with a low resistant deck and wide openings, windows and openings in the side induces a much more complex behavior. Rational approach is necessary to get a realistic understanding of the flux of forces and capture the complex behavior of such ships. Due to the large openings and discontinuities, racking and stress concentration are two major concerns. For architectural reason, pillars are often omitted in large public areas (theater, lounge, etc.). Today, 3D FEA is usually carried out to design large passenger vessels (Figures 18.54 and 18.55). Due to large opening in the side shells, the vertical stress distribution is not linear (Figure 18.35). This means that the basic beam bending formulation is no valid (equation 29). More general information related to passenger vessels is available in Chapter 37 – Passenger Ships and in reference 68. 18.6.9.5 Composite material Fiberglass boat building started in the 1960s. Today, designers are trying to plan composite construction of ships up to 100 meters in length. A comprehensive guide for the design of ship structures in composites is the Ship Structure Committee Report SSC-403 of Greene (96). Design methodology, materiel properties, micro and macro mechanic of composites and failures modes are deeply discussed. In addition to the classic failure modes of steel and aluminum structures presented in Subsection 18.6.1, composites are subject to specific failure modes. In compression, there are the crimping, skin wrinkling and dimpling of the honeycomb cores (Figure 18.56). In bending, instead of the traditional first yield bending moment, for composites, the design limit load corresponds to the first ply failure. The creep behavior and the long-term damage from


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water, UV and temperature, and their performance in fires are other specific structural problems of composites. A review of the performance of composite structures is proposed by Jensen et al (98). 18.6.9.6 Aluminum structures Compared to steel, the reduced specific weight of aluminum (2.70 kN/m3 for aluminum and 7.70 kN/m3 for steel) is a very interesting property for a ship designer. The yield stress of unwelded aluminum alloys can be comparable to mild steel (235 MPa) but changes drastically from one alloy to another (125 MPa for ALU 5083-O and 215 MPa for ALU 5083-H321). The modulus of elasticity of aluminum alloys is one-third of steel. The main difficulty for the use of aluminum use deals with its mechanical properties after welding. The yield stress of aluminum alloys may decrease significantly after welding (remains at 125 MPa for ALU 5083-O but drop to 140 MPa for ALU 5083-H321). The area close to a weld is called Heat Affected Zone (HAZ). It is characterized by reduced strength properties. HAZ is particularly important to assess the buckling and ultimate strength of welded components such as beam-column elements, stiffened panels, etc. For marine applications ALU 5083, 5086 and 6061 can be used. Nevertheless, the mechanical and strength properties of aluminum change a lot with the alloy composition and the production processing. Thus, the alloy selection must be done with care with regard to the yield strength before and after welding, the welding and extruding capabilities, the marine behavior, etc. Fire strength is another concerns when using aluminum alloys as it quickly loses its strength when the temperature rises. Despite the aforementioned shortcomings aluminum alloys will be more extensively use in the future for the de-

Figure 18.56 Potential Failure Modes of Sandwich Panels (100), (a) Face yielding/fracture, (b) Core shear failure, (c-d) Face wrinkling, (e) Buckling, (f) Shear crimping, (g) Face dimpling, (h) Local indentation.

sign of fast vessels, for which the structural weight is very important to reach higher speed (for high speed mono hull, catamaran and trimaran vessels). The good extruding capability of aluminum alloys has to be enhanced through scantling standardization. That helps to lower to production cost ($/man-hour) and compensate the initial higher material cost of aluminum, which is approximately 3 times higher that mild steel ($/kg). 18.6.9.7 Corrosion Corrosion does not present a structural design problem, as almost all the classification societies base their rules on a net scantling. This means that the thickness to consider in analysis (for empirical formulations up to complex FEA) is the reduced thickness (without corrosion allowance) and not the actual thickness. The difference between the reduced thickness and the actual one is usually fixed by the classification but can also change according to the owner requirements. This is an economic choice and not a structural problem. For bulk carriers, thickness reduction due to corrosion is generally assumed to be 5 mm for hold frames and 3 mm for side shell plating.

18.7 NUMERICAL ANALYSIS FOR STRUCTURAL DESIGN 18.7.1 Motivation for Numerical Analysis In most of the cases, a ship is a one of a kind product, even if limited series may exist in some cases. The design, study and production cycle is very short and major decision have to be taken very early in the project. It is well known that the cost of a late modification is very high and such a situation has to be avoided. Also experience-based design can be an obstacle to the introduction of innovation. Numerical analysis clearly is needed to improve the design (innovation) but also to control safety margins. Moreover, it gives access to local and detailed analysis, which is not possible with simplified methods. The concept of numerical mock up, used in aerospace and car industry has proven its efficiency. Shipbuilding is clearly moving in the same direction. 18.7.1.1 Static and quasi-static analysis Static and quasi-static analysis represents the traditional way to perform stress and strength analysis of a ship structure. Loads are assessed separately of the strength structure and, even if their origins are dynamic (flow induced), they are assumed to be static (do not change with the time). This assumption may be correct for the hydrostatic pressure but



