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- 1. 18.1 NOMENCLATURE For speciﬁc symbols, refer to the deﬁnitions contained in the various sections. ABS American Bureau of Shipping BEM Boundary Element Method BV Bureau Veritas DNV Det Norske Veritas FEA Finite Element Analysis FEM Finite Element Method IACS International Association of Classiﬁca- tion Societies ISSC International Ship & Offshore Structures Congress ISOPE International Offshore and Polar Engi- neering Conference ISUM Idealized Structural Unit method NKK Nippon Kaiji Kyokai PRADS Practical Design of Ships and Mobile Units, RINA Registro Italiano Navale SNAME Society of naval Architects and marine Engineers SSC Ship Structure Committee. a acceleration A area B breadth of the ship C wave coefﬁcient (Table 18.I) CB hull block coefﬁcient D depth of the ship g gravity acceleration m(x) longitudinal distribution of mass I(x) geometric moment of inertia (beam sec- tion x) L length of the ship M(x) bending moment at section x of a beam MT(x) torque moment at section x of a beam p pressure q(x) resultant of sectional force acting on a beam T draft of the ship V(x) shear at section x of a beam s,w (low case) still water, wave induced component v,h (low case) vertical, horizontal component w(x) longitudinal distribution of weight θ roll angle ρ density ω angular frequency 18.2 INTRODUCTION The purpose of this chapter is to present the fundamentals of direct ship structure analysis based on mechanics and strength of materials. Such analysis allows a rationally based design that is practical, efﬁcient, and versatile, and that has already been implemented in a computer program, tested, and proven. Analysis and Design are two words that are very often associated. Sometimes they are used indifferently one for the other even if there are some important differences be- tween performing a design and completing an analysis. 18-1 Chapter 18 Analysis and Design of Ship Structure Philippe Rigo and Enrico Rizzuto MASTER SET SDC 18.qxd Page 18-1 4/28/03 1:30 PM
- 2. Analysis refers to stress and strength assessment of the structure. Analysis requires information on loads and needs an initial structural scantling design. Output of the structural analysis is the structural response deﬁned in terms of stresses, deﬂections and strength. Then, the estimated response is compared to the design criteria. Results of this comparison as well as the objective functions (weight, cost, etc.) will show if updated (improved) scantlings are required. Design for structure refers to the process followed to se- lect the initial structural scantlings and to update these scant- lings from the early design stage (bidding) to the detailed design stage (construction). To perform analysis, initial de- sign is needed and analysis is required to design. This ex- plains why design and analysis are intimately linked, but are absolutely different. Of course design also relates to topology and layout deﬁnition. The organization and framework of this chapter are based on the previous edition of the Ship Design and Construction (1) and on the Chapter IV of Principles of Naval Architec- ture (2). Standard materials such as beam model, twisting, shear lag, etc. that are still valid in 2002 are partly duplicated from these 2 books. Other major references used to write this chapter are Ship Structural Design (3) also published by SNAME and the DNV 99-0394 Technical Report (4). The present chapter is intimately linked with Chapter 11 – Parametric Design, Chapter 17 – Structural Arrange- ment and Component Design and with Chapter 19 – Reli- ability-Based Structural Design. References to these chapters will be made in order to avoid duplications. In ad- dition, as Chapter 8 deals with classiﬁcation societies, the present chapter will focus mainly on the direct analysis methods available to perform a rationally based structural design, even if mention is made to standard formulations from Rules to quantify design loads. In the following sections of this chapter, steps of a global analysis are presented. Section 18.3 concerns the loads that are necessary to perform a structure analysis. Then, Sections 18.4, 18.5 and 18.6 concern, respectively, the stresses and deﬂections(basicshipresponses),thelimitstates,andthefail- ures modes and associated structural capacity. A review of theavailableNumericalAnalysisforStructuralDesign isper- formed in Section 18.7. Finally Design Criteria (Section 18.8) and Design Procedures (Section 18.9) are discussed. Structural modeling is discussed in Subsection 18.2.2 and moreextensivelyinSubsection18.7.2forﬁniteelementanaly- sis. Optimization is treated in Subsections 18.7.6 and 18.9.4. Ship structural design is a challenging activity. Hence Hughes (3) states: The complexities of modern ships and the demand for greater reliability, efﬁciency, and economy require a sci- entiﬁc, powerful, and versatile method for their structural design But, even with the development of numerical techniques, design still remains based on the designer’s experience and on previous designs. There are many designs that satisfy the strength criteria, but there is only one that is the optimum solution (least cost, weight, etc.). Ship structural analysis and design is a matter of com- promises: • compromise between accuracy and the available time to perform the design. This is particularly challenging at the preliminary design stage. A 3D Finite Element Method (FEM) analysis would be welcome but the time is not available. For that reason, rule-based design or simpliﬁed numerical analysis has to be performed. • to limit uncertainty and reduce conservatism in design, it is important that the design methods are accurate. On the other hand, simplicity is necessary to make repeated de- sign analyses efﬁcient. The results from complex analy- sesshouldbeveriﬁedbysimpliﬁedmethodstoavoiderrors and misinterpretation of results (checks and balances). • compromise between weight and cost or compromise between least construction cost, and global owner live cycle cost (including operational cost, maintenance, etc.), and • builder optimum design may be different from the owner optimum design. 18.2.1 Rationally Based Structural Design versus Rules-Based Design There are basically two schools to perform analysis and de- sign of ship structure. The ﬁrst one, the oldest, is called rule-based design. It is mainly based on the rules deﬁned by the classiﬁcation societies. Hughes (3) states: In the past, ship structural design has been largely empir- ical, based on accumulated experience and ship perform- ance, and expressed in the form of structural design codes or rules published by the various ship classiﬁcation soci- eties. These rules concern the loads, the strength and the design criteria and provide simpliﬁed and easy-to-use for- mulas for the structural dimensions, or “scantlings” of a ship. This approach saves time in the design ofﬁce and, since the ship must obtain the approval of a classiﬁcation society, it also saves time in the approval process. The second school is the Rationally Based Structural Design; it is based on direct analysis. Hughes, who could be considered as a father of this methodology, (3) further states: 18-2 Ship Design & Construction, Volume 1 MASTER SET SDC 18.qxd Page 18-2 4/28/03 1:30 PM
- 3. There are several disadvantages to a completely “rulebook” approach to design. First, the modes of structural failure are numerous, complex, and interdependent. With such simpliﬁed formulas the margin against failure remains un- known; thus one cannot distinguish between structural ad- equacy and over-adequacy. Second, and most important, these formulas involve a number of simplifying assump- tions and can be used only within certain limits. Outside of this range they may be inaccurate. For these reasons there is a general trend toward direct structural analysis. Even if direct calculation has always been performed, design based on direct analysis only became popular when numerical analysis methods became available and were cer- tiﬁed. Direct analysis has become the standard procedure in aerospace, civil engineering and partly in offshore in- dustries. In ship design, classiﬁcation societies preferred to offer updated rules resulting from numerical analysis cali- bration. For the designer, even if the rules were continuously changing, the design remained rule-based.There really were two different methodologies. Hopefully, in 2002 this is no longer true. The advantages of direct analysis are so obvious that classiﬁcation societies include, usually as an alternative, a direct analysis procedure (numerical packages based on the ﬁnite element method, seeTable 18.VIII, Subsection 18.7.5.2). In addition, for new vessel types or non-standard dimension, such direct proce- dure is the only way to assess the structural safety. There- fore it seems that the two schools have started a long merging procedure. Classiﬁcation societies are now encouraging and contributing greatly to the development of direct analysis and rationally based methods. Ships are very complex struc- tures compared with other types of structures. They are sub- ject to a very wide range of loads in the harsh environment of the sea. Progress in technologies related to ship design and construction is being made daily, at an unprecedented pace. A notable example is the fact that the efforts of a ma- jority of specialists together with rapid advances in com- puter and software technology have now made it possible to analyze complex ship structures in a practical manner using structural analysis techniques centering on FEM analysis. The majority of ship designers strive to develop rational and optimal designs based on direct strength analysis methods using the latest technologies in order to realize the shipowner’s requirements in the best possible way. When carrying out direct strength analysis in order to verify the equivalence of structural strength with rule re- quirements, it is necessary for the classiﬁcation society to clarify the strength that a hull structure should have with respect to each of the various steps taken in the analysis process, from load estimation through to strength evalua- tion. In addition, in order to make this a practical and ef- fective method of analysis, it is necessary to give careful consideration to more rational and accurate methods of di- rect strength analysis. Based on recognition of this need, extensive research has been conducted and a careful examination made, re- garding the strength evaluation of hull structures. The re- sults of this work have been presented in papers and reports regarding direct strength evaluation of hull structures (4,5). The ﬂow chart given in Figure 18.1 gives an overview of the analysis as deﬁned by a major classiﬁcation society. Note that a rationally based design procedure requires that all design decisions (objectives, criteria, priorities, con- straints…) must be made before the design starts. This is a major difﬁculty of this approach. 18.2.2 Modeling and Analysis General guidance on the modeling necessary for the struc- tural analysis is that the structural model shall provide re- sults suitable for performing buckling, yield, fatigue and Chapter 18: Analysis and Design of Ship Structure 18-3 Figure 18.1 Direct Structural Analysis Flow Chart Direct Load Analysis Design Load Study on Ocean Waves Effect on operationWave Load Response Response function of wave load Structural analysis by whole ship model Stress response function Short term estimation Long term estimation Design Sea State Design wave Wave impact load Structural response analysis Strength Assessment Yield strength Nonlinear influence in large waves Investigation on corrosion Buckling strength Ultimate strength Fatigue strength Modeling technique Direct structural analysis Stress Response in Waves Long term estimation Short term estimation MASTER SET SDC 18.qxd Page 18-3 4/28/03 1:30 PM
