Research ArticleAnalytical Method for Evaluating the Impact Response ofStiffeners in a Ship Side Shell Subjected to Bulbous Bow Collision

Min Zhang12 Zhijie Zhu 2 Yang Zeng 2 Jingxi Liu 23 and Zhiqiang Hu4

1School of Mechanical Engineering Wuhan Polytechnic University Wuhan 430048 China2School of Naval Architecture and Ocean Engineering Huazhong University of Science and Technology Wuhan 430074 China3Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE) Shanghai 200240 China4School of Engineering Newcastle University Newcastle upon Tyne NE1 7RU UK

Correspondence should be addressed to Zhijie Zhu 469087247qqcom

Received 23 November 2019 Revised 13 March 2020 Accepted 23 March 2020 Published 14 April 2020

Academic Editor Ali Ramazani

Copyright copy 2020 Min Zhang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper addresses a simplified analytical method for evaluating the impact responses of the stiffeners in a ship side shellsubjected to head-on collision by a bulbous bow e stiffeners are classified as the ldquocentral stiffenerrdquo and the ldquolateral stiffenerrdquoaccording to their relative position to the bulbous bow In analytical predictions it is assumed that the flexural bending of thecentral stiffener and plate occurs simultaneously However the deformation mode of the central stiffener outside the indentercontact region is simplified as linear to derive its deformation resistance e curved deformation mode of the lateral stiffener isproposed to calculate the deformation resistance and to consider the interaction effect with the plate which can cause the plate tofracture earlier Model tests with three specimens (one unstiffened plate for reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed to validate the proposed analytical method Resistance-penetration curves anddamage shapes for the three specimens are obtained e experimental results illustrate the effects of the stiffeners on thedeformation resistance and fracture initiation of the stiffened plate and the influence of stiffener tripping on the lateral resistanceMoreover the experimental and analytical predicted results correspond well suggesting that the proposed analytical method canaccurately predict the crashworthiness of a ship side shell subjected to bulbous bow collision

1 Introduction

Ship side shells are generally equipped with stiffened steelpanels to simplify fabrication additionally these panelshave an excellent strength-to-weight ratio During thesailing life of a ship the ship side may suffer various typesof loads Among the applied loadings the load from acollision with another ship can lead to serious conse-quences such as loss of structural integrity flooding of theship tank and severe oil pollution erefore accuratecrashworthiness assessment of ship side shells in thepredesign stage has been continuously studied byengineers

e commonly used approaches in ship collision in-vestigations are experiments numerical simulations andsimplified analytical methods [1] Experiments can provide

reliable data with respect to deformation and failure patternsand the characteristics of resistance-penetration responseswhich can be used to verify the other two methods Anumber of scaledmodel tests of stiffened plates punched by aspherical or conical indenter to fracture initiation have beenperformed [2ndash5] In all the tests the stiffeners can begenerally categorized into two types due to the differentdeformation driving factors Taking the experiments con-ducted by Kotildergesaar et al [4] as an example as shown inFigure 1 the stiffener immediately below the indenter theldquocentral stiffenerrdquo deforms due to the direct punch of theindenter e deformation of the stiffener away from theimpact position the ldquolateral stiffenerrdquo is driven by thedeformed plate In previous model tests the deformationpatterns of the lateral stiffener were similar For the centralstiffener different tripping extents can be observed

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 5025438 11 pageshttpsdoiorg10115520205025438

erefore the influence of central stiffener tripping on thelateral deformation resistance is particularly investigatedthrough model tests

Unlike experiments numerical simulations are low-costand easily repeatable with the help of powerful computerse numerical simulation method has the ability to predictthe collapse mode and reaction force of structures subjectedto collisions when provided with appropriate modelingparameters Until now numerous failure criteria consid-ering different factors (stress state loading path mesh sizestrain rate etc) that can influence plate fracture have beenproposed to predict the initial fracture of ship structures incollision and grounding analysis [6ndash8]

Compared with the first two methods the simplifiedanalytical method is the preferred tool in the predesign stagebecause this method can most rapidly assess the crash-worthiness of ship structures [9ndash11] Extensive studies havebeen conducted to estimate the large deformation resistanceand fracture initiation of an unstiffened plate subjected tolateral indentation by a spherical indenter [12ndash15] Ana-lytical methods for the stiffener components of the stiffenedplate were generally proposed in cases in which the stiffenedplates were punched by indenters with linear or rectangulartops [16ndash19] In these studies both the deformation modesof the plate and the attached stiffeners are treated as a linearform where the lateral resistances of the stiffeners are at-tributed to the rotation of the plastic hinges at the appliedload and the support and membrane tension over theplastically deformed region However the deformationmodes of the stiffeners are different in the cases of a stiffenedplate punched by a sphere As shown in Figure 1 the de-formation mode of the central stiffener is consistent withthat of the plate ie with a curved deformation profile and aspherical top In addition the deformation mode of thelateral stiffener is identical to that of the deformed plate ie

with a curved deformation profile Until now analyticalmethods to obtain the lateral indentation resistances forthese two different forms of stiffeners have seldom beenreferred to erefore the current study is intended topresent the deformation modes of the central and lateralstiffeners and the corresponding deformation resistances

Moreover fracture prediction of the ship side plate iscrucial for estimating energy dissipation and structural re-sistance Several analytical expressions have been proposedto obtain the critical penetration depth of an unstiffenedplate indented by a sphere [12ndash15] Nevertheless the addedstiffeners can lead to higher stiffness but reduce flexibilityand result in earlier fracture compared with the response ofan unstiffened plate [2] erefore an analytical solution forthe influence of stiffeners on the critical penetration depth ofthe plate should be investigated In summary the aim of thepresent analytical study is to build equations to predict thedeformation resistances of the attached stiffeners in thestiffened plate and the initial fracture of the stiffened plate

In this study simplified analytical methods are proposedto predict the deformation resistance and the critical pen-etration depth of a stiffened plate punched by a bulbous bowDeformation modes for the central and lateral stiffeners areproposed and the resistance-penetration relations are de-rived by theoretical calculations In addition the reductionin the critical penetration depth with the stiffener is derivedconsidering the interaction effect between the plate and thestiffener Moreover experimental tests are conducted onspecimens with different numbers of stiffeners quasistati-cally punched by a conical indenter e experimental re-sults validate the feasibility of the proposed method Finallysome conclusions are drawn

2 Analytical Predictions

is section presents analytical predictions for the large de-formation resistances of the central and lateral stiffeners and theinitial fracture of the stiffened plate in a typical ship bulbousbow-side collision scenario as shown in Figure 2 In developingthe analytical solutions several assumptions aremade as follows

(1) e ship side shell is assumed to undergo head-oncollision by a bulbous bow at the midspan betweenthe web girders

(2) e web girders are assumed to be stiff enough toconstrain the boundary of the outer side plate

(3) e bulbous bow is assumed to be rigid and theshape of the bulbous bow is simplified as conical

(4) e residual stress and initial deflection of thestiffened plate are not considered

Based on the assumptions theoretical deformationmodes and the derived formulae for the central stiffener andthe lateral stiffener are described in detail e deformationshape of the central stiffener is identical to that of the platebut the region not in contact with the indenter is treated aslinear to merely calculate the lateral resistance for simplicityIn particular a curved deformation mode for the lateralstiffener is proposed to consider its interaction effect with

Lateralstiffener

x

yCentralstiffener

Figure 1 Different types of stiffener [4]

2 Mathematical Problems in Engineering

the side plate which can influence the initial fracture of theship side plate

21 Large Deformation Resistance of the Stiffeners e an-alytical method to predict the deformation resistances of thecentral and lateral stiffeners is presented in this section Ingeneral the stiffener used for ship construction is the bulb-bar stiffener Nevertheless the resistance of the flat-barstiffener is analyzed for simplicity

211 Central Stiffener e movement of the central stiff-ener is driven by the indenter thus the deformation shape ofthe central stiffener is the same as that of the side plate In thewhole deformation process the central stiffener is assumedto maintain an in-plane deformation process ie tripping ofthe stiffener is ignored

Initially in the elastic stage the central stiffener mainlyexhibits a bending effect Assuming that the central stiffeneris placed in the x-w coordinate system as shown in Figure 3the external load work is equal to the strain energy

Fe cswcsmax

2 1113946

lcs

M2e cs(x)

2EIcs

dx (1)

where wcsmax is the maximum deflection of the centralstiffener lcs is the length of the central stiffener Fe_cs is theexternal load at the elastic stageMe_cs(x) is the moment in thestiffener applied by Fe_cs and can be obtained asMe_cs Fe_csx2 E is the elastic modulus and Ics is the moment of inertia Inthe pure bending state the neutral axis for the stiffened panelis located in the plate and the stiffener dominates the bendingeffect [20] Ics can be expressed as

Ics h3

s ts

3 (2)

where hs and ts are the height and thickness of the stiffenerrespectively

en according to the energymethod the instantaneousforce of the stiffener in the elastic stage can be obtained byintegrating equations (1) and (2)

Fe cs 2Eh3

s tswcsmax

l3s (3)

where ls is the half-length of the stiffenerIn the plastic stage the stiffener will experience bending and

tension simultaneouslyis deformationmode is also shown inFigure 3 According to Zhang et al [15] the deformation shapeof the curved stiffener can be expressed by a parabola

x(w) w2 minus 2wcsmaxw

2Rb cosφc minus 1( 1113857tanφc

(4)

where x and w are the horizontal and vertical distances fromany point on the plate to the plate boundary Rb is the radiusof the sphere and φc is the angle from the center of theindenter to point C as shown in Figure 3 Point C is theoutmost contact point between the plate and the indenter

Moreover wcsmax can be expressed as

wcsmax φc( 1113857

2 ls minus Rb sinφc( 1113857Rb 1 minus cosφc( 1113857tanφc + R2b 1 minus cosφc( 1113857

21113969

(5)

e actual deformation of the central stiffener illustrates thatthe central stiffener exhibits global bending and tension effectswhen punched by a sphere However the stiffener outside thecontact regionwith the indenter is treated as linear in the currentstudy for simplicity to obtain a large deformation resistanceus the global bending effect of the stiffener will concentrate inthe plastic hinges (see the dashed area in Figure 3)

e rotation angle of the stiffener c can be expressed as

tan c wc

xc

(6)

where wc and xc are the coordinate values at point C andthey can be expressed as

wc wcsmax minus Rb + Rb cosφc

xc ls minus Rb sinφc(7)

Striking ship

Ship side

Web girders impact location

Front view

2400

2400

Figure 2 Ship bow-side collision scenario

w

x

C

Rb

ls

φcwcsmax

γ

PlateCentral stiffener

Figure 3 Deformation mode of the central stiffener

Mathematical Problems in Engineering 3

Moreover wc and wcsmax are assumed to have the fol-lowing relation

wc ccswcsmax (8)

where ccs is the ratio of the deflection of point C to themaximum deflection of the central stiffener

us the relation between the indentation velocities ofthe point C _wc and the central stiffener _wcsmax can beexpressed as

_wc ccs _wcsmax (9)

en the angular velocity of rotational stiffener _c can beobtained

_c _wc

1 + tan2 c( 1113857xc

(10)

e bending energy rate of the central stiffener can beexpressed as

_Eb cs 4Mpsts _c (11)

where Mps is the plastic bending moment per unit thicknessand can be obtained as

Mps σ0sh

2s

2 (12)

where σ0s is the flow stress of the stiffener which is theaveraged value of the yield stress σys and ultimate tensionstress σus [21]

Moreover the tension strain εcs and tension strain rate_εcscan be approximated as

εcs 12(tan c)

2

_εcs tan c_wc

xc

(13)

us the rate of membrane tension of the stiffener canbe expressed as

_Em cs 2BScs

σ0ts _εcsdScs (14)

where Scs is the side of the area on which the central stiffenerexperiences tension and can be obtained from Scs xchs

