Analysis and Design of Ship Structure

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

Chapter

18

Analysis and Design of Ship Structure Philippe Rigo and Enrico Rizzuto

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

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

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

18.2

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

INTRODUCTION

The purpose of this chapter is to present the fundamentals of direct ship structure analysis based on mechanics and strength of materials. Such analysis allows a rationally based design that is practical, efficient, and versatile, and that has already been implemented in a computer program, tested, and proven. Analysis and Design are two words that are very often associated. Sometimes they are used indifferently one for the other even if there are some important differences between performing a design and completing an analysis.

18-1

MASTER SET SDC 18.qxd Page 18-2 4/28/03 1:30 PM

18-2

Ship Design & Construction, Volume 1

Analysis refers to stress and strength assessment of the structure. Analysis requires information on loads and needs an initial structural scantling design. Output of the structural analysis is the structural response defined in terms of stresses, deflections and strength. Then, the estimated response is compared to the design criteria. Results of this comparison as well as the objective functions (weight, cost, etc.) will show if updated (improved) scantlings are required. Design for structure refers to the process followed to select the initial structural scantlings and to update these scantlings from the early design stage (bidding) to the detailed design stage (construction). To perform analysis, initial design is needed and analysis is required to design. This explains why design and analysis are intimately linked, but are absolutely different. Of course design also relates to topology and layout definition. The organization and framework of this chapter are based on the previous edition of the Ship Design and Construction (1) and on the Chapter IV of Principles of Naval Architecture (2). Standard materials such as beam model, twisting, shear lag, etc. that are still valid in 2002 are partly duplicated from these 2 books. Other major references used to write this chapter are Ship Structural Design (3) also published by SNAME and the DNV 99-0394 Technical Report (4). The present chapter is intimately linked with Chapter 11 – Parametric Design, Chapter 17 – Structural Arrangement and Component Design and with Chapter 19 – Reliability-Based Structural Design. References to these chapters will be made in order to avoid duplications. In addition, as Chapter 8 deals with classification societies, the present chapter will focus mainly on the direct analysis methods available to perform a rationally based structural design, even if mention is made to standard formulations from Rules to quantify design loads. In the following sections of this chapter, steps of a global analysis are presented. Section 18.3 concerns the loads that are necessary to perform a structure analysis. Then, Sections 18.4, 18.5 and 18.6 concern, respectively, the stresses and deflections (basic ship responses), the limit states, and the failures modes and associated structural capacity. A review of the available Numerical Analysis for Structural Design is performed in Section 18.7. Finally Design Criteria (Section 18.8) and Design Procedures (Section 18.9) are discussed. Structural modeling is discussed in Subsection 18.2.2 and more extensively in Subsection 18.7.2 for finite element analysis. Optimization is treated in Subsections 18.7.6 and 18.9.4. Ship structural design is a challenging activity. Hence Hughes (3) states: The complexities of modern ships and the demand for greater reliability, efficiency, and economy require a sci-

entific, powerful, and versatile method for their structural design

But, even with the development of numerical techniques, design still remains based on the designer’s experience and on previous designs. There are many designs that satisfy the strength criteria, but there is only one that is the optimum solution (least cost, weight, etc.). Ship structural analysis and design is a matter of compromises: • compromise between accuracy and the available time to perform the design. This is particularly challenging at the preliminary design stage. A 3D Finite Element Method (FEM) analysis would be welcome but the time is not available. For that reason, rule-based design or simplified numerical analysis has to be performed. • to limit uncertainty and reduce conservatism in design, it is important that the design methods are accurate. On the other hand, simplicity is necessary to make repeated design analyses efficient. The results from complex analyses should be verified by simplified methods to avoid errors and misinterpretation of results (checks and balances). • compromise between weight and cost or compromise between least construction cost, and global owner live cycle cost (including operational cost, maintenance, etc.), and • builder optimum design may be different from the owner optimum design. 18.2.1 Rationally Based Structural Design versus Rules-Based Design There are basically two schools to perform analysis and design of ship structure. The first one, the oldest, is called rule-based design. It is mainly based on the rules defined by the classification societies. Hughes (3) states: In the past, ship structural design has been largely empirical, based on accumulated experience and ship performance, and expressed in the form of structural design codes or rules published by the various ship classification societies. These rules concern the loads, the strength and the design criteria and provide simplified and easy-to-use formulas for the structural dimensions, or “scantlings” of a ship. This approach saves time in the design office and, since the ship must obtain the approval of a classification society, it also saves time in the approval process.