- 4. vibration assessment of the relevant parts of the vessel. This is done by using a 3D model of the whole ship, supported by one or more levels of sub models. Several approaches may be applied such as a detailed 3D model of the entire ship or coarse meshed 3D model sup- ported by ﬁner meshed sub models. Coarse mesh can be used for determining stress results suited for yielding and buckling control but also to obtain the displacements to apply as boundary conditions for sub models with the purpose of determining the stress level in more detail. Strength analysis covers yield (allowable stress), buck- ling strength and ultimate strength checks of the ship. In ad- dition, speciﬁc analyses are requested for fatigue (Subsection 18.6.6), collision and grounding (Subsection 18.6.7) and vibration (Subsection 18.6.8). The hydrodynamic load model must give a good representation of the wetted sur- face of the ship, both with respect to geometry description and with respect to hydrodynamic requirements. The mass model, which is part of the hydrodynamic load model, must ensure a proper description of local and global moments of inertia around the global ship axes. Ultimate hydrodynamic loads from the hydrodynamic analysis should be combined with static loads in order to form the basis for the yield, buckling and ultimate strength checks.All the relevant load conditions should be examined to ensure that all dimensioning loads are correctly included. A ﬂow chart of strength analysis of global model and sub models is shown in Figure 18.2. 18.2.3 Preliminary Design versus Detailed Design For a ship structure, structural design consists of two dis- tinct levels: the Preliminary Design and the Detailed De- sign about which Hughes (3) states: The preliminary determines the location, spacing, and scant- lings of the principal structural members. The detailed de- signdeterminesthegeometryandscantlingsoflocalstructure (brackets, connections, cutouts, reinforcements, etc.). Preliminary design has the greatest inﬂuence on the structure design and hence is the phase that offers very large potential savings. This does not mean that detail de- sign is less important than preliminary design. Each level is equally important for obtaining an efﬁcient, safe and re- liable ship. During the detailed design there also are many bene- ﬁts to be gained by applying modern methods of engi- neering science, but the applications are different from preliminary design and the beneﬁts are likewise different. Since the items being designed are much smaller it is possible to perform full-scale testing, and since they are more repetitive it is possible to obtain the beneﬁts of mass production, standardization and so on. In fact, production aspects are of primary importance in detail design. Also, most of the structural items that come under de- tail design are similar from ship to ship, and so in-service experience provides a sound basis for their design. In fact, because of the large number of such items it would be in- efﬁcient to attempt to design all of them from ﬁrst princi- ples. Instead it is generally more efﬁcient to use design codes and standard designs that have been proven by ex- perience. In other words, detail design is an area where a rule-based approach is very appropriate, and the rules that are published by the various ship classiﬁcation societies contain a great deal of useful information on the design of local structure, structural connections, and other structural details. 18.3 LOADS Loads acting on a ship structure are quite varied and pecu- liar, in comparison to those of static structures and also of other vehicles. In the following an attempt will be made to review the main typologies of loads: physical origins, gen- eral interpretation schemes, available quantiﬁcation proce- 18-4 Ship Design & Construction, Volume 1 Figure 18.2 Strength Analysis Flow Chart (4) Structural model including necessary load definitions Hydrodynamic/static loads Load transfer to structural model Verified structural model Sub-models to be used in structural analysis Structural analysis Verification of response Verification of model/ loads Yes No Transfer of displacements/forces to sub-model? Verification of load transfer Structural drawings, mass description and loading conditions. MASTER SET SDC 18.qxd Page 18-4 4/28/03 1:30 PM
- 5. dures and practical methods for their evaluation will be sum- marized. 18.3.1 Classiﬁcation of Loads 18.3.1.1 Time Duration Static loads: These are the loads experienced by the ship in still water. They act with time duration well above the range of sea wave periods. Being related to a speciﬁc load con- dition, they have little and very slow variations during a voyage (mainly due to changes in the distribution of con- sumables on board) and they vary signiﬁcantly only during loading and unloading operations. Quasi-static loads: A second class of loads includes those with a period corresponding to wave actions (∼3 to 15 seconds). Falling in this category are loads directly in- duced by waves, but also those generated in the same fre- quency range by motions of the ship (inertial forces). These loads can be termed quasi-static because the structural re- sponse is studied with static models. Dynamic loads: When studying responses with fre- quency components close to the ﬁrst structural resonance modes, the dynamic properties of the structure have to be considered. This applies to a few types of periodic loads, generated by wave actions in particular situations (spring- ing) or by mechanical excitation (main engine, propeller). Also transient impulsive loads that excite free structural vi- brations (slamming, and in some cases sloshing loads) can be classiﬁed in the same category. High frequency loads: Loads at frequencies higher than the ﬁrst resonance modes (> 10-20 Hz) also are present on ships: this kind of excitation, however, involves more the study of noise propagation on board than structural design. Other loads:All other loads that do not fall in the above mentioned categories and need speciﬁc models can be gen- erally grouped in this class. Among them are thermal and accidental loads. A large part of ship design is performed on the basis of static and quasi-static loads, whose prediction procedures are quite well established, having been investigated for a long time. However, speciﬁc and imposing requirements can arise for particular ships due to the other load cate- gories. 18.3.1.2 Local and global loads Another traditional classiﬁcation of loads is based on the structural scheme adopted to study the response. Loads acting on the ship as a whole, considered as a beam (hull girder), are named global or primary loads and the ship structural response is accordingly termed global or primary response (see Subsection 18.4.3). Loads, deﬁned in order to be applied to limited struc- tural models (stiffened panels, single beams, plate panels), generally are termed local loads. The distinction is purely formal, as the same external forces can in fact be interpreted as global or local loads. For instance, wave dynamic actions on a portion of the hull, if described in terms of a bi-dimensional distribution of pres- sures over the wet surface, represent a local load for the hull panel, while, if integrated over the same surface, represent a contribution to the bending moment acting on the hull girder. This terminology is typical of simpliﬁed structural analy- ses, in which responses of the two classes of components are evaluated separately and later summed up to provide the total stress in selected positions of the structure. In a complete 3D model of the whole ship, forces on the structure are applied directly in their actual position and the result is a total stress distribution, which does not need to be decomposed. 18.3.1.3 Characteristic values for loads Structural veriﬁcations are always based on a limit state equation and on a design operational time. Main aspects of reliability-based structural design and analysis are (see Chapter 19): • the state of the structure is identiﬁed by state variables associated to loads and structural capacity, • state variables are stochastically distributed as a func- tion of time, and • the probability of exceeding the limit state surface in the design time (probability of crisis) is the element subject to evaluation. The situation to be considered is in principle the worst combination of state variables that occurs within the design time. The probability that such situation corresponds to an out crossing of the limit state surface is compared to a (low) target probability to assess the safety of the structure. This general time-variant problem is simpliﬁed into a time-invariant one. This is done by taking into account in the analysis the worst situations as regards loads, and, sep- arately, as regards capacity (reduced because of corrosion and other degradation effects). The simpliﬁcation lies in considering these two situations as contemporary, which in general is not the case. When dealing with strength analysis, the worst load sit- uation corresponds to the highest load cycle and is charac- terized through the probability associated to the extreme value in the reference (design) time. In fatigue phenomena, in principle all stress cycles con- tribute (to a different extent, depending on the range) to Chapter 18: Analysis and Design of Ship Structure 18-5 MASTER SET SDC 18.qxd Page 18-5 4/28/03 1:30 PM
- 6. damage accumulation. The analysis, therefore, does not re- gard the magnitude of a single extreme load application, but the number of cycles and the shape of the probability dis- tribution of all stress ranges in the design time. A further step towards the problem simpliﬁcation is rep- resented by the adoption of characteristic load values in place of statistical distributions. This usually is done, for example, when calibrating a Partial Safety Factor format for structural checks. Such adoption implies the deﬁnition of a single reference load value as representative of a whole probability distribution. This step is often performed by as- signing an exceeding probability (or a return period) to each variable and selecting the correspondent value from the sta- tistical distribution. The exceeding probability for a stochastic variable has the meaning of probability for the variable to overcome a given value, while the return period indicates the mean time to the ﬁrst occurrence. Characteristic values for ultimate state analysis are typ- ically represented by loads associated to an exceeding prob- ability of 10–8 . This corresponds to a wave load occurring, on the average, once every 108 cycles, that is, with a return period of the same order of the ship lifetime. In ﬁrst yield- ing analyses, characteristic loads are associated to a higher exceeding probability, usually in the range 10–4 to 10–6 . In fatigue analyses (see Subsection 18.6.6.2), reference loads are often set with an exceeding probability in the range 10–3 to 10–5 , corresponding to load cycles which, by effect of both amplitude and frequency of occurrence, contribute more to the accumulation of fatigue damage in the structure. On the basis of this, all design loads for structural analy- ses are explicitly or implicitly related to a low exceeding probability. 18.3.2 Deﬁnition of Global Hull Girder Loads The global structural response of the ship is studied with reference to a beam scheme (hull girder), that is, a mono- dimensional structural element with sectional characteris- tics distributed along a longitudinal axis. Actions on the beam are described, as usual with this scheme, only in terms of forces and moments acting in the transverse sections and applied on the longitudinal axis. Three components act on each section (Figure 18.3): a resultant force along the vertical axis of the section (con- tained in the plane of symmetry), indicated as vertical re- sultant force qV; another force in the normal direction, (local horizontal axis), termed horizontal resultant force qH and a moment mT about the x axis. All these actions are distrib- uted along the longitudinal axis x. Five main load components are accordingly generated along the beam, related to sectional forces and moment through equation 1 to 5: [1] [2] [3] [4] [5] Due to total equilibrium, for a beam in free-free condi- tions (no constraints at ends) all load characteristics have zero values at ends (equations 6). These conditions impose constraints on the distributions of qV, qH and mT. [6] Global loads for the veriﬁcation of the hull girder are ob- tained with a linear superimposition of still water and wave- induced global loads. They are used, with different characteristic values, in different types of analyses, such as ultimate state, ﬁrst yield- ing, and fatigue. 18.3.3 Still Water Global Loads Still water loads act on the ship ﬂoating in calm water, usu- ally with the plane of symmetry normal to the still water surface. In this condition, only a symmetric distribution of hydrostatic pressure acts on each section, together with ver- tical gravitational forces. If the latter ones are not symmetric, a sectional torque mTg(x) is generated (Figure 18.4), in addition to the verti- V (0) V (L) M (0) M (L) 0 V (0) V (L) M (0) M (L) 0 M (0) M (L) 0 V V V V H H H H T T = = = = = = = = = = M (x) m ( ) dT T 0 x = ∫ ξ ξ M (x) V ( ) dH H 0 x = ∫ ξ ξ V (x) q ) dH H 0 x = ∫ (ξ ξ M (x) V ( ) dV V 0 x = ∫ ξ ξ V (x) q ( ) dV V 0 x = ∫ ξ ξ 18-6 Ship Design & Construction, Volume 1 Figure 18.3 Sectional Forces and Moment MASTER SET SDC 18.qxd Page 18-6 4/28/03 1:30 PM