According to the upper bound theorem the equilibriumequation can be expressed as

Fp cs _wcsmax _Eb cs + _Em cs (15)

where Fp_cs is the resistance of the stiffener in the plasticstage

Finally the instantaneous resistance of the stiffener at theplastic stage can be derived by substituting (11) and (14) into(15)

Fp cs 2σ0shswcts

ls minus Rb sinφc( 1113857wcsmax

hs

1 + tan 2c+ wc1113888 1113889 (16)

212 Lateral Stiffener e deformation of the lateralstiffener is driven by the deformed plate It is assumed thatthe lateral stiffener deforms simultaneously with the sideplate us the overall movement of the lateral stiffener isthe superposition of the lateral deflection from the plate androtation with the plate Here it is assumed that the stiffenerswill remain perpendicular to the side plate until platefracture occurs

e deformation mode of the stiffener is shown inFigure 4e ends of the stiffener are welded to adjacent webgirders that can constrain the rotation of the stiffener locallyus plastic hinges will be generated at the ends of thestiffeners which can lead to an out-of-plane bending effectmarked by the red dashed lines in Figure 4 However thebending effect is neglected because the stiffener out-of-planebending moment is much smaller than its in-plane bendingmoment erefore only the membrane tension effect isconsidered for the lateral stiffener to predict its deformationresistance

As shown in Figure 4 the deformed stiffener is alsoplaced in the x-w rectangular coordinate system e de-formation shape of the stiffener can be expressed as

wls wlsmax sinπx

2ls1113888 1113889 (17)

where wls is the deflection of the stiffener and wlsmax is themaximum transverse deflection of the stiffener

According to (4) wlsmax can be obtained as

wlsmax w xls( 1113857 wcsmax

minus

w2csmax + 2Rb cosφc minus 1( 1113857tanφcxls

1113969

(18)

where xls is the initial horizontal distance between thestiffener and plate edge

Similar to the relation between wc and wcsmax for thecentral stiffener wlsmax and wcsmax have the followingrelations

wlsmax clswcsmax

_wlsmax cls _wcsmax(19)

where cls is the ratio of the maximum deflection between thelateral stiffener and the central stiffener

As the stiffener is assumed to be displaced vertically thetension strain εls can be approximated as

εls asymp12

zwls

zx1113888 1113889

2

w2lsmax

π2

8l2scos2

πx

2ls1113888 1113889 (20)

e strain rate of the stiffener _εls can be expressed as

_εls wlsmaxπ2

4l2scos2

πx

2ls1113888 1113889 _wlsmax (21)

e rate of membrane energy can be expressed as

_Em ls BSls

σ0sts _εlsdSls (22)

where Sls is the initial area of the lateral stiffener

4 Mathematical Problems in Engineering

According to the upper bound theorem the rate of workby the external load is equal to the rate of internal energydissipation e equilibrium can be expressed as

Fls middot _wcsmax _Em ls (23)

us the instantaneous resistance of the lateral stiffenerFls can be derived as

Fls σ0stshsπ2w2

lsmax4lswcsmax

(24)

22 Fracture Prediction of the Stiffened Plate Analyticalfracture prediction for the ship side plate under a bulbousbow striking scenario is crucial for estimating the criticalenergy dissipation Several equations were proposed tocalculate the critical penetration depth of an unstiffenedplate Recently an expression that was validated by anumber of experiments was proposed by Zhang et al [15]is expression is as follows

wp f c1

(7n + 076)b0Rb

1113969

(25)

where c1 is the correction coefficient and has been calibratedto be 05 n is the work hardening exponent of the platematerial and b0 is the half-width of the plate

In particular the effect of the lateral stiffener on theinitial fracture of the ship side plate is considered in thissection An analytical method is proposed to predict thefracture initiation of a stiffened side shell

Section 212 demonstrates that the deformation of thelateral stiffener is driven by the plate Actually the lateralstiffener interacts with the plate in the large deformationprocess us the lateral stiffener is able to restrain thedeformation of the plate and finally reduce the criticalpenetration depth of the plate Given the critical penetrationdepth of the plate wp f the critical penetration depth for thestiffened plate wsp f can be expressed as

wsp f wp f minus dwls (26)

where dwls is the penetration depth reduced by the lateralstiffeners

e cross section at the maximum deflection of thelateral stiffener is extracted to analyze the influence of thestiffener on the deflection of the plate Figure 5(a) shows theload state of the plate at the plate-stiffener intersection Atthe intersection angular discontinuity of the curved plateoccurs due to the vertical force Fs from the stiffener eplate also sustains tension from the adjacent plates whichare denoted as Fp1 and Fp2 Figure 5(b) depicts the internalforce of the stiffener where the infinitesimal fragment is ds

in length According to Section 212 the stiffener experi-ences a tension effect and the tension force FNs can beexpressed as

FNs σ0stshs (27)

us the vertical force Fs applied by the stiffener can beobtained as

Fs FNs

ds

Rs

(28)

where Rs is the radius of curvature of the deformed stiffenerAccording to (17) Rs can be obtained as

Rs 1 + _w2

ls(x)( 111385732

eurowls(x)1113868111386811138681113868

1113868111386811138681113868 (29)

Moreover the tension forces from the plate are assumedto be very similar and are expressed as

Fp1 Fp2 σ0ptpds (30)

where σ0p is the flow stress of the plate and tp is the thicknessof the plate

On the y-axis the resultant force should be zerous Fscan also be expressed as

Fs Fp1(sinφ minus sin(φ minus dφ)) (31)

Considering (28) and substituting (29) and (30) into(31) the increment in the rotation angle at the plate-stiffenerintersection can be approximated as

dφ wlsmaxπ2σ0shsts

4σ0pl2s tp

(32)

According to (4) the instantaneous angle φ at the plate-stiffener intersection satisfies the following relation

tan φ dw

dx

Rb cos φc minus 1( 1113857tan φc

1 minus 1cls( 1113857( 1113857wlsmax (33)

Based on (33) the lateral deflection of the plate limitedby the stiffener dwls can be approximated as

dwls φc( 1113857 wlsmax minus wp1113872 1113873wlsmax

Rb cosφc minus 1( 1113857tanφc

dφ (34)

Substituting (32) into (34) dwls can be further expressedas

dwls φc( 1113857 σ0shstsw

2lsmaxπ

2 wlsmax minus wcsmax( 1113857

4σ0pRbl2s tp cosφc minus 1( 1113857tanφc

(35)

3 Experimental Details

31 Penetration Test Design e quasistatic indentationexperiments were performed at Huazhong University ofScience and Technology e setup used in the experimentsis presented in Figure 6(a) e specimens were clampedbetween a bottom flange and an upper flange which weremade of Q345 steel with a thickness of 25mm ey werefixed together by M20 bolts e dimensions of the

ls

w

xwlsmax

Figure 4 Deformation mode of the lateral stiffener

Mathematical Problems in Engineering 5

experimental clamping system are illustrated in Figure 6(b)In addition the ends of the stiffeners were double-sidewelded on the bottom flange to restrain their freedom asshown in Figure 6(c) Current fixtures were proven toprovide clamped boundary constraints through validation

by numerical simulations with solid elements consideringall the fixtures Moreover as in previous studies a bulbousbow is generally treated as rigid and simplified as a conicalindenter defined by the top radius [2 14] as depicted inFigure 2 us the indenter shape was designed to be

Displacementtransducers

Forcetransducer

Conical indenter

Camera 1 Camera 2

(a)

Bottom flange

Specimen

Upper flange600

355

2525

2525

0 20

(b) (c)

Figure 6 Designed penetration test (a) Setup (b) Dimensions of the clamping system (c) End connections of the stiffeners

Fp1

Fp2

φ ndash dφ

Fs

y

xφ

Lateral stiffener

Plate

Tangent line

Horizontal line

φ

(a)

FNs FNs

ds

Rs

(b)

Figure 5 Load state for the components (a) Plate (b) Stiffener

6 Mathematical Problems in Engineering

conical e geometry and dimensions are shown inFigure 7

e initial distance between the indenter and thespecimen was approximately 20mme deformation of thespecimens was enforced at a rate of sim10mmmin [2ndash4] at themidspan by hydraulic cylinders with a 100 ton maximumcapacity A 100-ton load cell fixed between the hydrauliccylinder and the indenter and two displacement sensorsjointed on the indenter were utilized to obtain the force-timeand displacement-time curves respectively To visualize thedeformations 50times 50mm grids were drawn on the front andback sides of the specimens Moreover two cameras wereplaced under the bottom flange to capture the deformationprocess of the specimens

32 Specimens ree specimens were designed at one-fourthscale from the ship side as shown in Figure 8e unstiffenedplate (denoted as ldquoUSrdquo) was used as a reference to estimate theeffects of the stiffeners on the resistance and critical pene-tration depth of the ship side panel Stiffened plates with twoand three stiffeners (denoted as ldquo2FBrdquo and ldquo3FBrdquo respec-tively) were used to analyze the effects of the lateral stiffenerand the central stiffener respectively e dimensions of thespecimens are also illustrated in Figure 8 where the central600times 600mm square is the exposed area of the panels and thesurrounding areas with a width of 155mm are clamped to thespecimens In addition all the stiffeners were 55mm inheighte weld joint plate-stiffeners are alternatively double-sided filled welds with a size of sim30mm e selected elec-trodes were ASME (American Society of Mechanical Engi-neers) ER70S-6 with a diameter of 08mm e selection ofthe weld size and the electrode base material follows standardshipyard welding procedures Moreover the finished speci-mens were hammered at the plate-stiffener intersections by amallet to release the residual stress Furthermore replicatetests for each specimen were performed to ensure the reli-ability of the experimental results

e material used for the plates and stiffeners is grade Bnormal structural steel qualified by the CCS (China Classi-fication Society) considering the availability of the thin steelplate and the loading capacity of the hydraulic cylinderesesteel plates were from the same batch supplied by WISCO(Wuhan Iron and Steel (Group) Company) and were all315mm thick To obtain the mechanical properties of thesteel quasistatic tensile tests are conducted using threestandard tensile specimens and procedures e dimensionsof the machined tension test pieces are shown in Figure 9Based on the displacement-prescribed tensile tests performedon the universal testing machine the engineering stress-strainbehavior of the material can be obtained e tensile engi-neering stress-strain curve is presented in Figure 10 emechanical properties of the plate material are summarized inTable 1 where n is obtained according to Ref [15]

33 Results e experimentally measured resistance-pen-etration curves are shown in Figure 11 In addition thedeformation shapes when the plates are initially fracturedare shown in Figure 12 for the three specimensese curves

demonstrate that the lateral stiffener and the central stiffenercan both supply an extent of lateral resistance at differentpenetration depths compared with the unstiffened plate Inaddition the improved resistances due to the lateral andcentral stiffeners are close at different penetration depthse resistance improved by the central stiffener is moreremarkable than that by the lateral stiffener In addition theresistance-penetration curves indicate that the criticalpenetration depths for the three specimens are differentCompared with specimen US the reductions in the criticalpenetration depth of specimen 2FB and specimen 3FB are91mm and 78mm respectively is value for specimen2FB is larger because the horizontal distance between thelateral stiffener and the impact position is smaller which canlead to a stronger restriction effect in the plate