not when the dynamic wave loads are changed to static loads applied on the side plates of the hull. In the future, even if the assumption of static loads is not verified, static analysis will continue to be performed, as it is easier and faster to perform. In addition, tens of experience years have shown that they provide accurate results when stresses and deflections assessment are the main target (as defined in Section 18.4). Such analysis is also the standard procedure for fatigue assessment to determine the hot spot stress through fine mesh FEA. 18.7.1.2 Dynamic analysis When problems occur on a ship due to dynamic effects, it is very often late in the design and building stage and even in service, and corrective actions are costly. Simplified methods can only predict the first hull girder modes frequencies. Numerical finite element based simulation is mature enough to predict up to second propeller harmonic, the vibration level, giving a design tool to comply with ISO or ship owner requirements. Moreover, possible dynamic problems can be detected early enough in the design to allow for corrective actions. 18.7.1.3 Nonlinearities analysis Nonlinear structural analysis is mainly used to analyze buckling, ultimate strength and accidental or extreme situations (explosions, collisions, grounding, blast). The results of such costly and difficult analysis are often used to calibrate simplified methods or rules. But they are also very useful to understand possible failure modes and mechanical behavior under severe loads. 18.7.1.4 Emerging trends Like the automotive and aerospace industry, there is a clear trend towards the reduction of design cycle time. Numerical mock up or virtual ship approach (97), especially for one of a kind product, is clearly a way to achieve this. Required computing power is available and will no longer be a constraint. The first difficulty is to establish an efficient model of complex physical problems, associated with increasing demand for accuracy. The second difficulty is the manpower needed to prepare and check the models, which will be solved by the development of integrated solutions for ship description and modeling (99). Advances are expected in the field of FE-modeling. The trend is toward one structure description, one model and several applications. This is the field for multiphysics and coupling analysis. The base modeling will be re-used and adapted to perform successively,

• • • •

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static, fatigue and fracture analysis, buckling and ultimate strength analysis, vibration and acoustics analysis, and vulnerability assessment.

Progress is expected by the utilization of reliability methods already used in offshore industry, where uncertainties and dispersions of the loads, geometrical defaults, initial stresses and strains, material properties are defined as stochastic (non deterministic) data, leading to the calculation of a probability of failure. This philosophy can be applied to fatigue and ultimate strength, but also to dynamic response, leading to a more robust design, less sensitive to defaults, imperfections, uncertainties and stochastic nature of loads. Reliability-based analyses using probabilistic concept are presented in Chapter 19. In the future, safety aspects related to structural problems will also be tackled such as ultimate strength using nonlinear methods. Collision and grounding damages and improved design to increase ship safety will be studied by numerical simulation, whereas experimental approach is nearly impossible and/or too costly. Explicit codes, used in car crash simulation (101), will be adapted to specific aspects of ship structure (size and presence of fluid). In traditional sea keeping analysis, the ship is considered as a rigid body. In coupled problems such as slamming situations, this hypothesis is no more valid and a part of the energy is absorbed by ship deformation. Hydro-elasticity methods (102) aim taking into account the interaction of the flexible ship structure with the surrounding water. Nonlinear effects due to bow and aft part of the ship, ship velocity, diffraction radiation effects contribute to the complexity of the problem. The simulation of catamaran, trimaran and fast monohulls behavior need the development of new methods to take into account the high velocities and the complex 3D phenomena.

18.7.2 Finite Element Analysis The main aim of using the finite element method (FEM) in structural analysis is to obtain an accurate calculation of the stress response in the hull structure. Several types or levels of FE-models may be used in the analyses: • • • • •

global stiffness model, cargo hold model, frame and girder models, local structure models, and stress concentration models.

The model or sets of models applied is to give a proper representation of the following structure:


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• • • •


longitudinal plating, transverse bulkheads/frames, stringers/girders, and longitudinals or other structural stiffeners.

The finer mesh models are usually referred to as submodels. These models may be solved separately by transfer of boundary deformations/ boundary forces from the coarser model. This requires that the various mesh models are compatible, meaning that the coarser models have meshes producing deformations and/or forces applicable as boundary conditions for the finer mesh models. 18.7.2.1 Structural finite element models Global stiffness model: A relatively coarse mesh that is used to represent the overall stiffness and global stress distribution of the primary members of the total hull length. Typical models are shown in Figure 18.57. The mesh density of the model has to be sufficient to describe deformations and nominal stresses from the following effects:

The minimum element sizes to be used in a global structural model (coarse mesh) for 4–node elements (finer mesh divisions may of course be used and is welcomed, specially with regard to sub-models): • main model: 1 element between transverse frames/girders; 1element between structural deck levels and minimum three elements between longitudinal bulkheads, • girders: 3 elements over the height, and • plating: 1 element between 2 longitudinals.

Figure 18.57 Global Finite Element Model of Container Vessel Including a 4 Cargo Holds Sub-model (4).

• vertical hull girder bending including shear lag effects, • vertical shear distribution between ship side and bulkheads, • horizontal hull girder bending including shear lag effects, torsion of the hull girder, and • transverse shear and bending. Stiffened panels may be modeled by means of layered elements, anisotropic elements or frequently by a combination of plate and beam elements. It is important to have a good representation of the overall membrane panel stiffness in the longitudinal/transverse directions. Structure not contributing to the global strength of the vessel may be disregarded; the mass of these elements shall nevertheless be included (for vibration). The scantling is to be modeled with reduced scantling, that is, corrosion addition is to be deducted from the actual scantling. All girder webs should be modeled with shell elements. Flanges may be modeled using beam and truss elements. Web and flange properties are to be according to the real geometry. The performance of the model is closely linked to the type of elements and the mesh topology that is used. As a standard practice, it is recommended to use 4-node shell or membrane elements in combination with 2-node beam or truss elements are used. The shape of 4-node elements should be as rectangular as possible as skew elements will lead to inaccurate element stiffness properties. The element formulation of the 4-node elements requires all four nodes to be in the same plane. Double curved surfaces should therefore not be modeled with 4-node elements. 3-node elements should be used instead.

