Chapter 18 Analysis and Design of Ship Structure

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Philippe Rigo and Enrico Rizzuto

18.1 NOMENCLATURE m(x) longitudinal distribution of mass I(x) geometric moment of inertia (beam sec- For specific symbols, refer to the definitions contained in tion x) the various sections. L length of the ship ABS American Bureau of Shipping M(x) bending moment at section x of a beam BEM Boundary Element Method MT(x) torque moment at section x of a beam BV Bureau Veritas ppressure DNV Det Norske Veritas q(x) resultant of sectional force acting on a FEA Finite Element Analysis beam FEM Finite Element Method Tdraft of the ship IACS International Association of Classifica- V(x) shear at section x of a beam tion Societies s,w (low case) still water, wave induced component ISSC International Ship & Offshore Structures v,h (low case) vertical, horizontal component Congress w(x) longitudinal distribution of weight ISOPE International Offshore and Polar Engi- θ roll angle neering Conference ρ density ISUM Idealized Structural Unit method ω angular frequency NKK Nippon Kaiji Kyokai PRADS Practical Design of Ships and Mobile Units, 18.2 INTRODUCTION RINA Registro Italiano Navale SNAME Society of naval Architects and marine The purpose of this chapter is to present the fundamentals Engineers of direct ship structure analysis based on mechanics and SSC Ship Structure Committee. strength of materials. Such analysis allows a rationally based a acceleration design that is practical, efficient, and versatile, and that has Aarea already been implemented in a computer program, tested, Bbreadth of the ship and proven. Cwave coefficient (Table 18.I) Analysis and Design are two words that are very often

CB hull block coefﬁcient associated. Sometimes they are used indifferently one for Ddepth of the ship the other even if there are some important differences be- ggravity acceleration tween performing a design and completing an analysis.

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Analysis refers to stress and strength assessment of the entific, powerful, and versatile method for their structural structure. Analysis requires information on loads and needs design an initial structural scantling design. Output of the structural But, even with the development of numerical techniques, analysis is the structural response defined in terms of stresses, design still remains based on the designer’s experience and deflections and strength. Then, the estimated response is on previous designs. There are many designs that satisfy the compared to the design criteria. Results of this comparison strength criteria, but there is only one that is the optimum as well as the objective functions (weight, cost, etc.) will solution (least cost, weight, etc.). show if updated (improved) scantlings are required. Ship structural analysis and design is a matter of com- Design for structure refers to the process followed to se- promises: lect the initial structural scantlings and to update these scantlings from the early design stage (bidding) to the detailed ¥ compromise between accuracy and the available time to design stage (construction). To perform analysis, initial de- perform the design. This is particularly challenging at sign is needed and analysis is required to design. This ex- the preliminary design stage. A 3D Finite Element plains why design and analysis are intimately linked, but Method (FEM) analysis would be welcome but the time are absolutely different. Of course design also relates to is not available. For that reason, rule-based design or topology and layout definition. simplified numerical analysis has to be performed. The organization and framework of this chapter are based ¥ to limit uncertainty and reduce conservatism in design, it on the previous edition of the Ship Design and Construction is important that the design methods are accurate. On the (1) and on the Chapter IV of Principles of Naval Architec- other hand, simplicity is necessary to make repeated de- ture (2). Standard materials such as beam model, twisting, sign analyses efficient. The results from complex analy- shear lag, etc. that are still valid in 2002 are partly duplicated ses should be verified by simplified methods to avoid errors from these 2 books. Other major references used to write this and misinterpretation of results (checks and balances). chapter are Ship Structural Design (3) also published by ¥ compromise between weight and cost or compromise SNAME and the DNV 99-0394 Technical Report (4). between least construction cost, and global owner live The present chapter is intimately linked with Chapter cycle cost (including operational cost, maintenance, etc.), 11 Ð Parametric Design, Chapter 17 Ð Structural Arrange- and ment and Component Design and with Chapter 19 Ð Reli- ¥builder optimum design may be different from the owner ability-Based Structural Design. References to these optimum design. 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 18.2.1 Rationally Based Structural Design versus methods available to perform a rationally based structural Rules-Based Design design, even if mention is made to standard formulations There are basically two schools to perform analysis and de- from Rules to quantify design loads. sign of ship structure. The first one, the oldest, is called In the following sections of this chapter, steps of a global rule-based design. It is mainly based on the rules defined analysis are presented. Section 18.3 concerns the loads that by the classification societies. Hughes (3) states: are necessary to perform a structure analysis. Then, Sections In the past, ship structural design has been largely empir- 18.4, 18.5 and 18.6 concern, respectively, the stresses and ical, based on accumulated experience and ship perform- deflections (basic ship responses), the limit states, and the fail- ance, and expressed in the form of structural design codes ures modes and associated structural capacity. A review of or rules published by the various ship classification soci- the available Numerical Analysis for Structural Design is per- eties. These rules concern the loads, the strength and the formed in Section 18.7. Finally Design Criteria (Section design criteria and provide simplified and easy-to-use for- 18.8) and Design Procedures (Section 18.9) are discussed. mulas for the structural dimensions, or “scantlings” of a Structural modeling is discussed in Subsection 18.2.2 and ship. This approach saves time in the design office and, more extensively in Subsection 18.7.2 for finite element analy- since the ship must obtain the approval of a classification sis. Optimization is treated in Subsections 18.7.6 and 18.9.4. society, it also saves time in the approval process. Ship structural design is a challenging activity. Hence Hughes (3) states: The second school is the Rationally Based Structural Design; it is based on direct analysis. Hughes, who could The complexities of modern ships and the demand for be considered as a father of this methodology, (3) further greater reliability, efficiency, and economy require a sci- states: MASTER SET SDC 18.qxd Page 18-3 4/28/03 1:30 PM

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There are several disadvantages to a completely “rulebook” Hopefully, in 2002 this is no longer true. The advantages approach to design. First, the modes of structural failure of direct analysis are so obvious that classification societies are numerous, complex, and interdependent. With such include, usually as an alternative, a direct analysis procedure simplified formulas the margin against failure remains un- (numerical packages based on the finite element method, known; thus one cannot distinguish between structural ad- see Table 18.VIII, Subsection 18.7.5.2). In addition, for new equacy and over-adequacy. Second, and most important, vessel types or non-standard dimension, such direct proce- these formulas involve a number of simplifying assump- dure is the only way to assess the structural safety. There- tions and can be used only within certain limits. Outside fore it seems that the two schools have started a long merging of this range they may be inaccurate. procedure. Classification societies are now encouraging and For these reasons there is a general trend toward direct contributing greatly to the development of direct analysis structural analysis. and rationally based methods. Ships are very complex structures compared with other types of structures. They are sub- Even if direct calculation has always been performed, ject to a very wide range of loads in the harsh environment design based on direct analysis only became popular when of the sea. Progress in technologies related to ship design numerical analysis methods became available and were cer- and construction is being made daily, at an unprecedented tified. Direct analysis has become the standard procedure pace. A notable example is the fact that the efforts of a main aerospace, civil engineering and partly in offshore in- jority of specialists together with rapid advances in com- dustries. In ship design, classification societies preferred to puter and software technology have now made it possible to offer updated rules resulting from numerical analysis cali- analyze complex ship structures in a practical manner using bration. For the designer, even if the rules were continuously structural analysis techniques centering on FEM analysis. changing, the design remained rule-based. There really were The majority of ship designers strive to develop rational and two different methodologies. 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 Design Load verify the equivalence of structural strength with rule re- Direct Load Analysis quirements, it is necessary for the classification society to clarify the strength that a hull structure should have with Stress Response Study on Ocean Waves respect to each of the various steps taken in the analysis in Waves process, from load estimation through to strength evalua- Structural analysis by Effect on tion. In addition, in order to make this a practical and ef- whole ship model Wave Load Response operation fective method of analysis, it is necessary to give careful

Response function Stress response consideration to more rational and accurate methods of di- of wave load function rect strength analysis. Based on recognition of this need, extensive research Short term Design Short term has been conducted and a careful examination made, re- estimation estimation Sea State garding the strength evaluation of hull structures. The re-

Long term Long term sults of this work have been presented in papers and reports estimation estimation regarding direct strength evaluation of hull structures (4,5). The flow chart given in Figure 18.1 gives an overview Nonlinear influence Design wave Wave impact load of the analysis as defined by a major classification society. in large waves Note that a rationally based design procedure requires that all design decisions (objectives, criteria, priorities, con- Structural response analysis straints…) must be made before the design starts. This is a Modeling technique Direct structural Investigation on major difficulty of this approach. analysis corrosion

Strength Assessment 18.2.2 Modeling and Analysis Yield Buckling Ultimate Fatigue General guidance on the modeling necessary for the struc- strength strength strength strength tural analysis is that the structural model shall provide re- Figure 18.1 Direct Structural Analysis Flow Chart sults suitable for performing buckling, yield, fatigue and MASTER SET SDC 18.qxd Page 18-4 4/28/03 1:30 PM

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Structural drawings, to ensure that all dimensioning loads are correctly included. mass description and loading conditions. A ﬂow chart of strength analysis of global model and sub models is shown in Figure 18.2.

Structural model including necessary Hydrodynamic/static 18.2.3 Preliminary Design versus Detailed Design load definitions loads Verification For a ship structure, structural design consists of two dis- of model/ loads tinct levels: the Preliminary Design and the Detailed De- sign about which Hughes (3) states: Verified structural Load transfer to model structural model The preliminary determines the location, spacing, and scant- Verification lings of the principal structural members. The detailed de- of load transfer sign determines the geometry and scantlings of local structure Sub-models to be (brackets, connections, cutouts, reinforcements, etc.). Structural analysis used in structural analysis Preliminary design has the greatest influence on the Verification structure design and hence is the phase that offers very of response large potential savings. This does not mean that detail de- Transfer of sign is less important than preliminary design. Each level displacements/forces Yes is equally important for obtaining an efficient, safe and re- to sub-model? liable ship. During the detailed design there also are many bene- No fits to be gained by applying modern methods of engi- Figure 18.2 Strength Analysis Flow Chart (4) neering science, but the applications are different from preliminary design and the benefits are likewise different. Since the items being designed are much smaller it is vibration assessment of the relevant parts of the vessel. This possible to perform full-scale testing, and since they are is done by using a 3D model of the whole ship, supported more repetitive it is possible to obtain the benefits of mass by one or more levels of sub models. production, standardization and so on. In fact, production Several approaches may be applied such as a detailed aspects are of primary importance in detail design. 3D model of the entire ship or coarse meshed 3D model sup- Also, most of the structural items that come under de- ported by finer meshed sub models. tail design are similar from ship to ship, and so in-service Coarse mesh can be used for determining stress results experience provides a sound basis for their design. In fact, suited for yielding and buckling control but also to obtain because of the large number of such items it would be in- the displacements to apply as boundary conditions for sub efficient to attempt to design all of them from first princi- models with the purpose of determining the stress level in ples. Instead it is generally more efficient to use design more detail. codes and standard designs that have been proven by ex- Strength analysis covers yield (allowable stress), buck- perience. In other words, detail design is an area where a ling strength and ultimate strength checks of the ship. In ad- rule-based approach is very appropriate, and the rules that dition, specific analyses are requested for fatigue (Subsection are published by the various ship classification societies 18.6.6), collision and grounding (Subsection 18.6.7) and contain a great deal of useful information on the design of vibration (Subsection 18.6.8). The hydrodynamic load local structure, structural connections, and other structural model must give a good representation of the wetted sur- details. face of the ship, both with respect to geometry description and with respect to hydrodynamic requirements. The mass model, which is part of the hydrodynamic load model, must 18.3 LOADS ensure a proper description of local and global moments of inertia around the global ship axes. Loads acting on a ship structure are quite varied and pecu- Ultimate hydrodynamic loads from the hydrodynamic liar, in comparison to those of static structures and also of analysis should be combined with static loads in order to other vehicles. In the following an attempt will be made to form the basis for the yield, buckling and ultimate strength review the main typologies of loads: physical origins, gen- checks. All the relevant load conditions should be examined eral interpretation schemes, available quantification proce- MASTER SET SDC 18.qxd Page 18-5 4/28/03 1:30 PM

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dures and practical methods for their evaluation will be sum- Loads, defined in order to be applied to limited struc- marized. tural models (stiffened panels, single beams, plate panels), generally are termed local loads. The distinction is purely formal, as the same external 18.3.1 Classification of Loads forces can in fact be interpreted as global or local loads. For 18.3.1.1 Time Duration instance, wave dynamic actions on a portion of the hull, if Static loads: These are the loads experienced by the ship in described in terms of a bi-dimensional distribution of pres- still water. They act with time duration well above the range sures over the wet surface, represent a local load for the hull of sea wave periods. Being related to a specific load con- panel, while, if integrated over the same surface, represent dition, they have little and very slow variations during a a contribution to the bending moment acting on the hull voyage (mainly due to changes in the distribution of con- girder. sumables on board) and they vary significantly only during This terminology is typical of simplified structural analy- loading and unloading operations. ses, in which responses of the two classes of components Quasi-static loads: A second class of loads includes are evaluated separately and later summed up to provide those with a period corresponding to wave actions (∼3 to the total stress in selected positions of the structure. 15 seconds). Falling in this category are loads directly in- In a complete 3D model of the whole ship, forces on the duced by waves, but also those generated in the same fre- structure are applied directly in their actual position and the quency range by motions of the ship (inertial forces). These result is a total stress distribution, which does not need to loads can be termed quasi-static because the structural re- be decomposed. sponse is studied with static models. Dynamic loads: When studying responses with fre- 18.3.1.3 Characteristic values for loads quency components close to the first structural resonance Structural verifications are always based on a limit state modes, the dynamic properties of the structure have to be equation and on a design operational time. considered. This applies to a few types of periodic loads, Main aspects of reliability-based structural design and generated by wave actions in particular situations (spring- analysis are (see Chapter 19): ing) or by mechanical excitation (main engine, propeller). ¥ the state of the structure is identified by state variables Also transient impulsive loads that excite free structural vi- associated to loads and structural capacity, brations (slamming, and in some cases sloshing loads) can ¥ state variables are stochastically distributed as a func- be classified in the same category. tion of time, and High frequency loads: Loads at frequencies higher than ¥ the probability of exceeding the limit state surface in the the first resonance modes (> 10-20 Hz) also are present on design time (probability of crisis) is the element subject ships: this kind of excitation, however, involves more the to evaluation. study of noise propagation on board than structural design. Other loads: All other loads that do not fall in the above The situation to be considered is in principle the worst mentioned categories and need specific models can be gen- combination of state variables that occurs within the design erally grouped in this class. Among them are thermal and time. The probability that such situation corresponds to an accidental loads. out crossing of the limit state surface is compared to a (low) A large part of ship design is performed on the basis of target probability to assess the safety of the structure. static and quasi-static loads, whose prediction procedures This general time-variant problem is simplified into a are quite well established, having been investigated for a time-invariant one. This is done by taking into account in long time. However, specific and imposing requirements the analysis the worst situations as regards loads, and, sep- can arise for particular ships due to the other load cate- arately, as regards capacity (reduced because of corrosion gories. and other degradation effects). The simplification lies in considering these two situations as contemporary, which in 18.3.1.2 Local and global loads general is not the case. Another traditional classification of loads is based on the When dealing with strength analysis, the worst load sit- structural scheme adopted to study the response. uation corresponds to the highest load cycle and is charac- Loads acting on the ship as a whole, considered as a terized through the probability associated to the extreme beam (hull girder), are named global or primary loads and value in the reference (design) time. the ship structural response is accordingly termed global or In fatigue phenomena, in principle all stress cycles con- primary response (see Subsection 18.4.3). tribute (to a different extent, depending on the range) to MASTER SET SDC 18.qxd Page 18-6 4/28/03 1:30 PM

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damage accumulation. The analysis, therefore, does not re- resultant force along the vertical axis of the section (con- gard the magnitude of a single extreme load application, but tained in the plane of symmetry), indicated as vertical re-

the number of cycles and the shape of the probability dis- sultant force qV; another force in the normal direction, (local tribution of all stress ranges in the design time. horizontal axis), termed horizontal resultant force qH and a A further step towards the problem simplification is rep- moment mT about the x axis. All these actions are distrib- resented by the adoption of characteristic load values in uted along the longitudinal axis x. place of statistical distributions. This usually is done, for Five main load components are accordingly generated example, when calibrating a Partial Safety Factor format for along the beam, related to sectional forces and moment structural checks. Such adoption implies the definition of a through equation 1 to 5: single reference load value as representative of a whole x probability distribution. This step is often performed by as- = ξξ VVV(x)∫ q ( ) d [1] signing an exceeding probability (or a return period) to each 0 variable and selecting the correspondent value from the sta- x tistical distribution. M (x)= V (ξξ ) d [2] The exceeding probability for a stochastic variable has VV∫ the meaning of probability for the variable to overcome a 0 given value, while the return period indicates the mean time x = ξξ to the first occurrence. VHH(x)∫ q( ) d [3] Characteristic values for ultimate state analysis are typ- 0 ically represented by loads associated to an exceeding prob- x Ð8 ability of 10 . This corresponds to a wave load occurring, M (x)= ∫ V (ξξ ) d [4] on the average, once every 108 cycles, that is, with a return HH 0 period of the same order of the ship lifetime. In first yielding analyses, characteristic loads are associated to a higher x = ξξ exceeding probability, usually in the range 10Ð4 to 10Ð6. In M TT(x)∫ m ( ) d [5] fatigue analyses (see Subsection 18.6.6.2), reference loads 0 are often set with an exceeding probability in the range 10Ð3 Due to total equilibrium, for a beam in free-free condi- to 10Ð5, corresponding to load cycles which, by effect of both tions (no constraints at ends) all load characteristics have amplitude and frequency of occurrence, contribute more to zero values at ends (equations 6). the accumulation of fatigue damage in the structure. These conditions impose constraints on the distributions On the basis of this, all design loads for structural analy- of q ,q and m . ses are explicitly or implicitly related to a low exceeding V H T == = = probability. VVV(0) V (L) M V (0) M V (L) 0 == = = VHH(0) V (L) M H (0) M H (L) 0 [6] == 18.3.2 Definition of Global Hull Girder Loads M TT(0) M (L) 0 The global structural response of the ship is studied with Global loads for the verification of the hull girder are ob- reference to a beam scheme (hull girder), that is, a mono- tained with a linear superimposition of still water and wave- dimensional structural element with sectional characteris- induced global loads. tics distributed along a longitudinal axis. They are used, with different characteristic values, in Actions on the beam are described, as usual with this different types of analyses, such as ultimate state, first yield- scheme, only in terms of forces and moments acting in the ing, and fatigue. transverse sections and applied on the longitudinal axis. Three components act on each section (Figure 18.3): a 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

Figure 18.3 Sectional Forces and Moment mTg(x) is generated (Figure 18.4), in addition to the verti- MASTER SET SDC 18.qxd Page 18-7 4/28/03 1:30 PM

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cal load qSV(x), obtained as a difference between buoyancy At an even earlier stage of design, parametric formula- b(x) and weight w(x), as shown in equation 7 (2). tions can be used to derive directly reference values for still water hull girder loads. q (x)=− b(x) w(x) = gA (x) − m(x)g [7] SV I Common reference values for still water bending moment at mid-ship are provided by the major Classification where AI = transversal immersed area. Components of vertical shear and vertical bending can Societies (equation 8). be derived according to equations 1 and 2. There are no hor- C L2 B( 122.5− 15 C) (hogging) izontal components of sectional forces in equation 3 and ac- MN[]⋅ m= B s 2 ( + ) [8] cordingly no components of horizontal shear and bending C L B 45.5 65 CB (sagging) moment. As regards equation 5, only m , if present, is to Tg where C = wave parameter (Table 18.I). be accounted for, to obtain the torque. The formulations in equation 8 are sometimes explicitly reported in Rules, but they can anyway be indirectly de- 18.3.3.1 Standard still water bending moments rived from prescriptions contained in (6, 7). The first re- While buoyancy distribution is known from an early stage quirement (6) regards the minimum longitudinal strength of the ship design, weight distribution is completely defined modulus and provides implicitly a value for the total bend- only at the end of construction. Statistical formulations, cal- ing moment; the second one (7), regards the wave induced ibrated on similar ships, are often used in the design de- component of bending moment. velopment to provide an approximate quantification of Longitudinal distributions, depending on the ship type, weight items and their longitudinal distribution on board. are provided also. They can slightly differ among Class So- The resulting approximated weight distribution, together cieties, (Figure 18.5). with the buoyancy distribution, allows computing shear and bending moment. 18.3.3.2 Direct evaluation of still water global loads Classification Societies require in general a direct analysis of these types of load in the main loading conditions of the ship, such as homogenous loading condition at maximum draft, ballast conditions, docking conditions afloat, plus all other conditions that are relevant to the specific ship (non- homogeneous loading at maximum draft, light load at less than maximum draft, short voyage or harbor condition, 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. Figure 18.4 Sectional Resultant Forces in Still Water 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 approxi- (a) mated by a linear (trapezoidal) curve obtained by imposing

TABLE 18.I Wave Coefﬁcient Versus Length

(b) Ship Length L Wave Coefﬁcient C

90 ≤ L <300 m 10.75 Ð [(300 Ð L)/100]3/2 300 ≤ L <350 m 10.75 Figure 18.5 Examples of Reference Still Water Bending Moment Distribution 350 ≤ L 10.75 Ð [(300 Ð L)/150]3/2 (10). (a) oil tankers, bulk carriers, ore carriers, and (b) other ship types MASTER SET SDC 18.qxd Page 18-8 4/28/03 1:30 PM

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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 (Fig- ure 18.7), which generate an axial compressive force on the Figure 18.6 Weight Distribution Breakdown for Full Load Condition 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 counter- balanced by internal axial pressures, causing hull sagging Figure 18.7 Longitudinal Component of Pressure 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 Figure 18.8 Multi-hull Additional Still Water Loads (sketch) 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, the correspondence of area and barycenter of the trapezoid bending and torque in a transversal direction (see the sim- respectively to the total weight and center of gravity of the plified scheme of Figure 18.8, where S, B, and Q stand for

considered ship portion. shear, bending and torque; and L, T apply respectively to The procedure is usually applied separately for differ- longitudinal and transversal beams). ent types of weight items, grouping together the weights of the ship in lightweight conditions (always present on board) and those (cargo, ballast, consumables) typical of a load- 18.3.4 Wave Induced Global Loads ing condition (Figure 18.6). The prediction of the behaviour of the ship in waves represents a key point in the quantification of both global and 18.3.3.3 Uncertainties in the evaluation local loads acting on the ship. The solution of the seakeep- A significant contribution to uncertainties in the evaluation ing problem yields the loads directly generated by external of still water loads comes from the inputs to the procedure, pressures, but also provides ship motions and accelerations. in particular those related to quantification and location on The latter are directly connected to the quantification of in- board of weight items. ertial loads and provide inputs for the evaluation of other This lack of precision regards the weight distribution for types of loads, like slamming and sloshing. MASTER SET SDC 18.qxd Page 18-9 4/28/03 1:30 PM

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In particular, as regards global effects, the action of waves cation societies provide a statistically based reference values modiﬁes the pressure distribution along the wet hull sur- for the vertical component of wave-induced bending moment

face; the differential pressure between the situation in waves MWV,expressed as a function of main ship dimensions. and in still water generates, on the transverse section, ver- Such reference values for the midlength section of a ship

tical and horizontal resultant forces (bWV and bWH) and a with unrestricted navigation are yielded by equation 10 for moment component mTb. hog and sag cases (7) and corresponds to an extreme value Analogous components come from the sectional result- with a return period of about 20 years or an exceeding prob- ants of inertial forces and moments induced on the section ability of about 10Ð8 (once in the ship lifetime). by ship’s motions (Figure 18.9). 190 C L2 B C (hog) The total vertical and horizontal wave induced forces on MNm[]⋅ = B [10] WV −+2 the section, as well as the total torsional component, are 110 C L B ( CB 0 . 7) (sag) found summing up the components in the same direction (equations 9). Horizontal Wave-induced Bending Moment: Similar formulations are available for reference values of horizontal =− q WV(x) b WV (x) m(x)a V (x) wave induced bending moment, even though they are not =− as uniform among different Societies as for the main verti- q WH(x) b WH (x) m(x)a H (x) [9] cal component. m (x)=−θ m (x) I (x) TW Tb R In Table 18.II, examples are reported of reference values of horizontal bending moment at mid-length for ships where IR(x) is the rotational inertia of section x. The longitudinal distributions along the hull girder of hor- with unrestricted navigation. Simplified curves for the dis- izontal and vertical components of shear, bending moment tribution in the longitudinal direction are also provided. and torque can then be derived by integration (equations 1 Wave-induced Torque: A few reference formulations are to 5). given also for reference wave torque at midship (see ex- Such results are in principle obtained for each instanta- amples in Table 18.III) and for the inherent longitudinal neous wave pressure distribution, depending therefore, on distributions. time, on type and direction of sea encountered and on the ship geometrical and operational characteristics. 18.3.4.2 Static Wave analysis of global wave loads In regular (sinusoidal) waves, vertical bending moments A traditional analysis adopted in the past for evaluation of tend to be maximized in head waves with length close to wave-induced loads was represented by a quasi-static wave the ship length, while horizontal bending and torque com- approach. The ship is positioned on a freezed wave of given ponents are larger for oblique wave systems. characteristics in a condition of equilibrium between weight and static buoyancy. The scheme is analogous to the one de- 18.3.4.1 Statistical formulae for global wave loads scribed for still water loads, with the difference that the wa- Simplified, first approximation, formulations are available terline upper boundary of the immersed part of the hull is for the main wave load components, developed mainly on no longer a plane but it is a curved (cylindrical) surface. By the basis of past experience. definition, this procedure neglects all types of dynamic ef- Vertical wave-induced bending moment: IACS classifi- fects. Due to its limitations, it is rarely used to quantify wave loads. Sometimes, however, the concept of equivalent static wave is adopted to associate a longitudinal distribution of

TABLE 18.II Reference Horizontal Bending Moments ⋅ Class Society MWH [N m]