Moreover the influence of stiffener tripping on thelateral deformation resistance is evaluated Figure 13 showsthe experimental observations when fractures are initiallygenerated in the side plate e central stiffener in specimen3FB remains upright while the central stiffener in thereplicate test trips down Current experiments illustrate thatthe tripping extent of the stiffeners can vary greatly due tothe differences in the welding conditions and the relativespecimen-indenter impact locations In addition the cor-responding resistance-penetration responses for these twospecimens are shown in Figure 13 e compared curvesdemonstrate that the discrepancy of the resistance-pene-tration responses is small in the experiments which provesthat the tripping of the central stiffener has a slight influenceon the resistance response Clearly this conclusion is dif-ferent from that stated by Yu et al [20] e reason for thisdifference is attributed to the variations in the structuralforms of the central stiffener in these two studies e in-vestigated stiffener in Yu et al [20] is the T-profile stiffenerwhere the top flange will experience a remarkable membranetension effect Tripping of the T-profile stiffener can reducethe lateral deflection of the top flange thereby leading to asignificant decrease in resistance Meanwhile tripping of thestiffenerrsquos web will have little influence on the lateral re-sistance of the stiffened plate

4 Verification of the AnalyticalPrediction Method

In this section the proposed analytical method is verifiedwith respect to the large deformation resistance and the

250

R75

70deg

Figure 7 Dimensions of the conical indenter (dimensions in mm)

Mathematical Problems in Engineering 7

critical penetration depth of the stiffened plate by comparingthe analytically predicted resistance-penetration curves withthe experimental curves as shown in Figure 14 Moreoverthe workflow for obtaining the resistance-penetration re-lation for the stiffened plate is given in Figure 15

e current study proposes not only analytical predic-tions for the deformation resistance of the central stiffenerand the lateral stiffener but also a method to obtain thecritical penetration depth of the stiffened plate With theanalytical solutions for the large deformation resistance andthe critical penetration depth of the unstiffened plate in Ref[15] the analytical predictions for a stiffened plate can beobtained from large deformation to initial fracture ecompared resistance-penetration curves shown in Figure 14illustrate that the analytical method can adequately predict

0 30 60 90 1200

100

200

300

400(1007mm 3662kN)

(994mm 3092kN)

Resis

tanc

e (kN

)

Penetration (mm)

Specimen USSpecimen 2FBSpecimen 3FB

(1085mm 3155kN)

Figure 11 Experimental resistance-penetration responses

PL315oslash22

910

910

600

600

155115605

103 times 7

103 times 6 106106

(a)

FB55 times 315

200 times 3

(b)

FB55 times 315

150 times 4

(c)

Figure 8 Dimensions of the specimens (dimensions in mm) (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

G-Gauge lengthW-WidthT-ThicknessR-Radius of filletL-Overall lengthA-Length of reduced sectionB-Length of grip sectionC-Width of grip section

50mm125mm315mm125mm200mm85mm

~50mm20mm

B

G

A

CW

BL

TR

Figure 9 Dimensions of the standard tension tested piece (ASTME8) G gauge length (50mm) W width (125mm) T thickness(315mm) R radius of fillet (315mm) L overall length (200mm)A length of reduced section (85mm) B length of grid section(sim50mm) C width of grip section (20mm)

000 005 010 015 020 025 030 0350

70

140

210

280

350

420

Stre

ss (M

Pa)

Strain (ndash)

Figure 10 Engineering stress-strain curve

Table 1 Mechanical properties of material

Property Symbol Units SpecimensYoungrsquos modulus E GPa 207Poissonrsquos ratio v mdash 03Mass density ρ kgm3 7850Yield stress σY MPa 3028Ultimate tensile strength σu MPa 4084Fracture strain εf mdash 0306Strain-hardening index n mdash 02

8 Mathematical Problems in Engineering

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

the side plate which can influence the initial fracture of theship side plate

21 Large Deformation Resistance of the Stiffeners e an-alytical method to predict the deformation resistances of thecentral and lateral stiffeners is presented in this section Ingeneral the stiffener used for ship construction is the bulb-bar stiffener Nevertheless the resistance of the flat-barstiffener is analyzed for simplicity

211 Central Stiffener e movement of the central stiff-ener is driven by the indenter thus the deformation shape ofthe central stiffener is the same as that of the side plate In thewhole deformation process the central stiffener is assumedto maintain an in-plane deformation process ie tripping ofthe stiffener is ignored

Initially in the elastic stage the central stiffener mainlyexhibits a bending effect Assuming that the central stiffeneris placed in the x-w coordinate system as shown in Figure 3the external load work is equal to the strain energy

Fe cswcsmax

2 1113946

lcs

M2e cs(x)

2EIcs

dx (1)

where wcsmax is the maximum deflection of the centralstiffener lcs is the length of the central stiffener Fe_cs is theexternal load at the elastic stageMe_cs(x) is the moment in thestiffener applied by Fe_cs and can be obtained asMe_cs Fe_csx2 E is the elastic modulus and Ics is the moment of inertia Inthe pure bending state the neutral axis for the stiffened panelis located in the plate and the stiffener dominates the bendingeffect [20] Ics can be expressed as

Ics h3

s ts

3 (2)

where hs and ts are the height and thickness of the stiffenerrespectively

en according to the energymethod the instantaneousforce of the stiffener in the elastic stage can be obtained byintegrating equations (1) and (2)

Fe cs 2Eh3

s tswcsmax

l3s (3)

where ls is the half-length of the stiffenerIn the plastic stage the stiffener will experience bending and

tension simultaneouslyis deformationmode is also shown inFigure 3 According to Zhang et al [15] the deformation shapeof the curved stiffener can be expressed by a parabola

x(w) w2 minus 2wcsmaxw

2Rb cosφc minus 1( 1113857tanφc

(4)

where x and w are the horizontal and vertical distances fromany point on the plate to the plate boundary Rb is the radiusof the sphere and φc is the angle from the center of theindenter to point C as shown in Figure 3 Point C is theoutmost contact point between the plate and the indenter

Moreover wcsmax can be expressed as

wcsmax φc( 1113857

2 ls minus Rb sinφc( 1113857Rb 1 minus cosφc( 1113857tanφc + R2b 1 minus cosφc( 1113857

21113969

(5)

e actual deformation of the central stiffener illustrates thatthe central stiffener exhibits global bending and tension effectswhen punched by a sphere However the stiffener outside thecontact regionwith the indenter is treated as linear in the currentstudy for simplicity to obtain a large deformation resistanceus the global bending effect of the stiffener will concentrate inthe plastic hinges (see the dashed area in Figure 3)

e rotation angle of the stiffener c can be expressed as

tan c wc

xc

(6)

where wc and xc are the coordinate values at point C andthey can be expressed as

wc wcsmax minus Rb + Rb cosφc

xc ls minus Rb sinφc(7)

Striking ship

Ship side

Web girders impact location

Front view

2400

2400

Figure 2 Ship bow-side collision scenario

w

x

C

Rb

ls

φcwcsmax

γ

PlateCentral stiffener

Figure 3 Deformation mode of the central stiffener

Mathematical Problems in Engineering 3

Moreover wc and wcsmax are assumed to have the fol-lowing relation

wc ccswcsmax (8)

where ccs is the ratio of the deflection of point C to themaximum deflection of the central stiffener

us the relation between the indentation velocities ofthe point C _wc and the central stiffener _wcsmax can beexpressed as

_wc ccs _wcsmax (9)

en the angular velocity of rotational stiffener _c can beobtained

_c _wc

1 + tan2 c( 1113857xc

(10)

e bending energy rate of the central stiffener can beexpressed as

_Eb cs 4Mpsts _c (11)

where Mps is the plastic bending moment per unit thicknessand can be obtained as

Mps σ0sh

2s

2 (12)

where σ0s is the flow stress of the stiffener which is theaveraged value of the yield stress σys and ultimate tensionstress σus [21]

Moreover the tension strain εcs and tension strain rate_εcscan be approximated as

εcs 12(tan c)

2

_εcs tan c_wc

xc

(13)

us the rate of membrane tension of the stiffener canbe expressed as

_Em cs 2BScs

σ0ts _εcsdScs (14)

where Scs is the side of the area on which the central stiffenerexperiences tension and can be obtained from Scs xchs

According to the upper bound theorem the equilibriumequation can be expressed as

Fp cs _wcsmax _Eb cs + _Em cs (15)

where Fp_cs is the resistance of the stiffener in the plasticstage

Finally the instantaneous resistance of the stiffener at theplastic stage can be derived by substituting (11) and (14) into(15)

Fp cs 2σ0shswcts

ls minus Rb sinφc( 1113857wcsmax

hs

1 + tan 2c+ wc1113888 1113889 (16)

212 Lateral Stiffener e deformation of the lateralstiffener is driven by the deformed plate It is assumed thatthe lateral stiffener deforms simultaneously with the sideplate us the overall movement of the lateral stiffener isthe superposition of the lateral deflection from the plate androtation with the plate Here it is assumed that the stiffenerswill remain perpendicular to the side plate until platefracture occurs

e deformation mode of the stiffener is shown inFigure 4e ends of the stiffener are welded to adjacent webgirders that can constrain the rotation of the stiffener locallyus plastic hinges will be generated at the ends of thestiffeners which can lead to an out-of-plane bending effectmarked by the red dashed lines in Figure 4 However thebending effect is neglected because the stiffener out-of-planebending moment is much smaller than its in-plane bendingmoment erefore only the membrane tension effect isconsidered for the lateral stiffener to predict its deformationresistance

As shown in Figure 4 the deformed stiffener is alsoplaced in the x-w rectangular coordinate system e de-formation shape of the stiffener can be expressed as

wls wlsmax sinπx

2ls1113888 1113889 (17)

where wls is the deflection of the stiffener and wlsmax is themaximum transverse deflection of the stiffener

According to (4) wlsmax can be obtained as

wlsmax w xls( 1113857 wcsmax

minus

w2csmax + 2Rb cosφc minus 1( 1113857tanφcxls

1113969

(18)

where xls is the initial horizontal distance between thestiffener and plate edge

Similar to the relation between wc and wcsmax for thecentral stiffener wlsmax and wcsmax have the followingrelations

wlsmax clswcsmax

_wlsmax cls _wcsmax(19)

where cls is the ratio of the maximum deflection between thelateral stiffener and the central stiffener

As the stiffener is assumed to be displaced vertically thetension strain εls can be approximated as

εls asymp12

zwls

zx1113888 1113889

2

w2lsmax

π2

8l2scos2

πx

2ls1113888 1113889 (20)

e strain rate of the stiffener _εls can be expressed as

_εls wlsmaxπ2

4l2scos2

πx

2ls1113888 1113889 _wlsmax (21)

e rate of membrane energy can be expressed as

_Em ls BSls

σ0sts _εlsdSls (22)

where Sls is the initial area of the lateral stiffener

4 Mathematical Problems in Engineering

According to the upper bound theorem the rate of workby the external load is equal to the rate of internal energydissipation e equilibrium can be expressed as

Fls middot _wcsmax _Em ls (23)

us the instantaneous resistance of the lateral stiffenerFls can be derived as

Fls σ0stshsπ2w2

lsmax4lswcsmax

(24)

22 Fracture Prediction of the Stiffened Plate Analyticalfracture prediction for the ship side plate under a bulbousbow striking scenario is crucial for estimating the criticalenergy dissipation Several equations were proposed tocalculate the critical penetration depth of an unstiffenedplate Recently an expression that was validated by anumber of experiments was proposed by Zhang et al [15]is expression is as follows

wp f c1

(7n + 076)b0Rb

1113969

(25)

where c1 is the correction coefficient and has been calibratedto be 05 n is the work hardening exponent of the platematerial and b0 is the half-width of the plate

In particular the effect of the lateral stiffener on theinitial fracture of the ship side plate is considered in thissection An analytical method is proposed to predict thefracture initiation of a stiffened side shell