- 7. cal load qSV(x), obtained as a difference between buoyancy b(x) and weight w(x), as shown in equation 7 (2). [7] where AI = transversal immersed area. Components of vertical shear and vertical bending can be derived according to equations 1 and 2. There are no hor- izontal components of sectional forces in equation 3 and ac- cordingly no components of horizontal shear and bending moment. As regards equation 5, only mTg, if present, is to be accounted for, to obtain the torque. 18.3.3.1 Standard still water bending moments While buoyancy distribution is known from an early stage of the ship design, weight distribution is completely deﬁned only at the end of construction. Statistical formulations, cal- ibrated on similar ships, are often used in the design de- velopment to provide an approximate quantiﬁcation of weight items and their longitudinal distribution on board. The resulting approximated weight distribution, together with the buoyancy distribution, allows computing shear and bending moment. q (x) b(x) w(x) gA (x) m(x)gSV I= − = − At an even earlier stage of design, parametric formula- tions can be used to derive directly reference values for still water hull girder loads. Common reference values for still water bending mo- ment at mid-ship are provided by the major Classiﬁcation Societies (equation 8). [8] where C = wave parameter (Table 18.I). The formulations in equation 8 are sometimes explicitly reported in Rules, but they can anyway be indirectly de- rived from prescriptions contained in (6, 7). The ﬁrst re- quirement (6) regards the minimum longitudinal strength modulus and provides implicitly a value for the total bend- ing moment; the second one (7), regards the wave induced component of bending moment. Longitudinal distributions, depending on the ship type, are provided also. They can slightly differ among Class So- cieties, (Figure 18.5). 18.3.3.2 Direct evaluation of still water global loads Classiﬁcation Societies require in general a direct analysis of these types of load in the main loading conditions of the ship, such as homogenous loading condition at maximum draft, ballast conditions, docking conditions aﬂoat, plus all other conditions that are relevant to the speciﬁc ship (non- homogeneous loading at maximum draft, light load at less than maximum draft, short voyage or harbor condition, bal- last exchange at sea, etc.). The direct evaluation procedure requires, for a given loading condition, a derivation, section by section, of ver- tical resultants of gravitational (weight) and buoyancy forces, applied along the longitudinal axis x of the beam. To obtain the weight distribution w(x), the ship length is subdivided into portions: for each of them, the total weight andcenterofgravityisdeterminedsummingupcontributions from all items present on board between the two bounding sections. The distribution for w(x) is then usually approxi- mated by a linear (trapezoidal) curve obtained by imposing M N m C L B 122.5 15 C (hogging) C L B 45.5 65 C (sagging)s 2 B 2 B ⋅ −( ) + [ ] = ( ) Chapter 18: Analysis and Design of Ship Structure 18-7 Figure 18.4 Sectional Resultant Forces in Still Water Figure 18.5 Examples of Reference Still Water Bending Moment Distribution (10). (a) oil tankers, bulk carriers, ore carriers, and (b) other ship types TABLE 18.I Wave Coefﬁcient Versus Length Ship Length L Wave Coefﬁcient C 90 ≤ L <300 m 10.75 – [(300 – L)/100]3/2 300 ≤ L <350 m 10.75 350 ≤ L 10.75 – [(300 – L)/150]3/2 (a) (b) MASTER SET SDC 18.qxd Page 18-7 4/28/03 1:30 PM
- 8. the correspondence of area and barycenter of the trapezoid respectively to the total weight and center of gravity of the considered ship portion. The procedure is usually applied separately for differ- ent types of weight items, grouping together the weights of the ship in lightweight conditions (always present on board) and those (cargo, ballast, consumables) typical of a load- ing condition (Figure 18.6). 18.3.3.3 Uncertainties in the evaluation A signiﬁcant contribution to uncertainties in the evaluation of still water loads comes from the inputs to the procedure, in particular those related to quantiﬁcation and location on board of weight items. This lack of precision regards the weight distribution for the ship in lightweight condition (hull structure, machin- ery, outﬁtting) but also the distribution of the various com- ponents of the deadweight (cargo, ballast, consumables). Ship types like bulk carriers are more exposed to uncer- tainties on the actual distribution of cargo weight than, for example, container ships, where actual weights of single containers are kept under close control during operation. In addition, model uncertainties arise from neglecting the longitudinal components of the hydrostatic pressure (Fig- ure 18.7), which generate an axial compressive force on the hull girder. As the resultant of such components is generally below the neutral axis of the hull girder, it leads also to an addi- tional hogging moment, which can reach up to 10% of the total bending moment. On the other hand, in some vessels (in particular tankers) such action can be locally counter- balanced by internal axial pressures, causing hull sagging moments. All these compression and bending effects are neglected in the hull beam model, which accounts only for forces and moments acting in the transverse plane. This represents a source of uncertainties. Another approximation is represented by the fact that buoyancy and weight are assumed in a direction normal to the horizontal longitudinal axis, while they are actually ori- ented along the true vertical. Thisimpliesneglectingthestatictrimangleandtoconsider an approximate equilibrium position, which often creates the need for a few iterative corrections to the load curve qsv(x) in order to satisfy boundary conditions at ends (equations 6). 18.3.3.4 Other still water global loads In a vessel with a multihull conﬁguration, in addition to conventional still water loads acting on each hull consid- ered as a single longitudinal beam, also loads in the trans- versal direction can be signiﬁcant, giving rise to shear, bending and torque in a transversal direction (see the sim- pliﬁed scheme of Figure 18.8, where S, B, and Q stand for shear, bending and torque; and L, T apply respectively to longitudinal and transversal beams). 18.3.4 Wave Induced Global Loads The prediction of the behaviour of the ship in waves repre- sents a key point in the quantiﬁcation of both global and local loads acting on the ship. The solution of the seakeep- ing problem yields the loads directly generated by external pressures, but also provides ship motions and accelerations. The latter are directly connected to the quantiﬁcation of in- ertial loads and provide inputs for the evaluation of other types of loads, like slamming and sloshing. 18-8 Ship Design & Construction, Volume 1 Figure 18.6 Weight Distribution Breakdown for Full Load Condition Figure 18.7 Longitudinal Component of Pressure Figure 18.8 Multi-hull Additional Still Water Loads (sketch) MASTER SET SDC 18.qxd Page 18-8 4/28/03 1:30 PM
- 9. In particular, as regards global effects, the action of waves modiﬁes the pressure distribution along the wet hull sur- face; the differential pressure between the situation in waves and in still water generates, on the transverse section, ver- tical and horizontal resultant forces (bWV and bWH) and a moment component mTb. Analogous components come from the sectional result- ants of inertial forces and moments induced on the section by ship’s motions (Figure 18.9). The total vertical and horizontal wave induced forces on the section, as well as the total torsional component, are found summing up the components in the same direction (equations 9). [9] where IR(x) is the rotational inertia of section x. The longitudinal distributions along the hull girder of hor- izontal and vertical components of shear, bending moment and torque can then be derived by integration (equations 1 to 5). Such results are in principle obtained for each instanta- neous wave pressure distribution, depending therefore, on time, on type and direction of sea encountered and on the ship geometrical and operational characteristics. In regular (sinusoidal) waves, vertical bending moments tend to be maximized in head waves with length close to the ship length, while horizontal bending and torque com- ponents are larger for oblique wave systems. 18.3.4.1 Statistical formulae for global wave loads Simpliﬁed, ﬁrst approximation, formulations are available for the main wave load components, developed mainly on the basis of past experience. Vertical wave-induced bending moment: IACS classiﬁ- q (x) b (x) m(x)a (x) q (x) b (x) m(x)a (x) m (x) m (x) I (x) WV WV V WH WH H TW Tb R = − = − = − θ cation societies provide a statistically based reference values for the vertical component of wave-induced bending moment MWV, expressed as a function of main ship dimensions. Such reference values for the midlength section of a ship with unrestricted navigation are yielded by equation 10 for hog and sag cases (7) and corresponds to an extreme value with a return period of about 20 years or an exceeding prob- ability of about 10–8 (once in the ship lifetime). [10] HorizontalWave-induced Bending Moment: Similar for- mulations are available for reference values of horizontal wave induced bending moment, even though they are not as uniform among different Societies as for the main verti- cal component. In Table 18.II, examples are reported of reference val- ues of horizontal bending moment at mid-length for ships with unrestricted navigation. Simpliﬁed curves for the dis- tribution in the longitudinal direction are also provided. Wave-induced Torque: A few reference formulations are given also for reference wave torque at midship (see ex- amples in Table 18.III) and for the inherent longitudinal distributions. 18.3.4.2 Static Wave analysis of global wave loads A traditional analysis adopted in the past for evaluation of wave-induced loads was represented by a quasi-static wave approach. The ship is positioned on a freezed wave of given characteristics in a condition of equilibrium between weight and static buoyancy. The scheme is analogous to the one de- scribed for still water loads, with the difference that the wa- terline upper boundary of the immersed part of the hull is no longer a plane but it is a curved (cylindrical) surface. By deﬁnition, this procedure neglects all types of dynamic ef- fects. Due to its limitations, it is rarely used to quantify wave loads. Sometimes, however, the concept of equivalent static wave is adopted to associate a longitudinal distribution of M N m C L B C C L B C . (hog) (sag)WV B B ⋅[ ] = − +( ) 190 110 0 7 2 2 Chapter 18: Analysis and Design of Ship Structure 18-9 Figure 18.9 Sectional Forces and Moments in Waves TABLE 18.II Reference Horizontal Bending Moments Class Society MWH [N ⋅ m] ABS (8) 180 C1L2 DCB BV (9) RINA (10) 1600 L2.1 TCB DNV (11) 220 L9/4 (T + 0.3B)CB NKK (12) 320 L2C T L L− 35 / MASTER SET SDC 18.qxd Page 18-9 4/28/03 1:30 PM
