analysis method of ultimate hull girder strength under combined loads

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Ships and Offshore Structures

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Analysis method of ultimate hull girder strengthunder combined loads

Yoshiteru Tanaka, Hiroaki Ogawa, Akira Tatsumi & Masahiko Fujikubo

To cite this article: Yoshiteru Tanaka, Hiroaki Ogawa, Akira Tatsumi & Masahiko Fujikubo (2015)Analysis method of ultimate hull girder strength under combined loads, Ships and OffshoreStructures, 10:5, 587-598, DOI: 10.1080/17445302.2015.1045271

To link to this article: http://dx.doi.org/10.1080/17445302.2015.1045271

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Ships and Offshore Structures, 2015Vol. 10, No. 5, 587–598, http://dx.doi.org/10.1080/17445302.2015.1045271

Analysis method of ultimate hull girder strength under combined loads

Yoshiteru Tanakaa,∗, Hiroaki Ogawab, Akira Tatsumic and Masahiko Fujikuboc

aStructural Engineering Department, National Maritime Research Institute (NMRI), Tokyo, Japan; bStructural Research Group,Technical Research Center, Japan Marine United Corporation, Tsu, Japan; cDepartment of Naval Architecture and Ocean Engineering,

Osaka University, Suita, Japan

(Received 5 December 2014; accepted 10 April 2015)

The objective of this study is to propose an analysis method of ultimate hull girder strength under combined bendingand torsion. The hull girder is modelled by a series of thin-walled beam elements and the average stress–average strainrelationship of plate and stiffened panel elements under axial loads considering the effect of shear stress is implementedin the beam elements. First, a torsional moment is applied to the beam model for a whole model within the elastic range.Then, the ultimate bending strength of cross-sections is calculated applying Smith’s method to beam elements consideringthe warping and shear stresses. The proposed simplified method is applied to the progressive collapse tests of scale modelsunder combined loads. On the other hand, nonlinear explicit finite element method (FEM) is adopted for the analysis of thetest models. The effectiveness of the simplified method is discussed comparing with the results of experiments and FEManalysis.

Keywords: ship hull girder; ultimate strength; combined loads; Smith’s method; simplified analysis method; beam element

1. Introduction

When the ship hull is subjected to excessive longitudinalbending moment, buckling and yielding of plates andstiffeners take place progressively and the ultimate strengthof the cross-section is attained. The ultimate longitudinalbending strength is one of the most fundamental strengthof a ship hull girder. Caldwell (1965) first proposed an esti-mation method of the ultimate hull girder strength applyingthe condition of fully plastic section under bending byconsidering the influence of buckling. On the other hand,Smith (1977) presented a simplified calculation method totrack the progressive collapse behaviour of the ship hullgirder in bending. In the method, the entire cross-sectionreaches the ultimate state according to the progressivebuckling and plastic collapse of the members constitutingthe cross-section. This method has been widely used asthe so-called Smith’s method. Yao and Nikolov (1992)constructed a practical computer code for progressivecollapse analysis utilising this method. Moreover, theanalytical method for the ultimate strength calculation ofthe ship hull was proposed by Paik and Mansour (1995) andthis method was modified so as to be more general (Paiket al. 2013).

Recently, there is a growing demand for a containership, which is characterised as a hull girder with large opendecks. This type of ship has a relatively small torsional stiff-ness compared to the ships with closed cross-section and the

∗Corresponding author. Email: terry@nmri.go.jp

effect of torsion on the ultimate longitudinal strength maybe significant. However, the simplified methods mentionedabove cannot consider the influence of torsion. Therefore,some of the authors developed a simplified method of theultimate strength analysis of a hull girder under torsion aswell as bending (e.g. Tatsumi et al. 2011). In this method, ahull girder is divided by beam elements in the longitudinaldirection, and the warping as well as bending deformationis included in the formulation. The cross-section of a beamelement is divided into plate, stiffened panel and hard cor-ner elements based on Smith’s method. Therefore, the shiftof instantaneous neutral axis and shear centre can be au-tomatically considered by introducing the axial degree offreedom as well as the bending ones into the beam elements,and keeping the zero axial load condition. In this study, theaverage stress–average strain relationship of each elementis calculated using the formulae of the common structuralrules (CSR) [International Association of Classification So-cieties (IACS) 2006] considering the effect of shear stressdue to torsion on the yield strength.