Figure 18.58 Cargo Hold Model (Based on the Fine Mesh of the Frame Model), (4)

Figure 18.59 Frame and Girder Model (Web Frame), (4)



Cargo hold model: The model is used to analyze the deformation response and nominal stresses of the primary members of the midship area. The model will normally cover 1/2+1+1/2 cargo hold/tank length in the midship region. Typical models are shown in Figure 18.58. Frame and girder models: These models are used to analyze nominal stresses in the main framing/girder system (Figure 18.59). The element mesh is to be fine enough to describe stress increase in critical areas (such as bracket with continuous flange). This model may be included in the cargo hold model, or run separately with prescribed boundary deformations/forces. However, if sufficient computer capacity is available, it will normally be convenient to combine the two analyses into one model. Local structure analyses are used to analyze stresses in local areas. Stresses in laterally loaded local plates and stiffeners subjected to large relative deformations between girders/frames and bulkheads may be necessary to investigate along with stress increase in critical areas, such as brackets with continuous flanges. As an example, the areas to model are normally the following for a tanker: • longitudinals in double bottom and adjoining vertical bulkhead members, • deck longitudinals and adjoining vertical bulkhead members, • double side longitudinals and adjoining horizontal bulkhead members, • hatch corner openings, and • corrugations and supporting structure. The magnitude of the stiffener bending stress included in the stress results depends on the mesh division and the element type that is used. Figure 18.60 shows that the stiffener bending stress, using FEM, is dependent on the mesh size for 4-node shell elements. One element between floors results in zero stiffener bending. Two elements between floors result in a linear distribution with approximately zero bending in the middle of the elements. Stress concentration models are used for fatigue analyses of details were the geometrical stress concentration is unknown. A typical detail is presented Figure 18.61. Local FE analyses may be used for calculation of local geometric stresses at the hot spots and for determination of associated K-factors to be used in subsequent fatigue analyses (equation 63). The aim of the FE analysis is normally not to calculate directly the notch stress at a detail, but to calculate the geometric stress distribution in the region of the hot spot. These stresses can then be used either directly in the fatigue assessment of given details or as a basis for derivation of stress concentration factors. FE stress con-

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centration models are generally very sensitive to element type and mesh size. Several FEA benchmarks of such structural details were performed by ISSC technical committees (68,103). They assess the uncertainties of different FE packages associated with coarse and fine mesh models. Variation is usually around 10% but is sometime much larger. This implies that element sizes in the order of the plate thickness are to be used for the modeling. If solid modeling is used, the element size in way of the hot spot may have to be reduced to half the plate thickness in case the overall geometry of the weld is included in the model representation. 18.7.2.2. Uncertainties related to FEA An important issue in structural analysis is the verification of the analysis. The FEM is basically reliable but many sources of errors can appear, mainly induced by inappropriate modeling and wrong data. For this reason, different

Figure 18.60 Stiffener Bending Stress with FEM (from left to right: using 1, 2 or 8 elements), (4)

Figure 18.61 Stress Concentration Model of Hopper Tank Knuckle (4)


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levels of verification of the analysis should be performed in order to ensure trustworthiness of the analysis results. Verification must be achieved at the following steps: • basic input, • assumptions and simplifications made in modeling/ analysis, • models, • loads and load transfer, • analysis, • results, and • strength calculations. One important step in the verification is the understanding of the physics and check of deformations and stress flow against expected patterns/levels. However, all levels of verification are important in order to verify the results. Verifications of structural models: Assumptions and simplifications will have to be made for most structural models. These should be listed such that an evaluation of their influence on the results can be made. The boundary conditions for the global structural model should reflect simple supporting to avoid built in stresses. The fixation points should be located away from areas where stresses are of interest. Fixation points are often applied in the centerline close to the aft and the forward ends of the vessel. Verification of loads: Inaccuracy in the load transfer from the hydrodynamic analysis to the structural model is among the main error sources in this type of analysis. The load transfer can be checked on basis of the structural response or on basis on the load transfer itself. Verification of response: The response should be verified at several levels to ensure correctness of the analysis: • • • • • •

global displacement patterns/magnitude, local displacement patterns/magnitude, global sectional forces, stress levels and distribution, sub-model boundary displacement/forces, and reaction forces and moments.

18.7.2.3 FEM background Today the finite element method is studied worldwide in universities, in mechanical engineering, civil engineering, naval architecture, etc. Hundreds of papers are published yearly. Many commercial packages are available including pre and post processors and many books are published each year on the subject. Classification Societies also present technical reports and guidelines associated with their own direct analysis package (Table 18.VIII). It is not the purpose of this chapter to present the FE theory and a state of art. This topic is reviewed periodically by

ISSC. For instance, Sumi et al (68) presents finite element guidelines and a comprehensive review of the available software. Mesh modeling is discussed in ISSC’2000 by Porcari et al (103). Hughes (3) proposes in Chapter VI and VII of his book published by SNAME an easy way to learn FEM that does not require knowledge of variational calculus or of FEM. The Ship Structure Committee Reports (SSC 387 and 399) contains also Guideline for FEM (43,104). 18.7.3 Other Numerical Approaches As an alternative to FEA, two other approaches are presented, namely: the idealized Structural Unit Method (ISUM) and the Boundary Element Method (BEM). Both are general purpose oriented. Many others exist but they are usually dedicated to a special purpose. For instance, at the preliminary design stage, the LBR-5 package founded on the analytical solution of the governing differential equations of stiffened plates is a convenient alternative to standard FEA. Such an approach (30,105) allows structural design optimization to be performed at the earliest design stage but does not have the capability to perform detailed analysis including stress concentration and non-linear analysis. 18.7.3.1 Idealized structural unit method (ISUM) When subjected to extreme or accidental loading, ship structures can be involved in highly non-linear response associated with yielding, buckling, crushing and sometimes rupture of individual structural components. Quite accurate solutions of the non-linear structural response can be obtained by application of the conventional FEM. However, a weak feature of the conventional FEM is that it requires enormous modeling effort and computing time for non-linear analysis of large sized structures. Therefore, most efforts in the development of new non-linear finite element methods have focused on reducing modeling and computing times. The most obvious way to reduce modeling effort and computing time is to reduce the number of degrees of freedom so that the number of unknowns in the finite element stiffness equation decreases. Modeling the object structure with very large sized structural units is perhaps the best way to do that. Properly formulated structural units or super elements in such an approach can then be used to efficiently model the actual non-linear behavior of large structural units. The idealized structural unit method (ISUM), which is a type of simplified non-linear FEM, is one of such methods (106). Since ship structures are composed of several different types of structural members such as beams, columns, rectangular plates and stiffened panels, it is necessary in the ISUM approach to develop various ISUM units