- 10. pressures to extreme wave loads, derived, for example, from long term predictions based on other methods. 18.3.4.3 Linear methods for wave loads The most popular approach to the evaluation of wave loads is represented by solutions of a linearized potential ﬂow problem based on the so-called strip theory in the frequency domain (13). The theoretical background of this class of procedures is discussed in detail in PNA Vol. III (2). Here only the key assumptions of the method are pre- sented: • inviscid, incompressible and homogeneous ﬂuid in irro- tational ﬂow: Laplace equation 11 ∇2 Φ = 0 [11] where Φ = velocity potential • 2-dimensional solution of the problem • linearized boundary conditions: the quadratic compo- nent of velocity in the Bernoulli Equation is reformu- lated in linear terms to express boundary conditions: — on free surface: considered as a plane corresponding to still water: ﬂuid velocity normal to the free surface equal to velocity of the surface itself (kinematic con- dition); zero pressure, — on the hull: considered as a static surface, corre- sponding to the mean position of the hull: the com- ponent of the ﬂuid velocity normal to the hull surface is zero (impermeability condition), and • linear decomposition into additive independent compo- nents, separately solved for and later summed up (equa- tion 12). Φ = Φs + ΦFK + Φd + Φr [12] where: Φs = stationary component due to ship advancing in calm water Φr = radiation component due to the ship motions in calm water ΦFK = excitation component, due to the incident wave (undisturbed by the presence of the ship): Froude- Krylov Φd = diffraction component, due to disturbance in the wave potential generated by the hull This subdivision also enables the de-coupling of the ex- citation components from the response ones, thus avoiding a non-linear feedback between the two. Other key properties of linear systems that are used in the analysis are: • linear relation between the input and output amplitudes, and • superposition of effects (sum of inputs corresponds to sum of outputs). When using linear methods in the frequency domain, the input wave system is decomposed into sinusoidal com- ponents and a response is found for each of them in terms of amplitude and phase. The input to the procedure is represented by a spectral representation of the sea encountered by the ship. Responses, for a ship in a given condition, depend on the input sea char- acteristics (spectrum and spatial distribution respect to the ship course). The output consists of response spectra of point pres- sures on the hull and of the other derived responses, such as global loads and ship motions. Output spectra can be used to derive short and long-term predictions for the prob- ability distributions of the responses and of their extreme values (see Subsection 18.3.4.5). Despite the numerous and demanding simpliﬁcations at the basis of the procedure, strip theory methods, developed since the early 60s, have been validated over time in sev- eral contexts and are extensively used for predictions of wave loads. In principle, the base assumptions of the method are 18-10 Ship Design & Construction, Volume 1 TABLE 18.III Examples of Reference Values for Wave Torque Class Society Qw [N . m] (at mid-ship) ABS (bulk carrier) (e = vertical position of shear center) BV RINA 190 8 13 250 0 7 125 2 2 3 LB C . . L W − − 2700 0 5 0 1 0 13 0 142 2 0 5 LB T C . . . e D . TW . −( ) +[ ] − MASTER SET SDC 18.qxd Page 18-10 4/28/03 1:30 PM
- 11. valid only for small wave excitations, small motion re- sponses and low speed of the ship. In practice, the ﬁeld of successful applications extends far beyond the limits suggested by the preservation of re- alism in the base assumptions: the method is actually used extensively to study even extreme loads and for fast ves- sels. 18.3.4.4 Limits of linear methods for wave loads Due to the simpliﬁcations adopted on boundary conditions to linearize the problem of ship response in waves, results in terms of hydrodynamic pressures are given always up to the still water level, while in reality the pressure distribu- tion extends over the actual wetted surface. This represents a major problem when dealing with local loads in the side region close to the waterline. Another effect of basic assumptions is that all responses at a given frequency are represented by sinusoidal ﬂuctua- tions (symmetric with respect to a zero mean value).A con- sequence is that all the derived global wave loads also have the same characteristics, while, for example, actual values of vertical bending moment show marked differences be- tween the hogging and sagging conditions. Corrections to account for this effect are often used, based on statistical data (7) or on more advanced non-linear methods. A third implication of linearization regards the super- imposition of static and dynamic loads. Dynamic loads are evaluated separately from the static ones and later summed up: this results in an un-physical situation, in which weight forces (included only in static loads) are considered as act- ing always along the vertical axis of the ship reference sys- tem (as in still water). Actually, in a seaway, weight forces are directed along the true vertical direction, which depends on roll and pitch angles, having therefore also components in the longitudinal and lateral direction of the ship. This aspect represents one of the intrinsic non-lineari- ties in the actual system, as the direction of an external input force (weight) depends on the response of the system itself (roll and pitch angles). This effect is often neglected in the practice, where lin- ear superposition of still water and wave loads is largely fol- lowed. 18.3.4.5 Wave loads probabilistic characterization The most widely adopted method to characterize the loads in the probability domain is the so-called spectral method, used in conjunction with linear frequency-domain methods for the solution of the ship-wave interaction problem. From the frequency domain analysis response spectra Sy(ω) are derived, which can be integrated to obtain spec- tral moments mn of order n (equation 13). [13] This information is the basis of the spectral method, whose theoretical framework (main hypotheses, assump- tions and steps) is recalled in the following. If the stochastic process representing the wave input to the ship system is modeled as a stationary and ergodic Gaussian process with zero mean, the response of the sys- tem (load) can be modeled as a process having the same char- acteristics. The Parseval theorem and the ergodicity property es- tablish a correspondence between the area of the response spectrum (spectral moment of order 0: m0Y) and the vari- ance of its Gaussian probability distribution (14). This al- lows expressing the density probability distribution of the Gaussian response y in terms of m0Y (equation 14). [14] Equation 14 expresses the distribution of the ﬂuctuating response y at a generic time instant. From a structural point of view, more interesting data are represented by: • the probability distribution of the response at selected time instants, corresponding to the highest values in each zero-crossing period (peaks: variable p), • the probability distribution of the excursions between the highest and the lowest value in each zero-crossing period (range: variable r), and • the probability distribution of the highest value in the whole stationary period of the phenomenon (extreme value in period Ts, variable extrTs y). The aforementioned distributions can be derived from the underlying Gaussian distribution of the response (equa- tion 14) in the additional hypotheses of narrow band re- sponse process and of independence between peaks.The ﬁrst two probability distributions take the form of equations 15 and 16 respectively, both Rayleigh density distributions (see 14). The distribution in equation 16 is particularly interest- ing for fatigue checks, as it can be adopted to describe stress ranges of fatigue cycles. [15] [16]f r r m r mR ( ) = − 4 80 2 0 exp f p p m p mP ( ) = − 0 2 02 exp f (y) m eY Y y m Y= −( )1 2 0 22 0 2 π / m S ( )dny n y= ∫ ∞ ω ω ω 0 Chapter 18: Analysis and Design of Ship Structure 18-11 MASTER SET SDC 18.qxd Page 18-11 4/28/03 1:30 PM
- 12. The distribution for the extreme value in the stationary period Ts (short term extreme) can be modeled by a Pois- son distribution (in equation 17: expression of the cumula- tive distribution) or other equivalent distributions derived from the statistics of extremes. [17] Figure 18.10 summarizes the various short-term distri- butions. It is interesting to note that all the mentioned distribu- tions are expressed in terms of spectral moments of the re- sponse, which are available from a frequency domain solution of the ship motions problem. The results mentioned previously are derived for the period Ts in which the input wave system can be consid- ered as stationary (sea state: typically, a period of a few hours). The derived distributions (short-term predictions) are conditioned to the occurrence of a particular sea state, which is identiﬁed by the sea spectrum, its angular distri- bution around the main wave direction (spreading func- tion) and the encounter angle formed with ship advance direction. To obtain a long-term prediction, relative to the ship life (or any other design period Td which can be described as a series of stationary periods), the conditional hypothesis is to be removed from short-term distributions. In other words, the probability of a certain response is to be weighed by the probability of occurrence of the generating sea state (equa- tion18). [18] where: F(y) = probability for the response to be less than value y (unconditioned). F(ySi) = probability for the response to be less than value y, conditioned to occurrence of sea state Si (short term prediction). P(Si) = probability associated to the i-th sea state. n = total number of sea states, covering all combi- nations. Probability P(Si) can be derived from collections of sea data based on visual observations from commercial ships and/or on surveys by buoys. One of the most typical formats is the one contained in (15), where sea states probabilities are organized in bi-di- mensional histograms (scatter diagrams), containing classes F y F y S P(S )i i i n ( ) = ( )⋅ = ∑ 1 F p m m p m TextrT s s ( ) = − − exp exp 1 2 2 2 0 2 0∂ of signiﬁcant wave heights and mean periods. Such scatter diagrams are catalogued according to sea zones, such as shown in Figure 18.11 (the subdivision of the world atlas), and main wave direction. Seasonal characteristics are also available. The process described in equation 18 can be termed de- conditioning (that is removing the conditioning hypothesis). The same procedure can be applied to any of the variables studied in the short term and it does not change the nature of the variable itself. If a range distribution is processed, a long-term distribution for ranges of single oscillations is obtained (useful data for a fatigue analysis). If the distribution of variable extrTs y is de-conditioned, a weighed average of the highest peak in time Ts is achieved. In this case the result is further processed to get the distri- bution of the extreme value in the design time Td. This is done with an additional application of the concept of sta- tistics of extremes. In the hypothesis that the extremes of the various sea states are independent from each other, the extreme on time Td is given by equation 19: [19] where F(extrTd y) is the cumulative probability distribution for the highest response peak in time Td (long-term extreme distribution in time Td). 18.3.4.6 Uncertainties in long-term predictions The theoretical framework of the above presented spectral method, coupled to linear frequency domain methodolo- gies like those summarized in Subsection 18.3.4.3, allows the characterization, in the probability domain, of all the wave induced load variables of interest both for strength and fatigue checks. The results of this linear prediction procedure are af- fected by numerous sources of uncertainties, such as: F y F yextrTd extrTs Td/Ts ( ) = ( )[ ] 18-12 Ship Design & Construction, Volume 1 Figure 18.10 Short-term Distributions MASTER SET SDC 18.qxd Page 18-12 4/28/03 1:30 PM