2 ABS (8) 180 C1L DCB 2.1 BV (9) RINA (10) 1600 L TCB 9/4 DNV (11) 220 L (T + 0.3B)CB NKK (12) 320 L C TL− 35 / L Figure 18.9 Sectional Forces and Moments in Waves 2 MASTER SET SDC 18.qxd Page 18-10 4/28/03 1:30 PM

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TABLE 18.III Examples of Reference Values for Wave Torque . Class Society Qw [N m] (at mid-ship)

 05.  2 − 2 + − e  014.  ABS (bulk carrier) 2700LB T[]( CW 0 . 5) 0 . 1 0 . 13     D T  (e = vertical position of shear center)

 − 3  2 2 −  250 0.L 7  BV RINA 190LB C 8 . 13    W  125 

Φ pressures to extreme wave loads, derived, for example, from r = radiation component due to the ship motions in calm long term predictions based on other methods. water Φ FK = excitation component, due to the incident wave 18.3.4.3 Linear methods for wave loads (undisturbed by the presence of the ship): Froude- The most popular approach to the evaluation of wave loads Krylov Φ is represented by solutions of a linearized potential flow d = diffraction component, due to disturbance in the wave problem based on the so-called strip theory in the frequency potential generated by the hull domain (13). The theoretical background of this class of procedures This subdivision also enables the de-coupling of the ex- is discussed in detail in PNA Vol. III (2). citation components from the response ones, thus avoiding Here only the key assumptions of the method are pre- a non-linear feedback between the two. sented: Other key properties of linear systems that are used in the analysis are: ¥ inviscid, incompressible and homogeneous fluid in irro- tational flow: Laplace equation 11 ¥linear relation between the input and output amplitudes, and ∇2Φ = 0 [11] ¥ superposition of effects (sum of inputs corresponds to where Φ = velocity potential sum of outputs). ¥ 2-dimensional solution of the problem When using linear methods in the frequency domain, ¥ linearized boundary conditions: the quadratic compo- the input wave system is decomposed into sinusoidal com- nent of velocity in the Bernoulli Equation is reformu- ponents and a response is found for each of them in terms lated in linear terms to express boundary conditions: of amplitude and phase. — on free surface: considered as a plane corresponding The input to the procedure is represented by a spectral to still water: fluid velocity normal to the free surface representation of the sea encountered by the ship. Responses, equal to velocity of the surface itself (kinematic con- for a ship in a given condition, depend on the input sea char- dition); zero pressure, acteristics (spectrum and spatial distribution respect to the — on the hull: considered as a static surface, corre- ship course). sponding to the mean position of the hull: the com- The output consists of response spectra of point pres- ponent of the fluid velocity normal to the hull surface sures on the hull and of the other derived responses, such is zero (impermeability condition), and as global loads and ship motions. Output spectra can be used to derive short and long-term predictions for the prob- ¥ linear decomposition into additive independent compo- ability distributions of the responses and of their extreme nents, separately solved for and later summed up (equa- values (see Subsection 18.3.4.5). tion 12). Despite the numerous and demanding simplifications at Φ Φ Φ Φ Φ = s + FK + d + r [12] the basis of the procedure, strip theory methods, developed since the early 60s, have been validated over time in sev- where: eral contexts and are extensively used for predictions of Φ s = stationary component due to ship advancing in calm wave loads. water In principle, the base assumptions of the method are MASTER SET SDC 18.qxd Page 18-11 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-11

valid only for small wave excitations, small motion re- ∞ = ωωωn sponses and low speed of the ship. mS()dny ∫ y [13] In practice, the ﬁeld of successful applications extends 0 far beyond the limits suggested by the preservation of re- This information is the basis of the spectral method, alism in the base assumptions: the method is actually used whose theoretical framework (main hypotheses, assump- extensively to study even extreme loads and for fast ves- tions and steps) is recalled in the following. sels. If the stochastic process representing the wave input to the ship system is modeled as a stationary and ergodic 18.3.4.4 Limits of linear methods for wave loads Gaussian process with zero mean, the response of the sys- Due to the simpliﬁcations adopted on boundary conditions tem (load) can be modeled as a process having the same char- to linearize the problem of ship response in waves, results acteristics. in terms of hydrodynamic pressures are given always up to The Parseval theorem and the ergodicity property es- the still water level, while in reality the pressure distribu- tablish a correspondence between the area of the response

tion extends over the actual wetted surface. This represents spectrum (spectral moment of order 0: m0Y) and the vari- a major problem when dealing with local loads in the side ance of its Gaussian probability distribution (14). This al- region close to the waterline. lows expressing the density probability distribution of the

Another effect of basic assumptions is that all responses Gaussian response y in terms of m0Y (equation 14). at a given frequency are represented by sinusoidal fluctua- 1 −( ym2 / 2 2 ) tions (symmetric with respect to a zero mean value). A con- f (y) = e 0 Y [14] Y π sequence is that all the derived global wave loads also have 2 m 0 Y the same characteristics, while, for example, actual values of vertical bending moment show marked differences be- Equation 14 expresses the distribution of the fluctuating tween the hogging and sagging conditions. Corrections to response y at a generic time instant. account for this effect are often used, based on statistical From a structural point of view, more interesting data data (7) or on more advanced non-linear methods. are represented by: A third implication of linearization regards the super- ¥ the probability distribution of the response at selected imposition of static and dynamic loads. Dynamic loads are time instants, corresponding to the highest values in each evaluated separately from the static ones and later summed zero-crossing period (peaks: variable p), up: this results in an un-physical situation, in which weight ¥ the probability distribution of the excursions between forces (included only in static loads) are considered as act- the highest and the lowest value in each zero-crossing ing always along the vertical axis of the ship reference sys- period (range: variable r), and tem (as in still water). Actually, in a seaway, weight forces ¥ the probability distribution of the highest value in the are directed along the true vertical direction, which depends whole stationary period of the phenomenon (extreme on roll and pitch angles, having therefore also components value in period T ,variable extrTsy). in the longitudinal and lateral direction of the ship. s This aspect represents one of the intrinsic non-lineari- The aforementioned distributions can be derived from ties in the actual system, as the direction of an external input the underlying Gaussian distribution of the response (equa- force (weight) depends on the response of the system itself tion 14) in the additional hypotheses of narrow band re- (roll and pitch angles). sponse process and of independence between peaks. The first This effect is often neglected in the practice, where lin- two probability distributions take the form of equations 15 ear superposition of still water and wave loads is largely fol- and 16 respectively, both Rayleigh density distributions (see lowed. 14). The distribution in equation 16 is particularly interest- 18.3.4.5 Wave loads probabilistic characterization ing for fatigue checks, as it can be adopted to describe stress The most widely adopted method to characterize the loads ranges of fatigue cycles. in the probability domain is the so-called spectral method,  2  =−p p used in conjunction with linear frequency-domain methods fpP ( ) exp  [15] for the solution of the ship-wave interaction problem. m 0  2m 0  From the frequency domain analysis response spectra ω  2  Sy( ) are derived, which can be integrated to obtain spec- =−r r frR ( ) exp  [16] tral moments mn of order n (equation 13). 48m 0  m 0  MASTER SET SDC 18.qxd Page 18-12 4/28/03 1:30 PM

18-12 Ship Design & Construction, Volume 1

The distribution for the extreme value in the stationary

period Ts (short term extreme) can be modeled by a Pois- son distribution (in equation 17: expression of the cumulative distribution) or other equivalent distributions derived from the statistics of extremes.

  2   1 m 2 p extrTs =− −  Fp( ) exp∂ exp   Ts [17]  2 m 0  2m 0   Figure 18.10 summarizes the various short-term distributions. Figure 18.10 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. of signiﬁcant wave heights and mean periods. Such scatter The results mentioned previously are derived for the diagrams are catalogued according to sea zones, such as shown in Figure 18.11 (the subdivision of the world atlas), period Ts in which the input wave system can be considered as stationary (sea state: typically, a period of a few and main wave direction. Seasonal characteristics are also hours). The derived distributions (short-term predictions) available. are conditioned to the occurrence of a particular sea state, The process described in equation 18 can be termed de- which is identiﬁed by the sea spectrum, its angular distri- conditioning (that is removing the conditioning hypothesis). bution around the main wave direction (spreading func- The same procedure can be applied to any of the variables tion) and the encounter angle formed with ship advance studied in the short term and it does not change the nature direction. of the variable itself. If a range distribution is processed, a To obtain a long-term prediction, relative to the ship life long-term distribution for ranges of single oscillations is obtained (useful data for a fatigue analysis). (or any other design period Td which can be described as a extrTs series of stationary periods), the conditional hypothesis is If the distribution of variable y is de-conditioned, a weighed average of the highest peak in time T is achieved. to be removed from short-term distributions. In other words, s the probability of a certain response is to be weighed by the In this case the result is further processed to get the distribution of the extreme value in the design time T . This is probability of occurrence of the generating sea state (equa- d tion18). done with an additional application of the concept of statistics of extremes. n In the hypothesis that the extremes of the various sea ( ) = ⋅ Fy∑ FyS( ii) P(S ) [18] states are independent from each other, the extreme on time i = 1 Td is given by equation 19: where: Td/Ts FyFy( extrTd) = []( extrTs ) [19] F(y) = probability for the response to be less than value y (unconditioned). where F(extrTdy) is the cumulative probability distribution  F(y Si) = probability for the response to be less than value for the highest response peak in time Td (long-term extreme y, conditioned to occurrence of sea state Si (short distribution in time T ). term prediction). d

P(Si) = probability associated to the i-th sea state. n= total number of sea states, covering all combi- 18.3.4.6 Uncertainties in long-term predictions nations. The theoretical framework of the above presented spectral method, coupled to linear frequency domain methodolo-

Probability P(Si) can be derived from collections of sea data gies like those summarized in Subsection 18.3.4.3, allows based on visual observations from commercial ships and/or the characterization, in the probability domain, of all the on surveys by buoys. wave induced load variables of interest both for strength One of the most typical formats is the one contained in and fatigue checks. (15), where sea states probabilities are organized in bi-di- The results of this linear prediction procedure are af- mensional histograms (scatter diagrams), containing classes fected by numerous sources of uncertainties, such as: MASTER SET SDC 18.qxd Page 18-13 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-13

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

¥ sea description: as above mentioned, scatter diagrams Contrary to strength verifications of the hull girder, which are derived from direct observations on the field, which are nowadays largely based on ultimate limit states (for ex- are affected by a certain degree of indetermination. ample, in longitudinal strength: ultimate bending moment), In addition, simplified sea spectral shapes are adopted, checks on local structures are still in part implicitly based based on a limited number of parameters (generally, bi- on more conservative limit states (yield strength). parametric formulations based on significant wave and In many Rules, reference (characteristic) local loads, as mean wave period), well as the motions and accelerations on which they are ¥ model for the ship’s response: as briefly outlined in Sub- based, are therefore implicitly calibrated at an exceeding section 18.3.4.3, the model is greatly simplified, partic- probability higher than the 10Ð8 value adopted in global load ularly as regards fluid characteristics and boundary strength verifications. conditions. Numerical algorithms and specific procedures adopted for the solution also influence results, creating differences 18.3.6 External Pressure Loads even between theoretically equivalent methods, and Static and dynamic pressures generated on the wet surface ¥ the de-conditioning procedure adopted to derive long of the hull belong to external loads. They act as local trans- term predictions from short term ones can add further verse loads for the hull plating and supporting structures. uncertainties. 18.3.6.1 Static external pressures Hydrostatic pressure is related through equation 20 to the 18.3.5 Local Loads vertical distance between the free surface and the load point

As previously stated, local loads are applied to individual (static head hS). structural members like panels and beams (stiffeners or pri- ρ mary supporting members). pS = ghS [20]

They are once again traditionally divided into static and In the case of the external pressure on the hull, hS cor- dynamic loads, referred respectively to the situation in still responds to the local draft of the load point (reference is water and in a seaway. made to design waterline). MASTER SET SDC 18.qxd Page 18-14 4/28/03 1:30 PM

18-14 Ship Design & Construction, Volume 1

18.3.6.2 Dynamic pressures nal velocities can arise in the longitudinal and/or transver- The pressure distribution, as well as the wet portion of the sal directions, producing additional pressure loads (slosh- hull, is modified for a ship in a seaway with respect to the ing loads). still water (Figure 18.9). Pressures and areas of application If pitch or roll frequencies are close to the tank reso- are in principle obtained solving the general problem of nance frequency in the inherent direction (which can be ship motions in a seaway. evaluated on the basis of geometrical parameters and fill- Approximate distributions of the wave external pressure, ing ratio), kinetic energy tends to concentrate in the fluid to be added to the hydrostatic one, are adopted in Classifi- and sloshing phenomena are enhanced. cation Rules for the ship in various load cases (Figure 18.12). 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 pro- 18.3.7 Internal Loads—Liquid in Tanks cedures have been developed for the purpose. For more Liquid cargoes generate normal pressures on the walls of details see references 16 and 17. the containing tank. Such pressures represent a local trans- A further type of excitation is represented by impacts that versal load for plate, stiffeners and primary supporting mem- can occur on horizontal or sub-horizontal plates of the upper bers of the tank walls. part of the tank walls for high filling ratios and, at low filling levels, in vertical or sub-vertical plates of the lower part 18.3.7.1 Static internal pressure of the tank. For a ship in still water, gravitation acceleration g gener- Impact loads are very difficult to characterize, being re- ates a hydrostatic pressure, varying again according to equa- lated to a number of effects, such as: local shape and ve-

tion 20. The static head hS corresponds here to the vertical locity of the free surface, air trapping in the fluid and distance from the load point to the highest part of the tank, response of the structure. A complete model of the phe- increased to account for the vertical extension over that nomenon would require a very detailed two-phase scheme point of air pipes (that can be occasionally filled with liq- for the fluid and a dynamic model for the structure includ- uid) or, if applicable, for the ullage space pressure (the pres- ing hydro-elasticity effects. sure present at the free surface, corresponding for example Simplified distributions of sloshing and/or impact pres- to the setting pressure of outlet valves). sures are often provided by Classification Societies for structural verification (Figure 18.14). 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 on- board, 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 ac-

celeration aw. The total pressure (equation 21) can be evaluated in terms Figure 18.12 Example of Simpliﬁed Distribution of External Pressure (10)

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 Fig- θ ure 18.13 on roll angle r). ρ pf = aThT [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) MASTER SET SDC 18.qxd Page 18-15 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-15

18.3.7.3 Dry bulk cargo the flat part of the hull and the water free surface, presence In the case of a dry bulk cargo, internal friction forces arise and extension of air trapped between fluid and ship bottom within the cargo itself and between the cargo and the walls and structural dynamic behavior (18,19). of the hold. As a result, the component normal to the wall While slamming probability of occurrence can be stud- has a different distribution from the load corresponding to ied on the basis only of predictions of ship relative motions a liquid cargo of the same density; also additional tangen- (which should in principle include non-linear effects due to tial components are present. 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 18.3.8 Inertial Loads—Dry Cargo experimental point of view (18,19). To account for this effect, distributions for the components From a practical point of view, Class Societies prescribe, of cargo load are approximated with empirical formulations for ships with loading conditions corresponding to a low fore 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 Figure 18.14 Example of Simpliﬁed Distributions of Sloshing and Impact of the deck plate between stiffeners: in this case the load is Pressures (11) 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 mo- Figure 18.15 Scheme of Local Forces Transmitted by a Container to the tions relative to water, local angle formed at impact between Support System (8) MASTER SET SDC 18.qxd Page 18-16 4/28/03 1:30 PM

18-16 Ship Design & Construction, Volume 1

draft, local structural checks based on an additional exter- the stern region, thus generating an exciting force for the nal pressure. structure. Such additional pressure is formulated as a function of A second effect is due to axial and non axial forces and ship main characteristics, of local geometry of the ship moments generated by the propeller on the shaft and trans- (width of flat bottom, local draft) and, in some cases, of the mitted through the bearings to the hull (bearing forces). first natural frequency of flexural vibration of the hull girder. Due to the negative dynamic pressure generated by the The influence on global loads is accounted for by an ad- increased angle of attack, the local pressure on the back of ditional term for the vertical wave-induced bending mo- blade profiles can, for any rotation angle, fall below the ment, which can produce a significant increase (15% and vapor saturation pressure. In this case, a vapor sheet is gen- more) in the design value. erated on the back of the profile (cavitation phenomenon). A phenomenon quite similar to bottom slamming can The vapor filled cavity collapses as soon as the angle of at- occur also on the forebody of ships with a large bow flare. tack decreases in the propeller revolution and the local pres- In this case dynamic and (to a lesser extent) impulsive pressure rises again over the vapor saturation pressure. sures are generated on the sides of V-shaped fore sections. Cavitation further enhances pressure fluctuations, be- The phenomenon is likely to occur quite frequently on cause of the rapid displacement of the surrounding water ships prone to it, but with lower pressures than in bottom volume during the growing phase of the vapor bubble and slamming. The incremental effect on vertical bending mo- because of the following implosion when conditions for its ment can however be significant. existence are removed. A quantification of bow flare effects implies taking into All of the three mentioned types of excitation have their account the variation of the local breadth of the section as main components at the propeller rotational frequency, at a function of draft. It represents a typical non-linear effect the blade frequency, and at their first harmonics. In addi- (non-linearity due to hull geometry). tion to the above frequencies, the cavitation pressure field Slamming can also occur in the rear part of the ship, contains also other components at higher frequency, related when the flat part of the stern counter is close to surface. to the dynamics of the vapor cavity. Propellers with skewed blades perform better as regards 18.3.9.2 Springing induced pressure, because not all the blade sections pass si- Another phenomenon which involves the dynamic response multaneously in the region of the stern counter, where dis- of the hull girder is springing. For particular types of ships, turbances in the wake are larger; accordingly, pressure a coincidence can occur between the frequency of wave ex- fluctuations are distributed over a longer time period and citation and the natural frequency associated to the first peak values are lower. (two-node) flexural mode in the vertical plane, thus pro- Bearing forces and pressures induced on the stern counter ducing a resonance for that mode (see also Subsection by cavitating and non cavitating propellers can be calculated 18.6.8.2). with dedicated numerical simulations (18). The phenomenon has been observed in particular on Great Lakes vessels, a category of ships long and flexible, with com- 18.3.9.4 Main engine excitation paratively low resonance frequencies (1, Chapter VI). Another major source of dynamic excitation for the hull The exciting action has an origin similar to the case of girder is represented by the main engine. Depending on quasi-static wave bending moment and can be studied with general arrangement and on number of cylinders, diesel en- the same techniques, but the response in terms of deflec- gines generate internally unbalanced forces and moments, tion and stresses is magnified by dynamic effects. For re- mainly at the engine revolution frequency, at the cylinders cent developments of research in the field (see references firing frequency and inherent harmonics (Figure 18.16). 16 and 17). The excitation due to the first harmonics of low speed diesel engines can be at frequencies close to the first natu- 18.3.9.3 Propeller induced pressures and forces ral hull girder frequencies, thus representing a possible cause Due to the wake generated by the presence of the after part of a global resonance. of the hull, the propeller operates in a non-uniform incident In addition to frequency coincidence, also direction and velocity field. location of the excitation are important factors: for exam- Blade profiles experience a varying angle of attack dur- ple, a vertical excitation in a nodal point of a vertical flex- ing the revolution and the pressure field generated around ural mode has much less effect in exciting that mode than the blades fluctuates accordingly. the same excitation placed on a point of maximum modal The dynamic pressure field impinges the hull plating in deflection. MASTER SET SDC 18.qxd Page 18-17 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 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. = (φφ) FCWiL,T12/ F L,T Wi AV Wi Wi 2 [22] = (φφ) MCALVWiz12/ Mz Wi Wi Wi 2 [23] where: φ Wi = the angle formed by the direction of the wind relative to the ship φ φ φ CMz( Wi), CFL( Wi), CFT( Wi) are all coefficients depending Figure 18.16 Propeller, Shaft and Engine Induced Actions (20) on the shape of exposed part of the ship and on φ angle Wi AWi = the reference area for the surface of the ship exposed to wind, (usually the area of the cross section) In addition to low frequency hull vibrations, components V = the wind speed at higher frequencies from the same sources can give rise Wi to resonance in local structures, which can be predicted by The empirical formulas in equations 22 and 23 account suitable dynamic structural models (18,19). 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 18.3.10 Other Loads hull a similar action to the one of wind on the emerged part 18.3.10.1 Thermal loads (drag force). It can be described through coefficients and A ship experiences loads as a result of thermal effects, which variables analogous to those of equations 22 and 23. can be produced by external agents (the sun heating the Waves: Linear wave excitation has in principle a sinu- deck), or internal ones (heat transfer from/to heated or re- soidal time dependence (whose mean value is by definition frigerated cargo). zero). If ship motions in the wave direction are not con- What actually creates stresses is a non-uniform temper- strained (for example, if the anchor chain is not in tension) ature distribution, which implies that the warmer part of the the ship motion follows the excitation with similar time de- structure tends to expand while the rest opposes to this dependence and a small time lag. In this case the action on formation. A peculiar aspect of this situation is that the por- the mooring system is very small (a few percent of the other tion of the structure in larger elongation is compressed and actions). vice-versa, which is contrary to the normal experience. If the ship is constrained, significant loads arise on the It is very difficult to quantify thermal loads, the main mooring system, whose amplitude can be of the same order problems being related to the identification of the temper- of magnitude of the stationary forces due to the other actions. ature distribution and in particular to the model for con- In addition to the linear effects discussed above, non-lin- straints. Usually these loads are considered only in a ear wave actions, with an average value different from zero, qualitative way (1, Chapter VI). are also present, due to potential forces of higher order, formation of vortices, and viscous effects. These components 18.3.10.2 Mooring loads can be significant on off-shore floating structures, which For a moored vessel, loads are exerted from external actions often feature also complicated mooring systems: in those on the mooring system and from there to the local sup- cases the dynamic behavior of the mooring system is to be porting structure. The main contributions come by wind, included in the analysis, to solve a specific motion prob- waves and current. lem. For common ships, non-linear wave effects are usu- Wind: The force due to wind action is mainly directed in ally neglected. the direction of the wind (drag force), even if a limited com- A practical rule-of-thumb for taking into account wave ponent in the orthogonal direction can arise in particular sit- actions for a ship at anchor in non protected waters is to in- uations. The magnitude depends on the wind speed and on crease of 75 to 100% the sum of the other force components. extension and geometry of the exposed part of the ship. The Once the total force on the ship is quantified, the ten- action due to wind can be described in terms of two force sion in the mooring system (hawser, rope or chain) can be MASTER SET SDC 18.qxd Page 18-18 4/28/03 1:30 PM

18-18 Ship Design & Construction, Volume 1

derived by force decomposition, taking into account the integrated according to equations 1 and 2 to derive vertical angle formed with the external force in the horizontal and/or shear and bending moment. vertical plane. qVL(x) = w(x) Ð bL(x) Ð fC(x) [26] 18.3.10.3 Launching loads This computation is performed for various intermediate The launch is a unique moment in the life of the ship. For positions of the cradle during the launching in order to check a successful completion of this complex operation, a num- all phases. However, the most demanding situation for the ber of practical, organizational and technical elements are hull girder corresponds to the instant when pivoting starts. to be kept under control (as general reference see Reference In that moment the cradle force is concentrated close to 1, Chapter XVII). the bow, at the fore end of the cradle itself (on the fore pop- Here only the aspect of loads acting on the ship will be pet, if one is ﬁtted) and it is at the maximum value. discussed, so, among the various types of launch, only those A considerable sagging moment is present in this situ- which present peculiarities as regards ship loads will be ation, whose maximum value is usually lower than the de- considered: end launch and side launch. sign one, but tends to be located in the fore part of the ship, End Launch: In end launch, resultant forces and motions where bending strength is not as high as at midship. are contained in the longitudinal plane of the ship (Figure Furthermore, the ship at launching could still have tem- 18.17). porary openings or incomplete structures (lower strength) The vessel is subjected to vertical sectional forces dis- in the area of maximum bending moment.

tributed along the hull girder: weight w(x), buoyancy bL(x) Another matter of concern is the concentrated force at and the sectional force transmitted from the ground way to the fore end of the cradle, which can reach a signiﬁcant per- the cradle and from the latter to the ship’s bottom (in the centage of the total weight (typically 20Ð30%). It represents

following: sectional cradle force fC(x), with resultant FC). a strong local load and often requires additional temporary While the weight distribution and its resultant force internal strengthening structures, to distribute the force on (weight W) are invariant during launching, the other distri- a portion of the structure large enough to sustain it. butions change in shape and resultant: the derivation of Side Launch: In side launch, the main motion compo- launching loads is based on the computation of these two nents are directed in the transversal plane of the ship (see distributions. Figure 18.19, reproduced from reference 1, Chapter XVII). Such computation, repeated for various positions of the The vertical reaction from ground ways is substituted in cradle, is based on the global static equilibrium s (equa- a comparatively short time by buoyancy forces when the ship tions 24 and 25, in which dynamic effects are neglected: tilts and drops into water. quasi static approach). The kinetic energy gained during the tilting and drop- ping phases makes the ship oscillate around her ﬁnal posi- BT + FC Ð W = 0 [24]

xB BT + xF FC Ð xW W = 0 [25] where:

W, B T,FC = (respectively) weight, buoyancy and cradle force resultants

xW,xB,xF = their longitudinal positions In a ﬁrst phase of launching, when the cradle is still in contact for a certain length with the ground way, the buoy- Figure 18.17 End Launch: Sketch ancy distribution is known and the cradle force resultant and position is derived. In a second phase, beginning when the cradle starts to

rotate (pivoting phase: Figure 18.18), the position xF 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 18.18 Forces during Pivoting MASTER SET SDC 18.qxd Page 18-19 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-19

tion at rest. The amplitude of heave and roll motions and Governing equations for the problem are given by con- accelerations governs the magnitude of hull girder loads. servation of momentum and of energy. Within this frame- Contrary to end launch, trajectory and loads cannot be stud- work, time domain simulations can evaluate the magnitude ied as a sequence of quasi-static equilibrium positions, but of contact forces and the energy, which is absorbed by struc- need to be investigated with a dynamic analysis. ture deformation: these quantities, together with the response The problem is similar to the one regarding ship mo- characteristics of the structure (energy absorption capacity), tions in waves, (Subsection 18.3.4), with the difference that allow an evaluation of the damage penetration (21). here motions are due to a free oscillation of the system due Grounding: In grounding, dominant effects are forces and to an unbalanced initial condition and not to an external ex- motions in the vertical plane. citation. As regards forces, main components are contact forces, Another difference with respect to end launch is that developed at the first impact with the ground, then friction, both ground reaction (first) and buoyancy forces (later) are when the bow slides on the ground, and weight. always distributed along the whole length of the ship and From the point of view of energy, the initial kinetic en- are not concentrated in a portion of it. ergy is (a) dissipated in the deformation of the lower part of the bow (b) dissipated in friction of the same area against 18.3.10.4 Accidental loads the ground, (c) spent in deformation work of the ground (if Accidental loads (collision and grounding) are discussed soft: sand, gravel) and (d) converted into gravitational po- in more detail by ISSC (21). tential energy (work done against the weight force, which Collision: When defining structural loads due to colli- resists to the vertical raising of the ship barycenter). sions, the general approach is to model the dynamics of the In addition to soil characteristics, key parameters for the accident itself, in order to define trajectories of the unit(s) description are: slope and geometry of the ground, initial involved. speed and direction of the ship relative to ground, shape of In general terms, the dynamics of collision should be the bow (with/without bulb). formulated in six degrees of freedom, accounting for a num- The final position (grounded ship) governs the magni- ber of forces acting during the event: forces induced by pro- tude of the vertical reaction force and the distribution of peller, rudder, waves, current, collision forces between the shear and sagging moment that are generated in the hull units, hydrodynamic pressure due to motions. girder. Figure 18.20 gives an idea of the magnitude of Normally, theoretical models confine the analysis to grounding loads for different combinations of ground slopes components in the horizontal plane (3 degrees of freedom) and coefficients of friction for a 150 000 tanker (results of and to collision forces and motion-induced hydrodynamic simulations from reference 22). pressures. The latter are evaluated with potential methods In addition to numerical simulations, full and model of the same type as those adopted for the study of the re- scale tests are performed to study grounding events (21). sponse of the ship to waves. As regards collision forces, they can be described dif- ferently 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) Figure 18.20 Sagging Moments for a Grounded Ship: Simulation Results (22) MASTER SET SDC 18.qxd Page 18-20 4/28/03 1:30 PM

18-20 Ship Design & Construction, Volume 1

18.3.11 Combination of Loads 18.3.12.1 2D versus 3D models When dealing with the characterization of a set of loads Three-dimensional extensions of linear methods are avail- acting simultaneously, the interest lies in the definition of able; some non-linear methods have also 3-D features, while a total loading condition with the required exceeding prob- in other cases an intermediate approach is followed, with ability (usually the same of the single components). This boundary conditions formulated part in 2D, part in 3D. cannot be obtained by simple superposition of the characteristic values of single contributing loads, as the probabil- 18.3.12.2 Body boundary conditions ity that all design loads occur at the same time is much lower In linear methods, body boundary conditions are set with than the one associated to the single component. reference to the mean position of the hull (in still water). In the time domain, the combination problem is ex- Perturbation terms take into account, in the frequency or in pressed in terms of time shift between the instants in which the time domain, first order variations of hydrodynamic and characteristic values occur. hydrostatic coefficients around the still water line. In the probability domain, the complete formulation of Other non-linear methods account for perturbation terms the problem would imply, in principle, the definition of a of a higher order. In this case, body boundary conditions joint probability distribution of the various loads, in order are still linear (mean position of the hull), but second order to quantify the distribution for the total load. An approxi- variations of the coefficients are accounted for. mation would consist in modeling the joint distribution Mixed or blending procedures consist in linear methods through its first and second order moments, that is mean val- modified to include non-linear effects in a single compo- ues and covariance matrix (composed by the variances of nent of the velocity potential (while the other ones are treated the single variables and by the covariance calculated for linearly). In particular, they account for the actual geome- each couple of variables). However, also this level of sta- try of wetted hull (non-linear body boundary condition) in tistical characterization is difficult to obtain. the Froude-Krylov potential only. This effect is believed to As a practical solution to the problem, empirically based have a major role in the definition of global loads. load cases are defined in Rules by means of combination More evolved (and complex) methods are able to take coefficients (with values generally ≤ 1) applied to single properly into account the exact body boundary condition loads. Such load cases, each defined by a set of coefficients, (actual wetted surface of the hull). represent realistic and, in principle, equally probable combinations of characteristic values of elementary loads. 18.3.12.3 Free surface boundary conditions Structural checks are performed for all load cases. The Boundary conditions on free surface can be set, depending result of the verification is governed by the one, which turns on the various methods, with reference to: (a) a free stream out to be the most conservative for the specific structure. at constant velocity, corresponding to ship advance, (b) a This procedure needs a higher number of checks (which, on double body flow, accounting for the disturbance induced the other hand, can be easily automated today), but allows by the presence of a fully immersed double body hull on considering various load situations (defined with different the uniform flow, (c) the flow corresponding to the steady combinations of the same base loads), without choosing a advance of the ship in calm water, considering the free sur- priori the worst one. face 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 New Trends and Load Non-linearities A large part of research efforts is still devoted to a better 18.3.12.4 Fluid characteristics definition of wave loads. New procedures have been pro- All the methods above recalled are based on an inviscid posed in the last decades to improve traditional 2D linear fluid potential scheme. methods, overcoming some of the simplifications adopted Some results have been published of viscous flow mod- to treat the problem of ship motions in waves. For a com- els based on the solution of Reynolds Averaged Navier plete state of the art of computational methods in the field, Stokes (RANS) equations in the time domain. These meth- reference is made to (23). A very coarse classification of ods represent the most recent trend in the field of ship mo- the main features of the procedures reported in literature is tions and loads prediction and their use is limited to a few here presented (see also reference 24). research groups. MASTER SET SDC 18.qxd Page 18-21 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-21

18.4 STRESSES AND DEFLECTIONS ¥ Stresses in the plating of stiffened panel under lateral σ σ pressure may have different origins ( 2 and 2*). For a The reactions of structural components of the ship hull to σ stiffened panel, there is the stress ( 2) and deflection of external loads are usually measured by either stresses or the global bending of the orthotropic stiffened panels, deflections. Structural performance criteria and the associ- for example, the panel of bottom structure contained be- ated analyses involving stresses are referred to under the gen- tween two adjacent transverse bulkheads. The stiffener eral term of strength. The strength of a structural component and the attached plating bend under the lateral load and would be inadequate if it experiences a loss of load-carry- the plate develops additional plane stresses since the ing ability through material fracture, yield, buckling, or plate acts as a flange with the stiffeners. In longitudinally some other failure mechanism in response to the applied framed ships there is also a second type of secondary loading. Excessive deflection may also limit the structural σ stresses: 2* corresponds to the bending under the hy- effectiveness of a member, even though material failure drostatic pressure of the longitudinals between trans- does not occur, if that deflection results in a misalignment verse frames (web frames). For transversally framed or other geometric displacement of vital components of the σ panels, 2* may also exist and would correspond to the ship’s machinery, navigational equipment, etc., thus ren- bending of the equally spaced frames between two stiff dering the system ineffective. longitudinal girders. The present section deals with the determination of the ¥ A double bottom behaves as box girder but can bend lon- responses, in the form of stress and deflection, of structural gitudinally, transversally or both. This global bending in- members to the applied loads. Once these responses are duces stress (σ ) and deflection. In addition, there is also known it is necessary to determine whether the structure is 2 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 σ of load. The associated primary stresses ( 1) 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 dou- Figure 18.21 Primary (Hull), Secondary (Double Bottom and Stiffened Panels) ble bottom, double sides, etc., for double hull ships: and Tertiary (Plate) Structural Responses (1, 2) MASTER SET SDC 18.qxd Page 18-22 4/28/03 1:30 PM

18-22 Ship Design & Construction, Volume 1

σ the 2* stress that corresponds to the bending of the lon- tect deals principally with beam theory, plate theory, and gitudinals (for example, in the inner and outer bottom) combinations of both. between two transverse elements (floors). Tertiary response describes the out-of-plane deflection 18.4.2 Basic Structural Components and associated stress of an individual unstiffened plate panel Structural components are extensively discussed in Chap- included between 2 longitudinals and 2 transverse web ter 17 Ð Structure Arrangement Component Design. In this frames. The boundaries are formed by these components section, only the basic structural component used exten- (Figure 18.22). sively is presented. It is basically a stiffened panel. Primary and secondary responses induce in-plane mem- The global ship structure is usually referred to as being brane stresses, nearly uniformly distributed through the plate a box girder or hull girder. Modeling of this hull girder is thickness. Tertiary stresses, which result from the bending the first task of the designer. It is usually done by model- of the plate member itself vary through the thickness, but ing the hull girder with a series of stiffened panels. may contain a membrane component if the out-of-plane de- Stiffened panels are the main components of a ship. Al- flections are large compared to the plate thickness. most any part of the ship can be modeled as stiffened pan- In many instances, there is little or no interaction be- els (plane or cylindrical). tween the three (primary, secondary, tertiary) component This means that, once the ship’s main dimensions and stresses or deflections, and each component may be com- general arrangement are fixed, the remaining scantling de- puted by methods and considerations entirely independent velopment mainly deals with stiffened panels. of the other two. The resultant stress, in such a case, is then The panels are joined one to another by connecting lines obtained by a simple superposition of the three component (edges of the prismatic structures) and have longitudinal stresses (Subsection 18.4.7). An exception is the case of and transverse stiffening (Figures 18.23, 24 and 36). plate (tertiary) deflections, which are large compared to the thickness of plate. ¥ Longitudinal Stiffening includes In plating, each response induces longitudinal stresses, — longitudinals (equally distributed), used only for the transverse stresses and shear stresses. This is due to the design of longitudinally stiffened panels, Poisson’s Ratio. Both primary and secondary stresses are — girders (not equally distributed). bending stresses but in plating these stresses look like membrane stresses. ¥Transverse Stiffening includes (Figure 18.23) In stiffeners, only primary and secondary responses in- — transverse bulkheads (a), duce stresses in the direction of the members and shear — the main transverse framing also called web-frames stresses. Tertiary response has no effect on the stiffeners. (equally distributed; large spacing), used for longi- In Figure 18.21 (see also Figure 18.37) the three types of re- tudinally stiffened panels (b) and transversally stiff- σ σ σ sponse are shown with their associated stresses ( 1, 2, 2* ened panels (c). σ and 3). These considerations point to the inherent simplicity of the underlying theory. The structural naval archi- 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 uti- σ lized in computing the component of primary stress, 1, 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: ¥ the beam is prismatic, that is, all cross sections are the same and there is no openings or discontinuities, Figure 18.22 A Standard Stiffened Panel ¥plane cross sections remain plane after deformation, will MASTER SET SDC 18.qxd Page 18-23 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 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 concen- Figure 18.24 Behavior of an Elastic Beam under Shear Force and Bending trations are presented in Subsection 18.6.6.3 as they are Moment (2) linked to fatigue. The elastic linear bending equations, equations 27 and 28, are derived from basic mechanic principle presented at Figure 18.24. Hull Section Modulus: The plane section assumption to- EI (∂2w/∂x2) = M(x) [27] gether with elastic material behavior results in a longitudi- σ nal stress, 1, in the beam that varies linearly over the depth or of the cross section. EI (∂4w/∂x4) = q(x) [28] The simple beam theory for longitudinal strength calculations of a ship is based on the hypothesis (usually at- where: tributed to Navier) that plane sections remain plane and in w = deﬂection (Figure 18.24), in m the absence of shear, normal to the OXY plane (Figure E = modulus of elasticity of the material, in N/m2 18.24). This gives the well-known formula: I = moment of inertia of beam cross section about a  2  4 =−p p horizontal axis through its centroid, in m fpP ( ) exp  [29] M(x) = bending moment, in N.m m 0  2m 0  q(x) = load per unit length in N/m where: = ∂V(x)/∂x = ∂2M(x)/∂x2 M = bending moment (in N.m) = EI (∂4w/∂x4) σ = bending stress (in N/m2) MASTER SET SDC 18.qxd Page 18-24 4/28/03 1:30 PM

18-24 Ship Design & Construction, Volume 1

I = Sectional moment of Inertia about the neutral axis ordinates of the section-moduli curve yields stress values, (in m4) and by using both the hogging and sagging moment curves c = distance from the neutral axis to the extreme mem- four curves of stress can be obtained; that is, tension and com- ber (in m) pression values for both top and bottom extreme fibers. SM = section modulus (I/c) (in m3) It is customary, however, to assume the maximum bending moment to extend over the midship portion of the ship. For a given bending moment at a given cross section of Minimum section modulus most often occurs at the loca- a ship, at any part of the cross section, the stress may be ob- tion of a hatch or a deck opening. Accordingly, the classi- σ tained ( = M/SM = Mc/I) which is proportional to the dis- fication societies ordinarily require the maintenance of the tance c of that part from the neutral axis. The neutral axis midship scantlings throughout the midship four-tenths will seldom be located exactly at half-depth of the section; length. This practice maintains the midship section area of σ hence two values of c and will be obtained for each sec- structure practically at full value in the vicinity of maximum tion for any given bending moment, one for the top fiber shear as well as providing for possible variation in the pre- (deck) and one for the bottom fiber (bottom shell). cise location of the maximum bending moment. A variation on the above beam equations may be of im- Lateral Bending Combined with Vertical Bending: Up to portance in ship structures. It concerns beams composed of this point, attention has been focused principally upon the ver- two or more materials of different moduli of elasticity, for tical longitudinal bending response of the hull. As the ship example, steel and aluminum. In this case, the flexural rigid- moves through a seaway encountering waves from directions ∫ 2 ity, EI, is replaced by A E(z) z dA, where A is cross sec- other than directly ahead or astern, it will experience lateral tional area and E(z) the modulus of elasticity of an element bending loads and twisting moments in addition to the ver- of area dA located at distance z from the neutral axis. The tical loads. The former may be dealt with by methods that ∫ neutral axis is located at such height that A E(z) z dA = 0. are similar to those used for treating the vertical bending Calculation of Section Modulus: An important step in loads, noting that there will be no component of still water routine ship design is the calculation of the midship section bending moment or shear in the lateral direction. The twist- modulus. As defined in connection with equation 29, it in- ing or torsional loads will require some special consideration. dicates the bending strength properties of the primary hull Note that the response of the ship to the overall hull twisting structure. The section modulus to the deck or bottom is ob- loading should be considered a primary response. tained by dividing the moment of inertia by the distance The combination of vertical and horizontal bending mo- from the neutral axis to the molded deck line at side or to ment has as major effect to increase the stress at the ex- the base line, respectively. treme corners of the structure (equation 30). In general, the following items may be included in the calculation of the section modulus, provided they are continuous or effectively developed: ¥ deck plating (strength deck and other effective decks). (See Subsection 18.4.3.9 for Hull/Superstructure Inter- action). ¥ shell and inner bottom plating, ¥ deck and bottom girders, ¥plating and longitudinal stiffeners of longitudinal 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) MASTER SET SDC 18.qxd Page 18-25 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-25

ED: Correction on this equation is unclear. M M an element of side shell or deck plating may, in general be σ= v + h [30] subject to two other components of stress, a direct stress in (Icvv) (Ichh) the transverse direction and a shearing stress. where Mv,Iv,cv,and Mh,Ih,ch, correspond to the M, I, c This figure illustrates these as the stress resultants,de- defined in equation 29, for the vertical bending and the hor- fined as the stress multiplied by plate thickness. izontal bending respectively. The stress resultants (N/m) are given by the following For a given vertical bending (Mv), the periodical wave expressions: induced horizontal bending moment (M ) increases stresses, h N = t σ and Ns = t σ stress resultants, in N/m alternatively, on the upper starboard and lower portside, and x x s on the upper portside and lower starboard. This explains N = t τ shear stress resultant or shear flow, in N/m why these areas are usually reinforced. where: Empirical interaction formulas between vertical bend- σ σ ing, horizontal bending and shear related to ultimate strength x, s = stresses in the longitudinal and transverse direc- of hull girder are given in Subsection 18.6.5.2. tions, in N/m2 Transverse Stresses:With regards to the validity of the τ = shear stress, in N/m2 Navier Equation (equation 29), a significant improvement t = plate thickness, in m may be obtained by considering a longitudinal strength In many parts of the ship, the longitudinal stress, σ ,is member composed of thin plate with transverse framing. x the dominant component. There are, however, locations in This might, for example, represent a portion of the deck which the shear component becomes important and under structure of a ship that is subject to a longitudinal stress σ , x unusual circumstances the transverse component may, like- from the primary bending of the hull girder. As a result of wise, become important. A suitable procedure for estimat- the longitudinal strain, ε , which is associated with σ , there x x ing these other component stresses may be derived by will exist a transverse strain, ε . For the case of a plate that s considering the equations of static equilibrium of the ele- is free of constraint in the transverse direction, the two ment of plating (Figure 18.26). The static equilibrium con- strains will be of opposite sign and the ratio of their ab- ditions for a plate element subjected only to in-plane stress, solute values, given by | ε / ε | = ν, is a constant property s x that is, no plate bending, are: of the material. The quantity ν is called Poisson’s Ratio and, for steel and aluminum, has a value of approximately 0.3. ∂Nx / ∂x + ∂N / ∂s = 0 [33-a] Hooke’s Law, which expresses the relation between stress ∂Ns / ∂x + ∂N / ∂x = 0 [33-b] and strain in two dimensions, may be stated in terms of the plate strains (equation 31). This shows that the primary re- In these equations, s, is the transverse coordinate meas- σ sponse induces both longitudinal ( x) and transversal ured on the surface of the section from the x-axis as shown σ stresses ( s) in plating. in Figure 18.26. ε σ σ For vessels without continuous longitudinal bulkheads x = 1/E ( x Ð v S) [31] ε σ νσ S = 1/E ( S Ð x) As transverse plate boundaries are usually constrained (displacements not allowed), the transverse stress can be taken, in first approximation as: σ νσ s = x [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 preced- ing section permit evaluation the longitudinal component σ of the primary stress, x. In Figure 18.26, it can be seen that Figure 18.26 Shear Forces (2) MASTER SET SDC 18.qxd Page 18-26 4/28/03 1:30 PM

18-26 Ship Design & Construction, Volume 1

(single cell), having transverse symmetry and subject to a of the shear flows at two locations lying on a plane cutting bending moment in the vertical plane, the shear flow dis- the cell walls will still be given by equation 34, with m(s) tribution, N(s) is then given by: equal to the moment of the shaded area (Figure 18.28). However, the distribution of this sum between the two com- =  V(x)  ponents in bulkhead and side shell, requires additional in- N (s)   m (s) [34] I(x) formation for its determination. and the shear stress, τ ,at any point in the cross section is: This additional information may be obtained by considering the torsional equilibrium and deflection of the cel- V(x).m(s) lular section. The way to proceed is extensively explained t(s) = (in N / m2 ) [35] t(s) I(x) in Lewis (2).

where: 18.4.3.3 Shear stress associated with torsion V(x) = total shearing force (in N) in the hull for a given In order to develop the twisting equations, we consider a section x closed, single cell, thin-walled prismatic section subject s only to a twisting moment, M , which is constant along the m(s) = ∫ tszds() , in m3, is the first moment (or moment T o length as shown in Figure 18.29. The resulting shear stress = of area) about the neutral axis of the cross sectional may be assumed uniform through the plate thickness and area of the plating between the origin at the cen- is tangent to the mid-thickness of the material. Under these terline and the variable location designated by s. circumstances, the deflection of the tube will consist of a This is the crosshatched area of the section shown twisting of the section without distortion of its shape, and in Figure 18.26 the rate of twist, dθ/dx, will be constant along the length. 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 con- Figure 18.27 Shear Flow in Multicell Sections (1) tinuous longitudinal bulkheads, the expression for shear stress is more complex. Shear Flow in Multicell Sections: If the cross section of the ship shown in Figure 18.28 is subdivided into two or more closed cells by longitudinal bulkheads, tank tops, or decks, the problem of finding the shear flow in the 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 di- vides, but the relative proportions of the part in the bulkhead and the part in the deck are indeterminate. The sum Figure 18.28 Shear Flow in Multicell Sections (2) MASTER SET SDC 18.qxd Page 18-27 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-27

Now consider equilibrium of forces in the x-direction for 18.4.3.4 Twisting and warping the element dx.ds of the tube wall as shown in Figure 18.29. Torsional strength: Although torsion is not usually an im- Since there is no longitudinal load, there will be no longi- portant factor in ship design for most ships, it does result tudinal stress, and only the shear stresses at the top and bot- in significant additional stresses on ships, such as container tom edges need be considered in the expression for static ships, which have large hatch openings. These warping equilibrium. The shear flow, N = tτ, is therefore seen to be stresses can be calculated by a beam analysis, which takes constant around the section. into account the twisting and warping deflections. There

The magnitude of the moment,MT, may be computed can also be an interaction between horizontal bending and by integrating the moment of the elementary force arising torsion of the hull girder. Wave actions tending to bend the from this shear flow about any convenient axis. If r is the hull in a horizontal plane also induce torsion because of the distance from the axis, 0, perpendicular to the resultant shear open cross section of the hull, which results in the shear cen- flow at location s: ter being below the bottom of the hull. Combined stresses due to vertical bending, horizontal bending and torsion must ===Ω MrNdsNrdsNT ∫ ∫ 2 [36] be calculated. In order to increase the torsional rigidity of the contain- Here the symbol indicates that the integral is taken en- ership cross sections, longitudinal and transverse closed Ω 2 tirely around the section and, therefore, (m ) is the area box girders are introduced in the upper side and deck struc- enclosed by the mid-thickness line of the tubular cross sec- ture. tion. The constant shear flow, N (N/m), is then related to From previous studies, it has been established that spe- the applied twisting moment by: cial attention should be paid to the torsional rigidity distri- τ Ω bution along the hull. Usually, toward the ship’s ends, the N = . t = MT /2 [37] section moduli are justifiably reduced base on bending. On For uniform torsion of a closed prismatic section, the the contrary the torsional rigidity, especially in the forward angle of torsion is: hatches, should be gradually increased to keep the warping ML. stress as small as possible. θ= T (in radians) [38] GI Twisting of opened section: A lateral seaway could in- p duce severe twisting moment that is of the major importance where: for ships having large deck openings. The equations for the twist of a closed tube (equations 36 to 38) are applicable M = Twisting moment (torsion), in N.m T only to the computation of the torsional response of closed L = Length of the girder, in m thin-walled sections. I = Polar Inertia, in m4 p The relative torsional stiffness of closed and open sec- ν 2 G = E/2(1+ ), the shear Modulus, in N/m tions 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 18.29 Torsional Shear Flow (2). Figure 18.30 Twist of Open and Closed Tubes (2) MASTER SET SDC 18.qxd Page 18-28 4/28/03 1:30 PM

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of the open tube without longitudinal restraint. The resist- The angle between a deck beam and side frame tends to ance to twist of the entirely open section is given by the St. open on one side and to close on the other side at the top Venant torsion equation: and reverses its action at the bottom. The effect of the concentration of stiff and soft sections results in a distortion pat- M = G.J ∂θ/∂x (N.m) [39] T tern in the ship deck that is shown in Figure 18.31. The term where: snaking is sometimes used in referring to this behavior and relates to both twisting and racking. ∂θ/∂x = twist angle per unit length, in rad./m, which can be approximated by θ/L for uniform torsion and uni- 18.4.3.6 Effective breadth and shear lag form section. An important effect of the edge shear loading of a plate J = torsional constant of the section, in m4 s member is a resulting nonlinear variation of the longitudi- = 13/ ∫ tds3 for a thin walled open section nal stress distribution (Figure 18.32). In the real plate the 0 longitudinal stress decreases with increasing distance from 1 n = ∑ bt3 for a section composed of n different the shear-loaded edge, and this is called shear lag. This is 3 i i i=1 in contrast to the uniform stress distribution predicted in the beam flanges by the elementary beam equation 29. In = plates (bi= length, ti = thickness) many practical cases, the difference from the value pre- If warping resistance is present, that is, if the longitudi- dicted in equation 29 will be small. But in certain combi- nal displacement of the elemental strips shown in Figure nations of loading and structural geometry, the effect referred 18.30 is constrained, another component of torsional re- to by the term shear lag must be taken into consideration sistance is developed through the shear stresses that result if an accurate estimate of the maximum stress in the mem- from this warping restraint. This is added to the torque given ber is to be made. This may be conveniently done by defin- by equation 39. ing an effective breadth of the flange member. In ship structures, warping strength comes from four The ratio, be/b, of the effective breadth, be, to the real sources: breadth, b, is useful to the designer in determining the lon- 1. the closed sections of the structure between hatch open- gitudinal stress along the shear-loaded edge. It is a function ings, 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. Trans- verse bulkheads resist racking if the bulkhead spacing is close enough to prevent deﬂection of the shell or deck plating in Figure 18.31 Snaking Behavior of a Container Vessel (2). 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. Figure 18.32 Shear Lag Effect in a Deck (2) MASTER SET SDC 18.qxd Page 18-29 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-29

of the external loading applied and the boundary conditions w = k ( M L2/EI ) [41] along the plate edges, but not its thickness. Figure 18.33 where the dimensionless coefficient k may be taken, for first gives the effective breadth ratio at mid-length for column approximation, as 0.09 (2). loading and harmonic-shaped beam loading, together with Actual deflection in service is affected also by thermal a common approximation for both cases: influences, rigidity of structural components, and workmanship; furthermore, deflection due to shear is additive to b e = kL [40] b 6 b the bending deflection, though its amount is usually relatively small. The results are presented in a series of design charts, The same influences, which gradually increase nominal which are especially simple to use, and may be found in design stress levels, also increase flexibility. Additionally, Schade (26). draft limitations and stability requirements may force the A real situation in which such an alternating load dis- L/D ratio up, as ships get larger. In general, therefore, mod- tribution may be encountered is a bulk carrier loaded with ern design requires that more attention be focused on flex- a dense ore cargo in alternate holds, the remainder being ibility than formerly. empty. No specific limits on hull girder deflections are given in An example of the computation of the effective breadth the classification rules. The required minimum scantlings of bottom and deck plating for such a vessel is given in however, as well as general design practices, are based on Chapter VI of Taggart (1), using Figure 18.33. a limitation of the L/D ratio range. It is important to distinguish the effective breadth (equation 40) and the effective width (equations 54 and 55) pre- 18.4.3.8 Load diffusion into structure sented later in Subsection 18.6.3.2 for plate and stiffened The description of the computation of vertical shear and plate-buckling analysis. bending moment by integration of the longitudinal load distribution implies that the external vertical load is resisted 18.4.3.7 Longitudinal deflection directly by the vertical shear carrying members of the hull The longitudinal bending deflection of the ship girder is ob- girder such as the side shell or longitudinal bulkheads. In a tainable from the appropriate curvature equations (equa- longitudinally framed ship, such as a tanker, the bottom tions 27 and 28) by integrating twice. A semi-empirical pressures are transferred principally to the widely spaced approximation for bending deflection amidships is: transverse web frames or the transverse bulkheads where