There are a lot of papers (e.g. Paik et al. 2001),which discuss the importance of strength assessmentto large container ships under torsion. However, fewexperimental studies (Sun and Guedes Soares 2003)regarding the ultimate torsional strength of hull girders hasbeen reported. Therefore, in order to clarify the hull girdercollapse behaviour under combined bending and torsion, a

588 Y. Tanaka et al.

Figure 1. Deck plan of test models. (This figure is available in colour online.)

series of progressive collapse tests using scale models of acontainer ship was carried out. Three 1/13-scale three-holdmodels referring to a 5250 TEU (twenty-foot equivalentunit) container ship were employed in this experiment.

In this paper, the proposed simplified method is appliedto the test model. First, a torsional moment is applied toits beam model within the elastic range. Then, the ultimatebending strength of cross-sections is calculated applyingSmith’s method to beam elements considering the warpingand shear stresses. On the other hand, nonlinear explicitFEM is adopted for the analysis of the test models by us-ing LS-DYNA. The effectiveness of the present simplifiedanalysis method of the ultimate hull girder strength undercombined loads is discussed compared with the experimentand an LS-DYNA analysis.

2. Test method

2.1. Scale models of a container ship

In order to investigate the collapse mechanism and the ef-fect of the torsional moment on the longitudinal ultimatestrength of a ship, three scale models (from models 1 to3) were fabricated by referring to the hull structure of a5250 TEU container ship. The principal dimensions areLpp × B × D × d = 267.0 m × 39.8 m × 23.6 m × 12.5 m.The models were designed so as to keep the slenderness

ratio of the actual stiffened panel and their dimensions areshown in Figures 1 and 2 and listed in Table 1.

2.2. Material properties

Mechanical properties of all the materials used for the testmodels were examined by tensile tests and are listed inTable 2.

2.3. Initial imperfections

Measured values of initial imperfections are listed inTable 3. Initial deformations were measured by deflectiongauges along the panel centreline at the side shells andouter bottom shells of Bay-4. Though so-called thin-horsemode and buckling mode of stiffened panels were predom-inant in the components of initial deformations, W0max inTable 3 only indicates the maximum initial deformation inthe plates shown in Figure 2b. σ rc in Table 3 denotes theaverage compressive residual stresses in a cross-section ofstiffened panels, which were obtained from similar speci-mens by the so-called stress-release method.

2.4. Experimental setup

A series of collapse tests was conducted with the scalemodels applying the combined vertical bending moment,

Ships and Offshore Structures 589

Figure 2. Transverse cross-section of test models. (a) section A, (b) section B, (c) section C; (d) section D.

the shearing force and the torsional moment. The aft-endof Bay-6 was fixed to the rigid wall and the vertical forcesP1 and P2 were applied in the same or opposite directionat Bay-1 by the hydraulic jacks as shown in Figure 3. Theloads were applied in a step-by-step manner so that the loadscan be regarded as static ones. The strains on the plates, thedisplacements at the bilge corner and the reaction forcesand strokes of hydraulic jacks were measured.

It should be noted that the fixed boundary conditionhere was adopted as an experimental technique in order togenerate large warping stresses as much as possible and theanalysis of the test models are performed under this bound-ary condition (see Section 4). However, for the evaluationof actual ship structures under realistic combined loads, theanalysis only has to be performed under the condition that

the rigid-body displacement is constrained constituting aload system so as to satisfy the self-equilibrium. Hence, thepresent beam model can also be available.

2.5. Loading conditions

Generally, torsion of a ship hull girder induced by theoblique sea has strong correlation with the horizontal bend-ing (Mohammed et al. 2012 and Hirdaris et al. 2014). There-fore, the influence of the torsional moment on the ultimatehorizontal bending strength should be discussed for an ac-tual ship. However, in these experiments, the interactionbetween torsion and vertical bending was assumed for sim-plicity. Initial loading conditions of the progressive collapsetests for each model are listed in Table 4.