- 13. • sea description: as above mentioned, scatter diagrams are derived from direct observations on the ﬁeld, which are affected by a certain degree of indetermination. In addition, simpliﬁed sea spectral shapes are adopted, based on a limited number of parameters (generally, bi- parametric formulations based on signiﬁcant wave and mean wave period), • model for the ship’s response: as brieﬂy outlined in Sub- section 18.3.4.3, the model is greatly simpliﬁed, partic- ularly as regards ﬂuid characteristics and boundary conditions. Numerical algorithms and speciﬁc procedures adopted for the solution also inﬂuence results, creating differences even between theoretically equivalent methods, and • the de-conditioning procedure adopted to derive long term predictions from short term ones can add further uncertainties. 18.3.5 Local Loads As previously stated, local loads are applied to individual structural members like panels and beams (stiffeners or pri- mary supporting members). They are once again traditionally divided into static and dynamic loads, referred respectively to the situation in still water and in a seaway. Contrary to strength veriﬁcations of the hull girder, which are nowadays largely based on ultimate limit states (for ex- ample, in longitudinal strength: ultimate bending moment), checks on local structures are still in part implicitly based on more conservative limit states (yield strength). In many Rules, reference (characteristic) local loads, as well as the motions and accelerations on which they are based, are therefore implicitly calibrated at an exceeding probability higher than the 10–8 value adopted in global load strength veriﬁcations. 18.3.6 External Pressure Loads Static and dynamic pressures generated on the wet surface of the hull belong to external loads. They act as local trans- verse loads for the hull plating and supporting structures. 18.3.6.1 Static external pressures Hydrostatic pressure is related through equation 20 to the vertical distance between the free surface and the load point (static head hS). pS = ρghS [20] In the case of the external pressure on the hull, hS cor- responds to the local draft of the load point (reference is made to design waterline). Chapter 18: Analysis and Design of Ship Structure 18-13 Figure 18.11 Map of Sea Zones of the World (15) MASTER SET SDC 18.qxd Page 18-13 4/28/03 1:30 PM
- 14. 18.3.6.2 Dynamic pressures The pressure distribution, as well as the wet portion of the hull, is modiﬁed for a ship in a seaway with respect to the still water (Figure 18.9). Pressures and areas of application are in principle obtained solving the general problem of ship motions in a seaway. Approximate distributions of the wave external pressure, to be added to the hydrostatic one, are adopted in Classiﬁ- cation Rules for the ship in various load cases (Figure 18.12). 18.3.7 Internal Loads—Liquid in Tanks Liquid cargoes generate normal pressures on the walls of the containing tank. Such pressures represent a local trans- versal load for plate, stiffeners and primary supporting mem- bers of the tank walls. 18.3.7.1 Static internal pressure For a ship in still water, gravitation acceleration g gener- ates a hydrostatic pressure, varying again according to equa- tion 20. The static head hS corresponds here to the vertical distance from the load point to the highest part of the tank, increased to account for the vertical extension over that point of air pipes (that can be occasionally ﬁlled with liq- uid) or, if applicable, for the ullage space pressure (the pres- sure present at the free surface, corresponding for example to the setting pressure of outlet valves). 18.3.7.2 Dynamic internal pressure When the ship advances in waves, different types of mo- tions are generated in the liquid contained in a tank on- board, depending on the period of the ship motions and on the ﬁlling level: the internal pressure distribution varies ac- cordingly. In a completely full tank, ﬂuid internal velocities rela- tive to the tank walls are small and the acceleration in the ﬂuid is considered as corresponding to the global ship ac- celeration aw. The total pressure (equation 21) can be evaluated in terms of the total acceleration aT, obtained summing aw to grav- ity g. The gravitational acceleration g is directed according to the true vertical. This means that its components in the ship reference system depend on roll and pitch angles (in Fig- ure 18.13 on roll angle θr). pf = ρaThT [21] In equation 21, hT is the distance between the load point and the highest point of the tank in the direction of the total acceleration vector aT (Figure 18.13) If the tank is only partially ﬁlled, signiﬁcant ﬂuid inter- nal velocities can arise in the longitudinal and/or transver- sal directions, producing additional pressure loads (slosh- ing loads). If pitch or roll frequencies are close to the tank reso- nance frequency in the inherent direction (which can be evaluated on the basis of geometrical parameters and ﬁll- ing ratio), kinetic energy tends to concentrate in the ﬂuid and sloshing phenomena are enhanced. The resulting pressure ﬁeld can be quite complicated and speciﬁc simulations are needed for a detailed quantiﬁ- cation. Experimental techniques as well as 2D and 3D pro- cedures have been developed for the purpose. For more details see references 16 and 17. A further type of excitation is represented by impacts that can occur on horizontal or sub-horizontal plates of the upper part of the tank walls for high ﬁlling ratios and, at low ﬁll- ing levels, in vertical or sub-vertical plates of the lower part of the tank. Impact loads are very difﬁcult to characterize, being re- lated to a number of effects, such as: local shape and ve- locity of the free surface, air trapping in the ﬂuid and response of the structure. A complete model of the phe- nomenon would require a very detailed two-phase scheme for the ﬂuid and a dynamic model for the structure includ- ing hydro-elasticity effects. Simpliﬁed distributions of sloshing and/or impact pres- sures are often provided by Classiﬁcation Societies for struc- tural veriﬁcation (Figure 18.14). 18-14 Ship Design & Construction, Volume 1 Figure 18.13 Internal Fluid Pressure (full tank) Figure 18.12 Example of Simpliﬁed Distribution of External Pressure (10) MASTER SET SDC 18.qxd Page 18-14 4/28/03 1:30 PM
- 15. 18.3.7.3 Dry bulk cargo In the case of a dry bulk cargo, internal friction forces arise within the cargo itself and between the cargo and the walls of the hold. As a result, the component normal to the wall has a different distribution from the load corresponding to a liquid cargo of the same density; also additional tangen- tial components are present. 18.3.8 Inertial Loads—Dry Cargo To account for this effect, distributions for the components of cargo load are approximated with empirical formulations based on the material frictional characteristics, usually ex- pressed by the angle of repose for the bulk cargo, and on the slope of the wall. Such formulations cover both the static and the dynamic cases. 18.3.8.1 Unit cargo In the case of a unit cargo (container, pallet, vehicle or other) the local translational accelerations at the centre of gravity are applied to the mass to obtain a distribution of inertial forces. Such forces are transferred to the structure in dif- ferent ways, depending on the number and extension of con- tact areas and on typology and geometry of the lashing or supporting systems. Generally, this kind of load is modelled by one or more concentrated forces (Figure 18.15) or by a uniform load ap- plied on the contact area with the structure. The latter case applies, for example, to the inertial loads transmitted by tyred vehicles when modelling the response of the deck plate between stiffeners: in this case the load is distributed uniformly on the tyre print. 18.3.9 Dynamic Loads 18.3.9.1 Slamming and bow ﬂare loads When sailing in heavy seas, the ship can experience such large heave motions that the forebody emerges completely from the water. In the following downward fall, the bottom of the ship can hit the water surface, thus generating con- siderable impact pressures. The phenomenon occurs in ﬂat areas of the forward part of the ship and it is strongly correlated to loading condi- tions with a low forward draft. It affects both local structures (bottom panels) and the global bending behaviour of the hull girder with generation also of free vibrations at the ﬁrst vertical ﬂexural modes for the hull (whipping). A full description of the slamming phenomenon involves a number of parameters: amplitude and velocity of ship mo- tions relative to water, local angle formed at impact between the ﬂat part of the hull and the water free surface, presence and extension of air trapped between ﬂuid and ship bottom and structural dynamic behavior (18,19). While slamming probability of occurrence can be stud- ied on the basis only of predictions of ship relative motions (which should in principle include non-linear effects due to extreme motions), a quantiﬁcation of slamming pressure involves necessarily all the other mentioned phenomena and is very difﬁcult to attain, both from a theoretical and experimental point of view (18,19). From a practical point of view, Class Societies prescribe, for ships with loading conditions corresponding to a low fore Chapter 18: Analysis and Design of Ship Structure 18-15 Figure 18.14 Example of Simpliﬁed Distributions of Sloshing and Impact Pressures (11) Figure 18.15 Scheme of Local Forces Transmitted by a Container to the Support System (8) MASTER SET SDC 18.qxd Page 18-15 4/28/03 1:30 PM
- 16. draft, local structural checks based on an additional exter- nal pressure. Such additional pressure is formulated as a function of ship main characteristics, of local geometry of the ship (width of ﬂat bottom, local draft) and, in some cases, of the ﬁrst natural frequency of ﬂexural vibration of the hull girder. The inﬂuence on global loads is accounted for by an ad- ditional term for the vertical wave-induced bending mo- ment, which can produce a signiﬁcant increase (15% and more) in the design value. A phenomenon quite similar to bottom slamming can occur also on the forebody of ships with a large bow ﬂare. In this case dynamic and (to a lesser extent) impulsive pres- sures are generated on the sides of V-shaped fore sections. The phenomenon is likely to occur quite frequently on ships prone to it, but with lower pressures than in bottom slamming. The incremental effect on vertical bending mo- ment can however be signiﬁcant. A quantiﬁcation of bow ﬂare effects implies taking into account the variation of the local breadth of the section as a function of draft. It represents a typical non-linear effect (non-linearity due to hull geometry). Slamming can also occur in the rear part of the ship, when the ﬂat part of the stern counter is close to surface. 18.3.9.2 Springing Another phenomenon which involves the dynamic response of the hull girder is springing. For particular types of ships, a coincidence can occur between the frequency of wave ex- citation and the natural frequency associated to the ﬁrst (two-node) ﬂexural mode in the vertical plane, thus pro- ducing a resonance for that mode (see also Subsection 18.6.8.2). The phenomenon has been observed in particular on Great Lakes vessels, a category of ships long and ﬂexible, with com- paratively low resonance frequencies (1, Chapter VI). The exciting action has an origin similar to the case of quasi-static wave bending moment and can be studied with the same techniques, but the response in terms of deﬂec- tion and stresses is magniﬁed by dynamic effects. For re- cent developments of research in the ﬁeld (see references 16 and 17). 18.3.9.3 Propeller induced pressures and forces Due to the wake generated by the presence of the after part of the hull, the propeller operates in a non-uniform incident velocity ﬁeld. Blade proﬁles experience a varying angle of attack dur- ing the revolution and the pressure ﬁeld generated around the blades ﬂuctuates accordingly. The dynamic pressure ﬁeld impinges the hull plating in the stern region, thus generating an exciting force for the structure. A second effect is due to axial and non axial forces and moments generated by the propeller on the shaft and trans- mitted through the bearings to the hull (bearing forces). Due to the negative dynamic pressure generated by the increased angle of attack, the local pressure on the back of blade proﬁles can, for any rotation angle, fall below the vapor saturation pressure. In this case, a vapor sheet is gen- erated on the back of the proﬁle (cavitation phenomenon). The vapor ﬁlled cavity collapses as soon as the angle of at- tack decreases in the propeller revolution and the local pres- sure rises again over the vapor saturation pressure. Cavitation further enhances pressure ﬂuctuations, be- cause of the rapid displacement of the surrounding water volume during the growing phase of the vapor bubble and because of the following implosion when conditions for its existence are removed. All of the three mentioned types of excitation have their main components at the propeller rotational frequency, at the blade frequency, and at their ﬁrst harmonics. In addi- tion to the above frequencies, the cavitation pressure ﬁeld contains also other components at higher frequency, related to the dynamics of the vapor cavity. Propellers with skewed blades perform better as regards induced pressure, because not all the blade sections pass si- multaneously in the region of the stern counter, where dis- turbances in the wake are larger; accordingly, pressure ﬂuctuations are distributed over a longer time period and peak values are lower. Bearing forces and pressures induced on the stern counter by cavitating and non cavitating propellers can be calculated with dedicated numerical simulations (18). 18.3.9.4 Main engine excitation Another major source of dynamic excitation for the hull girder is represented by the main engine. Depending on general arrangement and on number of cylinders, diesel en- gines generate internally unbalanced forces and moments, mainly at the engine revolution frequency, at the cylinders ﬁring frequency and inherent harmonics (Figure 18.16). The excitation due to the ﬁrst harmonics of low speed diesel engines can be at frequencies close to the ﬁrst natu- ral hull girder frequencies, thus representing a possible cause of a global resonance. In addition to frequency coincidence, also direction and location of the excitation are important factors: for exam- ple, a vertical excitation in a nodal point of a vertical ﬂex- ural mode has much less effect in exciting that mode than the same excitation placed on a point of maximum modal deﬂection. 18-16 Ship Design & Construction, Volume 1 MASTER SET SDC 18.qxd Page 18-16 4/28/03 1:30 PM