The second school is the Rationally Based Structural Design; it is based on direct analysis. Hughes, who could be considered as a father of this methodology, (3) further states:

MASTER SET SDC 18.qxd Page 18-3 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure

There are several disadvantages to a completely “rulebook” approach to design. First, the modes of structural failure are numerous, complex, and interdependent. With such simplified formulas the margin against failure remains unknown; thus one cannot distinguish between structural adequacy and over-adequacy. Second, and most important, these formulas involve a number of simplifying assumptions and can be used only within certain limits. Outside of this range they may be inaccurate. For these reasons there is a general trend toward direct structural analysis.

Even if direct calculation has always been performed, design based on direct analysis only became popular when numerical analysis methods became available and were certified. Direct analysis has become the standard procedure in aerospace, civil engineering and partly in offshore industries. In ship design, classification societies preferred to offer updated rules resulting from numerical analysis calibration. For the designer, even if the rules were continuously changing, the design remained rule-based. There really were two different methodologies.

Design Load Direct Load Analysis Stress Response in Waves

Study on Ocean Waves

Effect on operation

Wave Load Response

Structural analysis by whole ship model Stress response function

Response function of wave load

Short term estimation

Design Sea State

Short term estimation

Long term estimation

Long term estimation

Nonlinear influence in large waves

Design wave

Wave impact load

Structural response analysis Modeling technique

Direct structural analysis

Investigation on corrosion

Strength Assessment Yield strength

Buckling strength

Ultimate strength

Fatigue strength

Figure 18.1 Direct Structural Analysis Flow Chart

18-3

Hopefully, in 2002 this is no longer true. The advantages of direct analysis are so obvious that classification societies include, usually as an alternative, a direct analysis procedure (numerical packages based on the finite element method, see Table 18.VIII, Subsection 18.7.5.2). In addition, for new vessel types or non-standard dimension, such direct procedure is the only way to assess the structural safety. Therefore it seems that the two schools have started a long merging procedure. Classification societies are now encouraging and contributing greatly to the development of direct analysis and rationally based methods. Ships are very complex structures compared with other types of structures. They are subject to a very wide range of loads in the harsh environment of the sea. Progress in technologies related to ship design and construction is being made daily, at an unprecedented pace. A notable example is the fact that the efforts of a majority of specialists together with rapid advances in computer and software technology have now made it possible to analyze complex ship structures in a practical manner using structural analysis techniques centering on FEM analysis. The majority of ship designers strive to develop rational and optimal designs based on direct strength analysis methods using the latest technologies in order to realize the shipowner’s requirements in the best possible way. When carrying out direct strength analysis in order to verify the equivalence of structural strength with rule requirements, it is necessary for the classification society to clarify the strength that a hull structure should have with respect to each of the various steps taken in the analysis process, from load estimation through to strength evaluation. In addition, in order to make this a practical and effective method of analysis, it is necessary to give careful consideration to more rational and accurate methods of direct strength analysis. Based on recognition of this need, extensive research has been conducted and a careful examination made, regarding the strength evaluation of hull structures. The results of this work have been presented in papers and reports regarding direct strength evaluation of hull structures (4,5). The flow chart given in Figure 18.1 gives an overview of the analysis as defined by a major classification society. Note that a rationally based design procedure requires that all design decisions (objectives, criteria, priorities, constraints…) must be made before the design starts. This is a major difficulty of this approach. 18.2.2 Modeling and Analysis General guidance on the modeling necessary for the structural analysis is that the structural model shall provide results suitable for performing buckling, yield, fatigue and

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

18-4

Ship Design & Construction, Volume 1

to ensure that all dimensioning loads are correctly included. A flow chart of strength analysis of global model and sub models is shown in Figure 18.2.

Structural drawings, mass description and loading conditions.

Verification of model/ loads

Structural model including necessary load definitions

Hydrodynamic/static loads

Verified structural model

Load transfer to structural model

Structural analysis

Sub-models to be used in structural analysis

Verification of load transfer

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

Yes

No

Figure 18.2 Strength Analysis Flow Chart (4)