for each type of structural member in advance. The non-linear behavior of each type of structural member is idealized and expressed in the form of a set of failure functions defining the necessary conditions for different failures which may take place in the corresponding ISUM unit, and sets of stiffness matrices representing the non-linear relationship between the nodal force vector and the nodal displacement vector until the limit state is reached. The ISUM super elements so developed are typically used within the framework of a non-linear matrix displacement procedure applying the incremental method. Figure 18.62 shows a cantilevers box girder and Figures 18.63 and 18.64 show typical FEM and ISUM models for the non-linear analysis. For a recent state-of-the-art review on ISUM theory and applications to ship structures, the reader is referred to Paik and Hughes (107). With the existing standard ISUM elements, the main difficulty is that computation of the post-collapse behavior in the structural elements beyond their ultimate strength as well as the flexural-torsional collapse behavior of stiffeners is not very successful. In fact, ISUM elements accommodating post-collapse behavior have previously been already developed but improvements are under development to better accommodate such behavior (107, 108). Usage of ISUM is limited to some specific problems and is not a general-purpose methodology. In contrast to FEM, for instance, it is necessary to formulate/develop ISUM elements specifically; by including buckling and collapse behavior for ultimate strength analysis or by including tearing and crushing for collision strength analysis. The former type element cannot be used for the purpose of latter type analysis and vice versa. ISUM is also not adequate for linear stress analysis. ISUM is very flexible, new closed form expressions of the ultimate strength can be directly utilized by replacing in the existing ISUM element the previous ultimate strength formulations with the new ones. 18.7.3.2 Boundary Element Method (BEM) In contrast to FEM, the boundary element method (BEM) is a type of semi-numerical method involving integral equations along the boundary of the integral domain (or volume). To solve a problem that involves the boundary integral equations, BEM typically uses an appropriate numerical integration technique so that the problem is discretized by dividing only the boundary of the integral domain into a number of segments or boundary elements, while the conventional FEM uses a mesh (finite elements) over the entire domain (or volume), that is, inside as well as its boundary. For a specific problem with a relatively simple

18-65

boundary domain, linear or flat boundary elements may be employed so that analytical solutions for the integral equations can be adopted, while higher degree boundary elements must be used for modeling an integral domain with more complex characteristics with the integration generally needing to be carried out numerically. Figure 18.65 shows typical FEM and BEM models for analysis of a pressure vessel (109). Since the publication of an early book on BEM, many engineering applications using BEM have been achieved. More recent developments of BEM together with the basic

Figure 18.62 Cantilever Box Girder

Figure 18.63 A Typical FEM Model for NonLinear Analysis of the Cantilever Box Girder

Figure 18.64 A Typical ISUM Model for Nonlinear Analysis of the Cantilever Box Girder


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idea may be found in Brebbia and Dominguez (109). While there are some problem areas to overcome in use of BEM for non-linear analysis, it has been recognized that BEM is a powerful alternative to FEM particularly for problems involving stress concentration or fracture mechanics, and for cases in which the integral domain extends to infinity. For example, to design the cathodic corrosion protection systems for ships, offshore structures and pipelines, it has been suggested that BEM should be employed, with the region of interest extending to infinity. BEM can also be applied to problems other than stress or temperature analysis, including fluid flow and diffusion (for example, for fluidstructure interaction, Subsection 18.6.8.2). Main advantages of BEM are due that very complex expressions of integral equations can be adopted, resulting in higher accuracy of the results. In this regard, BEM can be involved in the usage of more refined mathematical treatment than FEM. However, to calculate the integral equations using BEM, appropriate numerical techniques should be used, otherwise the integration results may not be accurate. For most linear problems, linear or flat boundary elements along the boundary of the integral domain can be used so that we don’t have to carry out numerical integration. If analytical solutions are available the required computing times will be very small and

(a)

the accuracy high. Nevertheless as the required computational times with the BEM is in general significant, BEM may be more appropriate for linear analysis of solids and for fluid mechanics problems.

18.7.4 Presentation of the Stress Result After performing an analysis, the presentation of the stress and deformation is very important. It should be based on stresses acting at the middle of element thickness, excluding plate-bending stress, in the form of ISO-stress contours in general. Numerical values should also be presented for highly stressed areas or locations where openings are not included in the model. The following results should be presented for parts of the vessel covered by the global model, such as, cargo hold model and frame and girder models: • deformed shape for each loading condition, • In-plane maximum normal stresses (σx and σy) in the global axis system, shear stresses (_) and equivalent von Mises stress (σe) of the following elements: — — — — — — —

bottom, inner bottom, deck, side shell, inner side including hopper tank top, longitudinal and transverse bulkheads, and longitudinal and transverse girders.