- 17. In addition to low frequency hull vibrations, components at higher frequencies from the same sources can give rise to resonance in local structures, which can be predicted by suitable dynamic structural models (18,19). 18.3.10 Other Loads 18.3.10.1 Thermal loads A ship experiences loads as a result of thermal effects, which can be produced by external agents (the sun heating the deck), or internal ones (heat transfer from/to heated or re- frigerated cargo). What actually creates stresses is a non-uniform temper- ature distribution, which implies that the warmer part of the structure tends to expand while the rest opposes to this de- formation.A peculiar aspect of this situation is that the por- tion of the structure in larger elongation is compressed and vice-versa, which is contrary to the normal experience. It is very difﬁcult to quantify thermal loads, the main problems being related to the identiﬁcation of the temper- ature distribution and in particular to the model for con- straints. Usually these loads are considered only in a qualitative way (1, Chapter VI). 18.3.10.2 Mooring loads For a moored vessel, loads are exerted from external actions on the mooring system and from there to the local sup- porting structure. The main contributions come by wind, waves and current. Wind: The force due to wind action is mainly directed in the direction of the wind (drag force), even if a limited com- ponent in the orthogonal direction can arise in particular sit- uations. The magnitude depends on the wind speed and on extension and geometry of the exposed part of the ship. The action due to wind can be described in terms of two force components; a longitudinal one FWiL, and a transverse one FWiT (equation 22), and a moment MWiz about the vertical axis (equation 23), all applied at the center of gravity. [22] [23] where: φWi = the angle formed by the direction of the wind rela- tive to the ship CMz(φWi), CFL(φWi), CFT(φWi) are all coefﬁcients depending on the shape of exposed part of the ship and on angle φWi AWi = the reference area for the surface of the ship exposed to wind, (usually the area of the cross section) VWi = the wind speed The empirical formulas in equations 22 and 23 account also for the tangential force acting on the ship surfaces par- allel to the wind direction. Current: The current exerts on the immersed part of the hull a similar action to the one of wind on the emerged part (drag force). It can be described through coefﬁcients and variables analogous to those of equations 22 and 23. Waves: Linear wave excitation has in principle a sinu- soidal time dependence (whose mean value is by deﬁnition zero). If ship motions in the wave direction are not con- strained (for example, if the anchor chain is not in tension) the ship motion follows the excitation with similar time de- pendence and a small time lag. In this case the action on the mooring system is very small (a few percent of the other actions). If the ship is constrained, signiﬁcant loads arise on the mooring system, whose amplitude can be of the same order of magnitude of the stationary forces due to the other actions. In addition to the linear effects discussed above, non-lin- ear wave actions, with an average value different from zero, are also present, due to potential forces of higher order, for- mation of vortices, and viscous effects. These components can be signiﬁcant on off-shore ﬂoating structures, which often feature also complicated mooring systems: in those cases the dynamic behavior of the mooring system is to be included in the analysis, to solve a speciﬁc motion prob- lem. For common ships, non-linear wave effects are usu- ally neglected. A practical rule-of-thumb for taking into account wave actions for a ship at anchor in non protected waters is to in- crease of 75 to 100% the sum of the other force components. Once the total force on the ship is quantiﬁed, the ten- sion in the mooring system (hawser, rope or chain) can be M C A L VWiz Mz Wi Wi Wi= ( )1 2 2/ φ φ F C A VWiL,T F L,T Wi Wi Wi= ( )1 2 2/ φ φ Chapter 18: Analysis and Design of Ship Structure 18-17 Figure 18.16 Propeller, Shaft and Engine Induced Actions (20) MASTER SET SDC 18.qxd Page 18-17 4/28/03 1:30 PM
- 18. derived by force decomposition, taking into account the angle formed with the external force in the horizontal and/or vertical plane. 18.3.10.3 Launching loads The launch is a unique moment in the life of the ship. For a successful completion of this complex operation, a num- ber of practical, organizational and technical elements are to be kept under control (as general reference see Reference 1, Chapter XVII). Here only the aspect of loads acting on the ship will be discussed, so, among the various types of launch, only those which present peculiarities as regards ship loads will be considered: end launch and side launch. End Launch: In end launch, resultant forces and motions are contained in the longitudinal plane of the ship (Figure 18.17). The vessel is subjected to vertical sectional forces dis- tributed along the hull girder: weight w(x), buoyancy bL(x) and the sectional force transmitted from the ground way to the cradle and from the latter to the ship’s bottom (in the following: sectional cradle force fC(x), with resultant FC). While the weight distribution and its resultant force (weight W) are invariant during launching, the other distri- butions change in shape and resultant: the derivation of launching loads is based on the computation of these two distributions. Such computation, repeated for various positions of the cradle, is based on the global static equilibrium s (equa- tions 24 and 25, in which dynamic effects are neglected: quasi static approach). BT + FC – W = 0 [24] xB BT + xF FC – xW W = 0 [25] where: W, BT, FC = (respectively) weight, buoyancy and cradle force resultants xW, xB, xF = their longitudinal positions In a ﬁrst phase of launching, when the cradle is still in contact for a certain length with the ground way, the buoy- ancy distribution is known and the cradle force resultant and position is derived. In a second phase, beginning when the cradle starts to rotate (pivoting phase: Figure 18.18), the position xF cor- responds steadily to the fore end of the cradle and what is unknown is the magnitude of FC and the actual aft draft of the ship (and consequently, the buoyancy distribution). The total sectional vertical force distribution is found as the sum of the three components (equation 26) and can be integrated according to equations 1 and 2 to derive vertical shear and bending moment. qVL(x) = w(x) – bL(x) – fC(x) [26] This computation is performed for various intermediate positions of the cradle during the launching in order to check all phases. However, the most demanding situation for the hull girder corresponds to the instant when pivoting starts. In that moment the cradle force is concentrated close to the bow, at the fore end of the cradle itself (on the fore pop- pet, if one is ﬁtted) and it is at the maximum value. A considerable sagging moment is present in this situ- ation, whose maximum value is usually lower than the de- sign one, but tends to be located in the fore part of the ship, where bending strength is not as high as at midship. Furthermore, the ship at launching could still have tem- porary openings or incomplete structures (lower strength) in the area of maximum bending moment. Another matter of concern is the concentrated force at the fore end of the cradle, which can reach a signiﬁcant per- centage of the total weight (typically 20–30%). It represents a strong local load and often requires additional temporary internal strengthening structures, to distribute the force on a portion of the structure large enough to sustain it. Side Launch: In side launch, the main motion compo- nents are directed in the transversal plane of the ship (see Figure 18.19, reproduced from reference 1, Chapter XVII). The vertical reaction from ground ways is substituted in a comparatively short time by buoyancy forces when the ship tilts and drops into water. The kinetic energy gained during the tilting and drop- ping phases makes the ship oscillate around her ﬁnal posi- 18-18 Ship Design & Construction, Volume 1 Figure 18.17 End Launch: Sketch Figure 18.18 Forces during Pivoting MASTER SET SDC 18.qxd Page 18-18 4/28/03 1:30 PM
- 19. tion at rest. The amplitude of heave and roll motions and accelerations governs the magnitude of hull girder loads. Contrary to end launch, trajectory and loads cannot be stud- ied as a sequence of quasi-static equilibrium positions, but need to be investigated with a dynamic analysis. The problem is similar to the one regarding ship mo- tions in waves, (Subsection 18.3.4), with the difference that here motions are due to a free oscillation of the system due to an unbalanced initial condition and not to an external ex- citation. Another difference with respect to end launch is that both ground reaction (ﬁrst) and buoyancy forces (later) are always distributed along the whole length of the ship and are not concentrated in a portion of it. 18.3.10.4 Accidental loads Accidental loads (collision and grounding) are discussed in more detail by ISSC (21). Collision: When deﬁning structural loads due to colli- sions, the general approach is to model the dynamics of the accident itself, in order to deﬁne trajectories of the unit(s) involved. In general terms, the dynamics of collision should be formulated in six degrees of freedom, accounting for a num- ber of forces acting during the event: forces induced by pro- peller, rudder, waves, current, collision forces between the units, hydrodynamic pressure due to motions. Normally, theoretical models conﬁne the analysis to components in the horizontal plane (3 degrees of freedom) and to collision forces and motion-induced hydrodynamic pressures. The latter are evaluated with potential methods of the same type as those adopted for the study of the re- sponse of the ship to waves. As regards collision forces, they can be described dif- ferently depending on the characteristics of the struck ob- ject (ship, platform, bridge pylon…) with different combinations of rigid, elastic or an elastic body models. Governing equations for the problem are given by con- servation of momentum and of energy. Within this frame- work, time domain simulations can evaluate the magnitude of contact forces and the energy, which is absorbed by struc- ture deformation: these quantities, together with the response characteristics of the structure (energy absorption capacity), allow an evaluation of the damage penetration (21). Grounding: In grounding, dominant effects are forces and motions in the vertical plane. As regards forces, main components are contact forces, developed at the ﬁrst impact with the ground, then friction, when the bow slides on the ground, and weight. From the point of view of energy, the initial kinetic en- ergy is (a) dissipated in the deformation of the lower part of the bow (b) dissipated in friction of the same area against the ground, (c) spent in deformation work of the ground (if soft: sand, gravel) and (d) converted into gravitational po- tential energy (work done against the weight force, which resists to the vertical raising of the ship barycenter). In addition to soil characteristics, key parameters for the description are: slope and geometry of the ground, initial speed and direction of the ship relative to ground, shape of the bow (with/without bulb). The ﬁnal position (grounded ship) governs the magni- tude of the vertical reaction force and the distribution of shear and sagging moment that are generated in the hull girder. Figure 18.20 gives an idea of the magnitude of grounding loads for different combinations of ground slopes and coefﬁcients of friction for a 150 000 tanker (results of simulations from reference 22). In addition to numerical simulations, full and model scale tests are performed to study grounding events (21). Chapter 18: Analysis and Design of Ship Structure 18-19 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 18-19 4/28/03 1:30 PM