Table 1. Dimensions of test models (in mm).

Model 1 Models 2 and 3

L1 900L2 1000L3 650L4 150LO 6550B1 180B2 190BO 3200D1 120D2 180D3 180DO 1800t1 5.94 5.87t2 4.35 4.48t3 3.13 3.14t4 2.28 2.13stiff. 1 50 × 5.97 50 × 5.89stiff. 2 50 × 2.92 50 × 2.89stiff. 3 50 × 2.92 50 × 2.89

For general container ships, the maximum value of thewave-induced torsional moment is at most 10% of that of thevertical bending moment (ClassNK 2012). Therefore, theinfluence of torsion was overestimated for the condition ofcombined loads chosen for the experiment. And this loadingcondition is a statically determinate problem to keep theratio between the imposed vertical bending and torsionalmoments constant within the elastic range. Therefore, thevarious combinations of loading condition are required forthe analysis of actual ships, since obtained results could

Table 2. Material properties.

E (GPa) σ 0.2 (MPa) σ u (MPa)

Model 1t1 195 231 351t2 175 203 338t3 183 175 302t4 194 240 332stiff. 1 202 761 780stiff. 2 and 3 207 727 730

Model 2 and 3t1 192 190 300t2 198 209 298t3 198 255 349t4 209 244 351stiff. 1 200 648 678stiff. 2 and 3 203 724 725

Note: E: Young’s modulus, σ 0.2: 0.2% proof stress, and σ u: maximumtensile stress.

Table 3. Initial imperfections.

W0max/tp σ rc/σ 0.2

Model t1 t2 t3 t4 t1 t2 t3 t4

1 0.47 0.44 2.23 2.89 0.22 0.18 0.28 0.272 0.24 0.55 2.35 3.13 0.57 0.30 0.24 0.253 0.48 0.43 2.22 3.09

Note: W0max: maximum value of initial deformation, tp: plate thicknes(t1 to t4) and σ rc: average value of compressive residual stress in plate.

Figure 3. Model setup. (This figure is available in colour online.)

Table 4. Initial loading conditions.

Loads (downward: + )

Model 1 P2 = –0.68 P1 (M/T = 0.5)Model 2 P2 = –P1 (M = 0)Model 3 P2 = P1 (T = 0)

Note: T: torsional moment,M: vertical bending moment (positive in hogging).

be valid only for that particular load combination at theconsidered cross-section. However, the main purpose ofthis study is introducing Smith’s method into general beamelement referring to the collapse behaviour obtained by theexperiments under combined loads.

2.6. Test results

The progressive collapse tests were conducted under thedisplacement control with hydraulic jacks according to theinitial loading conditions. The load–stroke relationships areshown in Figure 4. Figure 5 shows the relationships betweentorsional and vertical bending moments at fixed end gener-ated by jack loads. The collapse modes of the models areshown in Figure 6.

For the models 1 and 2, since the torsional momentwas predominant, shear buckling of side shell in the centralregion was observed first (see Figure 6a). Next, bucklingand yielding of bilge corner plates near the fixed end wascaused by warping stress (see Figure 6b), because the platethickness of the bottom side were thinner than that of thedeck side. As a result, the local stiffness of P1 side wasgradually reduced and the jack load of P1 side reached themaximum. Then the hatch corner adjacent to the closedsection was broken (see Figure 6c) and finally bucklingdeformation induced by warping stress extended widely tothe outer bottom and lower plates of side shell on P1 sideof Bay-5 (see Figure 6d).

The collapse mode of all test models except model 2almost coincided with that of the FEM analysis. The model3, loaded in bending, exhibited a typical bending collapseunder hogging.

2.7. Nonlinear finite element analysis

Elasto-plastic FEM analysis is carried out using LS-DYNAfor various loading conditions including tested ones. Theelement size is determined so as to be able to simulate thebuckling mode of stiffened plates with a sufficient accuracy.The true stress–plastic strain relationships obtained by thetensile tests for all the materials of test models, which wereapproximated by polygonal line, were introduced into theFEM analysis.