Section 212 demonstrates that the deformation of thelateral stiffener is driven by the plate Actually the lateralstiffener interacts with the plate in the large deformationprocess us the lateral stiffener is able to restrain thedeformation of the plate and finally reduce the criticalpenetration depth of the plate Given the critical penetrationdepth of the plate wp f the critical penetration depth for thestiffened plate wsp f can be expressed as

wsp f wp f minus dwls (26)

where dwls is the penetration depth reduced by the lateralstiffeners

e cross section at the maximum deflection of thelateral stiffener is extracted to analyze the influence of thestiffener on the deflection of the plate Figure 5(a) shows theload state of the plate at the plate-stiffener intersection Atthe intersection angular discontinuity of the curved plateoccurs due to the vertical force Fs from the stiffener eplate also sustains tension from the adjacent plates whichare denoted as Fp1 and Fp2 Figure 5(b) depicts the internalforce of the stiffener where the infinitesimal fragment is ds

in length According to Section 212 the stiffener experi-ences a tension effect and the tension force FNs can beexpressed as

FNs σ0stshs (27)

us the vertical force Fs applied by the stiffener can beobtained as

Fs FNs

ds

Rs

(28)

where Rs is the radius of curvature of the deformed stiffenerAccording to (17) Rs can be obtained as

Rs 1 + _w2

ls(x)( 111385732

eurowls(x)1113868111386811138681113868

1113868111386811138681113868 (29)

Moreover the tension forces from the plate are assumedto be very similar and are expressed as

Fp1 Fp2 σ0ptpds (30)

where σ0p is the flow stress of the plate and tp is the thicknessof the plate

On the y-axis the resultant force should be zerous Fscan also be expressed as

Fs Fp1(sinφ minus sin(φ minus dφ)) (31)

Considering (28) and substituting (29) and (30) into(31) the increment in the rotation angle at the plate-stiffenerintersection can be approximated as

dφ wlsmaxπ2σ0shsts

4σ0pl2s tp

(32)

According to (4) the instantaneous angle φ at the plate-stiffener intersection satisfies the following relation

tan φ dw

dx

Rb cos φc minus 1( 1113857tan φc

1 minus 1cls( 1113857( 1113857wlsmax (33)

Based on (33) the lateral deflection of the plate limitedby the stiffener dwls can be approximated as

dwls φc( 1113857 wlsmax minus wp1113872 1113873wlsmax

Rb cosφc minus 1( 1113857tanφc

dφ (34)

Substituting (32) into (34) dwls can be further expressedas

dwls φc( 1113857 σ0shstsw

2lsmaxπ

2 wlsmax minus wcsmax( 1113857

4σ0pRbl2s tp cosφc minus 1( 1113857tanφc

(35)

3 Experimental Details

31 Penetration Test Design e quasistatic indentationexperiments were performed at Huazhong University ofScience and Technology e setup used in the experimentsis presented in Figure 6(a) e specimens were clampedbetween a bottom flange and an upper flange which weremade of Q345 steel with a thickness of 25mm ey werefixed together by M20 bolts e dimensions of the

ls

w

xwlsmax

Figure 4 Deformation mode of the lateral stiffener

Mathematical Problems in Engineering 5

experimental clamping system are illustrated in Figure 6(b)In addition the ends of the stiffeners were double-sidewelded on the bottom flange to restrain their freedom asshown in Figure 6(c) Current fixtures were proven toprovide clamped boundary constraints through validation

by numerical simulations with solid elements consideringall the fixtures Moreover as in previous studies a bulbousbow is generally treated as rigid and simplified as a conicalindenter defined by the top radius [2 14] as depicted inFigure 2 us the indenter shape was designed to be

Displacementtransducers

Forcetransducer

Conical indenter

Camera 1 Camera 2

(a)

Bottom flange

Specimen

Upper flange600

355

2525

2525

0 20

(b) (c)

Figure 6 Designed penetration test (a) Setup (b) Dimensions of the clamping system (c) End connections of the stiffeners

Fp1

Fp2

φ ndash dφ

Fs

y

xφ

Lateral stiffener

Plate

Tangent line

Horizontal line

φ

(a)

FNs FNs

ds

Rs

(b)

Figure 5 Load state for the components (a) Plate (b) Stiffener

6 Mathematical Problems in Engineering

conical e geometry and dimensions are shown inFigure 7

e initial distance between the indenter and thespecimen was approximately 20mme deformation of thespecimens was enforced at a rate of sim10mmmin [2ndash4] at themidspan by hydraulic cylinders with a 100 ton maximumcapacity A 100-ton load cell fixed between the hydrauliccylinder and the indenter and two displacement sensorsjointed on the indenter were utilized to obtain the force-timeand displacement-time curves respectively To visualize thedeformations 50times 50mm grids were drawn on the front andback sides of the specimens Moreover two cameras wereplaced under the bottom flange to capture the deformationprocess of the specimens

32 Specimens ree specimens were designed at one-fourthscale from the ship side as shown in Figure 8e unstiffenedplate (denoted as ldquoUSrdquo) was used as a reference to estimate theeffects of the stiffeners on the resistance and critical pene-tration depth of the ship side panel Stiffened plates with twoand three stiffeners (denoted as ldquo2FBrdquo and ldquo3FBrdquo respec-tively) were used to analyze the effects of the lateral stiffenerand the central stiffener respectively e dimensions of thespecimens are also illustrated in Figure 8 where the central600times 600mm square is the exposed area of the panels and thesurrounding areas with a width of 155mm are clamped to thespecimens In addition all the stiffeners were 55mm inheighte weld joint plate-stiffeners are alternatively double-sided filled welds with a size of sim30mm e selected elec-trodes were ASME (American Society of Mechanical Engi-neers) ER70S-6 with a diameter of 08mm e selection ofthe weld size and the electrode base material follows standardshipyard welding procedures Moreover the finished speci-mens were hammered at the plate-stiffener intersections by amallet to release the residual stress Furthermore replicatetests for each specimen were performed to ensure the reli-ability of the experimental results

e material used for the plates and stiffeners is grade Bnormal structural steel qualified by the CCS (China Classi-fication Society) considering the availability of the thin steelplate and the loading capacity of the hydraulic cylinderesesteel plates were from the same batch supplied by WISCO(Wuhan Iron and Steel (Group) Company) and were all315mm thick To obtain the mechanical properties of thesteel quasistatic tensile tests are conducted using threestandard tensile specimens and procedures e dimensionsof the machined tension test pieces are shown in Figure 9Based on the displacement-prescribed tensile tests performedon the universal testing machine the engineering stress-strainbehavior of the material can be obtained e tensile engi-neering stress-strain curve is presented in Figure 10 emechanical properties of the plate material are summarized inTable 1 where n is obtained according to Ref [15]

33 Results e experimentally measured resistance-pen-etration curves are shown in Figure 11 In addition thedeformation shapes when the plates are initially fracturedare shown in Figure 12 for the three specimensese curves

demonstrate that the lateral stiffener and the central stiffenercan both supply an extent of lateral resistance at differentpenetration depths compared with the unstiffened plate Inaddition the improved resistances due to the lateral andcentral stiffeners are close at different penetration depthse resistance improved by the central stiffener is moreremarkable than that by the lateral stiffener In addition theresistance-penetration curves indicate that the criticalpenetration depths for the three specimens are differentCompared with specimen US the reductions in the criticalpenetration depth of specimen 2FB and specimen 3FB are91mm and 78mm respectively is value for specimen2FB is larger because the horizontal distance between thelateral stiffener and the impact position is smaller which canlead to a stronger restriction effect in the plate

Moreover the influence of stiffener tripping on thelateral deformation resistance is evaluated Figure 13 showsthe experimental observations when fractures are initiallygenerated in the side plate e central stiffener in specimen3FB remains upright while the central stiffener in thereplicate test trips down Current experiments illustrate thatthe tripping extent of the stiffeners can vary greatly due tothe differences in the welding conditions and the relativespecimen-indenter impact locations In addition the cor-responding resistance-penetration responses for these twospecimens are shown in Figure 13 e compared curvesdemonstrate that the discrepancy of the resistance-pene-tration responses is small in the experiments which provesthat the tripping of the central stiffener has a slight influenceon the resistance response Clearly this conclusion is dif-ferent from that stated by Yu et al [20] e reason for thisdifference is attributed to the variations in the structuralforms of the central stiffener in these two studies e in-vestigated stiffener in Yu et al [20] is the T-profile stiffenerwhere the top flange will experience a remarkable membranetension effect Tripping of the T-profile stiffener can reducethe lateral deflection of the top flange thereby leading to asignificant decrease in resistance Meanwhile tripping of thestiffenerrsquos web will have little influence on the lateral re-sistance of the stiffened plate

4 Verification of the AnalyticalPrediction Method

In this section the proposed analytical method is verifiedwith respect to the large deformation resistance and the

250

R75

70deg

Figure 7 Dimensions of the conical indenter (dimensions in mm)

Mathematical Problems in Engineering 7

critical penetration depth of the stiffened plate by comparingthe analytically predicted resistance-penetration curves withthe experimental curves as shown in Figure 14 Moreoverthe workflow for obtaining the resistance-penetration re-lation for the stiffened plate is given in Figure 15

e current study proposes not only analytical predic-tions for the deformation resistance of the central stiffenerand the lateral stiffener but also a method to obtain thecritical penetration depth of the stiffened plate With theanalytical solutions for the large deformation resistance andthe critical penetration depth of the unstiffened plate in Ref[15] the analytical predictions for a stiffened plate can beobtained from large deformation to initial fracture ecompared resistance-penetration curves shown in Figure 14illustrate that the analytical method can adequately predict

0 30 60 90 1200

100

200

300

400(1007mm 3662kN)

(994mm 3092kN)

Resis

tanc

e (kN

)

Penetration (mm)

Specimen USSpecimen 2FBSpecimen 3FB

(1085mm 3155kN)

Figure 11 Experimental resistance-penetration responses

PL315oslash22

910

910

600

600

155115605

103 times 7

103 times 6 106106

(a)

FB55 times 315

200 times 3

(b)

FB55 times 315

150 times 4

(c)

Figure 8 Dimensions of the specimens (dimensions in mm) (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

G-Gauge lengthW-WidthT-ThicknessR-Radius of filletL-Overall lengthA-Length of reduced sectionB-Length of grip sectionC-Width of grip section

50mm125mm315mm125mm200mm85mm

~50mm20mm

B

G

A

CW

BL

TR

Figure 9 Dimensions of the standard tension tested piece (ASTME8) G gauge length (50mm) W width (125mm) T thickness(315mm) R radius of fillet (315mm) L overall length (200mm)A length of reduced section (85mm) B length of grid section(sim50mm) C width of grip section (20mm)

000 005 010 015 020 025 030 0350

70

140

210

280

350

420

Stre

ss (M

Pa)

Strain (ndash)

Figure 10 Engineering stress-strain curve

Table 1 Mechanical properties of material

Property Symbol Units SpecimensYoungrsquos modulus E GPa 207Poissonrsquos ratio v mdash 03Mass density ρ kgm3 7850Yield stress σY MPa 3028Ultimate tensile strength σu MPa 4084Fracture strain εf mdash 0306Strain-hardening index n mdash 02

8 Mathematical Problems in Engineering

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

Moreover wc and wcsmax are assumed to have the fol-lowing relation

wc ccswcsmax (8)

where ccs is the ratio of the deflection of point C to themaximum deflection of the central stiffener

us the relation between the indentation velocities ofthe point C _wc and the central stiffener _wcsmax can beexpressed as

_wc ccs _wcsmax (9)

en the angular velocity of rotational stiffener _c can beobtained

_c _wc

1 + tan2 c( 1113857xc

(10)

e bending energy rate of the central stiffener can beexpressed as

_Eb cs 4Mpsts _c (11)

where Mps is the plastic bending moment per unit thicknessand can be obtained as

Mps σ0sh

2s

2 (12)

where σ0s is the flow stress of the stiffener which is theaveraged value of the yield stress σys and ultimate tensionstress σus [21]

Moreover the tension strain εcs and tension strain rate_εcscan be approximated as

εcs 12(tan c)