- 20. 18.3.11 Combination of Loads When dealing with the characterization of a set of loads acting simultaneously, the interest lies in the deﬁnition of a total loading condition with the required exceeding prob- ability (usually the same of the single components). This cannot be obtained by simple superposition of the charac- teristic values of single contributing loads, as the probabil- ity that all design loads occur at the same time is much lower than the one associated to the single component. In the time domain, the combination problem is ex- pressed in terms of time shift between the instants in which characteristic values occur. In the probability domain, the complete formulation of the problem would imply, in principle, the deﬁnition of a joint probability distribution of the various loads, in order to quantify the distribution for the total load. An approxi- mation would consist in modeling the joint distribution through its ﬁrst and second order moments, that is mean val- ues and covariance matrix (composed by the variances of the single variables and by the covariance calculated for each couple of variables). However, also this level of sta- tistical characterization is difﬁcult to obtain. As a practical solution to the problem, empirically based load cases are deﬁned in Rules by means of combination coefﬁcients (with values generally ≤ 1) applied to single loads. Such load cases, each deﬁned by a set of coefﬁcients, represent realistic and, in principle, equally probable com- binations of characteristic values of elementary loads. Structural checks are performed for all load cases. The result of the veriﬁcation is governed by the one, which turns out to be the most conservative for the speciﬁc structure. This procedure needs a higher number of checks (which, on the other hand, can be easily automated today), but allows considering various load situations (deﬁned with different combinations of the same base loads), without choosing a priori the worst one. 18.3.12 New Trends and Load Non-linearities A large part of research efforts is still devoted to a better deﬁnition of wave loads. New procedures have been pro- posed in the last decades to improve traditional 2D linear methods, overcoming some of the simpliﬁcations adopted to treat the problem of ship motions in waves. For a com- plete state of the art of computational methods in the ﬁeld, reference is made to (23). A very coarse classiﬁcation of the main features of the procedures reported in literature is here presented (see also reference 24). 18.3.12.1 2D versus 3D models Three-dimensional extensions of linear methods are avail- able; some non-linear methods have also 3-D features, while in other cases an intermediate approach is followed, with boundary conditions formulated part in 2D, part in 3D. 18.3.12.2 Body boundary conditions In linear methods, body boundary conditions are set with reference to the mean position of the hull (in still water). Perturbation terms take into account, in the frequency or in the time domain, ﬁrst order variations of hydrodynamic and hydrostatic coefﬁcients around the still water line. Other non-linear methods account for perturbation terms of a higher order. In this case, body boundary conditions are still linear (mean position of the hull), but second order variations of the coefﬁcients are accounted for. Mixed or blending procedures consist in linear methods modiﬁed to include non-linear effects in a single compo- nent of the velocity potential (while the other ones are treated linearly). In particular, they account for the actual geome- try of wetted hull (non-linear body boundary condition) in the Froude-Krylov potential only. This effect is believed to have a major role in the deﬁnition of global loads. More evolved (and complex) methods are able to take properly into account the exact body boundary condition (actual wetted surface of the hull). 18.3.12.3 Free surface boundary conditions Boundary conditions on free surface can be set, depending on the various methods, with reference to: (a) a free stream at constant velocity, corresponding to ship advance, (b) a double body ﬂow, accounting for the disturbance induced by the presence of a fully immersed double body hull on the uniform ﬂow, (c) the ﬂow corresponding to the steady advance of the ship in calm water, considering the free sur- face or (d) the incident wave proﬁle (neglecting the inter- action with the hull). Works based on fully non-linear formulations of the free surface conditions have also been published. 18.3.12.4 Fluid characteristics All the methods above recalled are based on an inviscid ﬂuid potential scheme. Some results have been published of viscous ﬂow mod- els based on the solution of Reynolds Averaged Navier Stokes (RANS) equations in the time domain. These meth- ods represent the most recent trend in the ﬁeld of ship mo- tions and loads prediction and their use is limited to a few research groups. 18-20 Ship Design & Construction, Volume 1 MASTER SET SDC 18.qxd Page 18-20 4/28/03 1:30 PM
- 21. 18.4 STRESSES AND DEFLECTIONS The reactions of structural components of the ship hull to external loads are usually measured by either stresses or deﬂections. Structural performance criteria and the associ- ated analyses involving stresses are referred to under the gen- eral term of strength. The strength of a structural component would be inadequate if it experiences a loss of load-carry- ing ability through material fracture, yield, buckling, or some other failure mechanism in response to the applied loading. Excessive deﬂection may also limit the structural effectiveness of a member, even though material failure does not occur, if that deﬂection results in a misalignment or other geometric displacement of vital components of the ship’s machinery, navigational equipment, etc., thus ren- dering the system ineffective. The present section deals with the determination of the responses, in the form of stress and deﬂection, of structural members to the applied loads. Once these responses are known it is necessary to determine whether the structure is adequate to withstand the demands placed upon it, and this requires consideration of the different failure modes asso- ciated to the limit states, as discussed in Sections 18.5 and 18.6 Although longitudinal strength under vertical bending moment and vertical shear forces is the ﬁrst important strength consideration in almost all ships, a number of other strength considerations must be considered. Prominent amongst these are transverse, torsional and horizontal bend- ing strength, with torsional strength requiring particular at- tention on open ships with large hatches arranged close together.All these are brieﬂy presented in this Section. More detailed information is available in Lewis (2) and Hughes (3), both published by SNAME, and Rawson (25). Note that the content of Section 18.4 is inﬂuenced mainly from Lewis (2). 18.4.1 Stress and Deﬂection Components The structural response of the hull girder and the associ- ated members can be subdivided into three components (Figure 18.21). Primary response is the response of the entire hull, when the ship bends as a beam under the longitudinal distribution of load.The associated primary stresses (σ1) are those, which are usually called the longitudinal bending stresses, but the general category of primary does not imply a direction. Secondary response relates to the global bending of stiff- ened panels (for single hull ship) or to the behavior of dou- ble bottom, double sides, etc., for double hull ships: • Stresses in the plating of stiffened panel under lateral pressure may have different origins (σ2 and σ2*). For a stiffened panel, there is the stress (σ2) and deﬂection of the global bending of the orthotropic stiffened panels, for example, the panel of bottom structure contained be- tween two adjacent transverse bulkheads. The stiffener and the attached plating bend under the lateral load and the plate develops additional plane stresses since the plate acts as a ﬂange with the stiffeners. In longitudinally framed ships there is also a second type of secondary stresses: σ2* corresponds to the bending under the hy- drostatic pressure of the longitudinals between trans- verse frames (web frames). For transversally framed panels, σ2* may also exist and would correspond to the bending of the equally spaced frames between two stiff longitudinal girders. • A double bottom behaves as box girder but can bend lon- gitudinally, transversally or both.This global bending in- duces stress (σ2) and deﬂection. In addition, there is also Chapter 18: Analysis and Design of Ship Structure 18-21 Figure 18.21 Primary (Hull), Secondary (Double Bottom and Stiffened Panels) and Tertiary (Plate) Structural Responses (1, 2) MASTER SET SDC 18.qxd Page 18-21 4/28/03 1:30 PM
- 22. the σ2* stress that corresponds to the bending of the lon- gitudinals (for example, in the inner and outer bottom) between two transverse elements (ﬂoors). Tertiary response describes the out-of-plane deﬂection and associated stress of an individual unstiffened plate panel included between 2 longitudinals and 2 transverse web frames. The boundaries are formed by these components (Figure 18.22). Primary and secondary responses induce in-plane mem- brane stresses, nearly uniformly distributed through the plate thickness. Tertiary stresses, which result from the bending of the plate member itself vary through the thickness, but may contain a membrane component if the out-of-plane de- ﬂections are large compared to the plate thickness. In many instances, there is little or no interaction be- tween the three (primary, secondary, tertiary) component stresses or deﬂections, and each component may be com- puted by methods and considerations entirely independent of the other two. The resultant stress, in such a case, is then obtained by a simple superposition of the three component stresses (Subsection 18.4.7). An exception is the case of plate (tertiary) deﬂections, which are large compared to the thickness of plate. In plating, each response induces longitudinal stresses, transverse stresses and shear stresses. This is due to the Poisson’s Ratio. Both primary and secondary stresses are bending stresses but in plating these stresses look like mem- brane stresses. In stiffeners, only primary and secondary responses in- duce stresses in the direction of the members and shear stresses. Tertiary response has no effect on the stiffeners. In Figure 18.21 (see also Figure 18.37) the three types of re- sponse are shown with their associated stresses (σ1, σ2, σ2* and σ3). These considerations point to the inherent sim- plicity of the underlying theory. The structural naval archi- tect deals principally with beam theory, plate theory, and combinations of both. 18.4.2 Basic Structural Components Structural components are extensively discussed in Chap- ter 17 – Structure Arrangement Component Design. In this section, only the basic structural component used exten- sively is presented. It is basically a stiffened panel. The global ship structure is usually referred to as being a box girder or hull girder. Modeling of this hull girder is the ﬁrst task of the designer. It is usually done by model- ing the hull girder with a series of stiffened panels. Stiffened panels are the main components of a ship. Al- most any part of the ship can be modeled as stiffened pan- els (plane or cylindrical). This means that, once the ship’s main dimensions and general arrangement are ﬁxed, the remaining scantling de- velopment mainly deals with stiffened panels. The panels are joined one to another by connecting lines (edges of the prismatic structures) and have longitudinal and transverse stiffening (Figures 18.23, 24 and 36). • Longitudinal Stiffening includes — longitudinals (equally distributed), used only for the design of longitudinally stiffened panels, — girders (not equally distributed). • Transverse Stiffening includes (Figure 18.23) — transverse bulkheads (a), — the main transverse framing also called web-frames (equally distributed; large spacing), used for longi- tudinally stiffened panels (b) and transversally stiff- ened panels (c). 