The finite element model is fixed at its aft end. Thehydraulic jack loads are applied through a rigid supporter

Figure 4. Load–stroke relationships. (a) Model 1 (M/T = 0.5).(b) Model 2 (M = 0). (c) Model 3 (T = 0).

Figure 5. History of applied torsional and vertical bending mo-ments.

Figure 6. Collapse modes of test models 1 and 2. (a) Shear buckling of side shell (b) buckling of bilge corner (c) break of hatch corner(d) collapse of Bay-5 side shell. (This figure is available in colour online.)

at the both side shells of fore-end part of the model by theprescribed vertical velocity (see Figure 7). The time historyof the prescribed velocity is determined so that the ratioof the reaction forces, P1 and P2 in the analysis coincided

with that recorded in the tests. The main purpose of theseanalyses is basically the comparison between the resultsof the FEM analysis of test models as three-dimensionalshell structures and those of the simplified method based

Figure 7. Collapse modes of model 1 obtained by FEA. (This figure is available in colour online.)

Figure 8. Coordinate system.

on Smith’s method. The collapse modes of model 1 afterits ultimate strength are shown in Figure 7. The calculatedcollapse modes of model 1 are in good agreement with thetest results as shown in Figure 6d. However, the calculatedresults show the higher ultimate strength than the test re-sults as shown in Figure 4. Further investigation is neededfor the better estimation by the FEM, in order to use theresults of the FEM as the reference solutions for the exam-ination of the proposed simplified method described in thenext chapter. Therefore, the FEM analysis of large-scalemodels considering the initial deformations and residualstresses will be performed in the future studies includingthe investigation about the methodology.

3. Simplified method of ultimate strength analysis ofhull girder under combined loads

3.1. Formulation of beam element

The coordinate system is defined as shown in Figure 8.Assuming that the cross-section remains undistorted in x–yplane, the displacements U, V and W in the x, y and zdirections at the arbitrary point (x, y, z) can be expressed as

U (x, y, z) = us (z) − (y − ys) θ (z) (1)

V (x, y, z) = vs (z) + (x − xs) θ (z) (2)

W (x, y, z) = w (z) − xu′s (z) − yv′

+ωns (x, y) θ ′ (z) , (3)

where us and vs are the displacements at the shear centre(xs, ys) in the x and y directions and w is the displacementat the gravity centre in the z direction. θ is the rotationabout the longitudinal axis at the shear centre, and ωns isthe associated warping function. A prime (′) denotes thedifferentiation with respect to the z-coordinate. The axialstrain in the z direction, εz, and the shear strain in the sz-plane, γ sz, can be expressed as

Figure 9. Nodal displacements.

εz = ∂W

∂z= w′ − xu′′

s − yv′′s + ωnsθ

′′ (4)

γsz = γxz

∂s+ γyz

∂ωns

∂s− (y − ys)

∂s+ (x − xs)

}θ ′. (5)

The general form of stress–strain relationship can beexpressed as

[d11 d12

d21 d22

] {εz

}, (6)

where σ z is the axial stress and τ sz the shear stress in theplane.

Consider a beam element ij of the length l as shown inFigure 9. The nodal displacement vectors are

{d}T = {{us} , {vs} , {θ} , {w}}T{us}T = {

usi u′si usj u′

}{vs}T = {

vsi v′si vsj v′

}{θ}T = {

θi θ ′i θj θ ′

}{w}T = {

}, (7)

and the corresponding nodal forces are

{F }T = {{Fu} , {Fv} , {Fθ } , {Fw}}T{Fu}T = {

Fxi Myi Fxj Myj

}{Fv}T = {

Fyi Mxi Fyj Mxj

}{Fθ }T = {

Ti Bi Tj Bj

}{Fw}T = {

Fzi Fzj

}, (8)

where {Fu} and {Fv} are the shear forces and bendingmoments, {Fθ} is the torsional moment and bi-moment

Figure 10. Division of cross-section (a) warping function calculation (b) for progressive collapse analysis.

about the axis at the shear centre and {Fw} is the axialforces.