2

_εcs tan c_wc

xc

(13)

us the rate of membrane tension of the stiffener canbe expressed as

_Em cs 2BScs

σ0ts _εcsdScs (14)

where Scs is the side of the area on which the central stiffenerexperiences tension and can be obtained from Scs xchs

According to the upper bound theorem the equilibriumequation can be expressed as

Fp cs _wcsmax _Eb cs + _Em cs (15)

where Fp_cs is the resistance of the stiffener in the plasticstage

Finally the instantaneous resistance of the stiffener at theplastic stage can be derived by substituting (11) and (14) into(15)

Fp cs 2σ0shswcts

ls minus Rb sinφc( 1113857wcsmax

hs

1 + tan 2c+ wc1113888 1113889 (16)

212 Lateral Stiffener e deformation of the lateralstiffener is driven by the deformed plate It is assumed thatthe lateral stiffener deforms simultaneously with the sideplate us the overall movement of the lateral stiffener isthe superposition of the lateral deflection from the plate androtation with the plate Here it is assumed that the stiffenerswill remain perpendicular to the side plate until platefracture occurs

e deformation mode of the stiffener is shown inFigure 4e ends of the stiffener are welded to adjacent webgirders that can constrain the rotation of the stiffener locallyus plastic hinges will be generated at the ends of thestiffeners which can lead to an out-of-plane bending effectmarked by the red dashed lines in Figure 4 However thebending effect is neglected because the stiffener out-of-planebending moment is much smaller than its in-plane bendingmoment erefore only the membrane tension effect isconsidered for the lateral stiffener to predict its deformationresistance

As shown in Figure 4 the deformed stiffener is alsoplaced in the x-w rectangular coordinate system e de-formation shape of the stiffener can be expressed as

wls wlsmax sinπx

2ls1113888 1113889 (17)

where wls is the deflection of the stiffener and wlsmax is themaximum transverse deflection of the stiffener

According to (4) wlsmax can be obtained as

wlsmax w xls( 1113857 wcsmax

minus

w2csmax + 2Rb cosφc minus 1( 1113857tanφcxls

1113969

(18)

where xls is the initial horizontal distance between thestiffener and plate edge

Similar to the relation between wc and wcsmax for thecentral stiffener wlsmax and wcsmax have the followingrelations

wlsmax clswcsmax

_wlsmax cls _wcsmax(19)

where cls is the ratio of the maximum deflection between thelateral stiffener and the central stiffener

As the stiffener is assumed to be displaced vertically thetension strain εls can be approximated as

εls asymp12

zwls

zx1113888 1113889

2

w2lsmax

π2

8l2scos2

πx

2ls1113888 1113889 (20)

e strain rate of the stiffener _εls can be expressed as

_εls wlsmaxπ2

4l2scos2

πx

2ls1113888 1113889 _wlsmax (21)

e rate of membrane energy can be expressed as

_Em ls BSls

σ0sts _εlsdSls (22)

where Sls is the initial area of the lateral stiffener

4 Mathematical Problems in Engineering

According to the upper bound theorem the rate of workby the external load is equal to the rate of internal energydissipation e equilibrium can be expressed as

Fls middot _wcsmax _Em ls (23)

us the instantaneous resistance of the lateral stiffenerFls can be derived as

Fls σ0stshsπ2w2

lsmax4lswcsmax

(24)

22 Fracture Prediction of the Stiffened Plate Analyticalfracture prediction for the ship side plate under a bulbousbow striking scenario is crucial for estimating the criticalenergy dissipation Several equations were proposed tocalculate the critical penetration depth of an unstiffenedplate Recently an expression that was validated by anumber of experiments was proposed by Zhang et al [15]is expression is as follows

wp f c1

(7n + 076)b0Rb

1113969

(25)

where c1 is the correction coefficient and has been calibratedto be 05 n is the work hardening exponent of the platematerial and b0 is the half-width of the plate

In particular the effect of the lateral stiffener on theinitial fracture of the ship side plate is considered in thissection An analytical method is proposed to predict thefracture initiation of a stiffened side shell

Section 212 demonstrates that the deformation of thelateral stiffener is driven by the plate Actually the lateralstiffener interacts with the plate in the large deformationprocess us the lateral stiffener is able to restrain thedeformation of the plate and finally reduce the criticalpenetration depth of the plate Given the critical penetrationdepth of the plate wp f the critical penetration depth for thestiffened plate wsp f can be expressed as

wsp f wp f minus dwls (26)

where dwls is the penetration depth reduced by the lateralstiffeners

e cross section at the maximum deflection of thelateral stiffener is extracted to analyze the influence of thestiffener on the deflection of the plate Figure 5(a) shows theload state of the plate at the plate-stiffener intersection Atthe intersection angular discontinuity of the curved plateoccurs due to the vertical force Fs from the stiffener eplate also sustains tension from the adjacent plates whichare denoted as Fp1 and Fp2 Figure 5(b) depicts the internalforce of the stiffener where the infinitesimal fragment is ds

in length According to Section 212 the stiffener experi-ences a tension effect and the tension force FNs can beexpressed as

FNs σ0stshs (27)

us the vertical force Fs applied by the stiffener can beobtained as

Fs FNs

ds

Rs

(28)

where Rs is the radius of curvature of the deformed stiffenerAccording to (17) Rs can be obtained as

Rs 1 + _w2

ls(x)( 111385732

eurowls(x)1113868111386811138681113868

1113868111386811138681113868 (29)

Moreover the tension forces from the plate are assumedto be very similar and are expressed as

Fp1 Fp2 σ0ptpds (30)

where σ0p is the flow stress of the plate and tp is the thicknessof the plate

On the y-axis the resultant force should be zerous Fscan also be expressed as

Fs Fp1(sinφ minus sin(φ minus dφ)) (31)

Considering (28) and substituting (29) and (30) into(31) the increment in the rotation angle at the plate-stiffenerintersection can be approximated as

dφ wlsmaxπ2σ0shsts

4σ0pl2s tp

(32)

According to (4) the instantaneous angle φ at the plate-stiffener intersection satisfies the following relation

tan φ dw

dx

Rb cos φc minus 1( 1113857tan φc

1 minus 1cls( 1113857( 1113857wlsmax (33)

Based on (33) the lateral deflection of the plate limitedby the stiffener dwls can be approximated as

dwls φc( 1113857 wlsmax minus wp1113872 1113873wlsmax

Rb cosφc minus 1( 1113857tanφc

dφ (34)

Substituting (32) into (34) dwls can be further expressedas

dwls φc( 1113857 σ0shstsw

2lsmaxπ

2 wlsmax minus wcsmax( 1113857

4σ0pRbl2s tp cosφc minus 1( 1113857tanφc

(35)

3 Experimental Details

31 Penetration Test Design e quasistatic indentationexperiments were performed at Huazhong University ofScience and Technology e setup used in the experimentsis presented in Figure 6(a) e specimens were clampedbetween a bottom flange and an upper flange which weremade of Q345 steel with a thickness of 25mm ey werefixed together by M20 bolts e dimensions of the

ls

w

xwlsmax

Figure 4 Deformation mode of the lateral stiffener

Mathematical Problems in Engineering 5

experimental clamping system are illustrated in Figure 6(b)In addition the ends of the stiffeners were double-sidewelded on the bottom flange to restrain their freedom asshown in Figure 6(c) Current fixtures were proven toprovide clamped boundary constraints through validation

by numerical simulations with solid elements consideringall the fixtures Moreover as in previous studies a bulbousbow is generally treated as rigid and simplified as a conicalindenter defined by the top radius [2 14] as depicted inFigure 2 us the indenter shape was designed to be

Displacementtransducers

Forcetransducer

Conical indenter

Camera 1 Camera 2

(a)

Bottom flange

Specimen

Upper flange600

355

2525

2525

0 20

(b) (c)

Figure 6 Designed penetration test (a) Setup (b) Dimensions of the clamping system (c) End connections of the stiffeners

Fp1

Fp2

φ ndash dφ

Fs

y

xφ

Lateral stiffener

Plate

Tangent line

Horizontal line

φ

(a)

FNs FNs

ds

Rs

(b)

Figure 5 Load state for the components (a) Plate (b) Stiffener

6 Mathematical Problems in Engineering

conical e geometry and dimensions are shown inFigure 7

e initial distance between the indenter and thespecimen was approximately 20mme deformation of thespecimens was enforced at a rate of sim10mmmin [2ndash4] at themidspan by hydraulic cylinders with a 100 ton maximumcapacity A 100-ton load cell fixed between the hydrauliccylinder and the indenter and two displacement sensorsjointed on the indenter were utilized to obtain the force-timeand displacement-time curves respectively To visualize thedeformations 50times 50mm grids were drawn on the front andback sides of the specimens Moreover two cameras wereplaced under the bottom flange to capture the deformationprocess of the specimens

32 Specimens ree specimens were designed at one-fourthscale from the ship side as shown in Figure 8e unstiffenedplate (denoted as ldquoUSrdquo) was used as a reference to estimate theeffects of the stiffeners on the resistance and critical pene-tration depth of the ship side panel Stiffened plates with twoand three stiffeners (denoted as ldquo2FBrdquo and ldquo3FBrdquo respec-tively) were used to analyze the effects of the lateral stiffenerand the central stiffener respectively e dimensions of thespecimens are also illustrated in Figure 8 where the central600times 600mm square is the exposed area of the panels and thesurrounding areas with a width of 155mm are clamped to thespecimens In addition all the stiffeners were 55mm inheighte weld joint plate-stiffeners are alternatively double-sided filled welds with a size of sim30mm e selected elec-trodes were ASME (American Society of Mechanical Engi-neers) ER70S-6 with a diameter of 08mm e selection ofthe weld size and the electrode base material follows standardshipyard welding procedures Moreover the finished speci-mens were hammered at the plate-stiffener intersections by amallet to release the residual stress Furthermore replicatetests for each specimen were performed to ensure the reli-ability of the experimental results

e material used for the plates and stiffeners is grade Bnormal structural steel qualified by the CCS (China Classi-fication Society) considering the availability of the thin steelplate and the loading capacity of the hydraulic cylinderesesteel plates were from the same batch supplied by WISCO(Wuhan Iron and Steel (Group) Company) and were all315mm thick To obtain the mechanical properties of thesteel quasistatic tensile tests are conducted using threestandard tensile specimens and procedures e dimensionsof the machined tension test pieces are shown in Figure 9Based on the displacement-prescribed tensile tests performedon the universal testing machine the engineering stress-strainbehavior of the material can be obtained e tensile engi-neering stress-strain curve is presented in Figure 10 emechanical properties of the plate material are summarized inTable 1 where n is obtained according to Ref [15]

33 Results e experimentally measured resistance-pen-etration curves are shown in Figure 11 In addition thedeformation shapes when the plates are initially fracturedare shown in Figure 12 for the three specimensese curves

demonstrate that the lateral stiffener and the central stiffenercan both supply an extent of lateral resistance at differentpenetration depths compared with the unstiffened plate Inaddition the improved resistances due to the lateral andcentral stiffeners are close at different penetration depthse resistance improved by the central stiffener is moreremarkable than that by the lateral stiffener In addition theresistance-penetration curves indicate that the criticalpenetration depths for the three specimens are differentCompared with specimen US the reductions in the criticalpenetration depth of specimen 2FB and specimen 3FB are91mm and 78mm respectively is value for specimen2FB is larger because the horizontal distance between thelateral stiffener and the impact position is smaller which canlead to a stronger restriction effect in the plate