• Axial stress of free flanges, • Deformations of supporting brackets for main frames including longitudinals connected to these when applicable, • Deformation of supports for longitudinals subject to large relative deformation when applicable. For parts of the vessel covered by the local model, the following stresses are to be presented: (b)

Figure 18.65 A Typical FEM/BEM Model for Analysis of the Pressure Vessel (109). (a) Typical BEM model, and (b) Typical FEM model.

• Equivalent stress of plate/membrane elements, • Axial stress of truss elements, • Axial forces, bending moments and shear forces for beam elements.

18.7.5 Relevant Structural Analysis Methods for Specific Design Stages Shipbuilding design offices face very challenging situations (especially for passenger and other complex ships). The products are one-of-a-kind or at least on short series and the resulting ships are designed and built within two years

Author: Please advise what symbold is needed.



for 20 to 30 years of operation. Another impact on design activities that is also challenging is that the design overlaps the production. To clarify the actual situation, a common view of the design workflow for a commercial ship in the shipyard is shown in Table 18.VII. 18.7.5.1 Basic design The Basic Design is the design activities performed before order. This phase does not overlap with the production but is very short and will become the technical basis for the contract. The shipyard must be sure that no technical problem will appear later on, to avoid extra costs not included in the contract. The structural analysis carried out in this phase must be as fast as possible because the allocated time is short. The most time consuming task for analysis is the data input. The more detailed are the data more accurate the results. There are three kinds of early analysis: 1. First principles methods: Very simplified geometric representation of the structure. These methods are dedicated to an assessment of the global behavior of the ship. They mainly use empirical or semi-empirical formulas.

TABLE 18.VII Timing of a Design Project Basic Design Concept Design Preliminary Design Contract Design

1 or 2 days About 1 week Months Receive Order

Production Design Complete Functional Design Production Design

1 or 2 months 6–10 months

TABLE 18.VIII Classification Society Tools Overview (110) Classification Society

Product

American Bureau of Shipping (ABS)

ABS Safe Hull

Bureau Veritas (BV)

VeriSTAR

Det Norske Veritas (DNV)

Electronic Rulebook & Nauticus HULL

Germanisher Lloyd (GL)

GL-Rules & POSEIDON

Korean Register of shipping (KRS)

KR-RULES, KR-TRAS

Lloyd’s Register of Shipping (LR)

Rulefinder, ShipRight

Nippon Kaiji Kyokai (NK)

PrimeShip BOSUN

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2. Two-dimensional (or almost 2D) geometry-based methods: These methods are based on one or more 2D views of the ship sections. The expected results may be: • Verification of main section scantlings, • Global strength assessment, • Global vibration levels prediction, • Ultimate strength determination, and • Early assessment of fatigue Two main approaches exist: — The main section of the ship is modeled a 2D way (including geometry and scantlings) then global, and possibly local, loadings are applied (bending moments, pressures, etc.). All major Classification Societies provide today the designer with such tools (Table 18.VIII). — Various significant sections are described as beam cross section properties (areas, inertias, etc.) and then the ship is represented by a beam with variable properties on which global loading is applied. 3. Simple three-dimensional models: These models are useful when a more detailed response is needed. The idea is to include main surfaces and actual scantlings (or from the main section when not available) in a 3D model that can be achieved in one or two weeks. This approach is mainly dedicated to novel ship designs for which the feedback is rather small. 18.7.5.2 Production design The most popular method for structural analysis at the production design stage remains the Finite Elements Analysis (FEA). This method is commonly used by Shipyards, Classification Societies, Research Institutes and Universities. It is very versatile and may be applied to various types of analysis: • global and local strength, • global and local vibration analysis (natural frequencies with or without external water, forced response to the propeller excitation, etc.), • ultimate strength, and • detailed stress for local fatigue assessment, • fatigue life cycle assessment, • analysis of various non-linearities (material, geometry, contact, etc.), and • collision and grounding studies. The two main approaches for solving the physical problem are: 1. implicit method is used to solve large problems (both linear and non linear) with a matrix-based method. This is


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the favored method for solving global and local linear strength and vibration problems. But it can also be applied to non linear calculations when the time step remains rather large (about 1/10 to 1 second), and 2. explicit method is mainly used for fast dynamics (as collision and grounding or explosion) where time step is quite smaller. This method allows using different formulations for structural elements (Lagrangian) and fluid elements (Eulerian). One interesting result from research that is being introduced today is the reliability approach (see Chapter 19). This approach introduces uncertainties within the model (non planar plates, residual stresses from welding, discrepancies in the thickness…) to provide the designer with a level of reliability for a given result instead of a deterministic value. For FEA models, the modeling time is usually assumed to be 70% of the overall calculation time and results exploitation 30%. The computation itself is regarded as negligible (excepted for explicit analysis). So the main efforts today are focused on reducing the modeling time. 18.7.6 Optimization Optimization is a field in which much research has been carried out over a long time. It is included today in many software tools and many designers are using it. The aim of optimization is to give the designers the opportunity to change design variables (such as thickness, number and cross section of stiffeners, shape or topology) to design a better structure for a given objective (lower weight or cost). Optimization can be performed both at basic and production design stages: • Basic Design: Even with simplified models, the designer can optimize the scantlings. It can be used for instance to find out the minimal scantlings for a novel ship for which the yard have a lack of feedback, • Production Design: Optimization can be used for three main purposes: — Scantlings optimization, which gives the user the minimum scantlings for a given structure. The number of longitudinals and the frame spacing for a given cargo hold/tank can also be optimized (105). — Shape optimization (111), which uses a given topology and scantlings to provide the user the minimum, required area of material (reducing holes in a plate for instance), and to improve the hull shape considering the fluid-structure interaction. — Topology optimization (112) which uses a given scantlings and allows the user to find out where to