Figure 18.33 Effective Breath Ratios at Midlength (1) MASTER SET SDC 18.qxd Page 18-30 4/28/03 1:30 PM

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they are transferred to the longitudinal bulkheads or side one-third). Further details on the design considerations for shell, again as localized shear forces. Thus, in reality, the deckhouses and superstructures may be found in Evans (27) loading q(x), applied to the side shell or the longitudinal and Taggart (1). bulkhead will consist of a distributed part due to the direct In addition to the overall bending, local stress concentra- transfer of load into the member from the bottom or deck tions may be expected at the ends of the house, since here the structure, plus a concentrated part at each bulkhead or web structure is transformed abruptly from that of a beam consist- frame. This leads to a discontinuity in the shear curve at the ing of the main hull alone to that of hull plus superstructure. bulkheads and webs. Recent works achieved in Norwegian University of Sci- ence & Technology have shown that the vertical stress dis- 18.4.3.9 Hull/superstructure interaction tribution in the side shell is not linear when there are large The terms superstructure and deckhouse refer to a structure openings in the side shell as it is currently the case for upper usually of shorter length than the entire ship and erected decks of passenger vessels. Approximated stress distribu- above the strength deck of the ship. If its sides are coplanar tions are presented at Figure 18.35. The reduced slope, θ, with the ship’s sides it is referred to as a superstructure. If for the upper deck has been found equal to 0.50 for a cata- its width is less than that of the ship, it is called a deckhouse. maran passenger vessel (28). 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 18.4.4 Secondary Response important localized effects. Two opposing schools of thought In the case of secondary structural response, the principal exist concerning the philosophy of design of such erections. objective is to determine the distribution of both in-plane 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 Figure 18.34 Three Interaction Levels between Superstructure and Hull (1) 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 super- z structure and the differential deﬂection between deckhouse σ=θ σ and hull for three different degrees of superstructure effec- r(zz) . ( )

tiveness are shown on Figure 18.34. Passenger deck The areas and inertias can be computed to account for σ(z) =()M z shear lag in decks and bottoms. If the erection material dif- Neutral axis I fers from that of the hull (aluminum on steel, for example) x

the geometric erection area Af and inertia If must be reduced according to the ratio of the respective material moduli; that Figure 18.35 Vertical Stress Distribution in Passenger Vessels having Large is, by multiplying by E (aluminum)/E (steel) (approximately Openings above the Passenger Deck MASTER SET SDC 18.qxd Page 18-31 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-31

and normal loading, deflection and stress over the length foundation theory, 3) grillage theory (intersecting beams), and and width dimensions of a stiffened panel. Remember that 4) the finite element method (FEM). the primary response involves the determination of only the Orthotropic plate theory refers to the theory of bending in-plane load, deflection, and stress as they vary over the of plates having different flexural rigidities in the two or- length of the ship. The secondary response, therefore, is thogonal directions. In applying this theory to panels hav- seen to be a two-dimensional problem while the primary ing discrete stiffeners, the structure is idealized by assuming response is essentially one-dimensional in character. that the structural properties of the stiffeners may be approximated by their average values, which are assumed to 18.4.4.1 Stiffened panels be distributed uniformly over the width or length of the A stiffened panel of structure, as used in the present con- plate. The deflections and stresses in the resulting contin- text, usually consists of a flat plate surface with its attached uum are then obtained from a solution of the orthotropic stiffeners, transverse frames and/or girders (Figure 18.36). plate deflection differential equation: When the plating is absent the module is a grid or grillage of beam members only, rather than a stiffened panel. ∂ 4 w ∂ 4 w ∂ 4 w a + a + a = p(x,y) [42] In principle, the solution for the deflection and stress in 1 ∂x 4 2 ∂∂xy22 3 ∂y 4 the stiffened panel may be thought of as a solution for the response of a system of orthogonal intersecting beams. where: A second type of interaction arises from the two-di- a1,a2,a3 = express the average flexural rigidity of the or- mensional stress pattern in the plate, which may be thought thotropic plate in the two directions of as forming a part of the flanges of the stiffeners. The plate w(x,y) = is the deflection of the plate in the normal di- contribution to the beam bending stiffness arises from the rection direct longitudinal stress in the plate adjacent to the stiff- p(x,y) = is the distributed normal pressure load per unit ener, modified by the transverse stress effects, and also from area the shear stress in the plane of the plate. The maximum secondary stress may be found in the plate itself, but more fre- Note that the behavior of the isotropic plate, that is, one quently it is found in the free flanges of the stiffeners, since having uniform flexural properties in all directions, is a spe- these flanges are at a greater distance than the plate mem- cial case of the orthotropic plate problem. The orthotropic ber from the neutral axis of the combined plate-stiffener. plate method is best suited to a panel in which the stiffen- At least four different procedures have been employed for ers are uniform in size and spacing and closely spaced. It obtaining the structural behavior of stiffened plate panels has been said that the application of this theory to cross- under normal loading, each embodying certain simplifying stiffened panels must be restricted to stiffened panels with assumptions: 1) orthotropic plate theory, 2) beam-on-elastic- 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 Figure 18.36 A Stiffened Panel with Uniformly Distributed Longitudinals, 4 bending behavior (w) and the inplane membrane behavior Webframes, and 3 Girders. (u and v) are analyzed simultaneously. Then, in addition to MASTER SET SDC 18.qxd Page 18-32 4/28/03 1:30 PM

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the flexural rigidity (bending), the inplane axial, torsional, water outside the ship or liquid or dry bulk cargo within. transverse shear and inplane shear rigidities of the stiffen- Such a loading is normal to and distributed over the surface ers in the both directions can also be considered. Finally, of the panel. In many cases, the proportions, orientation, and the approach is suited for stiffeners uniform in size and location of the panel are such that the pressure may be as- spacing, and closely spaced but also for individual mem- sumed constant over its area. bers, randomly distributed such as deck and bottom gird- As previously noted, the deflection response of an ers. These members considered through Heaviside functions isotropic plate panel is obtained as the solution of a special that allow replacing each individual member by a set of 3 case of the earlier orthotropic plate equation (equation 42), forces and 2 bending moment load lines. Figure 18.36 shows and is given by: a typical stiffened panel that can be considered. It includes ∂ 4 w ∂ 4 w ∂ 4 w p(x,y) uniformly distributed longitudinals and web frames, and + 2 + = [44] three prompt elements (girders). ∂x 4 ∂∂xy22 ∂y 4 D The beam on elastic foundation solution is suitable for a panel in which the stiffeners are uniform and closely spaced where: Et3 in one direction and sparser in the other one. Each of these D = plate flexural rigidity −ν members is treated individually as a beam on an elastic foun- 12(1 ) dation, for which the differential equation of deflection is, = Et3 / 12(1 Ð ν) t = the uniform plate thickness ∂ 4 w EI +=kw q(x) [43] p(x,y) = distributed unit pressure load ∂x 4 Appropriate boundary conditions are to be selected to where: represent the degree of fixity of the edges of the panel. w = is the deflection Stresses and deflections are obtained by solving this equa- I = is sectional moment of inertia of the longitudinal tion for rectangular plates under a uniform pressure distri- stiffener, including adjacent plating bution. Equation 44 is in fact a simplified case of the general k = is average spring constant per unit length of the one (equation 42). transverse stiffeners Information (including charts) on a plate subject to uni- q(x) = is load per unit length on the longitudinal member form load and concentrated load (patch load) is available in Hughes (3). The grillage approach models the cross-stiffened panel as a system of discrete intersecting beams (in plane frame), 18.4.5.2 Local deflections each beam being composed of stiffener and associated ef- Local deflections must be kept at reasonable levels in order fective plating. The torsional rigidity of the stiffened panel for the overall structure to have the proper strength and and the Poisson ratio effect are neglected. The validity of rigidity. Towards this end, the classification society rules may modeling the stiffened panel by an intersecting beam (or gril- contain requirements to ensure that local deflections are not lage) may be critical when the flexural rigidities of stiffen- excessive. ers are small compared to the plate stiffness. It is known Special requirements also apply to stiffeners. Tripping that the grillage approach may be suitable when the ratio brackets are provided to support the flanges, and they should of the stiffener flexural rigidity to the plate bending rigid- be in line with or as near as practicable to the flanges of struts. ity (EI/bD with I the moment of inertia of stiffener and D Special attention must be given to rigidity of members under the plate bending rigidity) is greater than 60 (31) otherwise compressive loads to avoid buckling. This is done by pro- if the bending rigidity of stiffener is smaller, an Orthotropic viding a minimum moment of inertia at the stiffener and as- Plate Theory has to be selected. sociated plating. The FEM approach is discussed in detail in section 18.7.2. 18.4.6 Transverse Strength 18.4.5 Tertiary Response Transverse strength refers to the ability of the ship struc- 18.4.5.1 Unstiffened plate ture to resist those loads that tend to cause distortion of the Tertiary response refers to the bending stresses and deflec- cross section. When it is distorted into a parallelogram shape tions in the individual panels of plating that are bounded by the effect is called racking. We recall that both the primary the stiffeners of a secondary panel. In most cases the load bending and torsional strength analyses are based upon the that induces this response is a fluid pressure from either the assumption of no distortion of the cross section. Thus, we MASTER SET SDC 18.qxd Page 18-33 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-33

see that there is an inherent relationship between transverse than comparative purposes. Ideally, the entire ship hull or strength and both longitudinal and torsional strength. Cer- at least a limited hold-model should be modeled. See Sub- tain structural members, including transverse bulkheads and section 18.7.2—Structural Finite Element Models (Figure deep web frames, must be incorporated into the ship in order 18.57). 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. 18.4.7 Superposition of Stresses For example, a portion of the bottom pressure loading on In plating, each response induces longitudinal stresses, trans- the hull is transferred via the center girder and the longitu- verse stresses and shear stresses. These stresses can be cal- dinals to the transverse bulkheads at the ends of theses lon- culated individually for each response. This is the traditional gitudinals. The bulkheads, in turn, transfer these loads as way followed by the classification societies. With direct vertical shears into the side shell. Thus some of the loads analysis such as finite element analysis (Subsection 18.7.2), acting on the transverse strength members are also the loads it is not always possible to separate the different responses. of concern in longitudinal strength considerations. If calculated individually, all the longitudinal stresses The general subject of transverse strength includes ele- have to be added. Similar cumulative procedure must be ments taken from both the primary and secondary strength achieved for the transverse stresses and the shear stresses. categories. The loads that cause effects requiring transverse At the end they are combined through a criteria, which is strength analysis may be of several different types, de- usually for ship structure, the von-Mises criteria (equation pending upon the type of ship, its structural arrangement, 45). mode of operation, and upon environmental effects. The standard procedure used by classification societies Typical situations requiring attention to the transverse considers that longitudinal stresses induced by primary re- strength are: sponse of the hull girder, can be assessed separately from the other stresses. Classification rules impose through al- ¥ ship out of water: on building ways or on construction lowable stress and minimal section modulus, a maximum or repair dry dock, longitudinal stress induced by the hull girder bending mo- ¥tankers having empty wing tanks and full centerline tanks ment. or vice versa, On the other hand, they recommend to combined stresses ¥ore carriers having loaded centerline holds and large from secondary response and tertiary response, in plating empty wing tanks, and in members. These are combined through the von Mises ¥ all types of ships: torsional and racking effects caused criteria and compared to the classification requirements. by asymmetric motions of roll, sway and yaw, and Such an uncoupled procedure is convenient to use but ¥ ships with structural features having particular sensitiv- does not reflect reality. Direct analysis does not follow this ity to transverse effects, as for instance, ships having approach. All the stresses, from the primary, secondary and largely open interior structure (minimum transverse bulk- tertiary responses are combined for yielding assessment. heads) such as auto carriers, containers and RO-RO ships. For buckling assessment, the tertiary response is discarded, As previously noted, the transverse structural response as it does not induce in-plane stresses. Nevertheless the lat- involves pronounced interaction between transverse and eral load can be considered in the buckling formulation longitudinal structural members. The principal loading con- (Subsection 18.6.3). Tertiary stresses should be added for sists of the water pressure distribution around the ship, and fatigue analysis. the weights and inertias of the structure and hold contents. Since all the methods of calculation of primary, sec- As a first approximation, the transverse response of such a ondary, and tertiary stress presuppose linear elastic behav- frame may be analyzed by a two-dimensional frame re- ior of the structural material, the stress intensities computed sponse procedure that may or may not allow for support by for the same member may be superimposed in order to ob- longitudinal structure. Such analysis can be easily performed tain a maximum value for the combined stress. In performing using 2D finite element analysis (FEA). Influence of lon- and interpreting such a linear superposition, several con- gitudinal girders on the frame would be represented by elas- siderations affecting the accuracy and significance of the re- tic attachments having finite spring constants (similar to sulting stress values must be borne in mind. equation 43). Unfortunately, such a procedure is very sen- First, the loads and theoretical procedures used in com- sitive to the spring location and the boundary conditions. puting the stress components may not be of the same ac- For this reason, a three-dimensional analysis is usually per- curacy or reliability. The primary loading, for example, may formed in order to obtain results that are useful for more be obtained using a theory that involves certain simplifica- MASTER SET SDC 18.qxd Page 18-34 4/28/03 1:30 PM

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tions in the hydrodynamics of ship and wave motion, and will not always be immediately obvious, but must be found the primary bending stress may be computed by simple by considering the combined stress effects at a number of beam theory, which gives a reasonably good estimate of the different locations and times. mean stress in deck or bottom but neglects certain localized The nominal stresses produced from the analysis will be effects such as shear lag or stress concentrations. a combination of the stress components shown in Figures Second, the three stress components may not necessar- 18.21 and 18.37. ily occur at the same instant in time as the ship moves through waves. The maximum bending moment amidships, 18.4.7.1 von Mises equivalent stress σ which results in the maximum primary stress, does not nec- The yield strength of the material, yield, is defined as the essarily occur in phase with the maximum local pressure measured stress at which appreciable nonlinear behavior on a midship panel of bottom structure (secondary stress) accompanied by permanent plastic deformation of the ma- or panel of plating (tertiary stress). terial occurs. The ultimate strength is the highest level of Third, the maximum values of primary, secondary, and stress achieved before the test specimen fractures. For most tertiary stress are not necessarily in the same direction or shipbuilding steels, the yield and tensile strengths in ten- even in the same part of the structure. In order to visualize sion and compression are assumed equal. this, consider a panel of bottom structure with longitudinal The stress criterion that must be used is one in which it framing. The forward and after boundaries of the panel will is possible to compare the actual multi-axial stress with the σ be at transverse bulkheads. The primary stress ( 1) will act material strength expressed in terms of a single value for in the longitudinal direction, as given by equation 29. It will the yield or ultimate stress. be nearly equal in the plating and the stiffeners, and will be For this purpose, there are several theories of material approximately constant over the length of a midship panel. failure in use. The one usually considered the most suitable There will be a small transverse component in the plating, for ductile materials such as ship steel is referred to as the due to the Poison coefficient, and a shear stress given by von Mises Theory: equation 35. The secondary stress will probably be greater 1 in the free flanges of the stiffeners than in the plating, since σσσσστ=+−22 + 22 exyxy( 3 ) [45] the combined neutral axis of the stiffener/plate combination is usually near the plate-stiffener joint. Secondary Consider a plane stress field in which the component stresses, which vary over the length of the panel, are usu- σ σ τ stresses are x, y and . The distortion energy states that ally subdivided into two parts in the case of single hull struc- σ ture. The first part ( 2) 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, σ * ( 2 ), is the stress resulting from the bending of the smaller panel of plating plus longitudinal stiffeners that is bounded σ by the deep web frames. The first of these components ( 2), as a result of the proportions of the panels of structure, is usually larger in the transverse than in the longitudinal di- σ * rection. The second ( 2 ) is predominantly longitudinal. σ The maximum tertiary stress ( 3) 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 res- olution of the stress pattern into principal stress components. From all these considerations, it is evident that, in many cases, the point in the structure having the highest stress level Figure 18.37 Definition of Stress Components (4) MASTER SET SDC 18.qxd Page 18-35 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-35

failure through yielding will occur if the equivalent von states. A more elaborate description of the failure modes and σ Mises stress, e,given by equation 45 exceeds the equiva- methods to assess the structural capabilities in relation to lent stress, σο, corresponding to yielding of the material test these failure modes is available in Subsection 18.6.1. specimen. The material yield strength may also be expressed Classically, the different limit states were divided in 2 σ σ σ through an equivalent stress at failure: 0 = yield (= y). major categories: the service limit state and the ultimate limit state. Today, from the viewpoint of structural design, 18.4.7.2 Permissible stresses (Yielding) it seems more relevant to use for the steel structures four In actual service, a ship may be subjected to bending in the types of limit states, namely: inclined position and to other forces, such as those, which 1. service or serviceability limit state, induce torsion or side bending in the hull girder, not to men- 2. ultimate limit state, tion the dynamic effects resulting from the motions of the 3. fatigue limit state, and ship itself. Heretofore it has been difficult to arrive at the 4. accidental limit state. minimum scantlings for a large ship’s hull by first principles alone, since the forces that the structure might be re- This classification has recently been adopted by ISO. quired to withstand in service conditions are uncertain. A service limit state corresponds to the situation where Accordingly, it must be assumed that the allowable stress the structure can no longer provide the service for which it includes an adequate factor of safety, or margin, for these was conceived, for example: excessive deck deflection, elas- uncertain loading factors. tic buckling in a plate, and local cracking due to fatigue. In practice, the margin against yield failure of the struc- Typically they relate to aesthetic, functional or maintenance ture is obtained by a comparison of the structure’s von Mises problem, but do not lead to collapse. σ equivalent stress, e,against the permissible stress (or al- An ultimate limit state corresponds to collapse/failure, σ lowable stress), 0,giving the result: including collision and grounding. A classic example of ultimate limit state is the ultimate hull bending moment (Fig- σ ≤σ = s ×σ [46] e 0 1 y ure 18.46). The ultimate limit state is symbolized by the where: higher point (C) of the moment-curvature curve (M-Φ). Fatigue can be either considered as a third limit state or, s = partial safety factor defined by classification societies, 1 classically, considered as a service limit state. Even if it is which depends on the loading conditions and method also a matter of discussion, yielding should be considered of analysis. For 20 years North Atlantic conditions as a service limit state. First yield is sometimes used to as- (seagoing condition), the s factor is usually taken be- 1 sess the ultimate state, for instance for the ultimate hull tween 0.85 and 0.95 bending moment, but basically, collapse occurs later. Most σ = minimum yield point of the considered steel (mild y of the time, vibration relates to service limit states. steel, high tensile steel, etc.) In practice, it is important to differentiate service, ulti- For special ship types, different permissible stresses may mate, fatigue and accidental limit states because the partial be specified for different parts of the hull structure. For ex- safety factors associated with these limit states are gener- ample, for LNG carriers, there are special strain require- ally different. ments in way of the bonds for the containment system, which in turn can be expressed as equivalent stress requirements. For local areas subjected to many cycles of load rever- 18.5.1 Basic Types of Failure Modes sal, fatigue life must be calculated and a reduced permissi- Ship structural failure may occur as a result of a variety of ble stress may be imposed to prevent fatigue failure (see causes, and the degree or severity of the failure may vary Subsection 18.6.6). from a minor esthetic degradation to catastrophic failure resulting in loss of the ship. Three major failure modes are defined: 18.5 LIMIT STATES AND FAILURE MODES 1. tensile or compressive yield of the material (plasticity), 2. compressive instability (buckling), and Avoidance of structural failure is the goal of all structural 3. fracture that includes ductile tensile rupture, low-cycle designers, and to achieve this goal it is necessary for the de- fatigue and brittle fracture. signer to be aware of the potential limit states, failure modes and methods of predicting their occurrence. This section Yield occurs when the stress in a structural member ex- presents the basic types of failure modes and associated limit ceeds a level that results in a permanent plastic deforma- MASTER SET SDC 18.qxd Page 18-36 4/28/03 1:30 PM

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tion of the material of which the member is constructed. This ocean structures is of such a nature that the cyclical stresses stress level is termed the material yield stress. At a some- may be of a relatively low level during the greater part of what higher stress, termed the ultimate stress,fracture of the time, with occasional periods of very high stress levels the material occurs. While many structural design criteria caused by storms. Exposure to such load conditions may are based upon the prevention of any yield whatsoever, it result in the occurrence of low-cycle fatigue cracks after an should be observed that localized yield in some portions of interval of a few years. These cracks may grow to serious a structure is acceptable. Yield must be considered as a serv- size if they are not detected and repaired. iceability limit state. Concerning brittle fracture, small cracks suddenly begin Instability and buckling failure of a structural member to grow and travel almost explosively through a major por- loaded in compression may occur at a stress level that is sub- tion of the structure. The term brittle fracture refers to the stantially lower than the material yield stress. The load at fact that below a certain temperature, the ultimate tensile which instability or buckling occurs is a function of mem- strength of steel diminishes sharply (lower impact energy). ber geometry and material elasticity modulus, that is, slen- The originating crack is usually found to have started as a derness, rather than material strength. The most common result of poor design or manufacturing practice. Fatigue example of an instability failure is the buckling of a simple (Subsection 18.6.6) is often found to play an important role column under a compressive load that equals or exceeds in the initiation and early growth of such originating cracks. the Euler Critical Load. A plate in compression also will The prevention of brittle fracture is largely a matter of ma- have a critical buckling load whose value depends on the terial selection and proper attention to the design of struc- plate thickness, lateral dimensions, edge support conditions tural details in order to avoid stress concentrations. The and material elasticity modulus. In contrast to the column, control of brittle fracture involves a combination of design however, exceeding this load by a small margin will not and inspection standards aimed toward the prevention of necessarily result in complete collapse of the plate but only stress concentrations, and the selection of steels having a in an elastic deflection of the central portion of the plate away high degree of notch toughness, especially at low tempera- from its initial plane. After removal of the load, the plate tures. Quality control during construction and in-service in- may return to its original un-deformed configuration (for spection form key elements in a program of fracture control. elastic buckling). The ultimate load that may be carried by In addition to these three failure modes, additional modes a buckled plate is determined by the onset of yielding at some are: point in the plate material or in the stiffeners, in the case of ¥ collision and grounding, and a stiffened panel. Once begun, yield may propagate rapidly ¥ vibration and noise. throughout the entire plate or stiffened panel with further increase in load. Collision and Grounding is discussed in Subsection Fatigue failure occurs as a result of a cumulative effect 18.6.7 and Vibration in Subsection 18.6.8. Vibration as well in a structural member that is exposed to a stress pattern al- as noise is not a failure mode, while it could fall into the ternating from tension to compression through many cy- serviceability limit state. cles. Conceptually, each cycle of stress causes some small but irreversible damage within the material and, after the accumulation of enough such damage, the ability of the 18.6 ASSESSMENT OF THE STRUCTURAL member to withstand loading is reduced below the level of CAPACITY the applied load. Two categories of fatigue damage are generally recognized and they are termed high-cycle and low- 18.6.1 Failure Modes Classification cycle fatigue. In high-cycle fatigue, failure is initiated in The types of failure that may occur in ship structures are the form of small cracks, which grow slowly and which generally those that are characteristic of structures made up may often be detected and repaired before the structure is of stiffened panels assembled through welding. Figure 18.38 endangered. High-cycle fatigue involves several millions presents the different structure levels: the global structure, of cycles of relatively low stress (less than yield) and is typ- usually a cargo hold (Level 1), the orthotropic stiffened ically encountered in machine parts rotating at high speed panel or grillage (Level 2) and the interframe longitudi- or in structural components exposed to severe and prolonged nally stiffened panel (Level 3) or its simplified modeling: vibration. Low-cycle fatigue involves higher stress levels, the beam-column (Level 3b). Level 4 (Figure 18.44a) is the up to and beyond yield, which may result in cracks being unstiffened plate between two longitudinals and two trans- initiated after several thousand cycles. verse frames (also called bare plate). The loading environment that is typical of ships and The word grillage should be reserve to a structure com- MASTER SET SDC 18.qxd Page 18-37 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure 18-37

posed of a grid of beams (without attached plating). When — plate induced failure (buckling) the grid is fixed on a plate, orthotropic stiffened panel seems — stiffener induced failure (buckling or yielding) to the authors more adequate to define a panel that is or- Mode IV and V: Instability of stiffeners (local buck- thogonally stiffened, and having thus orthotropic properties. ling, tripping—Figure 18.44c) The relations between the different failure modes and Mode VI: Gross Yielding structure levels can be summarized as follows: ¥ Level 4: Buckling collapse of unstiffened plate (bare plate, Figure 18.44a). ¥ Level 1: Ultimate bending moment, Mu, of the global structure (Figure 18.46). To avoid collapse related to the Mode I,a minimal rigid- ¥ Level 2: Ultimate strength of compressed orthotropic ity is generally imposed for the transverse frames so that an σ stiffened panels ( u), interframe panel collapse (Mode III) always occurs prior to overall buckling (Mode I). It is a simple and easy constraint σ = min [σ (mode i)], i = I to VI, u u to implement, thus avoiding any complex calculation of the 6 considered failure modes. overall buckling (mode I). ¥ Level 3: Note that the failure Mode III is influenced by the buck- Mode I: Overall buckling collapse (Figure 18.44d), ling of the bare plate (elementary unstiffened plate). Elas- Mode II: Plate/Stiffener Yielding tic buckling of theses unstiffened plates is usually not Mode III: Pult of interframe panels with a plate-stif considered as an ultimate limit state (failure mode), but ener combination (Figure 18.44b) using a beam-col- rather as a service limit state. Nevertheless, plate buckling umn model (Level 3b) or an orthotropic model (Level (Level 4) may significantly affect the ultimate strength of 3), considering: 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 (Sub- section 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). ¥Overall collapse of orthotropic panels (entire stiffened Figure 18.38 Structural Modeling of the Structure and its Components plate structure), MASTER SET SDC 18.qxd Page 18-38 4/28/03 1:30 PM