Moreover the influence of stiffener tripping on thelateral deformation resistance is evaluated Figure 13 showsthe experimental observations when fractures are initiallygenerated in the side plate e central stiffener in specimen3FB remains upright while the central stiffener in thereplicate test trips down Current experiments illustrate thatthe tripping extent of the stiffeners can vary greatly due tothe differences in the welding conditions and the relativespecimen-indenter impact locations In addition the cor-responding resistance-penetration responses for these twospecimens are shown in Figure 13 e compared curvesdemonstrate that the discrepancy of the resistance-pene-tration responses is small in the experiments which provesthat the tripping of the central stiffener has a slight influenceon the resistance response Clearly this conclusion is dif-ferent from that stated by Yu et al [20] e reason for thisdifference is attributed to the variations in the structuralforms of the central stiffener in these two studies e in-vestigated stiffener in Yu et al [20] is the T-profile stiffenerwhere the top flange will experience a remarkable membranetension effect Tripping of the T-profile stiffener can reducethe lateral deflection of the top flange thereby leading to asignificant decrease in resistance Meanwhile tripping of thestiffenerrsquos web will have little influence on the lateral re-sistance of the stiffened plate

4 Verification of the AnalyticalPrediction Method

In this section the proposed analytical method is verifiedwith respect to the large deformation resistance and the

250

R75

70deg

Figure 7 Dimensions of the conical indenter (dimensions in mm)

Mathematical Problems in Engineering 7

critical penetration depth of the stiffened plate by comparingthe analytically predicted resistance-penetration curves withthe experimental curves as shown in Figure 14 Moreoverthe workflow for obtaining the resistance-penetration re-lation for the stiffened plate is given in Figure 15

e current study proposes not only analytical predic-tions for the deformation resistance of the central stiffenerand the lateral stiffener but also a method to obtain thecritical penetration depth of the stiffened plate With theanalytical solutions for the large deformation resistance andthe critical penetration depth of the unstiffened plate in Ref[15] the analytical predictions for a stiffened plate can beobtained from large deformation to initial fracture ecompared resistance-penetration curves shown in Figure 14illustrate that the analytical method can adequately predict

0 30 60 90 1200

100

200

300

400(1007mm 3662kN)

(994mm 3092kN)

Resis

tanc

e (kN

)

Penetration (mm)

Specimen USSpecimen 2FBSpecimen 3FB

(1085mm 3155kN)

Figure 11 Experimental resistance-penetration responses

PL315oslash22

910

910

600

600

155115605

103 times 7

103 times 6 106106

(a)

FB55 times 315

200 times 3

(b)

FB55 times 315

150 times 4

(c)

Figure 8 Dimensions of the specimens (dimensions in mm) (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

G-Gauge lengthW-WidthT-ThicknessR-Radius of filletL-Overall lengthA-Length of reduced sectionB-Length of grip sectionC-Width of grip section

50mm125mm315mm125mm200mm85mm

~50mm20mm

B

G

A

CW

BL

TR

Figure 9 Dimensions of the standard tension tested piece (ASTME8) G gauge length (50mm) W width (125mm) T thickness(315mm) R radius of fillet (315mm) L overall length (200mm)A length of reduced section (85mm) B length of grid section(sim50mm) C width of grip section (20mm)

000 005 010 015 020 025 030 0350

70

140

210

280

350

420

Stre

ss (M

Pa)

Strain (ndash)

Figure 10 Engineering stress-strain curve

Table 1 Mechanical properties of material

Property Symbol Units SpecimensYoungrsquos modulus E GPa 207Poissonrsquos ratio v mdash 03Mass density ρ kgm3 7850Yield stress σY MPa 3028Ultimate tensile strength σu MPa 4084Fracture strain εf mdash 0306Strain-hardening index n mdash 02

8 Mathematical Problems in Engineering

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

According to the upper bound theorem the rate of workby the external load is equal to the rate of internal energydissipation e equilibrium can be expressed as

Fls middot _wcsmax _Em ls (23)

us the instantaneous resistance of the lateral stiffenerFls can be derived as

Fls σ0stshsπ2w2

lsmax4lswcsmax

(24)

22 Fracture Prediction of the Stiffened Plate Analyticalfracture prediction for the ship side plate under a bulbousbow striking scenario is crucial for estimating the criticalenergy dissipation Several equations were proposed tocalculate the critical penetration depth of an unstiffenedplate Recently an expression that was validated by anumber of experiments was proposed by Zhang et al [15]is expression is as follows

wp f c1

(7n + 076)b0Rb

1113969

(25)

where c1 is the correction coefficient and has been calibratedto be 05 n is the work hardening exponent of the platematerial and b0 is the half-width of the plate

In particular the effect of the lateral stiffener on theinitial fracture of the ship side plate is considered in thissection An analytical method is proposed to predict thefracture initiation of a stiffened side shell

Section 212 demonstrates that the deformation of thelateral stiffener is driven by the plate Actually the lateralstiffener interacts with the plate in the large deformationprocess us the lateral stiffener is able to restrain thedeformation of the plate and finally reduce the criticalpenetration depth of the plate Given the critical penetrationdepth of the plate wp f the critical penetration depth for thestiffened plate wsp f can be expressed as

wsp f wp f minus dwls (26)

where dwls is the penetration depth reduced by the lateralstiffeners

e cross section at the maximum deflection of thelateral stiffener is extracted to analyze the influence of thestiffener on the deflection of the plate Figure 5(a) shows theload state of the plate at the plate-stiffener intersection Atthe intersection angular discontinuity of the curved plateoccurs due to the vertical force Fs from the stiffener eplate also sustains tension from the adjacent plates whichare denoted as Fp1 and Fp2 Figure 5(b) depicts the internalforce of the stiffener where the infinitesimal fragment is ds

in length According to Section 212 the stiffener experi-ences a tension effect and the tension force FNs can beexpressed as

FNs σ0stshs (27)

us the vertical force Fs applied by the stiffener can beobtained as

Fs FNs

ds

Rs

(28)

where Rs is the radius of curvature of the deformed stiffenerAccording to (17) Rs can be obtained as

Rs 1 + _w2

ls(x)( 111385732

eurowls(x)1113868111386811138681113868

1113868111386811138681113868 (29)

Moreover the tension forces from the plate are assumedto be very similar and are expressed as

Fp1 Fp2 σ0ptpds (30)

where σ0p is the flow stress of the plate and tp is the thicknessof the plate

On the y-axis the resultant force should be zerous Fscan also be expressed as

Fs Fp1(sinφ minus sin(φ minus dφ)) (31)

Considering (28) and substituting (29) and (30) into(31) the increment in the rotation angle at the plate-stiffenerintersection can be approximated as

dφ wlsmaxπ2σ0shsts

4σ0pl2s tp

(32)

According to (4) the instantaneous angle φ at the plate-stiffener intersection satisfies the following relation

tan φ dw

dx

Rb cos φc minus 1( 1113857tan φc

1 minus 1cls( 1113857( 1113857wlsmax (33)

Based on (33) the lateral deflection of the plate limitedby the stiffener dwls can be approximated as

dwls φc( 1113857 wlsmax minus wp1113872 1113873wlsmax

Rb cosφc minus 1( 1113857tanφc

dφ (34)

Substituting (32) into (34) dwls can be further expressedas

dwls φc( 1113857 σ0shstsw

2lsmaxπ

2 wlsmax minus wcsmax( 1113857

4σ0pRbl2s tp cosφc minus 1( 1113857tanφc

(35)

3 Experimental Details

31 Penetration Test Design e quasistatic indentationexperiments were performed at Huazhong University ofScience and Technology e setup used in the experimentsis presented in Figure 6(a) e specimens were clampedbetween a bottom flange and an upper flange which weremade of Q345 steel with a thickness of 25mm ey werefixed together by M20 bolts e dimensions of the

ls

w

xwlsmax

Figure 4 Deformation mode of the lateral stiffener

Mathematical Problems in Engineering 5

experimental clamping system are illustrated in Figure 6(b)In addition the ends of the stiffeners were double-sidewelded on the bottom flange to restrain their freedom asshown in Figure 6(c) Current fixtures were proven toprovide clamped boundary constraints through validation

by numerical simulations with solid elements consideringall the fixtures Moreover as in previous studies a bulbousbow is generally treated as rigid and simplified as a conicalindenter defined by the top radius [2 14] as depicted inFigure 2 us the indenter shape was designed to be

Displacementtransducers

Forcetransducer

Conical indenter

Camera 1 Camera 2

(a)

Bottom flange

Specimen

Upper flange600

355

2525

2525

0 20

(b) (c)

Figure 6 Designed penetration test (a) Setup (b) Dimensions of the clamping system (c) End connections of the stiffeners

Fp1

Fp2

φ ndash dφ

Fs

y

xφ

Lateral stiffener

Plate

Tangent line

Horizontal line

φ

(a)

FNs FNs

ds

Rs

(b)

Figure 5 Load state for the components (a) Plate (b) Stiffener

6 Mathematical Problems in Engineering

conical e geometry and dimensions are shown inFigure 7

e initial distance between the indenter and thespecimen was approximately 20mme deformation of thespecimens was enforced at a rate of sim10mmmin [2ndash4] at themidspan by hydraulic cylinders with a 100 ton maximumcapacity A 100-ton load cell fixed between the hydrauliccylinder and the indenter and two displacement sensorsjointed on the indenter were utilized to obtain the force-timeand displacement-time curves respectively To visualize thedeformations 50times 50mm grids were drawn on the front andback sides of the specimens Moreover two cameras wereplaced under the bottom flange to capture the deformationprocess of the specimens

32 Specimens ree specimens were designed at one-fourthscale from the ship side as shown in Figure 8e unstiffenedplate (denoted as ldquoUSrdquo) was used as a reference to estimate theeffects of the stiffeners on the resistance and critical pene-tration depth of the ship side panel Stiffened plates with twoand three stiffeners (denoted as ldquo2FBrdquo and ldquo3FBrdquo respec-tively) were used to analyze the effects of the lateral stiffenerand the central stiffener respectively e dimensions of thespecimens are also illustrated in Figure 8 where the central600times 600mm square is the exposed area of the panels and thesurrounding areas with a width of 155mm are clamped to thespecimens In addition all the stiffeners were 55mm inheighte weld joint plate-stiffeners are alternatively double-sided filled welds with a size of sim30mm e selected elec-trodes were ASME (American Society of Mechanical Engi-neers) ER70S-6 with a diameter of 08mm e selection ofthe weld size and the electrode base material follows standardshipyard welding procedures Moreover the finished speci-mens were hammered at the plate-stiffener intersections by amallet to release the residual stress Furthermore replicatetests for each specimen were performed to ensure the reli-ability of the experimental results

e material used for the plates and stiffeners is grade Bnormal structural steel qualified by the CCS (China Classi-fication Society) considering the availability of the thin steelplate and the loading capacity of the hydraulic cylinderesesteel plates were from the same batch supplied by WISCO(Wuhan Iron and Steel (Group) Company) and were all315mm thick To obtain the mechanical properties of thesteel quasistatic tensile tests are conducted using threestandard tensile specimens and procedures e dimensionsof the machined tension test pieces are shown in Figure 9Based on the displacement-prescribed tensile tests performedon the universal testing machine the engineering stress-strainbehavior of the material can be obtained e tensile engi-neering stress-strain curve is presented in Figure 10 emechanical properties of the plate material are summarized inTable 1 where n is obtained according to Ref [15]

33 Results e experimentally measured resistance-pen-etration curves are shown in Figure 11 In addition thedeformation shapes when the plates are initially fracturedare shown in Figure 12 for the three specimensese curves

demonstrate that the lateral stiffener and the central stiffenercan both supply an extent of lateral resistance at differentpenetration depths compared with the unstiffened plate Inaddition the improved resistances due to the lateral andcentral stiffeners are close at different penetration depthse resistance improved by the central stiffener is moreremarkable than that by the lateral stiffener In addition theresistance-penetration curves indicate that the criticalpenetration depths for the three specimens are differentCompared with specimen US the reductions in the criticalpenetration depth of specimen 2FB and specimen 3FB are91mm and 78mm respectively is value for specimen2FB is larger because the horizontal distance between thelateral stiffener and the impact position is smaller which canlead to a stronger restriction effect in the plate