vibration assessment of the relevant parts of the vessel. This is done by using a 3D model of the whole ship, supported by one or more levels of sub models. Several approaches may be applied such as a detailed 3D model of the entire ship or coarse meshed 3D model supported by finer meshed sub models. Coarse mesh can be used for determining stress results suited for yielding and buckling control but also to obtain the displacements to apply as boundary conditions for sub models with the purpose of determining the stress level in more detail. Strength analysis covers yield (allowable stress), buckling strength and ultimate strength checks of the ship. In addition, specific analyses are requested for fatigue (Subsection 18.6.6), collision and grounding (Subsection 18.6.7) and vibration (Subsection 18.6.8). The hydrodynamic load model must give a good representation of the wetted surface of the ship, both with respect to geometry description and with respect to hydrodynamic requirements. The mass model, which is part of the hydrodynamic load model, must ensure a proper description of local and global moments of inertia around the global ship axes. Ultimate hydrodynamic loads from the hydrodynamic analysis should be combined with static loads in order to form the basis for the yield, buckling and ultimate strength checks. All the relevant load conditions should be examined

18.2.3 Preliminary Design versus Detailed Design For a ship structure, structural design consists of two distinct levels: the Preliminary Design and the Detailed Design about which Hughes (3) states: The preliminary determines the location, spacing, and scantlings of the principal structural members. The detailed design determines the geometry and scantlings of local structure (brackets, connections, cutouts, reinforcements, etc.). Preliminary design has the greatest influence on the structure design and hence is the phase that offers very large potential savings. This does not mean that detail design is less important than preliminary design. Each level is equally important for obtaining an efficient, safe and reliable ship. During the detailed design there also are many benefits to be gained by applying modern methods of engineering science, but the applications are different from preliminary design and the benefits are likewise different. Since the items being designed are much smaller it is possible to perform full-scale testing, and since they are more repetitive it is possible to obtain the benefits of mass production, standardization and so on. In fact, production aspects are of primary importance in detail design. Also, most of the structural items that come under detail design are similar from ship to ship, and so in-service experience provides a sound basis for their design. In fact, because of the large number of such items it would be inefficient to attempt to design all of them from first principles. Instead it is generally more efficient to use design codes and standard designs that have been proven by experience. In other words, detail design is an area where a rule-based approach is very appropriate, and the rules that are published by the various ship classification societies contain a great deal of useful information on the design of local structure, structural connections, and other structural details.

18.3

LOADS

Loads acting on a ship structure are quite varied and peculiar, in comparison to those of static structures and also of other vehicles. In the following an attempt will be made to review the main typologies of loads: physical origins, general interpretation schemes, available quantification proce-

MASTER SET SDC 18.qxd Page 18-5 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure

dures and practical methods for their evaluation will be summarized. 18.3.1 Classification of Loads 18.3.1.1 Time Duration Static loads: These are the loads experienced by the ship in still water. They act with time duration well above the range of sea wave periods. Being related to a specific load condition, they have little and very slow variations during a voyage (mainly due to changes in the distribution of consumables on board) and they vary significantly only during loading and unloading operations. Quasi-static loads: A second class of loads includes those with a period corresponding to wave actions (∼3 to 15 seconds). Falling in this category are loads directly induced by waves, but also those generated in the same frequency range by motions of the ship (inertial forces). These loads can be termed quasi-static because the structural response is studied with static models. Dynamic loads: When studying responses with frequency components close to the first structural resonance modes, the dynamic properties of the structure have to be considered. This applies to a few types of periodic loads, generated by wave actions in particular situations (springing) or by mechanical excitation (main engine, propeller). Also transient impulsive loads that excite free structural vibrations (slamming, and in some cases sloshing loads) can be classified in the same category. High frequency loads: Loads at frequencies higher than the first resonance modes (> 10-20 Hz) also are present on ships: this kind of excitation, however, involves more the study of noise propagation on board than structural design. Other loads: All other loads that do not fall in the above mentioned categories and need specific models can be generally grouped in this class. Among them are thermal and accidental loads. A large part of ship design is performed on the basis of static and quasi-static loads, whose prediction procedures are quite well established, having been investigated for a long time. However, specific and imposing requirements can arise for particular ships due to the other load categories. 18.3.1.2 Local and global loads Another traditional classification of loads is based on the structural scheme adopted to study the response. Loads acting on the ship as a whole, considered as a beam (hull girder), are named global or primary loads and the ship structural response is accordingly termed global or primary response (see Subsection 18.4.3).