The axial displacement w(z) is interpolated linearlywithin the beam element, and the horizontal and verticaldeflections us(z), vs(z) and the torsional angle θ (z) are inter-polated by cubic polynomials. Substituting those displace-ments expressed as a function of the nodal displacementsinto Equations (4) and (5), the axial and shear strain can beexpressed in the form

0−x [B2]

0−y [B2]

0ωns [B2]g (s) [B3]

]{d} .

Applying the principle of virtual work, the incrementalform of the stiffness equation is derived in the form

{F } = [K] {d} , (10)

where the stiffness equation [K] is given by

[K] = ∫V

⎡⎢⎢⎣

d11 [B1]−xd11 [B21]−yd11 [B21]ωnsd11 [B21]

−xd11 [B12]x2d11 [B2]xyd11 [B2]

−xωnsd11 [B2]−yd11 [B12]xyd11 [B2]y2d11 [B2]

−yωnsd11 [B2]

ωnsd11 [B12]−xωnsd11 [B2]−yωnsd11 [B2]

ω2nsd11 [B2] + g2d22[B2]

⎤⎥⎥⎦ dV.

For the elastic state, the stress–strain relationship ofEquation (6) at the arbitrary point is given by d11 = E,d22 = G and d12 = d21 = 0, where E is Young’s modulusand G shear modulus. On the other hand, in the progres-sive collapse behaviour, dij changes with the buckling andyielding of structural members. In the present analysis, d11

is changed based on Smith’s method while the rest are as-sumed to be same as the elastic ones. The applicability of

the present method is hence limited to the load case wherebending stresses are more dominant than shear stresses.

3.2. Warping function

The warping function ωns in Equation (11) is calculatedseparately before performing the collapse analysis usingthe beam elements described in Section 3.1. The elasticcross-section is assumed and Fujitani’s method (Park et al.1997) is applied. The thin-walled cross-section is dividedinto the plate elements as shown in Figure 10a. The stiffeneris not taken into account since the shear flow is carried onlyby the plate. Using the coordinates of the nodal points p andq, the coordinates of an arbitrary point within the straightelement pq can be expressed as

x (s) =(

1 − s

)xp + s

lsxq, 0 ≤ s ≤ ls

y (s) =(

1 − s

)yp + s

lsyq , 0 ≤ s ≤ ls , (12)

where ls is the length of the element pq. Equation (12) de-notes that x(s) and y(s) are linear function with respectto s and the equilibrium equation governing the warp-ing function ωns(s) of the thin-walled section is expressedby

∂2ωns

∂s2− y

∂s2+ x

∂s2= 0. (13)

Substituting Equation (12) into Equation (13), the sec-ond and the third terms of Equation (13) vanish for straight-line elements. Therefore, the warping function ωns(s) needto satisfy

∂2ωns

∂s2= 0 (14)

on the straight-line element. Equation (14) shows that thewarping function ωns(s) of the element pq is assumed to be

linearly changed within the element, that is, it should be alinear function about s. Therefore, ωns(s) at arbitrary points on the straight-line element can be expressed by using ωp

and ωq,

ωns (s) =(

1 − s

)ωp + s

lsωq , (0 ≤ s ≤ ls) , (15)

where ωp and ωq are the values of the warping function atboth ends of the element pq. As a result, even if a cross-section is divided by only straight-line elements, ωns(s) ofthe element pq gives the correct distribution of the warpingfunction not only for the open section but also for the closedsection or the open section including the closed sectionpartially.

In Saint-Venant’s torsional deformation, only shearstress and shear strain are induced in the cross-sectionunder the condition of zero axial force. The principle ofvirtual work with respect to a virtual warping displacementis therefore given by

∫τsz δγsz t ds = 0. (16)

Assuming the elastic state and substituting Equation (5)into Equation (16) yields

∫Glsθ

∂ωns

∂s− (y − ys)

∂s+ (x − xs)

× ∂ωns

∂st ds = 0. (17)

Substituting Equations (12) and (15) into Equation (17),the following equation for the value of the warping functionis obtained:

−Gtls

] {ωp

{ Gtls

(xpyq − xqyp

)−Gt

(xpyq − xqyp

Superposition of Equation (18) for the elements overthe cross-section gives a set of simultaneous equations forthe warping function for the cross-section.