put material. An academic example of topology optimization is given on Figure 18.66. Weight is the most usual objective function for structure optimization. Minimizing weight is of particular importance in deadweight carriers, in ships required to have a limited draft, and in fast fine lined ships, for example, passenger vessels. However, it is well know that the lowest weight solution is not usually the lowest acquisition cost. Today, cost is becoming the usual objective function for optimization (124). For the other ship types it is still desirable to minimize steel weight to reduce material cost but only when this can be done without increasing labor costs to an extent that exceeds the saving in material costs. On the other hand, a reduction in structural labor cost achieved by simplifying construction methods may still be worthwhile even if this is obtained at the expense of increasing the steel weight. Rigo (105) presents extensive review of ship structure optimization focusing on scantling optimization. Vanderplaats (113), and Sen and Yang (114) are standard reference books about optimization techniques. Catley et al (115), Hughes (3) and Chapter 11 of this book also contain valuable information on structure optimization. 18.7.6.1 Scantling optimization procedure A standard optimization problem is defined as follows: • Xi (i = 1, N), the N design variables, • F(Xi), the objective function to minimize, • Cj(Xi) ≤ CMj (j = 1, M), the M structural and geometrical constraints, • Xi min ≤ Xi ≤ Xi max upper and lower bounds of the Xi design variables: technological bounds (also called side constraints).

Figure 18.66 Topology Optimization



Constraints are linear or nonlinear functions, either explicit or implicit of the design variables (XI). These constraints are analytical translations of the limitations that the user wants to impose on the design variables themselves or to parameters like displacement, stress, ultimate strength, etc. Note that these parameters must be functions of the design variables. So it is possible to distinguish: Technological constraints (or side constraints) that provide the upper and lower bounds of the design variables. For example: Xi min = 4mm ≤ Xi ≤ Xi max = 40 mm, with: Xi min = a thickness limit dues to corrosion, Xi max = a technological limit of manufacturing or assembly. Geometrical constraints that impose relationships between design variables in order to guarantee a functional, feasible, reliable structure. They are generally based on good practice rules to avoid local strength failures (web or flange buckling, stiffener tripping, etc.), or to guarantee welding quality and easy access to the welds. For instance, welding a plate of 30 mm thick with one that is 5 mm thick is not recommended. Hence, the constraints can be 0.5 ≤ X2 / X1 ≤ 2 with X1, the web thickness of a stiffener and X2, the flange thickness. Structural constraints represent limit states in order to avoid yielding, buckling, cracks, etc. and to limit deflection, stress, etc. These constraints are based on solid-mechanics phenomena and modeled with rational equations. Rational equations mean a coherent and homogeneous group of analysis methods based on physics, solid mechanics, strength and stability treatises, etc. and that differ from empirical and parametric formulations. Such standard rational structural constraints can limit: • the deflection level (absolute or relative) in a point of the structure, • the stress level in an element: σx , σy, and σc = σvon Mises, • the safety level related to buckling, ultimate resistance, tripping, etc. For example: σ /σult ≤ 0.5. For each constraint, or solid-mechanics phenomenon, the selected behavior model is especially important since this model fixes the quality of the constraint modeling. These behavior models can be so complex that it is no longer possible to explicitly express the relation between the parameters being studied (stress, displacement, etc.) and the design variables (XI). This happens when one uses mathematical models (FEM, ISUM, BEM, etc.). In this case, one gener-

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ally uses a numeric procedure that consists of replacing the implicit function by an explicit approximated function adjusted in the vicinity of the initial values of the design variables (for instance using the first or second order Taylor series expansions). This way, the optimization process becomes an iterative analysis based on a succession of local approximations of the behavior models. At least one constraint should be defined for each failure mode and limit state considered in the Subsection 18.6.1. When going from the local to the general (Figure 18.38), there are three types of constraints: 1) constraints on stiffened panels and its components, 2) constraints on transverse frames and transversal stiffening, and 3) constraints on the global structure. Constraints on stiffened panels (Figure 18.22): Panels are limited by their lateral edges (junctions with other panels, AA’ and BB’) either by transverse bulkheads or transverse frames. These panels are orthotropic plates and shells supported on their four sides, laterally loaded (bending) and submitted, at their extremities, to in-plane loads (compression/tensile and shearing). Global buckling of panels (including the local transverse frames) must also be considered. Panel supports, in particular those corresponding to the reinforced frames, are assumed infinitely rigid. This means that they can distort themselves significantly only after the stiffened panel collapse. Constraints on the transverse frames (Figure 18.23): The frames take the lateral loads (pressure, dead weight, etc.) and are therefore submitted to combined loads (large bending and compression). The rigidity of these frames must be assured in order to respect the hypotheses on panel boundary conditions (undeformable supports). Constraints on the global structure (box girder/hull girder) (Figure 18.46): The ultimate strength of the global structure or a section (block) located between two rigid frames (or bulkheads) must be considered as well as the elastic bending moment of the hull girder (against yielding).