18-5

Loads, defined in order to be applied to limited structural models (stiffened panels, single beams, plate panels), generally are termed local loads. The distinction is purely formal, as the same external forces can in fact be interpreted as global or local loads. For instance, wave dynamic actions on a portion of the hull, if described in terms of a bi-dimensional distribution of pressures over the wet surface, represent a local load for the hull panel, while, if integrated over the same surface, represent a contribution to the bending moment acting on the hull girder. This terminology is typical of simplified structural analyses, in which responses of the two classes of components are evaluated separately and later summed up to provide the total stress in selected positions of the structure. In a complete 3D model of the whole ship, forces on the structure are applied directly in their actual position and the result is a total stress distribution, which does not need to be decomposed. 18.3.1.3 Characteristic values for loads Structural verifications are always based on a limit state equation and on a design operational time. Main aspects of reliability-based structural design and analysis are (see Chapter 19): • the state of the structure is identified by state variables associated to loads and structural capacity, • state variables are stochastically distributed as a function of time, and • the probability of exceeding the limit state surface in the design time (probability of crisis) is the element subject to evaluation. The situation to be considered is in principle the worst combination of state variables that occurs within the design time. The probability that such situation corresponds to an out crossing of the limit state surface is compared to a (low) target probability to assess the safety of the structure. This general time-variant problem is simplified into a time-invariant one. This is done by taking into account in the analysis the worst situations as regards loads, and, separately, as regards capacity (reduced because of corrosion and other degradation effects). The simplification lies in considering these two situations as contemporary, which in general is not the case. When dealing with strength analysis, the worst load situation corresponds to the highest load cycle and is characterized through the probability associated to the extreme value in the reference (design) time. In fatigue phenomena, in principle all stress cycles contribute (to a different extent, depending on the range) to

MASTER SET SDC 18.qxd Page 18-6 4/28/03 1:30 PM

18-6

Ship Design & Construction, Volume 1

damage accumulation. The analysis, therefore, does not regard the magnitude of a single extreme load application, but the number of cycles and the shape of the probability distribution of all stress ranges in the design time. A further step towards the problem simplification is represented by the adoption of characteristic load values in place of statistical distributions. This usually is done, for example, when calibrating a Partial Safety Factor format for structural checks. Such adoption implies the definition of a single reference load value as representative of a whole probability distribution. This step is often performed by assigning an exceeding probability (or a return period) to each variable and selecting the correspondent value from the statistical distribution. The exceeding probability for a stochastic variable has the meaning of probability for the variable to overcome a given value, while the return period indicates the mean time to the first occurrence. Characteristic values for ultimate state analysis are typically represented by loads associated to an exceeding probability of 10–8. This corresponds to a wave load occurring, on the average, once every 108 cycles, that is, with a return period of the same order of the ship lifetime. In first yielding analyses, characteristic loads are associated to a higher exceeding probability, usually in the range 10–4 to 10–6. In fatigue analyses (see Subsection 18.6.6.2), reference loads are often set with an exceeding probability in the range 10–3 to 10–5, corresponding to load cycles which, by effect of both amplitude and frequency of occurrence, contribute more to the accumulation of fatigue damage in the structure. On the basis of this, all design loads for structural analyses are explicitly or implicitly related to a low exceeding probability. 18.3.2 Definition of Global Hull Girder Loads The global structural response of the ship is studied with reference to a beam scheme (hull girder), that is, a monodimensional structural element with sectional characteristics distributed along a longitudinal axis. Actions on the beam are described, as usual with this scheme, only in terms of forces and moments acting in the transverse sections and applied on the longitudinal axis. Three components act on each section (Figure 18.3): a

Figure 18.3 Sectional Forces and Moment

resultant force along the vertical axis of the section (contained in the plane of symmetry), indicated as vertical resultant force qV; another force in the normal direction, (local horizontal axis), termed horizontal resultant force qH and a moment mT about the x axis. All these actions are distributed along the longitudinal axis x. Five main load components are accordingly generated along the beam, related to sectional forces and moment through equation 1 to 5: x

VV (x) =

∫ q V (ξ)

dξ

[1]

dξ

[2]

dξ

[3]

dξ

[4]

dξ

[5]

0

x

M V (x) =

∫ VV ( ξ ) 0

x

VH (x) =

∫ q H (ξ ) 0

x

M H (x) =

∫ VH ( ξ ) 0

x

M T (x) =

∫ m T (ξ) 0

Due to total equilibrium, for a beam in free-free conditions (no constraints at ends) all load characteristics have zero values at ends (equations 6). These conditions impose constraints on the distributions of qV, qH and mT. VV (0) = VV (L) = M V (0) = M V (L) = 0 VH (0) = VH (L) = M H (0) = M H (L) = 0 M T (0) = M T (L) = 0

[6]