3.3. Collapse analysis

For the progressive collapse analysis of a ship hull girderunder pure bending, Smith’s method has been widely em-ployed. The basic procedure of Smith’s method is as fol-lows:

(1) Subdivide the cross-section into elements com-posed of stiffened panels, attached plates or hardcorners as shown in Figure 10b;

Figure 11. Correction of average stress–average strain relation-ship considering the effect of shear stress.

(2) Derive average stress–average strain relationshipsof individual elements as shown by the solid lineof Figure 11 under the action of axial tension orcompression considering the influences of bucklingand yielding;

(3) Apply the curvature incrementally to the cross-section assuming that the cross-section is kept planeconsidering the average stress–average strain rela-tionships of all the elements. Then, calculate theincrement of the bending moment for the appliedcurvature increment;

(4) Obtain the bending moment–curvature relationshipof the entire cross-section by adding the incrementsof curvature and bending moment to their cumula-tive values.

In this paper, Smith’s method is applied to the hull girderunder the combined vertical bending and torsional mo-ments. Generally, the torsional moment acting on a shiphas stronger correlation with horizontal bending momentthan vertical one (Mohammed et al. 2012 and Hirdaris et al.2014). However, from the stand point of ultimate hull girderstrength, it is reasonable to assume for normal containerships that the hull girder bending collapses predominantlydue to bending with the effects of the torsional moment. Onthis assumption, the torsional moment is applied to the elas-tic beam model first. Then, the progressive vertical bendingcollapse analysis of the hull girder model is performedconsidering the effects of the warping and shear stressesinduced by the torsional moment in the first-step analysison the load effects and capacity. In addition, the horizontalbending moment can be simply introduced to this method,

because the collapse analysis is based on Smith’s method.The proposed method is outlined as follows:

(1) Divide a considered hull girder in a longitudinaldirection into beam elements with uniform section.Therefore, sufficient divisions are required for thenon-uniform parts of the hull girder. Next, calculatethe stiffness matrices of the beam elements based onthe general beam theory. Then, restrain the degreesof freedom of nodes so as to satisfy the boundaryconditions of the beam model;

(2) Subdivide the cross-sections of each beam elementsinto straight-line elements as shown in Figure 10ain order to calculate the warping functions individ-ually and superpose them into a whole matrix;

(3) Apply the values of the torsional moment corre-sponding to its distribution profile of the hull girderto the individual nodes of beam model. At thispoint, the application of bending moments, con-centrated loads or prescribed displacements to thebeam model is also available. Then, calculate thewarping and shear stresses of the subdivided plateelements in addition to the reaction forces and nodaldisplacements of the beam model;

(4) Derive average stress–average strain relationshipof subdivided elements according to step (2) ofSmith’s method. In this study, the relationships arecalculated by the CSR formula (IACS 2006);

(5) Set the warping stresses as initial stresses obtainedfrom step (3) to the subdivided elements before ex-ecuting progressive collapse analysis. In contrast,the influences of shear stresses are considered indi-rectly as shown in Figure 11;

(6) Apply bending moments, concentrated loads or pre-scribed displacements to the nodes of the beammodel incrementally. At this point, the collapseanalyses are executed for each beam elements ac-cording to steps (3) and (4) of Smith’s method andthe stresses and strains of subdivided elements arecalculated in addition to the reaction forces andnodal displacements of the beam model. Finally,obtain the applied loads–nodal displacement or ap-plied bending moment–curvature relationships.

Though the element subdivision of step (3) is appliedto the cross-section as shown in Figure 10b, the same sub-division is used for the calculation of the warping functionshown in Figure 10a.

One of the most important thing in the FE modellingis to make the axial nodal displacement uz at one sidefree under the condition of the zero-axial force increment(Nz = 0), which is the condition for the determinationof the location of the instantaneous neutral axis. Using thismodel, the shift of the instantaneous neutral axis can beautomatically considered. The calculated stress increments

are added to their cumulative values including the initialstresses due to torsion.