18.8

DESIGN CRITERIA

In ship design, the structural analysis phase is concerned with the prediction of the magnitude of the stresses and deflections that are developed in the structural members as a result of the action of the sea and other external and internal causes. Many of the failure mechanisms, particularly those that determine the ultimate strength and collapse of the structure, involve non-linear material and structural behavior that are beyond the range of applicability of the linear structural analysis procedures in Section 18.4, which are


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commonly used in design practice. Most of the available methods of non-linear structural analysis are briefly introduced in Sections 18.6 and 18.7. Sometimes, these methods are limited in their applicability to a narrow class of problems. One of the difficulties facing the structural designer is that linear analysis tools must often be used in predicting the behavior of a structure in which the ultimate capability is governed by non-linear phenomena. This is one of the important sources of uncertainty related to strength assessment. After performing an analysis, the adequacy or inadequacy of the member and/or the entire ship structure must then be judged through comparison with some kind of criterion of performance (Design Criteria). The conventional criteria that are commonly used today in ship structural design are usually stated in terms of acceptable levels of stress in comparison to the yield or ultimate strength of the material, or as acceptable stress levels compared to the critical buckling strength and ultimate strength of the structural member. Such criteria are, therefore, intended specifically for the prevention of yielding (hull girder, frames, longitudinals, etc), plate and stiffened plate buckling, plate and stiffened plate ultimate strength, ultimate strength of hull girder, fatigue, collision, grounding, vibration and many other failure modes specific to particular vessel types. Information related to the design criteria is given in Section 18.6 for each specific failure mode (see also Beghin et al (116)). 18.8.1 Structural Reliability as a Design Basis Three categories of design methodology are basically available. They are usually classified as: 1. deterministic method, 2. semiprobabilistic method, and 3. full probabilistic method. The deterministic method uses a global safety factor. It assumes that loads and strength are fully determined. This means that no aspect of randomness is considered. Everything is assumed to be deterministic. The global safety factor is compared to the ratio between the actual strength and the required strength. The full probabilistic method is an ideal approach assuming that all the randomness can be exactly considered within a global probabilistic approach. All the actual development in structural reliability and reliability analysis show the huge effort actually done to reach that aims. Chapter 19 presents in detail the reliability concept with examples of the reliability-based strength analysis of plates, stiffened panels, hull girder and fatigue. See also Mansour et al (42).

The semiprobabilistic method corresponds to the current practice used by codes and the major classifications societies. Load, strength, dimensions are random parameters but their distribution is basically not known. To overcome this, partial safety factor are used. Each safety factor corresponds to a load type, failure mode, etc. This is an intermediate step between the deterministic and the full probabilistic methods.

18.9

DESIGN PROCEDURE

It does not seem possible to unify all of the design procedures (117-122). They differ from country to country, from shipyard to shipyard and differ between naval ships, commercial ships and advanced high-speed catamaran passenger vessels. So, as an example of one feasible methodology, the design procedure for commercial vessel such as tanker, container, and VLCC is selected. It corresponds to the actual current shipyard procedure. This structural design procedure can be defined as follows: • receive general arrangement from the basic design group, • define structural arrangement based on the general arrangement, • determine initial scantling of structural members within design criteria (rule-based)., • check longitudinal and transverse strength, • change the structural arrangement or scantling, and • transfer the structural arrangement and scantling to the production design group. The structural design can also be classified according to available design tool: • use data of existing ship or past experience—expert system, (1st level) • use of a structural analysis software like FEM (2nd level) • use optimization software (3rd level) The adequacy of the relevant analysis method to use for a specific design stage is discussed in Subsection 18.7.5. Here the discussion concerns the procedure from a design point of view and not from the analysis point of view. 18.9.1 Initial Scantling At the basic design stage, principal dimensions, hull form, double bottom height, location of longitudinal bulkheads and transverse bulkheads, maximum still-water bending moment, etc. have already been determined to meet the owner’s requirements such as deadweight and ship’s speed. Such a



parametric design procedure presented in Chapter 11 is relevant for this stage. For the structural design stage, the structural arrangement is carried out to define the material property, plate breadth, stiffener spacing, stiffener type, slot type, shape of openings, and frame spacing. The initial scantling of longitudinal members such as plate thickness and section area of stiffener can be determined by applying the classification rules which give minimum required value to meet the bending, shear and buckling strength. As there are usually no suitable rules for the transverse members, the initial scantling of transverse members such as height and thickness of web, breadth and thickness of flange are determined by reference to similar ships or using empirical shipyard database. 18.9.2 Strength Assessment The purpose of the strength assessment is to validate the initial design, that is, to evaluate quantitatively the strength capability of the initial design. This problem was extensively presented in previous Sections 18.4, 18.5 and 18.6. In general, the longitudinal members are subjected to several kinds of stresses in the sea-going condition: primary, secondary and tertiary stresses (Subsection 18.4.1). As all these stresses act simultaneously, the superposition of these stresses should not exceed the allowable equivalent stress given by the classification rules (equations 45 and 46). There are two kinds of strength to design the longitudinal members. One is the local strength to avoid collapse, and the other is the longitudinal strength to consider the collapse of the ships’ hull girder. The local strength is automatically satisfied if the design is based on the classification rules. The hull girder longitudinal strength can be assessed with the hull section modulus (SM) at bottom and deck where the extreme stresses are taken place (equation 29). The hull section modulus is calculated easily by using available software. If the hull section modulus at bottom or deck part is bigger than the required value, this design can be considered as finished but this design might be too expensive. If the section modulus at the deck or at the bottom is less than the required value, the designer should change the initial scantlings. If the calculated hull section modulus at deck part is less than required, he can increase, step by step, the deck scantling (for example, 0.5 mm for the plate thickness) until the requirement is satisfied. The designer also has to modify the scantling (usually plate thickness) of transverse members, for which the stress exceeds the allowable value. The designer estimates the in-