Moreover the influence of stiffener tripping on thelateral deformation resistance is evaluated Figure 13 showsthe experimental observations when fractures are initiallygenerated in the side plate e central stiffener in specimen3FB remains upright while the central stiffener in thereplicate test trips down Current experiments illustrate thatthe tripping extent of the stiffeners can vary greatly due tothe differences in the welding conditions and the relativespecimen-indenter impact locations In addition the cor-responding resistance-penetration responses for these twospecimens are shown in Figure 13 e compared curvesdemonstrate that the discrepancy of the resistance-pene-tration responses is small in the experiments which provesthat the tripping of the central stiffener has a slight influenceon the resistance response Clearly this conclusion is dif-ferent from that stated by Yu et al [20] e reason for thisdifference is attributed to the variations in the structuralforms of the central stiffener in these two studies e in-vestigated stiffener in Yu et al [20] is the T-profile stiffenerwhere the top flange will experience a remarkable membranetension effect Tripping of the T-profile stiffener can reducethe lateral deflection of the top flange thereby leading to asignificant decrease in resistance Meanwhile tripping of thestiffenerrsquos web will have little influence on the lateral re-sistance of the stiffened plate

4 Verification of the AnalyticalPrediction Method

In this section the proposed analytical method is verifiedwith respect to the large deformation resistance and the

250

R75

70deg

Figure 7 Dimensions of the conical indenter (dimensions in mm)

Mathematical Problems in Engineering 7

critical penetration depth of the stiffened plate by comparingthe analytically predicted resistance-penetration curves withthe experimental curves as shown in Figure 14 Moreoverthe workflow for obtaining the resistance-penetration re-lation for the stiffened plate is given in Figure 15

e current study proposes not only analytical predic-tions for the deformation resistance of the central stiffenerand the lateral stiffener but also a method to obtain thecritical penetration depth of the stiffened plate With theanalytical solutions for the large deformation resistance andthe critical penetration depth of the unstiffened plate in Ref[15] the analytical predictions for a stiffened plate can beobtained from large deformation to initial fracture ecompared resistance-penetration curves shown in Figure 14illustrate that the analytical method can adequately predict

0 30 60 90 1200

100

200

300

400(1007mm 3662kN)

(994mm 3092kN)

Resis

tanc

e (kN

)

Penetration (mm)

Specimen USSpecimen 2FBSpecimen 3FB

(1085mm 3155kN)

Figure 11 Experimental resistance-penetration responses

PL315oslash22

910

910

600

600

155115605

103 times 7

103 times 6 106106

(a)

FB55 times 315

200 times 3

(b)

FB55 times 315

150 times 4

(c)

Figure 8 Dimensions of the specimens (dimensions in mm) (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

G-Gauge lengthW-WidthT-ThicknessR-Radius of filletL-Overall lengthA-Length of reduced sectionB-Length of grip sectionC-Width of grip section

50mm125mm315mm125mm200mm85mm

~50mm20mm

B

G

A

CW

BL

TR

Figure 9 Dimensions of the standard tension tested piece (ASTME8) G gauge length (50mm) W width (125mm) T thickness(315mm) R radius of fillet (315mm) L overall length (200mm)A length of reduced section (85mm) B length of grid section(sim50mm) C width of grip section (20mm)

000 005 010 015 020 025 030 0350

70

140

210

280

350

420

Stre

ss (M

Pa)

Strain (ndash)

Figure 10 Engineering stress-strain curve

Table 1 Mechanical properties of material

Property Symbol Units SpecimensYoungrsquos modulus E GPa 207Poissonrsquos ratio v mdash 03Mass density ρ kgm3 7850Yield stress σY MPa 3028Ultimate tensile strength σu MPa 4084Fracture strain εf mdash 0306Strain-hardening index n mdash 02

8 Mathematical Problems in Engineering

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

experimental clamping system are illustrated in Figure 6(b)In addition the ends of the stiffeners were double-sidewelded on the bottom flange to restrain their freedom asshown in Figure 6(c) Current fixtures were proven toprovide clamped boundary constraints through validation

by numerical simulations with solid elements consideringall the fixtures Moreover as in previous studies a bulbousbow is generally treated as rigid and simplified as a conicalindenter defined by the top radius [2 14] as depicted inFigure 2 us the indenter shape was designed to be

Displacementtransducers

Forcetransducer

Conical indenter

Camera 1 Camera 2

(a)

Bottom flange

Specimen

Upper flange600

355

2525

2525

0 20

(b) (c)

Figure 6 Designed penetration test (a) Setup (b) Dimensions of the clamping system (c) End connections of the stiffeners

Fp1

Fp2

φ ndash dφ

Fs

y

xφ

Lateral stiffener

Plate

Tangent line

Horizontal line

φ

(a)

FNs FNs

ds

Rs

(b)

Figure 5 Load state for the components (a) Plate (b) Stiffener

6 Mathematical Problems in Engineering

conical e geometry and dimensions are shown inFigure 7

e initial distance between the indenter and thespecimen was approximately 20mme deformation of thespecimens was enforced at a rate of sim10mmmin [2ndash4] at themidspan by hydraulic cylinders with a 100 ton maximumcapacity A 100-ton load cell fixed between the hydrauliccylinder and the indenter and two displacement sensorsjointed on the indenter were utilized to obtain the force-timeand displacement-time curves respectively To visualize thedeformations 50times 50mm grids were drawn on the front andback sides of the specimens Moreover two cameras wereplaced under the bottom flange to capture the deformationprocess of the specimens

32 Specimens ree specimens were designed at one-fourthscale from the ship side as shown in Figure 8e unstiffenedplate (denoted as ldquoUSrdquo) was used as a reference to estimate theeffects of the stiffeners on the resistance and critical pene-tration depth of the ship side panel Stiffened plates with twoand three stiffeners (denoted as ldquo2FBrdquo and ldquo3FBrdquo respec-tively) were used to analyze the effects of the lateral stiffenerand the central stiffener respectively e dimensions of thespecimens are also illustrated in Figure 8 where the central600times 600mm square is the exposed area of the panels and thesurrounding areas with a width of 155mm are clamped to thespecimens In addition all the stiffeners were 55mm inheighte weld joint plate-stiffeners are alternatively double-sided filled welds with a size of sim30mm e selected elec-trodes were ASME (American Society of Mechanical Engi-neers) ER70S-6 with a diameter of 08mm e selection ofthe weld size and the electrode base material follows standardshipyard welding procedures Moreover the finished speci-mens were hammered at the plate-stiffener intersections by amallet to release the residual stress Furthermore replicatetests for each specimen were performed to ensure the reli-ability of the experimental results

e material used for the plates and stiffeners is grade Bnormal structural steel qualified by the CCS (China Classi-fication Society) considering the availability of the thin steelplate and the loading capacity of the hydraulic cylinderesesteel plates were from the same batch supplied by WISCO(Wuhan Iron and Steel (Group) Company) and were all315mm thick To obtain the mechanical properties of thesteel quasistatic tensile tests are conducted using threestandard tensile specimens and procedures e dimensionsof the machined tension test pieces are shown in Figure 9Based on the displacement-prescribed tensile tests performedon the universal testing machine the engineering stress-strainbehavior of the material can be obtained e tensile engi-neering stress-strain curve is presented in Figure 10 emechanical properties of the plate material are summarized inTable 1 where n is obtained according to Ref [15]

33 Results e experimentally measured resistance-pen-etration curves are shown in Figure 11 In addition thedeformation shapes when the plates are initially fracturedare shown in Figure 12 for the three specimensese curves

demonstrate that the lateral stiffener and the central stiffenercan both supply an extent of lateral resistance at differentpenetration depths compared with the unstiffened plate Inaddition the improved resistances due to the lateral andcentral stiffeners are close at different penetration depthse resistance improved by the central stiffener is moreremarkable than that by the lateral stiffener In addition theresistance-penetration curves indicate that the criticalpenetration depths for the three specimens are differentCompared with specimen US the reductions in the criticalpenetration depth of specimen 2FB and specimen 3FB are91mm and 78mm respectively is value for specimen2FB is larger because the horizontal distance between thelateral stiffener and the impact position is smaller which canlead to a stronger restriction effect in the plate

Moreover the influence of stiffener tripping on thelateral deformation resistance is evaluated Figure 13 showsthe experimental observations when fractures are initiallygenerated in the side plate e central stiffener in specimen3FB remains upright while the central stiffener in thereplicate test trips down Current experiments illustrate thatthe tripping extent of the stiffeners can vary greatly due tothe differences in the welding conditions and the relativespecimen-indenter impact locations In addition the cor-responding resistance-penetration responses for these twospecimens are shown in Figure 13 e compared curvesdemonstrate that the discrepancy of the resistance-pene-tration responses is small in the experiments which provesthat the tripping of the central stiffener has a slight influenceon the resistance response Clearly this conclusion is dif-ferent from that stated by Yu et al [20] e reason for thisdifference is attributed to the variations in the structuralforms of the central stiffener in these two studies e in-vestigated stiffener in Yu et al [20] is the T-profile stiffenerwhere the top flange will experience a remarkable membranetension effect Tripping of the T-profile stiffener can reducethe lateral deflection of the top flange thereby leading to asignificant decrease in resistance Meanwhile tripping of thestiffenerrsquos web will have little influence on the lateral re-sistance of the stiffened plate

4 Verification of the AnalyticalPrediction Method

In this section the proposed analytical method is verifiedwith respect to the large deformation resistance and the

250

R75

70deg

Figure 7 Dimensions of the conical indenter (dimensions in mm)

Mathematical Problems in Engineering 7

critical penetration depth of the stiffened plate by comparingthe analytically predicted resistance-penetration curves withthe experimental curves as shown in Figure 14 Moreoverthe workflow for obtaining the resistance-penetration re-lation for the stiffened plate is given in Figure 15

e current study proposes not only analytical predic-tions for the deformation resistance of the central stiffenerand the lateral stiffener but also a method to obtain thecritical penetration depth of the stiffened plate With theanalytical solutions for the large deformation resistance andthe critical penetration depth of the unstiffened plate in Ref[15] the analytical predictions for a stiffened plate can beobtained from large deformation to initial fracture ecompared resistance-penetration curves shown in Figure 14illustrate that the analytical method can adequately predict

0 30 60 90 1200

100

200

300

400(1007mm 3662kN)

(994mm 3092kN)

Resis

tanc

e (kN

)

Penetration (mm)

Specimen USSpecimen 2FBSpecimen 3FB

(1085mm 3155kN)

Figure 11 Experimental resistance-penetration responses

PL315oslash22

910

910

600

600

155115605

103 times 7

103 times 6 106106

(a)

FB55 times 315

200 times 3

(b)

FB55 times 315

150 times 4

(c)

Figure 8 Dimensions of the specimens (dimensions in mm) (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

G-Gauge lengthW-WidthT-ThicknessR-Radius of filletL-Overall lengthA-Length of reduced sectionB-Length of grip sectionC-Width of grip section

50mm125mm315mm125mm200mm85mm

~50mm20mm

B

G

A

CW

BL

TR

Figure 9 Dimensions of the standard tension tested piece (ASTME8) G gauge length (50mm) W width (125mm) T thickness(315mm) R radius of fillet (315mm) L overall length (200mm)A length of reduced section (85mm) B length of grid section(sim50mm) C width of grip section (20mm)

000 005 010 015 020 025 030 0350

70

140

210

280

350

420

Stre

ss (M

Pa)

Strain (ndash)

Figure 10 Engineering stress-strain curve

Table 1 Mechanical properties of material

Property Symbol Units SpecimensYoungrsquos modulus E GPa 207Poissonrsquos ratio v mdash 03Mass density ρ kgm3 7850Yield stress σY MPa 3028Ultimate tensile strength σu MPa 4084Fracture strain εf mdash 0306Strain-hardening index n mdash 02