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¥ Collapse of interframe longitudinally stiffened panel, 18.6.2 Yielding including torsional-flexural (lateral-torsional) buckling As explained in Subsection 18.5.1 yield occurs when the of stiffeners (also called tripping). stress in a structural component exceeds the yield stress. It is necessary to distinguish between first yield state and 18.6.1.2 Frame failure modes fully plastic state. In bending, first yield corresponds to the Service limit state (Subsection 18.4.6). situation when stress in the extreme fiber reaches the yield ≤ ≤ stress. If the bending moment continues to increase the yield ¥ Upper and lower bounds (Xmin X Xmax), ¥Minimal rigidity to guarantee rigid supports to the in- area is growing. The final stage corresponds to the Plastic terframe panels (between two transverse frames). Moment (Mp), where, both the compression and tensile sides ¥ Allowable stresses under the resultant forces (bending, are fully yielded (as shown on Figure 18.47). shear, torsion) Yield can be assessed using basic bending theory, equation 29, up to complex 3D nonlinear FE analysis. Design — Elastic analysis, criteria related to first yield is the von Mises equivalent — Elasto-plastic analysis. stress (equation 45). ¥Fatigue (Subsection 18.6.6) Yielding is discussed in detail in Section 18.4. Ultimate limit state 18.6.3 Buckling and Ultimate Strength of Plates ¥Frame bucklings: These failures modes are considered A ship stiffened plate structure can become unstable if ei- as ultimate limit states rather than a service limit state. ther buckling or collapse occurs and may thus fail to per- If one of them appears, the assumption of rigid supports form its function. Hence plate design needs to be such that is no longer valid and the entire stiffened panel can reach instability under the normal operation is prevented (Figure the ultimate limit state. 18.44a). The phenomenon of buckling is normally divided — Buckling of the compressed members, into three categories, namely elastic buckling, elastic-plas- — Local buckling (web, flange). tic buckling and plastic buckling, the last two being called inelastic buckling. Unlike columns, thin plating buckled in 18.6.1.3 Hull Girder Collapse modes the elastic regime may still be stable since it can normally Service limit state sustain further loading until the ultimate strength is reached, even if the in-plane stiffness significantly decreases after the ¥ Allowable stresses and first yield (Subsection 18.4.3.1), inception of buckling. In this regard, the elastic buckling of ¥ Deflection of the global structure and relative deflec- plating between stiffeners may be allowed in the design, tions of components and panels (Subsection 18.4.3.7). sometimes intentionally in order to save weight. Since sig- Ultimate limit state nificant residual strength of the plating is not expected after buckling occurs in the inelastic regime, however, inelastic ¥ Global ultimate strength (of the hull girder/box girder). buckling is normally considered to be the ultimate strength This can be done by considering an entire cargo hold or of the plate. only the part between two transverse web frames (Sub- The buckling and ultimate strength of the structure de- section 18.6.5). Collapse of frames is assumed to only pends on a variety of influential factors, namely geomet- appear after the collapse of panels located between these ric/material properties, loading characteristics, fabrication frames. This means that it is sufficient to verify the box related imperfections, boundary conditions and local dam- girder ultimate strength between two frames to be pro- age related to corrosion, fatigue cracking and denting. tected against a more general collapse including, for instance, one or more frame spans. This approach can be 18.6.3.1 Direct Analysis un-conservative if the frames are not stiff enough. In estimating the load-carrying capacity of plating between ¥ Collision and grounding (Subsection 18.6.7), which is stiffeners, it is usually assumed that the stiffeners are sta- in fact an accidental limit state. ble and fail only after the plating. This means that the stiff- A relevant comparative list of the limit states was de- eners should be designed with proper proportions that help fined by the Ship Structure Committee Report No 375 (32). 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. MASTER SET SDC 18.qxd Page 18-39 4/28/03 1:30 PM

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σ τ Four load components, namely longitudinal compres- B = critical buckling strength (that is, B for sion/tension, transverse compression/tension, edge shear and shear stress) σ σ lateral pressure loads, are typically considered to act on ship F = Y for4 normal stress σ √ plating between stiffeners, as shown in Figure 18.39, while = Y 3 for shear stress σ the in-plane bending effects on plate buckling are also some- Y = material yield stress times accounted for. In actual ship structures, lateral pres- In ship rules and books, equation 47 may appear with sure loading arises from water pressure and cargo weight. somewhat different constants depending on the structural The still water magnitude of water pressure depends on the proportional limit assumed. The above form assumes a struc- vessel draft, and the still water value of cargo pressure is de- tural proportional limit of a half the applicable yield value. termined by the amount and density of cargo loaded. For axial tensile loading, the critical strength may be These still water pressure values may be augmented by considered to equal the material yield stress (σ ). wave action and vessel motion. Typically the larger in-plane Y Under single types of loads, the critical plate buckling loads are caused by longitudinal hull girder bending, both strength must be greater than the corresponding applied in still water and in waves at sea, which is the source of the stress component with the relevant margin of safety. For primary stress as previously noted in Subsection 18.4.3. combined biaxial compression/tension and edge shear, the The elastic plate buckling strength components under σ σ σ σ following type of critical buckling strength interaction cri- single types of loads, that is, xE for xav, yE for yav and τ τ terion would need to be satisfied, for example: E for av, can be calculated by taking into account the re- c lated effects arising from in-plane bending, lateral pressure,  σ  c σ σ  σ   τ  c  xav  −+α xav yav yav +  av  ≤ η σ σ σ  σ  τ B [48] cut-outs, edge conditions and welding induced residual  xB  xB yB  yB   B  stresses. The critical (elastic-plastic) buckling strength compo- where: nents under single types of loads, that is, σ for σ , σ xB xav yB η = usage factor for buckling strength, which is typically for σ and τ for τ ,are typically calculated by plasticity B yav B av the inverse of the conventional partial safety factor. correction of the corresponding elastic buckling strength η = 1.0 is often taken for direct strength calculation, while using the Johnson-Ostenfeld formula, namely: B it is taken less than 1.0 for practical design in accor- σσσ≤  EEFfor 05. dance with classification society rules.  σ = B   σ  [47] Compressive stress is taken as negative while tensile σ 1 − F  for σσ> 05. α σ σ F  4σ  EF stress is taken as positive and = 0 if both xav and yav are  E α σ σ compressive, and = 1 if either xav or yav or both are ten- where: sile. The constant c is often taken as c = 2. Figure 18.40 shows a typical example of the axial mem- σ E = elastic plate buckling strength brane 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 non- uniform 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 mid- Figure 18.39 A Simply Supported Rectangular Plate Subject to Biaxial dle of the plate element will initially yield by the action of Compression/tension, Edge Shear and Lateral Pressure Loads bending. However, as long as it is possible to redistribute MASTER SET SDC 18.qxd Page 18-40 4/28/03 1:31 PM

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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: σσσσσ2 −+=22 x max x max y min y min Y [49a] (b) Yielding at transverse edges: σσσσ2 −+=2 σ2 x min x min y max y max Y [49b] Figure 18.40 Membrane Stress Distribution Inside the Plate Element under Predomianntly Longitudinal Compressive Loads; (a) Before buckling, (b) After The maximum and minimum membrane stresses of equa- buckling, unloaded edges move freely in plane, (c) After buckling, unloaded tions 49a and 49b can be expressed in terms of applied edges kept straight 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 the applied loads to the straight plate boundaries by the differential equation. membrane action, the plate element will not collapse. Col- On the other hand, the plate ultimate edge shear strength, τ τ τ τ σ lapse will then occur when the most stressed boundary lo- u , is often taken u = B (equation 47, with B instead of B). cations yield, since the plate element can not keep the Also, an empirical formula obtained by curve fitting based boundaries straight any further, resulting in a rapid increase on nonlinear finite element solutions may be utilized (33). of lateral plate deflection (33). Because of the nature of ap- The effect of lateral pressure loads on the plate ultimate edge plied axial compressive loading, the possible yield loca- shear strength may in some cases need to be accounted for. MASTER SET SDC 18.qxd Page 18-41 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure 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  c σ σ  σ   τ  c  xav  −+α xav yav yav +  av  ≤ η σ σ σ  σ  τ u [50]  xu  xu yu  yu   u  where: α and c = variables deﬁned in equation 48 η u = usage factors for the ultimate limit state σ σ σ xu and yu = solutions of equation 49a with regard to xav σ and equation 49b with regard to yav,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 Figure 18.42 Compressive Buckling Coefficient for Plates in Compression; for restraints along the plate/stiffener junctions, the real elastic 5 Configurations (2) (A, B, C, D and E) where Boundary Conditions of Unloaded buckling strength with rotational restraints would of course Edges are: SS: Simply Supported, C: Clamped, and F: Free 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. σ The theoretical solution for critical buckling stress, B , in the elastic range has been found for a number of cases ≤ compression (a > b), kc = 4, and for wide plate (a b) in of interest. For rectangular plate subject to compressive in- compression, kc = (1 + b2 / a2)2,for simply supported edges. plane stress in one direction: τ For shear force, the critical buckling shear stress, B, can also be obtain by equation 51 and the buckling coefficient π 2 Et  2 σ = k [51] for simply supported edges is: Bc12() 1 − ν2  b  kc = 5.34 + 4(b/a)2 [53] α Here kc is a function of the plate aspect ratio, = a/b, the boundary conditions on the plate edges and the type of Figure 18.42 presents, kc,versus the aspect ratio, a/b, for loading. If the load is applied uniformly to a pair of oppo- different configurations of rectangular plates in compression. site edges only, and if all four edges are simply supported, For the simplified prediction of the plate ultimate strength then k is given by: under uniaxial compressive loads, one of the most common ap- c proaches is to assume that the plate will collapse if the maxi-  m α  2 mum compressive stress at the plate corner reaches the material k =+ [52] yield stress, namely σ = σ for σ or σ = σ for σ . c  α m  x max Y xav y max Y yav This assumption is relevant when the unloaded edges where m is the number of half-waves of the deflected plate move freely in plane as that shown in Figure 40(b). Another in the longitudinal direction, which is taken as an integer approximate method is to use the plate effective width con- satisfying the condition α= m(m + 1). For long plate in cept, which provides the plate ultimate strength components MASTER SET SDC 18.qxd Page 18-42 4/28/03 1:31 PM

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σ σ under uniaxial compressive stresses ( xu and yu), as fol- compression do not occur simultaneously. For instance, low: DNV (4) recommends: σ σ ¥maximum compression, σ , in a plate field and phase xu b eu yu a eu x ==and [54] σ τ σ b σ a angle associated with y, (buckling control), Y Y σ ¥maximum compression, y, in a plate field and phase σ τ where aeu and beu are the plate effective length and width at angle associated with x, (buckling control), the ultimate limit state,respectively. ¥absolute maximum shear stress, τ, in a plate field and σ σ While a number of the plate effective width expressions phase angle associated with x, y (buckling control), have been developed, a typical approach is exemplified by and σ Faulkner, who suggests an empirical effective width (beu /b) ¥maximum equivalent von Mises stress, e, at given po- formula for simply supported steel plates, as follows, sitions (yield control). ¥for longitudinal axial compression (34), σ σ In order to get x and y, the following stress compo- 11for β < nents may normally be considered for the buckling control: b  eu =  cc σ = stress from primary response, and 12−≥β [55a] 1 b  for 1 σ = stress from secondary response (that is, double  β β 2 2 bottom bending). ¥ for transverse axial compression (35), As the lateral bending effects should be normally in- a   cluded in the buckling strength formulation, stresses from eu =+09..b 19 −09 . σ * 1  [55b] local bending of stiffeners (secondary response), 2 , and a β 22a β  β  σ local bending of plate (tertiary response), 3,must there- where: fore not to be included in the buckling control. If FE-analy- σ σ sis is performed the local plate bending stress, 3, can easily β = b Y is the plate slenderness be excluded using membrane stresses. tE E = the Young’s modulus t = the plate thickness 18.6.4 Buckling and Ultimate Strength of Stiffened c1 ,c2 = typically taken as c1 = 2 and c2 = 1 Panels The plate ultimate strength components under uniaxial For the structural capacity analysis of stiffened panels, it is compressive loads are therefore predicted by substituting presumed that the main support members including longi- the plate effective width formulae (equation 55a) into equa- tudinal girders, transverse webs and deep beams are de- tion 54. signed with proper proportions and stiffening systems so More charts and formulations are available in many that their instability is prevented prior to the failure of the books, for example, Bleich (36), ECCS-56 (37), Hughes stiffened panels they support. (3) and Lewis (2). In addition, the design strength of plate In many ship stiffened panels, the stiffeners are usually (unstiffened panels) is detailed in Chapter 19, Subsection attached in one direction alone, but for generality, the de- 19.5.4.1, including an example of reliability-based design sign criteria often consider that the panel can have stiffen- and alternative equations to equations 56 and 57. ers in one direction and webs or girders in the other, this arrangement corresponds to a typical ship stiffened panels 18.6.3.3 Design criteria (Figure 18.43a). The stiffeners and webs/girders are at- When a single load component is involved, the buckling or tached to only one side of the panel. ultimate strength must be greater than the corresponding ap- The number of load components acting on stiffened steel plied stress component with an appropriate target partial panels are generally of four types, namely biaxial loads, that safety factor. In a multiple load component case, the struc- is compression or tension, edge shear, biaxial in-plane bend- tural safety check is made with equation 48 against buck- ing and lateral pressure, as shown in Figure 18.43. When the ling and equation 50 against ultimate limit state being panel size is relatively small compared to the entire structure, satisfied. the influence of in-plane bending effects may be negligible. To ensure that the possible worst condition is met (buck- However, for a large stiffened panel such as that in side ling and yield) for the ship, several stress combination must shell of ships, the effect of in-plane bending may not be be considered, as the maximum longitudinal and transverse negligible, since the panel may collapse by failure of stiff- MASTER SET SDC 18.qxd Page 18-43 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure 18-43

eners which are loaded by largest added portion of axial different from that of the plate. It is therefore necessary to compression due to in-plane bending moments. take into account this effect in the structural capacity for- When the stiffeners are relatively small so that they mulations, at least approximately. buckle together with the plating, the stiffened panel typi- For analysis of the ultimate strength capacity of stiffened cally behaves as an orthotropic plate. In this case, the av- panels which are supported by longitudinal girders, trans- erage values of the applied axial stresses may be used by verse webs and deep beams, it is often assumed that the neglecting the inﬂuence of in-plane bending. When the stiff- panel edges are simply supported, with zero deﬂection and eners are relatively stiff so that the plating between stiffen- zero rotational restraints along four edges, with all edges ers buckles before failure of the stiffeners, the ultimate kept straight. strength is eventually reached by failure of the most highly This idealization may provide somewhat pessimistic, stressed stiffeners. In this case, the largest values of the axial but adequate predictions of the ultimate strength of stiffened compressive or tensile stresses applied at the location of the panels supported by heavy longitudinal girders, transverse stiffeners are used for the failure analysis of the stiffeners. webs and deep beams (or bulkheads). In stiffened panels of ship structures, material properties of Today, direct non-linear strength assessment methods the stiffeners including the yield stress are in some cases using recognized programs is usual (38). The model should

(a) (a)

(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). MASTER SET SDC 18.qxd Page 18-44 4/28/03 1:31 PM

18-44 Ship Design & Construction, Volume 1

be capable of capturing all relevant buckling modes and with experimental and/or FE analysis are available (43-45). detrimental interactions between them. The fabrication re- An example of reliability-based assessment of the stiff- lated initial imperfections in the form of initial deflections ened panel strength is presented in Chapter 19. Formula- (plates, stiffeners) and residual stresses can in some cases tions of Herzog, Hughes and Adamchack are also discussed. significantly affect (usually reduce) the ultimate strength of the panel so that they should be taken into account in the 18.6.4.2 Simplified models strength computations as parameters of influence. Existing simplified methods for predicting the ultimate strength of stiffened panels typically use one or more of the 18.6.4.1 Direct analysis following approaches: The primary modes for the ultimate limit state of a stiffened ¥orthotropic plate approach, panel subject to predominantly axial compressive loads may ¥plate-stiffener combination approach (or beam-column be categorized as follows (Figure 18.44): approach), and ¥ Mode I: Overall collapse after overall buckling, ¥grillage approach. ¥Mode II: Plate induced failure—yielding of the plate- These approaches are similar to those presented in Sub- stiffener combination at panel edges, section 18.4.4.1 for linear analysis. All have the same back- ¥Mode III: Plate induced failure—flexural buckling fol- ground but, here, the buckling and the ultimate strength is lowed by yielding of the plate-stiffener combination at considered. mid-span, In the orthotropic plate approach, the stiffened panel is ¥ Mode IV: Stiffener induced failure—local buckling of idealized as an equivalent orthotropic plate by smearing the stiffener web, stiffeners into the plating. The orthotropic plate theory will ¥ Mode V: Stiffener induced failure—tripping of stiffener, then be useful for computation of the panel ultimate strength and for the overall grillage collapse mode (Mode I, Figure ¥Mode VI: Gross yielding. 18.44d), (31,46,48). Calculation of the ultimate strength of the stiffened panel The plate-stiffener combination approach (also called under combined loads taking into account all of the possi- beam-column approach) models the stiffened panel behav- ble failure modes noted above is not straightforward, be- ior by that of a single “beam” consisting of a stiffener to- cause of the interplay of the various factors previously noted gether with the attached plating, as representative of the such as geometric and material properties, loading, fabri- stiffened panel (Figure 18.38, level 3b). The beam is con- cation related initial imperfections (initial deflection and sidered to be subjected to axial and lateral line loads. The welding induced residual stresses) and boundary conditions. torsional rigidity of the stiffened panel, the Poisson ratio ef- As an approximation, the collapse of stiffened panels is then fect and the effect of the intersecting beams are all neg- usually postulated to occur at the lowest value among the lected. The beam-column approach is useful for the various ultimate loads calculated for each of the above col- computation of the panel ultimate strength based on Mode lapse patterns. III, which is usually an important failure mode that must be This leads to the easier alternative wherein one calcu- considered in design. The degree of accuracy of the beam- lates the ultimate strengths for all collapse modes mentioned column idealization may become an important considera- above separately and then compares them to find the min- tion when the plate stiffness is relatively large compared to imum value which is then taken to correspond to the real the rigidity of stiffeners and/or under significant biaxial panel ultimate strength. The failure mode of stiffened pan- loading. els is a broad topic that cannot be covered totally within this Stiffened panels are asymmetric in geometry about the chapter. Many simplified design methods have of course plate-plane. This necessitates strength control for both plate been previously developed to estimate the panel ultimate induced failure and stiffener-induced failure. strength, considering one or more of the failure modes Plate induced failure: Deflection away from the plate as- among those mentioned above. Some of those methods have sociated with yielding in compression at the connection be- been reviewed by the ISSC’2000 (39). On the other hand, tween plate and stiffener. The characteristic buckling a few authors provide a complete set of formulations that strength for the plate is to be used. cover all the feasible failure modes noted previously, namely, Stiffener induced failure: Deflection towards the plate as- Dowling et al (40), Hughes (3), Mansour et al (41,42), and sociated with yielding in compression in top of the stiffener more recently Paik (38). or torsional buckling of the stiffener. Assessment of different formulations by comparison Various column strength formulations have been used as MASTER SET SDC 18.qxd Page 18-45 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure 18-45

the basis of the beam-column approach, three of the more and common types being the following: σ λ ==a σ Y Y ¥ Johnson-Ostenfeld (or Bleich-Ostenfeld) formulation, πr E σE ¥Perry-Robertson formulation, and ¥ empirical formulations obtained by curve ﬁtting exper- where:

imental or numerical data. r = radius4 of gyration √ A stocky panel that has a high elastic buckling strength = I / A, (m) I = inertia, (m4) will not buckle in the elastic regime and will reach the ulti- A = cross section of the plate-stiffener combination with full mate limit state with a certain degree of plasticity. In most attached plating, (m2) design rules of classification societies, the so-called John- t = plate thickness, (m) son-Ostenfeld formulation is used to account for this behav- a = span of the stiffeners, (m) ior (equation 47). On the other hand, in the so-called b = spacing between 2 longitudinals, (m) Perry-Robertson formulation, the strength expression assumes that the stiffener with associated plating will collapse Note that A, I, r, ... refer to the full section of the plate- as a beam-column when the maximum compressive stress in stiffener combination, that is, without considering an ef- the extreme fiber reaches the yield strength of the material. fective plating. In empirical approaches, the ultimate strength formula- Figure 18.45 compares the Johnson-Ostenfeld formula tions are developed by curve fitting based on mechanical (equation 47), the Perry-Robertson formula and the Paik- collapse test results or numerical solutions. Even if limited Thayamballi empirical formula (equation 56) for on the col- to a range of applicability (load types, slenderness ranges, umn ultimate strength for a plate-stiffener combination assumed level of initial imperfections, etc.) they are very varying the column slenderness ratios, with selected initial useful for preliminary design stage, uncertainty assessment eccentricity and plate slenderness ratios. In usage of the and as constraint in optimization package. While a vast num- Perry-Roberson formula, the lower strength as obtained ber of empirical formulations (sometimes called column from either plate induced failure or stiffener-induced fail- curves) for ultimate strength of simple beams in steel framed ure is adopted herein. Interaction between bending axial 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 σ strength of columns ( u) must be less than the elastic col- σ umn buckling strength ( E), the Paik-Thayamballi empirical formula for a plate-stiffener combination is given by: σ u = 1 σ [56] Y 0..995+++ 0 936λβ2 017 .2 0 .188 λβλ22 −0 .067 4

and σ σ u ≤=1 E σ σ Y λ2 Y with Figure 18.45 A Comparison of the Ultimate Strength Formulations for η βσ= b Y Plate-stiffener Combinations under Axial Compression ( relates to the t E initial deﬂection) MASTER SET SDC 18.qxd Page 18-46 4/28/03 1:31 PM

18-46 Ship Design & Construction, Volume 1

compression and lateral pressure can, within the same fail- comprehensive works performed by the Special Task Com- ure mode (Flexural Buckling—Mode III), leads to three-fail- mittees of ISSC 2000. Yao (51) contains an historical re- ure scenario: plate induced failure, stiffener induced failure view and a state of art on this matter.

or a combined failure of stiffener and plating (see Chapter Computation of Mu depends closely on the ultimate 19 Ð Figure 19.11 ). strength of the structure’s constituent panels, and particularly on the ultimate strength in compressed panels or components. 18.6.4.3 Design criteria Figure 18.46 shows that in sagging, the deck is compressed σ σ σ The ultimate strength based design criteria of stiffened pan- ( deck) and reaches the ultimate limit state when deck = u. els can also be deﬁned by equation 50, but using the corre- On the other hand, the bottom is in tensile and reaches its ul- σ σ σ sponding stiffened panel ultimate strength and stress timate limit state after complete yielding, bottom = 0 ( 0 parameters. Either all of the six design criteria, that is, against being the yield stress). individual collapse modes I to VI noted above, or a single de- Basically, there exist two main approaches to evaluate sign criterion in terms of the real (minimum) ultimate strength the hull girder ultimate strength of a ship’s hull under lon- components must be satisﬁed. For stiffened panels follow- gitudinal bending moments. One, the approximate analy- ing Mode I behavior, the safety check is similar to a plate, sis,is to calculate the ultimate bending moment directly

using average applied stress components. The applied axial (Mu, point C on Figure 18.46), and the other is to perform stress components for safety evaluation of the stiffened panel progressive collapse analysis on a hull girder and obtain, φ following Modes IIÐVI behavior will use the maximum axial both, Mu and the curves M- . stresses at the most highly stressed stiffeners. The ﬁrst approach, approximate analysis,requires an assumption on the longitudinal stress distribution. Figure 18.47 shows several distributions corresponding to differ- 18.6.5 Ultimate Bending Moment of Hull Girder ent methods. On the other hand, the progressive collapse Ultimate hull girder strength relates to the maximum load analysis does not need to know in advance this distribution. that the hull girder can support before collapse. These loads Accordingly, to determine the global ultimate bending

induce vertical and horizontal bending moment, torsional moment (Mu), one must know in advance moment, vertical and horizontal shear forces and axial force. ¥ the ultimate strength of each compressed panel (σ ), and For usual seagoing vessels axial force can be neglected. As u ¥ the average stress-average strain relationship (σ−ε), to the maximun shear forces and maximum bending moment perform a progressive collapse analysis. do not occur at the same place, ultimate hull girder strength should be evaluated at different locations and for a range of For an approximate assessment, such as the Caldwell bending moments and shear forces. method, only the ultimate strength of each compressed panel σ The ultimate bending moment (Mu) refers to a combined ( u) is required. 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 18.6.5.1 Direct analysis the feasible loading cases that induce hull girder collapse. The direct analysis corresponds to the Progressive collapse

Today, Mu is considered as being a relevant design case. analysis. The methods include the typical numerical analy- 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

(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. MASTER SET SDC 18.qxd Page 18-47 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure 18-47

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 non- linearities can be considered. A 3D analysis of a hold or a ship’s section is funda- mentally possible but very difficult to perform. This is because a ship’s hull is too large and complicated for such kind of analysis. Nevertheless, since 1983 results of FEM analy- Figure 18.48 The Smith’s Progressive Collapse Method ses 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 (a) 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 sup- plemented 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 (b) 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 Figure 18.49 Inﬂuence of Element Average Stress-Average Strain Curves 18.49a. (σ−ε) on Progressive Collapse Behavior. (a) Average stress-average strain Step 3: To perform progressive collapse analysis, Figure relationships of element, and (b) moment-curvature relationship of cross- 18.49b. section. MASTER SET SDC 18.qxd Page 18-48 4/28/03 1:31 PM