The axial stiffness d11 in the stress–strain relationshipof Equation (6) corresponds to the tangent modulus of theaverage stress–average strain relationship considered at step(2) of Smith’s method. In this study, the average stress–average strain relationships, given in the formulae of theCSR (IACS 2006) are employed. The effect of shear stressis considered by reducing the yield strength, σ Y, to theeffective yield strength, σ Ye, based on Mises’s yield criterionas

1 − 3

. (19)

The solid line in Figure 11 illustrates the average stress–average strain relationship of subdivided elements given bythe CSR formulae. When the shear stress is considered, theeffective yield strength is reduced as shown by the dashedline.

The warping function ωns and the location of instanta-neous shear centre change due to the buckling and yieldingof the members. Here, the function ωns calculated for theelastic cross-section is used throughout the collapse anal-ysis. The shift of the instantaneous shear centre can beconsidered by giving the condition of zero increment ofthe axial force and the torsional moment in the bendingcollapse analysis.

It should be noted that the shear stresses due to tor-sion and the influence of the pertinent longitudinal warpingstrain onto a primary longitudinal strain were not introducedinto the present version. However, they will be treated inthe forthcoming version of the proposed method.

4. Application of the simplified method and FEM totest models

The proposed method is adopted for the analysis of thetest model 3. The model is idealised by the 20 thin-walledbeam elements as shown in Figure 12 and the cross-sectionsshown in Figure 2 are subdivided by plate, stiffened paneland hard corner elements according to Smith’s method. Thebeam model is provided for bending and torsional analysesin elastic states and progressive collapse analysis under thecombined torsional and bending moment.

Figures 13 and 14 show the distribution of the verticaldisplacement and the torsional angle when the model isapplied to the vertical bending and torsional moments of1.0 × 103 kNm, respectively. For the bending analysis, thevertical displacement calculated by the proposed methodgives a rough agreement with FEM results considering withand without transverse bulkheads. In contrast, the torsionalangle obtained by the proposed method is larger than thatof the FEM result. This fact suggests that the effect of the

Figure 12. Idealising the test model as beam elements.

transverse bulkhead is to be considered in the proposedmethod. The effect of transverse bulkheads will be intro-duced into the proposed method in the forthcoming paper.

Next, the progressive collapse analysis by the proposedmethod and FEM are applied to the test model 3 to comparewith the test result. Figure 15 shows the bending moment–vertical displacement relationship obtained by them. Theultimate strengths of numerical analyses are 10% largerthan that of the experimental result. In addition, the verticaldisplacement of test results is much larger than the numer-

Figure 13. Distribution of vertical displacement by bending.

Figure 14. Distribution of torsional angle by torsion.

Figure 15. Relationship between the bending moment and thevertical displacement.

ical analyses. This is because the joint looseness betweenthe test model and supporter was not excluded.

Finally, the test model is provided for the progressivecollapse analysis under combined loads. For the proposedsimplified method, the torsional moment is applied first inthe elastic state and then progressive collapse analysis isexecuted by adding increments of the vertical displacementat the fore end.

Figure 16 shows the loci of the bending and tor-sional moments at the fixed end obtained by the FEMfor different magnitude of initial torsional moment. Thebending moment M and the torsional moment T arenon-dimensionalised by the ultimate longitudinal bendingstrength MU ( = 7.20 × 103 kNm) and the ultimate tor-sional strength TU ( = 2.55 × 103 kNm), respectively. Theultimate bending strengths calculated by the proposed sim-plified method are plotted in the figure.

When the bending moment is predominant comparedto the torsional moment, the ultimate strength calculated by

Figure 16. Ultimate strength interaction relationships betweenthe vertical bending and torsion.

the proposed simplified method is in good agreement withthat by LS-DYNA. With increase in the applied torsionalmoment, however, the present method cannot accuratelypredict the ultimate bending strength. This is because it isassumed that the buckling of subdivided elements of thecross-section is not induced by the shear stresses due to theelastic torsional analysis and because the plastic interac-tions between shear and normal stresses are not taken intoaccount, that is, it is assumed that d12 and d21 of Equation (6)remain zero in the inelastic condition. Although there aresuch a clear limit and some points to be improved in theapplication, the proposed method gives a reasonable esti-mate of the effect of the torsional moment on the hull-girderbending strength, provided that the bending moment is pre-dominant, as is often the case of the hull girder collapse.