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

MASTER SET SDC 18.qxd Page 18-7 4/28/03 1:30 PM

Chapter 18: Analysis and Design of Ship Structure

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

[7]

where AI = transversal immersed area. Components of vertical shear and vertical bending can be derived according to equations 1 and 2. There are no horizontal components of sectional forces in equation 3 and accordingly no components of horizontal shear and bending moment. As regards equation 5, only mTg, if present, is to be accounted for, to obtain the torque. 18.3.3.1 Standard still water bending moments While buoyancy distribution is known from an early stage of the ship design, weight distribution is completely defined only at the end of construction. Statistical formulations, calibrated on similar ships, are often used in the design development to provide an approximate quantification of weight items and their longitudinal distribution on board. The resulting approximated weight distribution, together with the buoyancy distribution, allows computing shear and bending moment.

Figure 18.4 Sectional Resultant Forces in Still Water

(a)

18-7

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

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

where C = wave parameter (Table 18.I). The formulations in equation 8 are sometimes explicitly reported in Rules, but they can anyway be indirectly derived from prescriptions contained in (6, 7). The first requirement (6) regards the minimum longitudinal strength modulus and provides implicitly a value for the total bending moment; the second one (7), regards the wave induced component of bending moment. Longitudinal distributions, depending on the ship type, are provided also. They can slightly differ among Class Societies, (Figure 18.5). 18.3.3.2 Direct evaluation of still water global loads Classification Societies require in general a direct analysis of these types of load in the main loading conditions of the ship, such as homogenous loading condition at maximum draft, ballast conditions, docking conditions afloat, plus all other conditions that are relevant to the specific ship (nonhomogeneous loading at maximum draft, light load at less than maximum draft, short voyage or harbor condition, ballast exchange at sea, etc.). The direct evaluation procedure requires, for a given loading condition, a derivation, section by section, of vertical resultants of gravitational (weight) and buoyancy forces, applied along the longitudinal axis x of the beam. To obtain the weight distribution w(x), the ship length is subdivided into portions: for each of them, the total weight and center of gravity is determined summing up contributions from all items present on board between the two bounding sections. The distribution for w(x) is then usually approximated by a linear (trapezoidal) curve obtained by imposing

TABLE 18.I Wave Coefficient Versus Length (b)

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

Ship Length L

Wave Coefficient C

90 ≤ L 0.5 σ F 

where: σE = elastic plate buckling strength

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

[47]

18-39

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

c

  τ av  +  τ  B

c

  ≤ η B [48] 

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

MASTER SET SDC 18.qxd Page 18-40 4/28/03 1:31 PM

18-40

Ship Design & Construction, Volume 1

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

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

[49a]

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

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

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

[49b]

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

MASTER SET SDC 18.qxd Page 18-41 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure

18-41

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

c

  τ av  +  τ  u

c

  ≤ η u [50] 

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

2 π2E  t 12 (1 − ν 2 )  b 

[51]

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

[52]

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

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

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

[53]

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

MASTER SET SDC 18.qxd Page 18-42 4/28/03 1:31 PM

18-42

Ship Design & Construction, Volume 1

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

[54]

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

[55a]

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

[55b]

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

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

MASTER SET SDC 18.qxd Page 18-43 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure

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

18-43

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

(a) (a)

(b)

(b)

(c)

(d)

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

mode III).

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

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

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

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

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

18-44

Ship Design & Construction, Volume 1

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

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

MASTER SET SDC 18.qxd Page 18-45 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure

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

1 0.995

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

18-45

and λ=

a πr

σ

Y = E

σY σE

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

[56]

and σu σ 1 ≤ 2 = E σY σY λ with

Figure 18.45 A Comparison of the Ultimate Strength Formulations for

b β= t

Y σ E

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

MASTER SET SDC 18.qxd Page 18-46 4/28/03 1:31 PM

18-46

Ship Design & Construction, Volume 1

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

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

(a)

(b)

(c)

(d)

(e)

(f)

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

Caldwell Modified (f) Plastic Bending Moment.

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

Chapter 18: Analysis and Design of Ship Structure

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

18-47

Figure 18.48 The Smith’s Progressive Collapse Method

(a)

(b)

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

MASTER SET SDC 18.qxd Page 18-48 4/28/03 1:31 PM

18-48

Ship Design & Construction, Volume 1

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

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

MASTER SET SDC 18.qxd Page 18-49 4/28/03 1:31 PM

Chapter 18: Analysis and Design of Ship Structure

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

18-49

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

a  Mh   + α  M hu 

b

  =1 

[57]

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