18.4.3 Primary Response 18.4.3.1 Beam Model and Hull Section Modulus The structural members involved in the computation of pri- mary stress are, for the most part, the longitudinally contin- uous members such as deck, side, bottom shell, longitudinal bulkheads, and continuous or fully effective longitudinal primary or secondary stiffening members. Elementary beam theory (equation 29) is usually uti- lized in computing the component of primary stress, σ1, and deﬂection due to vertical or lateral hull bending loads. In assessing the applicability of this beam theory to ship struc- tures, it is useful to restate the underlying assumptions: • the beam is prismatic, that is, all cross sections are the same and there is no openings or discontinuities, • plane cross sections remain plane after deformation, will 18-22 Ship Design & Construction, Volume 1 Figure 18.22 A Standard Stiffened Panel MASTER SET SDC 18.qxd Page 18-22 4/28/03 1:30 PM
- 23. not deform in their own planes, and merely rotate as the beam deﬂects. • transverse (Poisson) effects on strain are neglected. • the material behaves elastically: the elasticity modulus in tension and compression is equal. • Shear effects and bending (stresses, strains) are not cou- pled. For torsional deformation, the effect of secondary shear and axial stresses due to warping deformations are neglected. Since stress concentrations (deck openings, side ports, etc.) cannot be avoided in a highly complex structure such as a ship, their effects must be included in any comprehen- sive stress analysis. Methods dealing with stress concen- trations are presented in Subsection 18.6.6.3 as they are linked to fatigue. The elastic linear bending equations, equations 27 and 28, are derived from basic mechanic principle presented at Figure 18.24. EI (∂2 w/∂x2 ) = M(x) [27] or EI (∂4 w/∂x4 ) = q(x) [28] where: w = deﬂection (Figure 18.24), in m E = modulus of elasticity of the material, in N/m2 I = moment of inertia of beam cross section about a horizontal axis through its centroid, in m4 M(x) = bending moment, in N.m q(x) = load per unit length in N/m = ∂V(x)/∂x = ∂2 M(x)/∂x2 = EI (∂4 w/∂x4 ) Hull Section Modulus: The plane section assumption to- gether with elastic material behavior results in a longitudi- nal stress, σ1, in the beam that varies linearly over the depth of the cross section. The simple beam theory for longitudinal strength cal- culations of a ship is based on the hypothesis (usually at- tributed to Navier) that plane sections remain plane and in the absence of shear, normal to the OXY plane (Figure 18.24). This gives the well-known formula: [29] where: M = bending moment (in N.m) σ = bending stress (in N/m2 ) f p p m p mP ( ) = − 0 2 02 exp Chapter 18: Analysis and Design of Ship Structure 18-23 Figure 18.23 Types of Stiffening (Longitudinal and Transverse) Figure 18.24 Behavior of an Elastic Beam under Shear Force and Bending Moment (2) MASTER SET SDC 18.qxd Page 18-23 4/28/03 1:30 PM
- 24. I = Sectional moment of Inertia about the neutral axis (in m4 ) c = distance from the neutral axis to the extreme mem- ber (in m) SM = section modulus (I/c) (in m3 ) For a given bending moment at a given cross section of a ship, at any part of the cross section, the stress may be ob- tained (σ = M/SM = Mc/I) which is proportional to the dis- tance c of that part from the neutral axis. The neutral axis will seldom be located exactly at half-depth of the section; hence two values of c and σ will be obtained for each sec- tion for any given bending moment, one for the top ﬁber (deck) and one for the bottom ﬁber (bottom shell). A variation on the above beam equations may be of im- portance in ship structures. It concerns beams composed of two or more materials of different moduli of elasticity, for example, steel and aluminum. In this case, the ﬂexural rigid- ity, EI, is replaced by ∫A E(z) z2 dA, where A is cross sec- tional area and E(z) the modulus of elasticity of an element of area dA located at distance z from the neutral axis. The neutral axis is located at such height that ∫A E(z) z dA = 0. Calculation of Section Modulus: An important step in routine ship design is the calculation of the midship section modulus. As deﬁned in connection with equation 29, it in- dicates the bending strength properties of the primary hull structure. The section modulus to the deck or bottom is ob- tained by dividing the moment of inertia by the distance from the neutral axis to the molded deck line at side or to the base line, respectively. In general, the following items may be included in the calculation of the section modulus, provided they are con- tinuous or effectively developed: • deck plating (strength deck and other effective decks). (See Subsection 18.4.3.9 for Hull/Superstructure Inter- action). • shell and inner bottom plating, • deck and bottom girders, • plating and longitudinal stiffeners of longitudinal bulk- heads, • all longitudinals of deck, sides, bottom and inner bot- tom, and • continuous longitudinal hatch coamings. In general, only members that are effective in both tension and compression are assumed to act as part of the hull girder. Theoretically, a thorough analysis of longitudinal strength would include the construction of a curve of section moduli throughout the length of the ship as shown in Figure 18.25. Dividingtheordinatesofthemaximumbending-moments curve (the envelope curve of maxima) by the corresponding ordinates of the section-moduli curve yields stress values, and by using both the hogging and sagging moment curves fourcurvesofstresscanbeobtained;thatis,tensionandcom- pression values for both top and bottom extreme ﬁbers. It is customary, however, to assume the maximum bend- ing moment to extend over the midship portion of the ship. Minimum section modulus most often occurs at the loca- tion of a hatch or a deck opening. Accordingly, the classi- ﬁcation societies ordinarily require the maintenance of the midship scantlings throughout the midship four-tenths length. This practice maintains the midship section area of structure practically at full value in the vicinity of maximum shear as well as providing for possible variation in the pre- cise location of the maximum bending moment. Lateral Bending Combined with Vertical Bending: Up to thispoint,attentionhasbeenfocusedprincipallyuponthever- tical longitudinal bending response of the hull. As the ship moves through a seaway encountering waves from directions other than directly ahead or astern, it will experience lateral bending loads and twisting moments in addition to the ver- tical loads. The former may be dealt with by methods that are similar to those used for treating the vertical bending loads, noting that there will be no component of still water bending moment or shear in the lateral direction. The twist- ing or torsional loads will require some special consideration. Note that the response of the ship to the overall hull twisting loading should be considered a primary response. The combination of vertical and horizontal bending mo- ment has as major effect to increase the stress at the ex- treme corners of the structure (equation 30). 18-24 Ship Design & Construction, Volume 1 Figure 18.25 Moment of Inertia and Section Modulus (1) MASTER SET SDC 18.qxd Page 18-24 4/28/03 1:30 PM
- 25. [30] where Mv, Iv, cv, and Mh, Ih, ch, correspond to the M, I, c deﬁned in equation 29, for the vertical bending and the hor- izontal bending respectively. For a given vertical bending (Mv), the periodical wave induced horizontal bending moment (Mh) increases stresses, alternatively, on the upper starboard and lower portside, and on the upper portside and lower starboard. This explains why these areas are usually reinforced. Empirical interaction formulas between vertical bend- ing, horizontal bending and shear related to ultimate strength of hull girder are given in Subsection 18.6.5.2. Transverse Stresses: With regards to the validity of the Navier Equation (equation 29), a signiﬁcant improvement may be obtained by considering a longitudinal strength member composed of thin plate with transverse framing. This might, for example, represent a portion of the deck structure of a ship that is subject to a longitudinal stress σx, from the primary bending of the hull girder. As a result of the longitudinal strain, εx, which is associated with σx, there will exist a transverse strain, εs. For the case of a plate that is free of constraint in the transverse direction, the two strains will be of opposite sign and the ratio of their ab- solute values, given by | εs / εx | = ν, is a constant property of the material. The quantity ν is called Poisson’s Ratio and, for steel and aluminum, has a value of approximately 0.3. Hooke’s Law, which expresses the relation between stress and strain in two dimensions, may be stated in terms of the plate strains (equation 31). This shows that the primary re- sponse induces both longitudinal (σx) and transversal stresses (σs) in plating. εx = 1/E ( σx – v σS) [31] εS = 1/E ( σS – ν σx) As transverse plate boundaries are usually constrained (displacements not allowed), the transverse stress can be taken, in ﬁrst approximation as: σs = ν σx [32] Equation 32 is only valid to assess the additional stresses in a given direction induced by the stresses in the perpen- dicular direction computed, for instance, with the Navier equation (equation 29). 18.4.3.2 Shear stress associated to shear forces The simple beam theory expressions given in the preced- ing section permit evaluation the longitudinal component of the primary stress, σx. In Figure 18.26, it can be seen that σ = ( ) + ( ) M I c M I c v v v h h h an element of side shell or deck plating may, in general be subject to two other components of stress, a direct stress in the transverse direction and a shearing stress. This ﬁgure illustrates these as the stress resultants, de- ﬁned as the stress multiplied by plate thickness. The stress resultants (N/m) are given by the following expressions: Nx = t σx and Ns = t σs stress resultants, in N/m N = t τ shear stress resultant or shear ﬂow, in N/m where: σx, σs = stresses in the longitudinal and transverse direc- tions, in N/m2 τ = shear stress, in N/m2 t = plate thickness, in m In many parts of the ship, the longitudinal stress, σx, is the dominant component. There are, however, locations in which the shear component becomes important and under unusual circumstances the transverse component may, like- wise, become important. A suitable procedure for estimat- ing these other component stresses may be derived by considering the equations of static equilibrium of the ele- ment of plating (Figure 18.26). The static equilibrium con- ditions for a plate element subjected only to in-plane stress, that is, no plate bending, are: ∂Nx / ∂x + ∂N / ∂s = 0 [33-a] ∂Ns / ∂x + ∂N / ∂x = 0 [33-b] In these equations, s, is the transverse coordinate meas- ured on the surface of the section from the x-axis as shown in Figure 18.26. For vessels without continuous longitudinal bulkheads Chapter 18: Analysis and Design of Ship Structure 18-25 Figure 18.26 Shear Forces (2) ED: Correction on this equation is unclear. MASTER SET SDC 18.qxd Page 18-25 4/28/03 1:30 PM

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