The future work to be conducted is to clarify the limitin the application to the real ships considering the responseof the bending and torsional moments in waves.

5. Conclusions

Three 1/13-scale models of a container ship were providedfor the progressive collapse tests so as to evaluate the ef-fect of the torsional moment on the ultimate strength of thecontainer ship. The results have been compared with thoseof the elasto-plastic FEM analysis using LS-DYNA. On theother hand, in order to develop a practical method to esti-mate the ultimate longitudinal strength of a ship hull girdersubjected to combined torsion and bending, a simplifiedcalculation method of the ultimate hull girder strength hasbeen presented and applied to test models. Those modelsare idealised as thin-walled beams. First, the warping andshear stresses of subdivided elements of each cross-sectionare calculated by the elastic torsional analyses based on Fu-jitani’s method (Park et al. 1997). Next, the average stress–average strain relationships of the subdivided elements arederived according to the formulae of CSR (IACS 2006).At this time, the buckling of the elements is taken intoaccount and the effects of the shear stresses obtained bythe torsional analysis are considered by reducing the yieldstrength of them. Then, the warping stresses are set to thesubdivided elements as initial stresses and the progressivecollapse analysis is performed based on Smith’s method(Smith 1977) under the above conditions.

The following conclusions can be drawn from thepresent experiment and analysis:

(1) The calculated results by the explicit FEM programLS-DYNA show higher ultimate strength than thetest results for the three loading conditions, how-ever, their behaviours are roughly in accordancewith the test results;

(2) When bending moment is predominant for the testmodels as compared with the torsional moment, theultimate strength calculated by the proposed sim-

plified analysis method is roughly in good agree-ment with that by LS-DYNA. However, the presentmethod needs to improve the treatment about ini-tial imperfections and the buckling of subdividedelement due to the shear stress induced by torsion.

Disclosure statement

No potential conflict of interest was reported by the authors.

FundingThis work was supported by JSPS Grant-in-Aid for ScientificResearch (A) [grant number 23246150].

ReferencesCaldwell JB. 1965. Ultimate longitudinal strength. Trans RINA.

107:411–430.Class NK. 2012. Guidelines for container carrier struc-

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Hirdaris SE, Bai W, Dessi D, Ergin A, Gu X, Hermundstad OA,Huijsmans R, Iijima K, Nielsen UD, Parunov J, Fonseca N,Papanikolaou A, Argyriadis K. 2014. Loads for use in thedesign of ships and offshore structures. Ocean Eng. 78:131–174.

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Paik JK, Kim DK, Park DH, Kim HB, Mansour AE, Caldwell JB.2013. Modified Paik-Mansour formula for ultimate strengthcalculations of ship hulls. Ships Offshore Struct. 8(3–4):245–260.

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Paik JK, Thayamballi AK, Pedersen PT, Park YI. 2001. Ultimatestrength of ship hulls under torsion. Ocean Eng. 28:1097–1133.

Park SW, Fujii D, Fujitani Y. 1997. A finite element analysisof discontinuous thin-walled beams considering nonuniformshear warping deformation. Comput Struct. 65:17–27.

Smith CS. 1977. Influence of local compressive failure on ulti-mate longitudinal strength of a ship’s hull. Proceedings of theInternational Symposium on Practical Design in Shipbuilding(PRADS), Tokyo, Japan; p. 73–79.

Sun HH, Guedes Soares C. 2003. An experimental study of ulti-mate torsional strength of a ship-type hull girder with a largedeck opening. Marine Struct. 16(1):51–67.

Tatsumi A, Ogawa H, Tanaka Y, Fujikubo M, Iijima K. 2011.Progressive collapse analysis of a ship hull girder under com-bined bending and torsion. Proceedings of the 25th Asian-Pacific Technical Exchange and Advisory Meeting on MarineStructures (TEAM2011), Incheon, Korea.

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analysis method of ultimate hull girder strength under combined loads

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