18-48 Ship Design & Construction, Volume 1

In Step 1, the cross-section of a hull girder is divided An interesting well-studied ship that reached its ultimate into elements composed of a longitudinal stiffener and at- bending moment is the Energy Concentration (63). It fre- tached plating. In Step 2, the average stress-average strain quently is used as a reference case (benchmark) by authors relationship (σ−ε) of this stiffener element is derived under to validate methods. the axial load considering the influences of buckling and Figure 18.49 shows typical average stress-average strain yielding. Step 3 can be explained as follows: relationships, and the associated bending moment-curvature relationships (M-φ). Four typical σ−ε curves are con- ¥axial rigidities of individual elements are calculated using sidered, which are: the average stress-average strain relationships (σ−ε), •flexural rigidity of the cross-section is evaluated using Case A: Linear relationship (elastic). The M-φ relationship the axial rigidities of elements, is free from the influences of yielding and buck- ¥vertical and horizontal curvatures of the hull girder are ling, and is linear. applied incrementally with the assumption that the plane Case B: Bi-linear relationship (elastic-perfectly plastic, cross-section remains plane and that the bending occurs without buckling). about the instantaneous neutral axis of the cross-section, Case C: With buckling but without strength reduction be- ¥ the corresponding incremental bending moments are yond the ultimate strength. evaluated and so the strain and stress increments in in- Case D: With buckling and a strength reduction beyond dividual elements, and the ultimate strength (actual behavior). ¥ incremental curvatures and bending moments of the cross-section as well as incremental strains and stresses In Case B, where yielding takes place but no buckling, of elements are summed up to provide their cumulative the deck initially undergoes yielding and then the bottom. values. With the increase in curvature, yielded regions spread in the side shell plating and the longitudinal bulkheads towards Figure 18.48 shows that the σ−ε curves are used to es- the plastic neutral axis. timate the bending moment carried by the complete trans- In this case, the maximum bending moment is the fully

verse section (Mi). The contribution of each element (dM) plastic bending moment (Mp) of the cross-section and its depends on its location in the section, and speciﬁcally on absolute value is the same both in the sagging and the hog-

its distance from the current position of the neutral axis (Yi). ging conditions. The contribution will then also depend on the strain that is For Cases C and D, the element strength is limited by applied to it, since ε = Ðy φ,where φ is the hull curvature plate buckling, stiffener flexural buckling, tripping, etc. For and y is the distance from the neutral axis (simple beam as- Case C, it is assumed that the structural components can con- sumption). The average stress-average strain curve (σ-ε) tinue to carry load after attaining their ultimate strength. σ φ will then provide an estimate of the longitudinal stress ( i) The collapse behavior (M- curve) is similar to that of Case acting on the section. Individual moments about the neu- B, but the ultimate strength is different in the sagging and tral axis are then summed to give the total bending moment the hogging conditions, since the buckling collapse strength φ for a particular curvature i. is different in the deck and the bottom. The accuracy of the calculated ultimate bending mo- Case D is the actual case; the capacity of each structural ment depends on the accuracy of the average stress-aver- member decreases beyond its ultimate strength. In this case, age strain relationships of individual elements. Main the bending moment shows a peak value for a certain value difficulties concern the modeling of initial imperfections of the curvature. This peak value is defined as the ultimate

(deflection and welding residual stress) and the boundary longitudinal bending moment of the hull girder (Mu). conditions (multi-span model, interaction between adjacent Shortcomings and limitations of the Smith’s method re- elements, etc.). lates to the fact that a typical analysis concerns one frame Many formulations and methods to calculate these av- spacing of a whole cargo hold and not a complete 3D hold. erage stress-average strain relationships are available: As simple linear beam theory is used, deviations such Adamchack (56), Beghin et al (57), Dow et al (58), Gordo as shear lag, warping and racking are thus ignored. This and Guedes Soares (59,60) and, Yao and Nikolov (61,62). method may be a little un-conservative if the structure is The FEM can even be used to get these curves (Smith 54). predominantly subjected to lateral pressure loads as well as For most of the methods, typical element types are: plate axial compression, and if it is not realized that the trans- element, beam-column element (stiffener and attached plate) verse frames can deflect/fail and significantly affect the stiff- and hard corner. ened plate structure and hull girder bending capacity. MASTER SET SDC 18.qxd Page 18-49 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure 18-49

18.6.5.2 Simplified models tion equations to predict the ultimate strength. Each load Caldwell (64) was the first who tried to theoretically eval- component is supposed to act separately. These methods uate the ultimate hull girder strength of a ship subjected to were reviewed by ISSC (68) and are often formulated as longitudinal bending. He introduced a so-called Plastic De- equation 57. sign considering the influence of buckling and yielding of a b structural members composing a ship’s hull (Figure 18.47).  M   M   v  + α  h  = 1 [57] He idealised a stiffened cross-section of a ship’s hull to  M vu   M hu  an unstiffened cross-section with equivalent thickness. If buckling takes place at the compression side of bending, where: compressive stress cannot reach the yield stress, and the fully Mv and Mh = vertical and horizontal bending moments plastic bending moment (Mp) cannot be attained. Caldwell Mvu and Mhu = ultimate vertical and horizontal bending mo- introduced a stress reduction factor in the compression side ments of bending, and the bending moment produced by the reduced a, b and α = empirical constants stress was considered as the ultimate hull girder strength. Several authors have proposed improvements for the For instance, Mansour et al (47) proposes a=1, b=2 and α Caldwell formulation (65). Each of them is characterized = 0.8 based on analysis on one container, one tanker and by an assumed stress distribution (Figure 18.47). Such meth- 2 cruisers, and Gordo and Soares (60) 1.5

18-50 Ship Design & Construction, Volume 1

TABLE 18.IV Sensitivity Factors for Ultimate Strength but they are time consuming and there is large uncertainty Assessment of Hull Girder. of using simplified methods. With the introduction of higher tensile steels in hull struc- Model Capability Impact tures, at first in deck and bottom to increase hull girder strength, and later in local structures, the fatigue problem Plate buckling H became more imminent. The fatigue strength does not in- Stiffened plate buckling H crease according to the yield strength of the steel. In fact, Post buckling behavior H fatigue is found to be independent of the yield strength. The Plate welding residual stress H higher stress levels in modern hull structures using higher tensile steel have therefore led to a growing number of fa- M-φ curve (post collapse prediction) H tigue crack problems. Plate initial deflection M To ensure that the structure will fulfill its intended func- Stiffener initial deflection M tion, fatigue assessment should be carried out for each in- Stiffener welding residual stress M dividual type of structural detail that is subjected to extensive Multi-span model (instead of single span) H dynamic loading. It should be noted that every welded joint (see Figure 19.12 Ð Chapter 19) and attachment or other form of stress concentration is po- tentially 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, ing, σ−ε curves, curvature incrementation), the global un- and so on, are given in: Almar-Naess (73), DNV (4), Fricke certainty on the ultimate bending moment is usually large et al (74), Maddox (75), Niemi (76), NRC (77) and Peter- (55). A bias of 10 to 15% must be considered as acceptable. shagen et al (78). Reliability-based fatigue procedure is pre- For intact hull the design criteria for M ,defined by clas- u sented by Ayyub and Assakkaf in Chapter 19. These authors sification societies, is given by: also have contributed to this section. ≤ MS + s1 Mw s2 MU [58] 18.6.6.2 Basic fatigue theories where: Fatigue analyses can be performed based on: s = the partial safety factor for load (typically 1.10) 1 ¥ simplified analytical expressions, s = the material partial safety factor (typically 0.85) 2 ¥ more refined analysis where loadings/load effects are M = still water moment S calculated by numerical analysis, and M = design wave moment (20 year return period) w ¥a combination of simplified and refined techniques.` There are generally two major technical approaches for 18.6.6 Fatigue and Fracture fatigue life assessment of welded joints the Fracture Me- 18.6.6.1 General chanics Approach and the Characteristic S-N Curves Ap- Design criteria stated expressly in terms of fatigue damage proach. resistance were in the past seldom employed in ship struc- The Fracture Mechanics Approach is based on crack tural design although cumulative fatigue criteria have been growth data assuming that the crack initiation already ex- used in offshore structure design. It was assumed that fa- ists. The initiation phase is not modeled as it is assumed that tigue resistance is implicitly included in the conventional the lifetime can be predicted only using fracture mechan- safety factors or acceptable stress margins based on past ics method of the growing cracks (after initiation). The frac- experience. ture mechanics approach is obviously more detailed than Today, fatigue considerations become more and more the S-N curves approach. It involves examining crack growth important in the design of details such as hatch corners, re- and determining the number of load cycles that are needed inforcements for openings in structural members and so on. for small initial defects to grow into cracks large enough to Since the ship-loading environment consists in large part cause fractures. The growth rate is proportional to the stress of alternating loads, ship structures are highly sensitive to range, S (or ∆σ) that is expressed in terms of a stress in- fatigue failures. Since 1990, fatigue is maybe the most sen- tensity factor, K,which accounts for the magnitude of the sitive point at the detailed design stage. Tools are available stress, current crack size, and weld and joint details. The MASTER SET SDC 18.qxd Page 18-51 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure 18-51

basic equation that governs crack growth (79) is known as logN = log (∆A) Ð m log (∆σ)[62] the Paris Law is: where: da = C .(K)∆ m [59] ∆ = fatigue damage ratio (≤ 1) dN log(∆A) = intercept of the S-N curve of the Log N axis where: Ð1 / m = slope of the S-N curve, (≅3 ≤ m ≤≅7) Ð S = mean of the Miner’s equivalent stress range S ,de- a = crack size, e e ﬁned at Table 18.V N = number of fatigue cycles (fatigue life), k = fatigue stress uncertainty factor ∆K= S.Y(a). πa , range of stress intensity factor, (K S Ð max ∆σ = k . S (or the constant amplitude stress range for fail- Ð K ) S e min ure at N cycles) C, m = crack propagation parameters, N = fatigue life, or number of loading cycles expected dur- S = constant amplitude stress range, ing the life of a detail ∆σ σ σ = = max Ð min Y(a) = function of crack geometry. The Miner’s equivalent stress range, Se, can be evaluated based on the models provided in Table 18.V (83). The Fatigue life prediction based on the fracture mechanics most reﬁned model would start with a scatter diagram of approach shall be computed according to the following sea-states, information on ship’s routes and operating char- equation: 1 a da N = [60] mm∫ CS. a 0 Y 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 Mechan- ics 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 Figure 18.50 A Typical S-N Curve 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). ∆A N = [61] Figure 18.51 Comparison between the Characteristic S-N Curve and Fracture m m kSS e Mechanics Approach MASTER SET SDC 18.qxd Page 18-52 4/28/03 1:31 PM

18-52 Ship Design & Construction, Volume 1

acteristics, and use of a ship response computer program to Environmental conditions: The long-term distribution provide a detailed history of stress ranges over the service of load responses for fatigue analyses may be estimated life of the ship. For such model, the wave exceedance dia- using the wave climate, represented by the distribution of gram (deterministic method) and the spectral method (prob- Hs and Ts, representing the sea operation conditions. As abilistic method) can be employed (Table 18.V). guidance to the choice between these data sets, one should S-N curves are obtained from fatigue tests and are avail- consider the average wave environment the vessel is ex- able in different design codes for various structural details pected to encounter during its design life. The world wide in bridges, ships, and offshore structures. The design S-N sailing routes will therefore normally apply. For shuttle curves are based on the mean-minus-two-standard-devia- tankers and vessels that will sail frequently on the North At- tion curves for relevant experimental data (Figure 18.50). lantic, or in other harsh environments, the wave data given They are thus associated with a 97.6% probability of sur- in accordance with this should be applied. For vessels that vival. Some classification societies use 90%. will sail in more smooth sailing routes, less harsh environ- In practice, the actual probabilities of failure associated mental data may be applied. This should be decided upon with fatigue design lives is usually higher due to uncer- for each case. tainties associated with the calculated stresses, the various Geometrical imperfections: The fatigue life of a welded S-N curve correction factors, and the critical value of the joint is much dependent on the local stress concentrations cumulative fatigue damage ratio, ∆. factors arising from surface imperfections during the fab- Cumulative damage: The damage may either be calculated rication process, consisting of weld discontinuities and geo- on basis of the long-term stress range distribution using metrical deviations. Surface weld discontinuities are weld Weibull parameters (simplified method), or on summation of toe undercuts, cracks, overlaps, incomplete penetration, etc. damage from each short-term distribution in the scatter dia- Geometrical imperfections are defined as misalignment, an- gram (probabilistic and deterministic methods, Table 18.V). gular distortion, excessive weld reinforcement and other- The stress range (S or ∆σ): The procedure for the fa- wise poor weld shapes. tigue analysis is based on the assumption that it is only nec- Effect of grinding of welds: For welded joints involving essary to consider the ranges of cyclic principal stresses in potential fatigue cracking from the weld toe an improve- determining the fatigue endurance. However, some reduc- ment in strength by a factor of at least 2 on fatigue life can tion in the fatigue damage accumulation can be credited be obtained by controlled local machining or grinding of when parts of the stress cycle range are in compression. the weld toe. Note that grinding of welds should not be used Fatigue areas: The potential for fatigue damage is de- as a “design tool”, but rather as a mean to lower the fatigue pendent on weather conditions, ship type, corrosion level, damage when special circumstances have made it necessary. location on ship, structural detail and weld geometry and This should be used as a reserve if the stress in special areas workmanship. The potential danger of fatigue damage will turns out to be larger than estimated at an earlier stage of also vary according to crack location and number of po- the design. tential damage points. Fatigue strength assessment shall normally be carried out for: 18.6.6.3 Stress concentration and hot spot stress The stress level obtained from a structural analysis, such as ¥longitudinal and transverse element in: FEA, will depend on the fineness of the model. The differ- — bottom/inner bottom (side), ent analysis models described in Subsection 18.7.2 will — longitudinal and transverse bulkheads. therefore lead to different levels of result processing in order to complete the fatigue calculations. ¥ strength deck in the midship region and forebody, and In order to correctly determine the stresses to be used in ¥ other highly stressed structural details in the midship re- fatigue analyses, it is important to note the definition of the gion and forebody, like panel knuckles. different stress categories (Figure 18.52). Time at sea: Vessel response may differ significantly for Nominal stresses are those, typically, derived from coarse different loading conditions. It is therefore of major im- mesh FE models. Stress concentrations resulting from the portance to include response for actual loading conditions. gross shape of the structure, for example, shear lag effects, Since fatigue is a result of numerous cyclic loads, only the have to be included in the nominal stresses derived from most frequent loading conditions are included in the fatigue stress analysis. analysis. These will normally be ballast and full load con- Geometric stresses include nominal stresses and stresses dition. Under certain circumstances, other loading condi- due to structural discontinuities and presence of attach- tions may be used. ments, but excluding stresses due to presence of welds. MASTER SET SDC 18.qxd Page 18-53 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure 18-53

Stresses derived from fine mesh FE models are geometric the notch stress. This can be done by multiplication of K- stresses. Effects caused by fabrication imperfections as mis- factors arising from different causes. The resulting K-fac- alignment of structural parts, are normally not included in tor to be used for calculation of notch stress is: FEA, and must be separately accounted for, using, for in- K = K1 . K2 . K3 . K4 . K5 [65] stance (equation 65). Hot spot stress is the greatest value of the extrapolation where: to the weld toe of the geometric stress distribution imme- K = stress concentration factor due to the gross geometry diately outside the region affected by the geometry of the 1 of the detail considered weld (Figure 18.52). K = stress concentration factor due to the weld geometry Notch stress is the total stress at the weld toe (hot spot 2 (notch factor); K = 1.5 if not stated otherwise location) and includes the geometric stress and the stress 2 K = additional stress concentration factor due to eccen- due to the presence of the weld. The notch stress may be 3 tricity tolerance calculated by multiplying the hot spot stress by a stress con- K = additionally stress concentration factor due to angu- centration factor, or more precisely the theoretical notch 4 lar mismatch factor, K (equation 65). 2 K = additional stress concentration factor for un-symmet- FE may be used to directly determine the notch stress. 5 rical stiffeners on laterally loaded panels, applicable However, because of the small notch radius and the steep when the nominal stress is derived from simple beam stress gradient at a weld, a very fine mesh is needed. analyses In practice, the stress concentration factors (K-factors) may be determined based on fine mesh FE analyses, or, al- Fatigue cracks are assumed to be independent of princi- ternatively, from the selection of factors for typical details. pal stress direction within 45° of the normal to the weld toe. The notch stress range governs the fatigue life of a de- Hot spot stress extrapolation procedure: The hot spot tail. For components other than smooth specimens the notch stress extrapolation procedure (Figure 18.52) is only to be stress is obtained by multiplication of the nominal stress by used for stresses that are derived from stress concentration K-factors (equation 63). The K-factors in this document are models (fine mesh). Nominal stresses found from other thus defined as models should be multiplied with appropriate stress con- σ centration factors (equation 65). The stress extrapolation = notch K σ [63] procedure is specific to each classification societies (74). nominal Today, there is unfortunately no standard procedure. The relation between the notch stress range to be used together with the S-N-curve and the nominal stress range 18.6.6.4 Direct analysis is Several S-N fatigue approaches exists, they all have advantages and disadvantages. The different approaches are ==∆∆σσ = ∆ σ SKnotch. nominal [64] therefore suitable for different areas. Load effects, accu- All stress risers have to be considered when evaluating racy 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 Figure 18.52 Definition of Stress Categories (4) the surface region are of major importance. This is due to MASTER SET SDC 18.qxd Page 18-54 4/28/03 1:31 PM

18-54 Ship Design & Construction, Volume 1

TABLE 18.V Commonly Used Expressions for Evaluating tions by use of load/stress ratios, Hi (equation 66). The load Miner’s Equivalent Stress Range (S ), (83) e transfer functions, Hi, normally include the global hull girder bending sectional forces and moments, the pressures for all 1. Wave Exceedance Diagram (Deterministic Method) panels of the 3-D diffraction model, the internal tank pres-

n bbn sures. m =→=m m m The stress transfer functions, H ,are combined to a total SfSSfSe ∑∑i i eii i i i stress transfer function, Hσ,by a linear complex summation of the different transfer functions (4), as: Si = stress range = Fi = fraction of cycles in the ith stress block HAHσ ∑ ii [66] i nb = number of stress block where: 2. Spectral Method (Probabilistic Method) Ai = stress per unit axial force deﬁned as the local stress m response in the considered detail due to a unit sec- (22)  m  Smm = λγσ( ) Γ + 1 ∑ f m tional load for load component i. e f  2  ii i Η 0 i σ = total transfer function for the combined local stress, H = transfer function for the load component i, that is, axial λ i (m) = rainﬂow correction force, bending moments, twisting and lateral load. Γ(.) = gamma function This approach enables the use of separate load factors on γ ι = fraction of time in ith sea-state each load component and thus includes loads non-linearities. fi = frequency of wave loading in ith sea-state Few load cases have to be analyzed and it is possible to use

σι = RMS of stress process in ith sea-state simplified formulas for the area of interest but errors are easily made in the combination of stresses, manual definition of 3. Weibull Model for Stress Ranges (Simplified Method) extra load cases may cause errors and simplifications are usually made in loading. Suitable areas are components where n n bb geometric stress concentration factors, K1,are available (lon- m =→=m m m SfSSfSe ∑∑i i eii gitudinals, plating, cut-outs and standard hopper knuckles) i i and areas where side pressure is of importance. The simplified design wave approach (Weibull Model, Sd = stress range that is exceeded on the average once out of N stress cycles Table 18.V) is a simplification to the previous component d based stochastic fatigue analyses. In this simplified ap- Γ(.) = gamma function proach, the extreme load response effect over a specified k = Weibull shape parameter number of load cycles, for example, 104 cycles, is deter- ∆σ Nd = total number of stress ranges in design life mined. The resulting stress range, , 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 de- the fact that all load effects result in one set of combined fine the long-term stress range distribution (Table 18.V). stresses, making it difficult to modify the stress caused by The Weibull shape parameter, k, for the stress response one of the load effects. should be determined from the long-term distribution of the The approach is suitable for areas where the stress con- dominating load calculated in the hydrodynamic analysis. centration factors are unknown (knuckles, bracket and flange This simplified approach only requires the considera- terminations of main girder, stiffeners subjected to large tion of one load case. It is easy and fast to perform but it relative deformations). can only be used if one load dominates the response and the results are very sensitive to selection of design wave. 18.6.6.5 Simplified models Suitable areas concern components where one load is dom- The stress component based stochastic fatigue analysis: inating the response, that is, deck areas and other areas with- The idea of the stress component based fatigue analysis is out local loading. to change the direct load transfer functions calculated from the hydrodynamic load program into stress transfer func- MASTER SET SDC 18.qxd Page 18-55 4/28/03 1:31 PM

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18.6.6.6 Design criteria structures against collision and grounding (85). For the ac- The standard fatigue design criterion is basically the ex- cidental limit state design, the integrity of a structure can pected lifetime before that significant damage appears be checked in two steps. In the first step, the structural per- (cracks). It usually is taken as being 20 years. Then, the de- formance against design accident events will be assessed, signer’s target is to design structural details for which the while post-accident effects such as likely oil outflow are fatigue failure happens after, for instance, 20 years. If it evaluated in the second step. happens before, the fixing cost is very high and induces The primary concern of the accidental limit state design owner losses. If the first failure only happens after 30 years in such cases is to maintain the water tightness of ship com- or later, the structural detail scantlings were globally over- partments, the containment of dangerous or pollutant car- estimated, the hull weight too high and, therefore, that the goes, and the integrity of critical spaces (reactor compart- owner had lost payload during 20 years. ments of nuclear powered ships or tanks in LNG ships) at Partial safety factors, additional stress concentration fac- the greatest possible levels, and to minimize the release/out- tors and the stress extrapolation procedure are typically de- flow of cargo. To facilitate a rescue mission, it is also nec- fined by the classifications societies. essary 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 Collision and Grounding 18.6.7.1 Present design approaches 18.6.7.3 Simplified models The OPA 90 and equivalent IMO requirements must be sat- Since the response of ships in collision or grounding acci- isfied in structural design of ships carrying dangerous or pol- dent includes relatively complicated behavior such as crush- lutant cargoes, for example, chemicals, bulk oil, liquefied ing, tearing and yielding, existing simplified methods are gas. The primary requirements are to arrange a double bot- not always adequate. However, many simplified models tom of a required minimum height, and double sides of a useful for predicting accident induced structural damages required minimum width. In this context, to reduce the out- and residual strength of damaged ship structures have been flow of pollutant cargoes in ship collision or grounding ac- developed and continue to be successfully used. Simplified cident, OPA 90 and IMO both require that the minimum models for collision are rather different from those of vertical height, h, of each double bottom ballast tank or void grounding since both are different in the nature of the me- space is not to be less than 2.0 m or B/15 (B = ship’s beam), chanics involved. As it is impossible to describe them in a whichever is the lesser, but in no case is the height to be limited space, valuable references are Ohtsubo et al (86), less than 1.0 m. OPA and IMO also require that the mini- and Kaminski et al (39). mum 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 18.6.7.4 Design criteria the ship in tonnes. In no case is w to be less than 1.0 (m). The structural design criteria for ship collisions and ground- More detailed information is available in Chapter 29 on Oil ing are based on limiting accidental consequences such as Tanker. structural damage, fire and explosion, and environmental pollution, and to make sure that the main safety functions 18.6.7.2 Direct analysis of ship structures are not impaired to a significant extent dur- To reduce the probability of outflow of hazardous cargo in ing any accidental event or within a certain time period ship collisions and grounding, the kinetic energy loss dur- thereafter. ing the accident should be entirely absorbed by damage of Structural performance of a ship against collision or outer structures, that is, before the inner shell in contact grounding can be measured by: with the cargo can rupture. Of crucial importance, then, is ¥ energy absorption capability, how to arrange or make the scantlings of strength members ¥maximum penetration in an accident, in the implicated ship structures such that the initial kinetic ¥spillage amount of hazardous cargo, for example, crude energy is effectively consumed and the structural perform- oil, and ance against an accident will be maximized. For this pur- ¥ hull girder ultimate strength of damaged ships (Section pose, the structural crashworthiness of ships in collisions 18.6.5). and grounding must be analyzed using accurate and efficient procedures (84). Design acceptance criteria may be based on the follow- Figure 18.53 shows direct design procedures of ship ing parameters (87): MASTER SET SDC 18.qxd Page 18-56 4/28/03 1:31 PM

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

¥minimum distance of cargo containment from the outer 18.6.8 Vibration shell, 18.6.8.1 Present Vibration Design Approaches ¥ ship speed above which a critical event (breaching of The traditional design methodology for vibration is based on cargo containment) happens, rules, defined by classification societies. Vibrations are not ¥allowable quantity of oil outflow, and explicitly covered by class rules but their prediction is needed ¥minimum values of section modulus or ultimate hull to achieve a good design. Ship structures are excited by nu- girder strength. merous dynamic oscillating forces. Excitation may originate And the design results must satisfy: within the ship or outside the ship by external forces. Reci- procating machinery such as large main propulsion diesel ¥ cargo tanks/holds are not breached in an accident so that produce important forces at low frequency. Pressure fluctu- there will be no danger of pollution, or ations due to propeller at blade rate frequency induce pres- ¥ if the cargo tanks are breached, the oil outflow follow- sure variation on the ship’s hull. Varying hull pressures ing an accident is limited, and/or associated with waves belong also to external excitations. All ¥ the ship has adequate residual hull girder strength so that these forces can be approximated by a combination of har- it will survive an accident and will not break apart, min- monic forces. If their frequencies coincide with the structure imizing a second chance of pollution. eigen frequencies, resonant behavior will happen. MASTER SET SDC 18.qxd Page 18-57 4/28/03 1:31 PM