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creased thickness according to the difference between the actual stress and allowable stress. If the difference is small, it is not necessary to perform a new strength assessment and the design may be completed with only small changes. If the difference is large, the design should be drastically changed and it will be necessary to analyze the structure again (see previous step in this Subsection). Then, the designer has to check the transverse strength by comparing the actual stresses in the transverse frames with the allowable stresses given by the classification rules. The actual stresses such as equivalent stress and shear stress can be obtained using commercial FEA packages. If the stress in some of elements exceeds the allowable stress, the designer should increase the initial scantling. These changes are performed at the third step Structural Design using the results of the Strength Assessment and by comparison with the design criteria. 18.9.3 Structural Design If all of local scantlings are determined by the rule minimum values, and if the longitudinal strength satisfies the rule strength requirement, the design is completed. But, even if this design is strong enough, it might be too heavy and/or too expensive and it should be refined. In practice, refining an already feasible design is a difficult task and requires experience. The designer can change the structural arrangement, especially the dimensions such as frame spacing, and material properties to better fit with the longitudinal strength requirements. This work has to be done in agreement with the basic design team. Instead of the trial and error procedure discussed above, an automatic optimization technique can be used to obtain the minimum weight and/or cost for the longitudinal and transverse structural member. The object function(s) can be structural weight and/or fabrication cost, using either a single object function approach or a multiple objective function method. The design variables can be longitudinal and transverse spacing, deck/bottom scantlings for the longitudinal and transverse members (web height and thickness, flange width and thickness). The constraints and limitations of the optimization process can be the range of each design variable as well as the required hull section modulus and minimum deck/bottom scantlings for the longitudinal members, and allowable bending and shear stresses for the transverse members (see Optimization in Subsection 18.7.6). 18.9.4 A Generic Design Framework By comparison with the previous standard procedure, Figure 18.67 shows a new generic and advanced design method-


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ology where the performance of the system, the manufacturing process of the system and the associated life cycle costs are considered in an integrated fashion (120). Designing ship structures systems involves achieving simultaneous, though sometimes competing, objectives. The structure must perform its function while conforming to structural, economic and production constraints. The present design framework consists of establishing the structural system and composite subsystems, which optimally satisfy the topology, shape, loading and performance constraints while simultaneously considering the manufacturing or fabrication processes in a cost effective manner. The framework is used within a computerized virtual environment in which CAD product models, physics-based models, production process models and cost models are used simultaneously by a designer or design team. The performance of the product or process is in general judged by some time independent parameter, which is referred to as a response metric (R). Specifications for the system must be established in terms of these Response Metrics. The formulation of the design problem is thus the same whether the product or process systems (or both) are considered. The general framework consists of a system definition module, a simulation module and a design module.

Operational Requirements ParametersZ

System Definition Model Parameters Y Environmental Model Product Model Process Model Parameters U ParametersV ParametersW

Simulation Based Design Translator Simulation Parameters T Design Variables X

Simulations Simulation Response S(T ,X ,time)

Design Criteria Constraints G(T,X,Y,Z)

Response Metrics R [S(T ,X )]

Objective Function F(R,T,X,Y,Z)

Yes No

Is Design Space Feasible?

Optimization Steepest Descent Convex Linearization

Design Assessment Min (F) ? R
No

Conditions Satisfied ? Yes

Yes

Redesign?

No

Stop

Figure 18.67 A Generic Design Framework (120)

The system definition module [Y(U,V,W)] is used to build an environmental model [U], a product model [V] and a process model [W]. The system definition module receives operational requirements [Z] such as owner’s requirements. These operational parameters are presumed fixed throughout the design. They of course can eventually be changed if no acceptable design is established, but presumably any design would have operational parameters, which would not be sacrificed. The environmental model [U] includes the still water and wave loading conditions and the product model [V] contains the production information, for example. The process model [W] is built to consider or define the fabrication sequence. A translator (simulation based design translator) assigns some [Y] model parameters to the simulation parameters [T] and design variables [X]. These parameters are selected based on the available simulation tools [S] that require specific data ([T],[X] and time). The simulation module [S(T, X, time)] is used to produce simulation responses such as Response Metrics [R[S(T, X)]]. The time is needed to consider the dynamic effects and actual dynamic load conditions [U]. The optimum design module includes the Design Criteria, the Design Assessment and the Optimization components. The design criteria module provides constraints [G(T, X, Y, Z)] and objective functions [F(R, T, X, Y, Z)]. These are used to assess the design through the Design Assessment component of the module (for example R≤G). The constraints are obtained by considering not only the simulation parameters [T] and the design variables [X] but also the operational requirements [Z] and the system definition parameter [Y]. Also, the objective function [F] is calculated using the response metrics [R], the operational requirements [Z], the system definition parameter [Y] as well as the design variables [X] and simulation parameters [T]. Based on the results of the Design Assessment (Min(F) and R≤G) several strategies for the design procedure (iterations) can be followed: • if the object function does not reach its minimum value or the response metrics do not satisfy the constraints, an optimization algorithm (steepest descent, dual approach and convex linearization, evolutionary strategies, etc.) is adopted to find a new set of design variables. Standard algorithms are presented in (113,114,123): — if the optimizer fails to find an improved solution (unfeasible design space), it is required to change the simulation parameter values [T] and/or design variables selection [X] or even to modify the Model Parameters [Y].



— otherwise, the design space is feasible, and a change of design variable values [X] is performed based on the optimizer solution (in other words a new iteration). • if the object function reaches its minimum value and the response metrics satisfy the constraints, two alternatives are examined: — change the operational requirements parameters [Z], repeat the previous procedure and to compare with other alternative designs, or — end the design procedure.

19.

20. 21.

22. 23.

24.

18.10

REFERENCES

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