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It is of prime importance to avoid global main hull vi- placement of the surrounding fluid. Therefore imparting ki- brations. If they do occur, the remedial action will proba- netic energy in the fluid. This phenomenon can be taken bly be very costly. So, during early design, the hull girder into account for the hull girder modes and frequencies cal- frequencies must be compared to wave excitation (spring- culation as added mass terms. Various methods can be used ing risk), and to propeller and engine excitation. Table 18.VI for the determination of added mass term. Lumped mass ap- gives some typical values of the first hull girder frequen- proach is the simplest one (89) but is only valid for simple cies in Hz of some ship types. prismatic slender shapes, and for a single mode. Fluid fi- Hull girder frequencies and modes should be computed nite and semi-infinite elements or boundary integral for- using approximate empirical formulae (88), simple beam mulation lead to the calculation of more accurate added models for long prismatic structures (VLCC, container ships, mass matrices (90), especially for complex hull forms and etc.) associated with lumped added mass models, or using appendices study (rudder). Added mass matrices associated 3D finite element models for complex ships (RO-RO, cruise with 3D finite element model of the structure, allow for an ship), LNG, and short and non-prismatic structures (tug, accurate determination of hull girder modes and frequen- catamaran, etc.). cies. Added mass terms may also be needed for the vibrations of tank walls. The corresponding methods and 18.6.8.2 Fluid structure interaction associated software are available for industrial usage (Fig- Fluid structure interaction is evidenced in the dynamic be- ure 18.54) and numerical simulations are today predictable havior of ships. As a first approximation, the ship is con- with good accuracy (91). Figure 18.54 shows a fluid-struc- sidered as a rigid body, for the sea keeping analyses (wave ture coupled FE-model of a 230 m long passenger vessel induced motions and loads). using 150 000 degrees of freedom. Wave vibration induced: An early determination of hull A difficult coupled problem is the fluid impact occur- girder vibration modes and frequencies is important to avoid ring in slamming or due to sloshing in tanks. The local de- serious problems that would be difficult to solve at a later formation of the impacted shells and plating influences the 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-

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

Large Order Cruise Fast (mode) ship monohull LNG VLCC Frigate Tug

1 1.0 Hz 1.8 0.9 0.8 1.9 7.0 2 1.5 Hz 2.9 2.0 1.7 3.8 13 3 2.6 Hz — — — 5.8 — Figure 18.54 Fluid/Structure FE-Model of a Passenger Vessel (Principia 4 3.2 Hz — — — 7.8 — Marine, France) MASTER SET SDC 18.qxd Page 18-58 4/28/03 1:31 PM

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pressures and fluid velocities. Moreover, air trapped in such ious parts of the ship can only be performed by simulation an impact may have a cushioning effect, softening its sever- in the time domain based on 3D detailed finite element mod- ity. The numerical simulation of those heavily coupled prob- els (Figure 18.55). The main difficulty is the determination lems still belongs to the research domain, though its of the time and space dependent slamming forces. industrial importance for the design of ship structures (92). 18.6.8.4 Simplified models 18.6.8.3 Direct analysis Unfortunately, they are of little use for simplified vibration Vibration problems are critical for passenger ships with typ- predictions. Beam models associated to database can be ically a 12-Hertz blade excitation. Ship owners demand very used for an approximate determination of hull girder modes low vertical velocity levels incabins and public areas (less and frequencies at early stage of the project. Decks zones than 1.2 mm/s in the 5-25 Hz frequency band). and equipment frequencies may also be estimated by for- Numerical simulation using 3D finite element models is mulas given by reference books (94). the only method to predict ship response (including the var- Dedicated software has also been written for the study ious frequency modes) to pressure fluctuation on the ship of shafting, including journal and bearing stiffness and hull. Such simulation is now used as a design tool to select whirling effect (95). appropriate scantlings of decks, location of pillars, detect possible resonance, and select the number of propeller 18.6.8.5 Design criteria blades. The main difficulty is to perform this analysis early The most effective way to control vibration resides in the enough in a very short design cycle. reduction of the excitation. This can be achieved by bal- Local analyses also have to be performed, based on fi- ancing all forces in reciprocating and rotary machinery and nite element models to check the potential risk of vibration using special mounts. Hydrodynamic forces can be reduced of local areas, when local modes can be considered as de- by improving the flow around the propeller and siting it coupled from global hull girder modes. Decks, superstruc- clear of the hull. Propulsion using pods can dramatically re- ture, appendices (rudder, radar mast, etc.) can be analyzed duce pressure fluctuations. Excitation frequencies can also to check scantling and avoid the risk of resonance. be modified by changing the number of propeller blades. Slamming impacts generate impulsive response of the A good design, ensuring continuity of vertical bulkheads, hull girder (whipping), which affects comfort and fatigue. avoiding cantilevered and stiff or mass discontinuities, con- Prediction of stress fluctuations and vibration levels in var- tributes to improving the dynamic behavior of the ship. The

Figure 18.55 Hull Girder Vibration—Mode #3 (Principia Marine-France) MASTER SET SDC 18.qxd Page 18-59 4/28/03 1:31 PM

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second action consists in avoiding resonance by modifica- ular effects in the structural design of a main hull structure. tion of the hull scantlings, and addition of pillars, in order The general characteristics of container ships are detailed to increase or lower the eigen frequencies. in Chapter 36 Ð Container Ships. Reduction of unavoidable vibration levels can be achieved for local vibrations by dynamic isolation for equip- 18.6.9.3 Bulk carriers ments, passive damping solutions (floating floors on ab- Casualty of bulk carriers was very high in the early 1990s. sorbing material), and dynamic energy absorbers. All these The main reasons were a lack of maintenance, excessive cor- curative actions are usually difficult, costly, only applica- rosion and fatigue (77). Weak point of these ships is the ble for local vibrations and nearly impossible for vibrations lower part of the side plate at the junction with the bilge due to global modes. Local modes determination is diffi- hopper. Now, classification societies are aware about this cult at early stage of the design mainly due to the uncer- problem and had updated their rules and associated struc- tainty on mass distribution, non-structural mass (outfitting tural details. The general design practice on bulk carriers is and equipments) being of the some order of magnitude as detailed in Chapter 33 Ð Bulk Carriers. the steelwork part. 18.6.9.4 Passenger vessels Ship strength analysis is based on a beam model. The com- 18.6.9 Special Considerations plexity of large passenger ships, with a low resistant deck and wide openings, windows and openings in the side in- In addition to the considerations for LNG tank, container duces a much more complex behavior. Rational approach ship, bulk carrier and passenger vessel, special considera- is necessary to get a realistic understanding of the flux of tions are available in Volume II of this book. Moreover, forces and capture the complex behavior of such ships. ISSC committees 1997 and 2000 also provide valuable in- Due to the large openings and discontinuities, racking and formation on specific ship types, that is, high-speed vessels stress concentration are two major concerns. For archi- and ships sailing in ice conditions. tectural reason, pillars are often omitted in large public areas (theater, lounge, etc.). Today, 3D FEA is usually car- 18.6.9.1 LNG Tanks ried out to design large passenger vessels (Figures 18.54 General information on such ships is available in Chapter and 18.55). Due to large opening in the side shells, the ver- 32 – Liquefied Gas Carriers. These ships contain usually a tical stress distribution is not linear (Figure 18.35). This double hull (sides and bottom). Major structural concerns means that the basic beam bending formulation is no valid deal with the tanks themselves and with their support legs. (equation 29). More general information related to pas- Dilatation, tightness and thermal isolation are important as- senger vessels is available in Chapter 37 Ð Passenger Ships pects. There are several patented concepts: independent and in reference 68. tanks, membrane tanks, semi-membranes tanks and integral tanks. Excepted for the integral tanks, the tanks are self- 18.6.9.5 Composite material supporting and are not essential to the hull strength. When Fiberglass boat building started in the 1960s. Today, de- supported by legs, these legs require a particular attention. signers are trying to plan composite construction of ships Integral tanks form a structural part of the ship’s hull and up to 100 meters in length. A comprehensive guide for the are influenced in the same manner by wave loads. design of ship structures in composites is the Ship Struc- ture Committee Report SSC-403 of Greene (96). Design 18.6.9.2 Container ships methodology, materiel properties, micro and macro me- The design of container ships of 5000 and 6000 TEU hav- chanic of composites and failures modes are deeply dis- ing a beam of 40m has increased the standard torsional prob- cussed. lem of ships having a large open deck. Torsional strength In addition to the classic failure modes of steel and alu- and limitation of the equivalent stress (equation 45) at the minum structures presented in Subsection 18.6.1, compos- hatch corners are the major issues in the evaluation of the ites are subject to specific failure modes. strength of main hull structure. Use of multicell structures In compression, there are the crimping, skin wrinkling in side shell and double bottom is recommended. More- and dimpling of the honeycomb cores (Figure 18.56). In over, the torsional moment distribution must be assessed bending, instead of the traditional first yield bending mo- with care. ment, for composites, the design limit load corresponds to As hatch covers are not considered as hull strength mem- the first ply failure. bers, omission of hatch covers does not impose any partic- The creep behavior and the long-term damage from MASTER SET SDC 18.qxd Page 18-60 4/28/03 1:31 PM

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water, UV and temperature, and their performance in fires sign of fast vessels, for which the structural weight is very are other specific structural problems of composites. A re- important to reach higher speed (for high speed mono hull, view of the performance of composite structures is pro- catamaran and trimaran vessels). The good extruding ca- posed by Jensen et al (98). pability of aluminum alloys has to be enhanced through scantling standardization. That helps to lower to produc- 18.6.9.6 Aluminum structures tion cost ($/man-hour) and compensate the initial higher Compared to steel, the reduced specific weight of aluminum material cost of aluminum, which is approximately 3 times (2.70 kN/m3 for aluminum and7.70 kN/m3 for steel) is a very higher that mild steel ($/kg). interesting property for a ship designer. The yield stress of unwelded aluminum alloys can be comparable to mild steel 18.6.9.7 Corrosion (235 MPa) but changes drastically from one alloy to an- Corrosion does not present a structural design problem, as other (125 MPa for ALU 5083-O and 215 MPa for ALU almost all the classification societies base their rules on a 5083-H321). The modulus of elasticity of aluminum alloys net scantling. This means that the thickness to consider in is one-third of steel. analysis (for empirical formulations up to complex FEA) The main difficulty for the use of aluminum use deals is the reduced thickness (without corrosion allowance) and with its mechanical properties after welding. The yield stress not the actual thickness. The difference between the reduced of aluminum alloys may decrease significantly after weld- thickness and the actual one is usually fixed by the classi- ing (remains at 125 MPa for fication but can also change according to the owner re- ALU 5083-O but drop to 140 MPa for ALU 5083-H321). quirements. This is an economic choice and not a structural The area close to a weld is called Heat Affected Zone (HAZ). problem. It is characterized by reduced strength properties. HAZ is For bulk carriers, thickness reduction due to corrosion particularly important to assess the buckling and ultimate is generally assumed to be 5 mm for hold frames and 3 mm strength of welded components such as beam-column ele- for side shell plating. ments, 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 18.7 NUMERICAL ANALYSIS FOR STRUCTURAL and the production processing. Thus, the alloy selection DESIGN must be done with care with regard to the yield strength before and after welding, the welding and extruding capabil- 18.7.1 Motivation for Numerical Analysis ities, the marine behavior, etc. In most of the cases, a ship is a one of a kind product, even Fire strength is another concerns when using aluminum if limited series may exist in some cases. The design, study alloys as it quickly loses its strength when the temperature and production cycle is very short and major decision have rises. to be taken very early in the project. It is well known that Despite the aforementioned shortcomings aluminum al- the cost of a late modification is very high and such a situ- loys will be more extensively use in the future for the de- ation 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 Figure 18.56 Potential Failure Modes of Sandwich Panels (100), (a) Face and, even if their origins are dynamic (ﬂow induced), they yielding/fracture, (b) Core shear failure, (c-d) Face wrinkling, (e) Buckling, (f) are assumed to be static (do not change with the time). This Shear crimping, (g) Face dimpling, (h) Local indentation. assumption may be correct for the hydrostatic pressure but MASTER SET SDC 18.qxd Page 18-61 4/28/03 1:31 PM

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not when the dynamic wave loads are changed to static loads ¥ static, fatigue and fracture analysis, applied on the side plates of the hull. ¥buckling and ultimate strength analysis, In the future, even if the assumption of static loads is not ¥ vibration and acoustics analysis, and verified, static analysis will continue to be performed, as it ¥ vulnerability assessment. is easier and faster to perform. In addition, tens of experi- Progress is expected by the utilization of reliability meth- ence years have shown that they provide accurate results ods already used in offshore industry, where uncertainties when stresses and deflections assessment are the main tar- and dispersions of the loads, geometrical defaults, initial get (as defined in Section 18.4). stresses and strains, material properties are defined as sto- Such analysis is also the standard procedure for fatigue chastic (non deterministic) data, leading to the calculation assessment to determine the hot spot stress through fine of a probability of failure. This philosophy can be applied mesh FEA. to fatigue and ultimate strength, but also to dynamic response, leading to a more robust design, less sensitive to 18.7.1.2 Dynamic analysis defaults, imperfections, uncertainties and stochastic nature When problems occur on a ship due to dynamic effects, it of loads. Reliability-based analyses using probabilistic con- is very often late in the design and building stage and even cept are presented in Chapter 19. in service, and corrective actions are costly. Simplified meth- In the future, safety aspects related to structural prob- ods can only predict the first hull girder modes frequencies. lems will also be tackled such as ultimate strength using non- Numerical finite element based simulation is mature enough linear methods. Collision and grounding damages and to predict up to second propeller harmonic, the vibration improved design to increase ship safety will be studied by level, giving a design tool to comply with ISO or ship owner numerical simulation, whereas experimental approach is requirements. Moreover, possible dynamic problems can nearly impossible and/or too costly. Explicit codes, used in be detected early enough in the design to allow for correc- car crash simulation (101), will be adapted to specific as- tive actions. pects of ship structure (size and presence of fluid). In traditional sea keeping analysis, the ship is considered as a 18.7.1.3 Nonlinearities analysis rigid body. In coupled problems such as slamming situa- Nonlinear structural analysis is mainly used to analyze buck- tions, this hypothesis is no more valid and a part of the en- ling, ultimate strength and accidental or extreme situations ergy is absorbed by ship deformation. Hydro-elasticity (explosions, collisions, grounding, blast). The results of methods (102) aim taking into account the interaction of the such costly and difficult analysis are often used to calibrate flexible ship structure with the surrounding water. Nonlin- simplified methods or rules. But they are also very useful ear effects due to bow and aft part of the ship, ship veloc- to understand possible failure modes and mechanical be- ity, diffraction radiation effects contribute to the complexity havior under severe loads. 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 com- 18.7.1.4 Emerging trends plex 3D phenomena. Like the automotive and aerospace industry, there is a clear trend towards the reduction of design cycle time. Numeri- cal mock up or virtual ship approach (97), especially for one of a kind product, is clearly a way to achieve this. Required 18.7.2 Finite Element Analysis computing power is available and will no longer be a con- The main aim of using the finite element method (FEM) in straint. The first difficulty is to establish an efficient model structural analysis is to obtain an accurate calculation of the of complex physical problems, associated with increasing stress response in the hull structure. Several types or levels demand for accuracy. The second difficulty is the manpower of FE-models may be used in the analyses: needed to prepare and check the models, which will be ¥ global stiffness model, solved by the development of integrated solutions for ship ¥ cargo hold model, description and modeling (99). ¥frame and girder models, Advances are expected in the field of FE-modeling. The ¥ local structure models, and trend is toward one structure description, one model and sev- ¥ stress concentration models. eral applications. This is the field for multiphysics and coupling analysis. The base modeling will be re-used and The model or sets of models applied is to give a proper adapted to perform successively, representation of the following structure: MASTER SET SDC 18.qxd Page 18-62 4/28/03 1:31 PM

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¥longitudinal plating, The minimum element sizes to be used in a global struc- ¥transverse bulkheads/frames, tural model (coarse mesh) for 4–node elements (finer mesh ¥ stringers/girders, and divisions may of course be used and is welcomed, specially ¥longitudinals or other structural stiffeners. with regard to sub-models): The finer mesh models are usually referred to as sub- ¥ main model: 1 element between transverse frames/gird- models. These models may be solved separately by trans- ers; 1element between structural deck levels and mini- fer of boundary deformations/ boundary forces from the mum three elements between longitudinal bulkheads, coarser model. This requires that the various mesh models ¥girders: 3 elements over the height, and are compatible, meaning that the coarser models have ¥plating: 1 element between 2 longitudinals. 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. Typi- cal models are shown in Figure 18.57. The mesh density of Figure 18.57 Global Finite Element Model of Container Vessel Including a 4 the model has to be sufficient to describe deformations and Cargo Holds Sub-model (4). nominal stresses from the following effects: ¥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 dis- regarded; 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 de- Figure 18.58 Cargo Hold Model (Based on the Fine Mesh of the Frame ducted from the actual scantling. Model), (4) 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 18.59 Frame and Girder Model (Web Frame), (4) MASTER SET SDC 18.qxd Page 18-63 4/28/03 1:31 PM

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Cargo hold model:The model is used to analyze the de- centration models are generally very sensitive to element formation response and nominal stresses of the primary type and mesh size. members of the midship area. The model will normally Several FEA benchmarks of such structural details were cover 1/2+1+1/2 cargo hold/tank length in the midship re- performed by ISSC technical committees (68,103). They as- gion. Typical models are shown in Figure 18.58. sess the uncertainties of different FE packages associated Frame and girder models:These models are used to an- with coarse and fine mesh models. Variation is usually alyze nominal stresses in the main framing/girder system around 10% but is sometime much larger. (Figure 18.59). The element mesh is to be fine enough to This implies that element sizes in the order of the plate describe stress increase in critical areas (such as bracket thickness are to be used for the modeling. If solid model- with continuous flange). This model may be included in the ing is used, the element size in way of the hot spot may cargo hold model, or run separately with prescribed bound- have to be reduced to half the plate thickness in case the ary deformations/forces. However, if sufficient computer overall geometry of the weld is included in the model rep- capacity is available, it will normally be convenient to com- resentation. bine the two analyses into one model. Local structure analyses are used to analyze stresses in 18.7.2.2. Uncertainties related to FEA local areas. Stresses in laterally loaded local plates and stiff- An important issue in structural analysis is the verification eners subjected to large relative deformations between gird- of the analysis. The FEM is basically reliable but many ers/frames and bulkheads may be necessary to investigate sources of errors can appear, mainly induced by inappro- along with stress increase in critical areas, such as brack- priate modeling and wrong data. For this reason, different ets 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, Figure 18.60 Stiffener Bending Stress with FEM (from left to right: using 1, 2 ¥hatch corner openings, and or 8 elements), (4) ¥ 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- Figure 18.61 Stress Concentration Model of Hopper Tank Knuckle (4) MASTER SET SDC 18.qxd Page 18-64 4/28/03 1:31 PM

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levels of verification of the analysis should be performed ISSC. For instance, Sumi et al (68) presents finite element in order to ensure trustworthiness of the analysis results. Ver- guidelines and a comprehensive review of the available soft- ification must be achieved at the following steps: ware. Mesh modeling is discussed in ISSC’2000 by Por- cari et al (103). Hughes (3) proposes in Chapter VI and VII ¥ basic input, of his book published by SNAME an easy way to learn ¥ assumptions and simplifications made in modeling/ FEM that does not require knowledge of variational calcu- analysis, lus or of FEM. The Ship Structure Committee Reports (SSC ¥ models, 387 and 399) contains also Guideline for FEM (43,104). ¥ loads and load transfer, ¥ analysis, ¥results, and 18.7.3 Other Numerical Approaches ¥ strength calculations. As an alternative to FEA, two other approaches are pre- One important step in the verification is the understanding sented, namely: the idealized Structural Unit Method (ISUM) of the physics and check of deformations and stress flow and the Boundary Element Method (BEM). Both are gen- against expected patterns/levels. However, all levels of ver- eral purpose oriented. Many others exist but they are usu- ification are important in order to verify the results. ally dedicated to a special purpose. For instance, at the Verifications of structural models:Assumptions and sim- preliminary design stage, the LBR-5 package founded on the plifications will have to be made for most structural mod- analytical solution of the governing differential equations of els. These should be listed such that an evaluation of their stiffened plates is a convenient alternative to standard FEA. influence on the results can be made. Such an approach (30,105) allows structural design opti- The boundary conditions for the global structural model mization to be performed at the earliest design stage but does should reflect simple supporting to avoid built in stresses. The not have the capability to perform detailed analysis includ- fixation points should be located away from areas where ing stress concentration and non-linear analysis. stresses are of interest. Fixation points are often applied in the centerline close to the aft and the forward ends of the vessel. 18.7.3.1 Idealized structural unit method (ISUM) Verification of loads: Inaccuracy in the load transfer from When subjected to extreme or accidental loading, ship struc- the hydrodynamic analysis to the structural model is among tures can be involved in highly non-linear response associ- the main error sources in this type of analysis. The load ated with yielding, buckling, crushing and sometimes transfer can be checked on basis of the structural response rupture of individual structural components. Quite accurate or on basis on the load transfer itself. solutions of the non-linear structural response can be ob- Verification of response: The response should be veri- tained by application of the conventional FEM. However, fied at several levels to ensure correctness of the analysis: a weak feature of the conventional FEM is that it requires enormous modeling effort and computing time for non-lin- ¥ global displacement patterns/magnitude, ear analysis of large sized structures. Therefore, most ef- ¥ local displacement patterns/magnitude, forts in the development of new non-linear finite element ¥ global sectional forces, methods have focused on reducing modeling and comput- ¥ stress levels and distribution, ing times. ¥ sub-model boundary displacement/forces, and The most obvious way to reduce modeling effort and ¥reaction forces and moments. computing time is to reduce the number of degrees of freedom so that the number of unknowns in the finite element 18.7.2.3 FEM background stiffness equation decreases. Modeling the object structure Today the finite element method is studied worldwide in uni- with very large sized structural units is perhaps the best way versities, in mechanical engineering, civil engineering, naval to do that. Properly formulated structural units or super el- architecture, etc. Hundreds of papers are published yearly. ements in such an approach can then be used to efficiently Many commercial packages are available including pre and model the actual non-linear behavior of large structural post processors and many books are published each year on units. The idealized structural unit method (ISUM), which the subject. Classification Societies also present technical is a type of simplified non-linear FEM, is one of such meth- reports and guidelines associated with their own direct ods (106). Since ship structures are composed of several analysis package (Table 18.VIII). different types of structural members such as beams, It is not the purpose of this chapter to present the FE the- columns, rectangular plates and stiffened panels, it is nec- ory and a state of art. This topic is reviewed periodically by essary in the ISUM approach to develop various ISUM units MASTER SET SDC 18.qxd Page 18-65 4/28/03 1:31 PM

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for each type of structural member in advance. The non-lin- boundary domain, linear or flat boundary elements may be ear behavior of each type of structural member is idealized employed so that analytical solutions for the integral equa- and expressed in the form of a set of failure functions defin- tions can be adopted, while higher degree boundary ele- ing the necessary conditions for different failures which ments must be used for modeling an integral domain with may take place in the corresponding ISUM unit, and sets more complex characteristics with the integration gener- of stiffness matrices representing the non-linear relationship ally needing to be carried out numerically. Figure 18.65 between the nodal force vector and the nodal displacement shows typical FEM and BEM models for analysis of a pres- vector until the limit state is reached. The ISUM super el- sure vessel (109). ements so developed are typically used within the frame- Since the publication of an early book on BEM, many work of a non-linear matrix displacement procedure engineering applications using BEM have been achieved. applying the incremental method. More recent developments of BEM together with the basic 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 im- Figure 18.62 Cantilever Box Girder provements 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 Figure 18.63 A Typical FEM Model for NonLinear Analysis of the Cantilever the ultimate strength can be directly utilized by replacing Box Girder 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 Figure 18.64 A Typical ISUM Model for Nonlinear Analysis of the boundary. For a speciﬁc problem with a relatively simple Cantilever Box Girder MASTER SET SDC 18.qxd Page 18-66 4/28/03 1:31 PM

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idea may be found in Brebbia and Dominguez (109). While the accuracy high. Nevertheless as the required computa- there are some problem areas to overcome in use of BEM tional times with the BEM is in general significant, BEM for non-linear analysis, it has been recognized that BEM is may be more appropriate for linear analysis of solids and a powerful alternative to FEM particularly for problems in- for fluid mechanics problems. volving stress concentration or fracture mechanics, and for cases in which the integral domain extends to infinity. For example, to design the cathodic corrosion protection sys- 18.7.4 Presentation of the Stress Result tems for ships, offshore structures and pipelines, it has been After performing an analysis, the presentation of the stress suggested that BEM should be employed, with the region and deformation is very important. It should be based on of interest extending to infinity. BEM can also be applied stresses acting at the middle of element thickness, exclud- to problems other than stress or temperature analysis, in- ing plate-bending stress, in the form of ISO-stress contours cluding fluid flow and diffusion (for example, for fluid- in general. Numerical values should also be presented for structure interaction, Subsection 18.6.8.2). highly stressed areas or locations where openings are not Main advantages of BEM are due that very complex ex- included in the model. pressions of integral equations can be adopted, resulting in The following results should be presented for parts of higher accuracy of the results. the vessel covered by the global model, such as, cargo hold In this regard, BEM can be involved in the usage of more model and frame and girder models: refined mathematical treatment than FEM. However, to calculate the integral equations using BEM, appropriate nu- ¥ deformed shape for each loading condition, Author: σ σ merical techniques should be used, otherwise the integration ¥ In-plane maximum normal stresses ( x and y) in the Please results may not be accurate. For most linear problems, lin- global axis system, shear stresses (_) and equivalent von advise σ what ear or flat boundary elements along the boundary of the in- Mises stress ( e) of the following elements: symbold tegral domain can be used so that we don’t have to carry — bottom, is out numerical integration. If analytical solutions are avail- — inner bottom, needed. able the required computing times will be very small and — deck, — side shell, — inner side including hopper tank top, — longitudinal and transverse bulkheads, and — longitudinal and transverse girders. (a) ¥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: ¥ Equivalent stress of plate/membrane elements, (b) ¥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 Figure 18.65 A Typical FEM/BEM Model for Analysis of the products are one-of-a-kind or at least on short series and Pressure Vessel (109). (a) Typical BEM model, and (b) Typical FEM model. the resulting ships are designed and built within two years MASTER SET SDC 18.qxd Page 18-67 4/28/03 1:31 PM

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for 20 to 30 years of operation. Another impact on design 2. Two-dimensional (or almost 2D) geometry-based meth- activities that is also challenging is that the design overlaps ods:These methods are based on one or more 2D views the production. To clarify the actual situation, a common of the ship sections. The expected results may be: view of the design workflow for a commercial ship in the ¥ Verification of main section scantlings, shipyard is shown in Table 18.VII. ¥ Global strength assessment, ¥ Global vibration levels prediction, 18.7.5.1 Basic design ¥ Ultimate strength determination, and The Basic Design is the design activities performed before ¥ Early assessment of fatigue order. This phase does not overlap with the production but is very short and will become the technical basis for the Two main approaches exist: contract. The shipyard must be sure that no technical prob- — The main section of the ship is modeled a 2D way lem will appear later on, to avoid extra costs not included (including geometry and scantlings) then global, and in the contract. The structural analysis carried out in this possibly local, loadings are applied (bending mo- phase must be as fast as possible because the allocated time ments, pressures, etc.). All major Classification So- is short. The most time consuming task for analysis is the cieties provide today the designer with such tools data input. The more detailed are the data more accurate the (Table 18.VIII). results. There are three kinds of early analysis: — Various significant sections are described as beam 1. First principles methods:Very simplified geometric rep- cross section properties (areas, inertias, etc.) and then resentation of the structure. These methods are dedicated the ship is represented by a beam with variable prop- to an assessment of the global behavior of the ship. They erties on which global loading is applied. mainly use empirical or semi-empirical formulas. 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 TABLE 18.VII Timing of a Design Project the main section when not available) in a 3D model that can be achieved in one or two weeks. This approach is Basic Design mainly dedicated to novel ship designs for which the Concept Design 1 or 2 days feedback is rather small. Preliminary Design About 1 week Contract Design Months 18.7.5.2 Production design Receive Order The most popular method for structural analysis at the production design stage remains the Finite Elements Analysis Production Design (FEA). This method is commonly used by Shipyards, Classi- Complete Functional Design 1 or 2 months fication Societies, Research Institutes and Universities. It is Production Design 6Ð10 months very versatile and may be applied to various types of analysis: ¥ global and local strength, ¥ global and local vibration analysis (natural frequencies TABLE 18.VIII Classification Society Tools Overview (110) with or without external water, forced response to the propeller excitation, etc.), Classification Society Product ¥ ultimate strength, and American Bureau of Shipping (ABS) ABS Safe Hull ¥ detailed stress for local fatigue assessment, ¥fatigue life cycle assessment, Bureau Veritas (BV) VeriSTAR ¥ analysis of various non-linearities (material, geometry, Det Norske Veritas (DNV) Electronic Rulebook & contact, etc.), and Nauticus HULL ¥ collision and grounding studies. Germanisher Lloyd (GL) GL-Rules & POSEIDON The two main approaches for solving the physical prob- Korean Register of shipping (KRS) KR-RULES, KR-TRAS lem are: Lloyd’s Register of Shipping (LR) Rulefinder, ShipRight 1. implicit method is used to solve large problems (both lin- Nippon Kaiji Kyokai (NK) PrimeShip BOSUN ear and non linear) with a matrix-based method. This is MASTER SET SDC 18.qxd Page 18-68 4/28/03 1:31 PM

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the favored method for solving global and local linear put material. An academic example of topology op- strength and vibration problems. But it can also be ap- timization is given on Figure 18.66. plied to non linear calculations when the time step re- Weight is the most usual objective function for structure mains rather large (about 1/10 to 1 second), and optimization. Minimizing weight is of particular impor- 2. explicit method is mainly used for fast dynamics (as col- tance in deadweight carriers, in ships required to have a lision and grounding or explosion) where time step is limited draft, and in fast fine lined ships, for example, pas- quite smaller. This method allows using different for- senger vessels. However, it is well know that the lowest mulations for structural elements (Lagrangian) and fluid weight solution is not usually the lowest acquisition cost. elements (Eulerian). Today, cost is becoming the usual objective function for op- One interesting result from research that is being intro- timization (124). duced today is the reliability approach (see Chapter 19). For the other ship types it is still desirable to minimize This approach introduces uncertainties within the model steel weight to reduce material cost but only when this can (non planar plates, residual stresses from welding, dis- be done without increasing labor costs to an extent that ex- crepancies in the thickness…) to provide the designer with ceeds the saving in material costs. On the other hand, a re- a level of reliability for a given result instead of a deter- duction in structural labor cost achieved by simplifying ministic value. construction methods may still be worthwhile even if this For FEA models, the modeling time is usually assumed is obtained at the expense of increasing the steel weight. to be 70% of the overall calculation time and results ex- Rigo (105) presents extensive review of ship structure ploitation 30%. The computation itself is regarded as neg- optimization focusing on scantling optimization. Vander- ligible (excepted for explicit analysis). So the main efforts plaats (113), and Sen and Yang (114) are standard reference today are focused on reducing the modeling time. books about optimization techniques. Catley et al (115), Hughes (3) and Chapter 11 of this book also contain valu- 18.7.6 Optimization able information on structure optimization. Optimization is a field in which much research has been car- 18.7.6.1 Scantling optimization procedure ried out over a long time. It is included today in many soft- A standard optimization problem is defined as follows: ware tools and many designers are using it. The aim of

optimization is to give the designers the opportunity to ¥Xi (i = 1, N), the N design variables, change design variables (such as thickness, number and ¥ F(Xi), the objective function to minimize, ≤ cross section of stiffeners, shape or topology) to design a ¥Cj(Xi) CMj (j = 1, M), the M structural and geomet- better structure for a given objective (lower weight or cost). rical constraints, ≤ ≤ Optimization can be performed both at basic and pro- ¥Xi min Xi Xi max upper and lower bounds of the Xi de- duction design stages: sign variables: technological bounds (also called side constraints). ¥ 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 Figure 18.66 Topology Optimization MASTER SET SDC 18.qxd Page 18-69 4/28/03 1:31 PM

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Constraints are linear or nonlinear functions, either ex- ally uses a numeric procedure that consists of replacing the plicit or implicit of the design variables (XI). These con- implicit function by an explicit approximated function ad- straints are analytical translations of the limitations that the justed in the vicinity of the initial values of the design vari- user wants to impose on the design variables themselves or ables (for instance using the ﬁrst or second order Taylor to parameters like displacement, stress, ultimate strength, series expansions). This way, the optimization process be- etc. Note that these parameters must be functions of the de- comes an iterative analysis based on a succession of local sign variables. approximations of the behavior models. So it is possible to distinguish: At least one constraint should be deﬁned for each failure mode and limit state considered in the Subsection 18.6.1. Technological constraints (or side constraints) that provide When going from the local to the general (Figure 18.38), the upper and lower bounds of the design variables. For ex- there are three types of constraints: 1) constraints on stiff- ample: ened panels and its components, 2) constraints on trans- ≤ ≤ Xi min = 4mm Xi Xi max = 40 mm, verse frames and transversal stiffening, and 3) constraints on the global structure. with: Constraints on stiffened panels (Figure 18.22): Panels

Xi min = a thickness limit dues to corrosion, are limited by their lateral edges (junctions with other pan- Xi max = a technological limit of manufacturing or assembly. els, AA’ and BB’) either by transverse bulkheads or transverse frames. These panels are orthotropic plates and shells Geometrical constraints that impose relationships between supported on their four sides, laterally loaded (bending) and design variables in order to guarantee a functional, feasi- submitted, at their extremities, to in-plane loads (compres- ble, reliable structure. They are generally based on good sion/tensile and shearing). practice rules to avoid local strength failures (web or flange Global buckling of panels (including the local transverse buckling, stiffener tripping, etc.), or to guarantee welding frames) must also be considered. Panel supports, in partic- quality and easy access to the welds. For instance, welding ular those corresponding to the reinforced frames, are as- a plate of 30 mm thick with one that is 5 mm thick is not sumed infinitely rigid. This means that they can distort recommended. Hence, the constraints can be 0.5 ≤ X2 / X1 themselves significantly only after the stiffened panel col- ≤ 2 with X1, the web thickness of a stiffener and X2, the lapse. flange thickness. Constraints on the transverse frames (Figure 18.23): The Structural constraints represent limit states in order to avoid frames take the lateral loads (pressure, dead weight, etc.) yielding, buckling, cracks, etc. and to limit deflection, stress, and are therefore submitted to combined loads (large bend- etc. These constraints are based on solid-mechanics phe- ing and compression). The rigidity of these frames must be nomena and modeled with rational equations. Rational equa- assured in order to respect the hypotheses on panel bound- tions mean a coherent and homogeneous group of analysis ary conditions (undeformable supports). methods based on physics, solid mechanics, strength and Constraints on the global structure (box girder/hull stability treatises, etc. and that differ from empirical and girder) (Figure 18.46): The ultimate strength of the global parametric formulations. Such standard rational structural structure or a section (block) located between two rigid constraints can limit: frames (or bulkheads) must be considered as well as the elastic bending moment of the hull girder (against yielding). ¥ the deflection level (absolute or relative) in a point of the structure, σ σ σ σ ¥ the stress level in an element: x , y, and c = von Mises, ¥ the safety level related to buckling, ultimate resistance, 18.8 DESIGN CRITERIA tripping, etc. For example: σ /σ ≤ 0.5. ult In ship design, the structural analysis phase is concerned For each constraint, or solid-mechanics phenomenon, with the prediction of the magnitude of the stresses and de- the selected behavior model is especially important since flections that are developed in the structural members as a this model fixes the quality of the constraint modeling. These result of the action of the sea and other external and inter- behavior models can be so complex that it is no longer pos- nal causes. Many of the failure mechanisms, particularly sible to explicitly express the relation between the param- those that determine the ultimate strength and collapse of eters being studied (stress, displacement, etc.) and the design the structure, involve non-linear material and structural be- variables (XI). This happens when one uses mathematical havior that are beyond the range of applicability of the lin- models (FEM, ISUM, BEM, etc.). In this case, one gener- ear structural analysis procedures in Section 18.4, which are MASTER SET SDC 18.qxd Page 18-70 4/28/03 1:31 PM

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commonly used in design practice. Most of the available The semiprobabilistic method corresponds to the cur- methods of non-linear structural analysis are briefly intro- rent practice used by codes and the major classifications so- duced in Sections 18.6 and 18.7. Sometimes, these meth- cieties. Load, strength, dimensions are random parameters ods are limited in their applicability to a narrow class of but their distribution is basically not known. To overcome problems. this, partial safety factor are used. Each safety factor cor- One of the difficulties facing the structural designer is that responds to a load type, failure mode, etc. This is an inter- linear analysis tools must often be used in predicting the be- mediate step between the deterministic and the full havior of a structure in which the ultimate capability is gov- probabilistic methods. erned by non-linear phenomena. This is one of the important sources of uncertainty related to strength assessment. After performing an analysis, the adequacy or inade- 18.9 DESIGN PROCEDURE quacy of the member and/or the entire ship structure must then be judged through comparison with some kind of cri- It does not seem possible to unify all of the design proce- terion of performance (Design Criteria). The conventional dures (117-122). They differ from country to country, from criteria that are commonly used today in ship structural de- shipyard to shipyard and differ between naval ships, com- sign are usually stated in terms of acceptable levels of stress mercial ships and advanced high-speed catamaran passen- in comparison to the yield or ultimate strength of the ma- ger vessels. So, as an example of one feasible methodology, terial, or as acceptable stress levels compared to the criti- the design procedure for commercial vessel such as tanker, cal buckling strength and ultimate strength of the structural container, and VLCC is selected. It corresponds to the ac- member. Such criteria are, therefore, intended specifically tual current shipyard procedure. for the prevention of yielding (hull girder, frames, longitu- This structural design procedure can be defined as fol- dinals, etc), plate and stiffened plate buckling, plate and lows: stiffened plate ultimate strength, ultimate strength of hull ¥ receive general arrangement from the basic design group, girder, fatigue, collision, grounding, vibration and many ¥ define structural arrangement based on the general other failure modes specific to particular vessel types. In- arrangement, formation related to the design criteria is given in Section ¥ determine initial scantling of structural members within 18.6 for each specific failure mode (see also Beghin et al design criteria (rule-based)., (116)). ¥check longitudinal and transverse strength, ¥change the structural arrangement or scantling, and 18.8.1 Structural Reliability as a Design Basis ¥transfer the structural arrangement and scantling to the production design group. Three categories of design methodology are basically available. They are usually classified as: The structural design can also be classified according to available design tool: 1. deterministic method, 2. semiprobabilistic method, and ¥ use data of existing ship or past experience—expert sys- 3. full probabilistic method. tem, (1st level) ¥ use of a structural analysis software like FEM (2nd level) The deterministic method uses a global safety factor. It ¥ use optimization software (3rd level) assumes that loads and strength are fully determined. This means that no aspect of randomness is considered. Every- The adequacy of the relevant analysis method to use for thing is assumed to be deterministic. The global safety fac- a specific design stage is discussed in Subsection 18.7.5. tor is compared to the ratio between the actual strength and Here the discussion concerns the procedure from a design the required strength. point of view and not from the analysis point of view. 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 devel- 18.9.1 Initial Scantling opment in structural reliability and reliability analysis show At the basic design stage, principal dimensions, hull form, the huge effort actually done to reach that aims. Chapter 19 double bottom height, location of longitudinal bulkheads and presents in detail the reliability concept with examples of the transverse bulkheads, maximum still-water bending mo- reliability-based strength analysis of plates, stiffened pan- ment, etc. have already been determined to meet the owner’s els, hull girder and fatigue. See also Mansour et al (42). requirements such as deadweight and ship’s speed. Such a MASTER SET SDC 18.qxd Page 18-71 4/28/03 1:31 PM

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parametric design procedure presented in Chapter 11 is rel- creased thickness according to the difference between the evant for this stage. actual stress and allowable stress. If the difference is small, For the structural design stage, the structural arrangement it is not necessary to perform a new strength assessment is carried out to define the material property, plate breadth, and the design may be completed with only small changes. stiffener spacing, stiffener type, slot type, shape of open- If the difference is large, the design should be drastically ings, and frame spacing. The initial scantling of longitudi- changed and it will be necessary to analyze the structure nal members such as plate thickness and section area of again (see previous step in this Subsection). stiffener can be determined by applying the classification Then, the designer has to check the transverse strength rules which give minimum required value to meet the bend- by comparing the actual stresses in the transverse frames ing, shear and buckling strength. As there are usually no suit- with the allowable stresses given by the classification rules. able rules for the transverse members, the initial scantling The actual stresses such as equivalent stress and shear stress of transverse members such as height and thickness of web, can be obtained using commercial FEA packages. If the breadth and thickness of flange are determined by reference stress in some of elements exceeds the allowable stress, the to similar ships or using empirical shipyard database. 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 18.9.2 Strength Assessment the design criteria. 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 18.9.3 Structural Design presented in previous Sections 18.4, 18.5 and 18.6. If all of local scantlings are determined by the rule mini- In general, the longitudinal members are subjected to mum values, and if the longitudinal strength satisfies the rule several kinds of stresses in the sea-going condition: pri- strength requirement, the design is completed. But, even if mary, secondary and tertiary stresses (Subsection 18.4.1). this design is strong enough, it might be too heavy and/or As all these stresses act simultaneously, the superposition too expensive and it should be refined. In practice, refining of these stresses should not exceed the allowable equiva- an already feasible design is a difficult task and requires ex- lent stress given by the classification rules (equations 45 perience. The designer can change the structural arrange- and 46). ment, especially the dimensions such as frame spacing, and There are two kinds of strength to design the longitudi- material properties to better fit with the longitudinal strength nal members. One is the local strength to avoid collapse, requirements. This work has to be done in agreement with and the other is the longitudinal strength to consider the the basic design team. collapse of the ships’ hull girder. The local strength is au- Instead of the trial and error procedure discussed above, tomatically satisfied if the design is based on the classifi- an automatic optimization technique can be used to obtain cation rules. The hull girder longitudinal strength can be the minimum weight and/or cost for the longitudinal and assessed with the hull section modulus (SM) at bottom and transverse structural member. The object function(s) can be deck where the extreme stresses are taken place (equation structural weight and/or fabrication cost, using either a sin- 29). The hull section modulus is calculated easily by using gle object function approach or a multiple objective func- available software. tion method. The design variables can be longitudinal and If the hull section modulus at bottom or deck part is big- transverse spacing, deck/bottom scantlings for the longitu- ger than the required value, this design can be considered dinal and transverse members (web height and thickness, as finished but this design might be too expensive. If the flange width and thickness). The constraints and limitations section modulus at the deck or at the bottom is less than the of the optimization process can be the range of each design required value, the designer should change the initial scant- variable as well as the required hull section modulus and lings. minimum deck/bottom scantlings for the longitudinal mem- If the calculated hull section modulus at deck part is less bers, and allowable bending and shear stresses for the trans- than required, he can increase, step by step, the deck scant- verse members (see Optimization in Subsection 18.7.6). ling (for example, 0.5 mm for the plate thickness) until the requirement is satisfied. The designer also has to modify the scantling (usually 18.9.4 A Generic Design Framework plate thickness) of transverse members, for which the stress By comparison with the previous standard procedure, Fig- exceeds the allowable value. The designer estimates the in- ure 18.67 shows a new generic and advanced design method- MASTER SET SDC 18.qxd Page 18-72 4/28/03 1:31 PM

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ology where the performance of the system, the manufac- The system definition module [Y(U,V,W)] is used to turing process of the system and the associated life cycle build an environmental model [U], a product model [V] and costs are considered in an integrated fashion (120). De- a process model [W]. The system definition module receives signing ship structures systems involves achieving simul- operational requirements [Z] such as owner’s requirements. taneous, though sometimes competing, objectives. The These operational parameters are presumed fixed through- structure must perform its function while conforming to out the design. structural, economic and production constraints. The pres- They of course can eventually be changed if no accept- ent design framework consists of establishing the structural able design is established, but presumably any design would system and composite subsystems, which optimally satisfy have operational parameters, which would not be sacrificed. the topology, shape, loading and performance constraints The environmental model [U] includes the still water and while simultaneously considering the manufacturing or fab- wave loading conditions and the product model [V] con- rication processes in a cost effective manner. tains the production information, for example. The process The framework is used within a computerized virtual model [W] is built to consider or define the fabrication se- environment in which CAD product models, physics-based quence. A translator (simulation based design translator) models, production process models and cost models are assigns some [Y] model parameters to the simulation pa- used simultaneously by a designer or design team. The per- rameters [T] and design variables [X]. formance of the product or process is in general judged by These parameters are selected based on the available some time independent parameter, which is referred to as simulation tools [S] that require specific data ([T],[X] and a response metric (R). Specifications for the system must time). be established in terms of these Response Metrics. The for- The simulation module [S(T, X, time)] is used to pro- mulation of the design problem is thus the same whether duce simulation responses such as Response Metrics [R[S(T, the product or process systems (or both) are considered. X)]]. The time is needed to consider the dynamic effects and The general framework consists of a system definition actual dynamic load conditions [U]. module, a simulation module and a design module. The optimum design module includes the Design Cri- teria, 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 Operational Requirements are used to assess the design through the Design Assess- Parameters Z ment component of the module (for example R≤G). The constraints are obtained by considering not only the simu- System Definition Model Parameters Y lation parameters [T] and the design variables [X] but also Environmental Model Product Model Process Model the operational requirements [Z] and the system definition Parameters U Parameters V Parameters W 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 de- Simulation Based Design Translator Simulation Parameters T sign variables [X] and simulation parameters [T]. Design Variables X Based on the results of the Design Assessment (Min(F) and R≤G) several strategies for the design procedure (iter- Simulations ations) can be followed: Simulation Response S(T ,X ,time) Design Criteria Constraints G(T,X,Y,Z) Response Metrics R [S(T ,X )] ¥ if the object function does not reach its minimum value Objective Function F(R,T,X,Y,Z) or the response metrics do not satisfy the constraints, an Yes optimization algorithm (steepest descent, dual approach No Is Design Space Design Assessment Feasible? Min (F) ? and convex linearization, evolutionary strategies, etc.) is R < G ? adopted to find a new set of design variables. Standard algorithms are presented in (113,114,123): Optimization Steepest Descent No Conditions Satisfied ? — if the optimizer fails to find an improved solution (un- Convex Linearization Yes feasible design space), it is required to change the simulation parameter values [T] and/or design vari- Stop Yes Redesign? No ables selection [X] or even to modify the Model Pa- Figure 18.67 A Generic Design Framework (120) rameters [Y]. MASTER SET SDC 18.qxd Page 18-73 4/28/03 1:31 PM

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— otherwise, the design space is feasible, and a change Committee II.2),” Proceedings of 13th ISSC, Moan & Berge of design variable values [X] is performed based on (Eds.), Pergamon, Norway, 1, 1997 the optimizer solution (in other words a new itera- 19. Temarel, P., et al. “Dynamic Response (Report of ISSC Com- tion). mittee II.2),” Proceedings of 14th ISSC, Ohtsubo & Sumi (Eds.), Elsevier, Japan, 1, 2000 ¥ if the object function reaches its minimum value and the 20. “Vibration Control in Ships,” A/S. VERITEC H¿vik, Nor- response metrics satisfy the constraints, two alternatives way, 1985 are examined: 21. Kaminski, M.L., et al. “Ultimate Strength (Report of ISSC Committee III.1),” Proceedings of 14th ISSC, Ohtsubo & — change the operational requirements parameters [Z], Sumi (Eds.), Elsevier, Japan, 1, 2000 repeat the previous procedure and to compare with 22. Pedersen, P. T., “Ship Grounding and Hull Girder Strength” other alternative designs, or Marine Structures, 7, 1994 — end the design procedure. 23. Beck R. F. and Reed A. M., “Modern Seakeeping Computa- tions for Ships” Proc. 23rd Symposium Naval Hydrodynam- ics Val de Reuil, France, 2000 24. Jensen, J. J. et al., “Extreme Hull Girder Loading,” Report 18.10 REFERENCES of Special Task Committee VI.1 Proc. 14th International Ship and Offshore Structures Congress, Ohtsubo and Sumi (Edi- 1. Taggart R., Ship Design and Construction,SNAME, New tors), 2: 261Ð320, 2000 York, 1980 25. Rawson, K. J., Tupper E. C., Basic Ship Theory (Fourth edi- 2. Lewis, E. V., Principles of Naval Architecture (2nd revision), tion), 1 & 2, Longman Scientic & Technical, Essex, UK, vol.1, SNAME, 1988 1994 3. Hughes O. F., Ship Structural Design: A Rationally -Based, 26. Schade, H. A., “The Effective Breath of Stiffened Plating Computer-Aided Optimization Approach,SNAME, New Jer- Under Bending Loads,” Transactions SNAME, 61, 1951 sey, 1988 27. Evans, H. J., Ship Structural Design Concepts—Second Cycle, 4. DnV 99Ð0394, Calculation Procedures for Direct Global Cornell Maritime Press, First Edition, Maryland, 1983 Structural Analysis, Det Norske Veritas, Technical Report, 28. Heggelund, S. E., Moan, T. and Omar, S., “Global Structural 1999 Analysis of Large Catamarans,” Proceedings Fifth Confer- 5. Arai H., “Evolution of Classification Rules for Ships,” In Re- ence on Fast Sea Transportation, FAST’99, SNAME, Seat- cent Advances in Marine Structures, ISSC’2000 Pre-Con- tle: 757Ð771, 1999 gress Symposium, Society of Naval Architects of Japan, 29. Rigo, P., “Stiffened Sheathings of Orthotropic Cylindrical Tokyo: 8.1Ð8.22, 2000 Shells,” Journal of Structural Engineering, ASCE, 118 (4): 6. IACS Unified Requirement S7 “Minimum Longitudinal 926Ð943, 1992 Strength Standards,” 1989 30. Rigo, P. and Fleury, C., “Scantling Optimization Based on 7. IACS Unified Requirement S11 “Longitudinal Strength Stan- Convex Linearizations and a Dual Approach,” Marine Struc- dard,” 1993 8. ABS Rules for Building and Classing Steel Vessels, 2000 tures, Elsevier Science Ltd., 14 (6): 631Ð649, 2001 9. BV Rules for Steel Ships, 2001 31. Mansour,A. E., “Gross Panel Strength under Combined Load- 10. RINA Rules, 2001 ing,” Ship Structure Committee, SSC-270, NTIS, Washing- 11. DNV Rules for Classification of Ships, 2001 ton DC, 1977 12. NKK Rules and Guidance for the Survey and Construction 32. Hughes, O., Nikolaidis, E., Ayyub, B., White, G. and Hess, of Steel Ships, 2001 P.,“Uncertainty in Strength Models for Marine Structures,” 13. Salvensen, N., Tuck, E. O. & Faltinsen, O., “Ship Motions Ship Structure Committee (375), NTIS, Washington DC, and Sea Loads”, Transactions SNAME, 78: 250Ð287, 1970 1994 14. Ochi, M.K., “Applied Probability & Stochastic Processes,” 33. Paik, J. K., Thayamballi, A. and Kim, B., “Advanced Ulti- John Wiley & Sons, 1990 mate Strength Formulations for Ship Plating under Com- 15. GWS, “Global Wave Statistics” British Maritime Technol- bined Biaxial Compression/Tension, Edge Shear and Lateral ogy Ltd. Feltham, 1986 Pressure Loads,” Marine Technology, 38, (1): 9Ð25, 2001 16. Guedes Soares, C., et al. “Loads (Report of ISSC Commit- 34. Faulkner, D., “A Review of Effective Plating for use in the tee I.2),” Proceedings of 13th ISSC, Moan & Berge (Eds.), Analysis of Stiffened Plating in Bending and Compression,” Pergamon, Norway, 1, 1997 Journal of Ship Research, 18 (1): 1Ð17, 1975 17. Guedes Soares, C., et al. “Loads (Report of ISSC Commit- 35. Faulkner, D.,Adamchak, J., Snyder, G. and Vetter, M., “Syn- tee I.2),” Proceedings of 14th ISSC, Ohtsubo & Sumi (Eds.), thesis of Welded Grillages to withstand Compression and Elsevier, Japan, 1, 2000 ” Normal Loads,” Computers & Structures, Vol.3, 1973, 18. Chung, T. Y., et al. “Dynamic Response (Report of ISSC pp.221Ð246. MASTER SET SDC 18.qxd Page 18-74 4/28/03 1:31 PM

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