8 Mathematical Problems in Engineering

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

conical e geometry and dimensions are shown inFigure 7

e initial distance between the indenter and thespecimen was approximately 20mme deformation of thespecimens was enforced at a rate of sim10mmmin [2ndash4] at themidspan by hydraulic cylinders with a 100 ton maximumcapacity A 100-ton load cell fixed between the hydrauliccylinder and the indenter and two displacement sensorsjointed on the indenter were utilized to obtain the force-timeand displacement-time curves respectively To visualize thedeformations 50times 50mm grids were drawn on the front andback sides of the specimens Moreover two cameras wereplaced under the bottom flange to capture the deformationprocess of the specimens

32 Specimens ree specimens were designed at one-fourthscale from the ship side as shown in Figure 8e unstiffenedplate (denoted as ldquoUSrdquo) was used as a reference to estimate theeffects of the stiffeners on the resistance and critical pene-tration depth of the ship side panel Stiffened plates with twoand three stiffeners (denoted as ldquo2FBrdquo and ldquo3FBrdquo respec-tively) were used to analyze the effects of the lateral stiffenerand the central stiffener respectively e dimensions of thespecimens are also illustrated in Figure 8 where the central600times 600mm square is the exposed area of the panels and thesurrounding areas with a width of 155mm are clamped to thespecimens In addition all the stiffeners were 55mm inheighte weld joint plate-stiffeners are alternatively double-sided filled welds with a size of sim30mm e selected elec-trodes were ASME (American Society of Mechanical Engi-neers) ER70S-6 with a diameter of 08mm e selection ofthe weld size and the electrode base material follows standardshipyard welding procedures Moreover the finished speci-mens were hammered at the plate-stiffener intersections by amallet to release the residual stress Furthermore replicatetests for each specimen were performed to ensure the reli-ability of the experimental results

e material used for the plates and stiffeners is grade Bnormal structural steel qualified by the CCS (China Classi-fication Society) considering the availability of the thin steelplate and the loading capacity of the hydraulic cylinderesesteel plates were from the same batch supplied by WISCO(Wuhan Iron and Steel (Group) Company) and were all315mm thick To obtain the mechanical properties of thesteel quasistatic tensile tests are conducted using threestandard tensile specimens and procedures e dimensionsof the machined tension test pieces are shown in Figure 9Based on the displacement-prescribed tensile tests performedon the universal testing machine the engineering stress-strainbehavior of the material can be obtained e tensile engi-neering stress-strain curve is presented in Figure 10 emechanical properties of the plate material are summarized inTable 1 where n is obtained according to Ref [15]

33 Results e experimentally measured resistance-pen-etration curves are shown in Figure 11 In addition thedeformation shapes when the plates are initially fracturedare shown in Figure 12 for the three specimensese curves

demonstrate that the lateral stiffener and the central stiffenercan both supply an extent of lateral resistance at differentpenetration depths compared with the unstiffened plate Inaddition the improved resistances due to the lateral andcentral stiffeners are close at different penetration depthse resistance improved by the central stiffener is moreremarkable than that by the lateral stiffener In addition theresistance-penetration curves indicate that the criticalpenetration depths for the three specimens are differentCompared with specimen US the reductions in the criticalpenetration depth of specimen 2FB and specimen 3FB are91mm and 78mm respectively is value for specimen2FB is larger because the horizontal distance between thelateral stiffener and the impact position is smaller which canlead to a stronger restriction effect in the plate

Moreover the influence of stiffener tripping on thelateral deformation resistance is evaluated Figure 13 showsthe experimental observations when fractures are initiallygenerated in the side plate e central stiffener in specimen3FB remains upright while the central stiffener in thereplicate test trips down Current experiments illustrate thatthe tripping extent of the stiffeners can vary greatly due tothe differences in the welding conditions and the relativespecimen-indenter impact locations In addition the cor-responding resistance-penetration responses for these twospecimens are shown in Figure 13 e compared curvesdemonstrate that the discrepancy of the resistance-pene-tration responses is small in the experiments which provesthat the tripping of the central stiffener has a slight influenceon the resistance response Clearly this conclusion is dif-ferent from that stated by Yu et al [20] e reason for thisdifference is attributed to the variations in the structuralforms of the central stiffener in these two studies e in-vestigated stiffener in Yu et al [20] is the T-profile stiffenerwhere the top flange will experience a remarkable membranetension effect Tripping of the T-profile stiffener can reducethe lateral deflection of the top flange thereby leading to asignificant decrease in resistance Meanwhile tripping of thestiffenerrsquos web will have little influence on the lateral re-sistance of the stiffened plate

4 Verification of the AnalyticalPrediction Method

In this section the proposed analytical method is verifiedwith respect to the large deformation resistance and the

250

R75

70deg

Figure 7 Dimensions of the conical indenter (dimensions in mm)

Mathematical Problems in Engineering 7

critical penetration depth of the stiffened plate by comparingthe analytically predicted resistance-penetration curves withthe experimental curves as shown in Figure 14 Moreoverthe workflow for obtaining the resistance-penetration re-lation for the stiffened plate is given in Figure 15

e current study proposes not only analytical predic-tions for the deformation resistance of the central stiffenerand the lateral stiffener but also a method to obtain thecritical penetration depth of the stiffened plate With theanalytical solutions for the large deformation resistance andthe critical penetration depth of the unstiffened plate in Ref[15] the analytical predictions for a stiffened plate can beobtained from large deformation to initial fracture ecompared resistance-penetration curves shown in Figure 14illustrate that the analytical method can adequately predict

0 30 60 90 1200

100

200

300

400(1007mm 3662kN)

(994mm 3092kN)

Resis

tanc

e (kN

)

Penetration (mm)

Specimen USSpecimen 2FBSpecimen 3FB

(1085mm 3155kN)

Figure 11 Experimental resistance-penetration responses

PL315oslash22

910

910

600

600

155115605

103 times 7

103 times 6 106106

(a)

FB55 times 315

200 times 3

(b)

FB55 times 315

150 times 4

(c)

Figure 8 Dimensions of the specimens (dimensions in mm) (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

G-Gauge lengthW-WidthT-ThicknessR-Radius of filletL-Overall lengthA-Length of reduced sectionB-Length of grip sectionC-Width of grip section

50mm125mm315mm125mm200mm85mm

~50mm20mm

B

G

A

CW

BL

TR

Figure 9 Dimensions of the standard tension tested piece (ASTME8) G gauge length (50mm) W width (125mm) T thickness(315mm) R radius of fillet (315mm) L overall length (200mm)A length of reduced section (85mm) B length of grid section(sim50mm) C width of grip section (20mm)

000 005 010 015 020 025 030 0350

70

140

210

280

350

420

Stre

ss (M

Pa)

Strain (ndash)

Figure 10 Engineering stress-strain curve

Table 1 Mechanical properties of material

Property Symbol Units SpecimensYoungrsquos modulus E GPa 207Poissonrsquos ratio v mdash 03Mass density ρ kgm3 7850Yield stress σY MPa 3028Ultimate tensile strength σu MPa 4084Fracture strain εf mdash 0306Strain-hardening index n mdash 02

8 Mathematical Problems in Engineering

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

critical penetration depth of the stiffened plate by comparingthe analytically predicted resistance-penetration curves withthe experimental curves as shown in Figure 14 Moreoverthe workflow for obtaining the resistance-penetration re-lation for the stiffened plate is given in Figure 15

e current study proposes not only analytical predic-tions for the deformation resistance of the central stiffenerand the lateral stiffener but also a method to obtain thecritical penetration depth of the stiffened plate With theanalytical solutions for the large deformation resistance andthe critical penetration depth of the unstiffened plate in Ref[15] the analytical predictions for a stiffened plate can beobtained from large deformation to initial fracture ecompared resistance-penetration curves shown in Figure 14illustrate that the analytical method can adequately predict

0 30 60 90 1200

100

200

300

400(1007mm 3662kN)

(994mm 3092kN)

Resis

tanc

e (kN

)

Penetration (mm)

Specimen USSpecimen 2FBSpecimen 3FB

(1085mm 3155kN)

Figure 11 Experimental resistance-penetration responses

PL315oslash22

910

910

600

600

155115605

103 times 7

103 times 6 106106

(a)

FB55 times 315

200 times 3

(b)

FB55 times 315

150 times 4

(c)

Figure 8 Dimensions of the specimens (dimensions in mm) (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

G-Gauge lengthW-WidthT-ThicknessR-Radius of filletL-Overall lengthA-Length of reduced sectionB-Length of grip sectionC-Width of grip section

50mm125mm315mm125mm200mm85mm

~50mm20mm

B

G

A

CW

BL

TR

Figure 9 Dimensions of the standard tension tested piece (ASTME8) G gauge length (50mm) W width (125mm) T thickness(315mm) R radius of fillet (315mm) L overall length (200mm)A length of reduced section (85mm) B length of grid section(sim50mm) C width of grip section (20mm)

000 005 010 015 020 025 030 0350

70

140

210

280

350

420

Stre

ss (M

Pa)

Strain (ndash)

Figure 10 Engineering stress-strain curve

Table 1 Mechanical properties of material

Property Symbol Units SpecimensYoungrsquos modulus E GPa 207Poissonrsquos ratio v mdash 03Mass density ρ kgm3 7850Yield stress σY MPa 3028Ultimate tensile strength σu MPa 4084Fracture strain εf mdash 0306Strain-hardening index n mdash 02

8 Mathematical Problems in Engineering

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

0 30 60 90 1200

100

200

300

400Re

sista

nce (

kN)

Penetration (mm)

Specimen 3FBReplicate test

Specimen 3FB

Replicate test

Platefracture

Figure 13 Experimental results for specimen 3FB and the replicate test

0 30 60 90 1200

80

160

240

320

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(a)

0 30 60 90 1200

80

160

240

320

400

Resis

tanc

e (kN

)

Penetration (mm)

ExperimentalAnalytical

(b)

Figure 14 Comparison of analytical and experimental resistance-penetration responses (a) Specimen 2FB (b) Specimen 3FB

(a) (b) (c)

Figure 12 Deformation shapes of specimens when plates are initially fractured (a) Specimen US (b) Specimen 2FB (c) Specimen 3FB

Mathematical Problems in Engineering 9

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

the resistance due to the large deformation and fractureinitiation of the stiffened plate is demonstrates that theproposed analytical method can accurately estimate thecrashworthiness of a ship side plate impacted by a bulbousbow

In particular the current study considers the influence ofthe stiffener on the critical penetration depth of the plateus the solution of the critical penetration depth for thestiffened plate is described in detail First the critical pen-etration depth for the plate can be calculated according to(25) en the penetration depth reduced by the lateralstiffener is calculated based on (35) Finally the criticalpenetration depth of the stiffened plate can be determinedwhen satisfying the equation in Figure 15 According to thesimilarity of the resistance-penetration curves the proposedanalytical method can predict the initial fracture of thestiffened plate with flat-bar stiffeners

5 Conclusions

is paper assesses the effects of stiffeners on the crash-worthiness of the ship side shell impacted by a bulbous bowAnalytical expressions are presented to calculate the largedeformation resistances for the lateral and central flat-barstiffeners and the critical penetration depth for the stiffenedplate e deformation shape of the central stiffener isidentical to that of the plate However the deformationmode of the central stiffener outside the contact region withthe indenter is treated as linear for simplicity to calculate thedeformation resistance In addition the deformation modeof the lateral stiffener is treated as a sine curve to obtain itsdeformation resistance and the reduction in the criticalpenetration depth of the plate due to the tension effect

Model tests with three specimens (one unstiffened platefor reference and two stiffened plates) quasistaticallypunched by a conical indenter are performed e resis-tance-penetration curves and the damage shapes are ob-tained through the experiments e experimental resultsillustrate that the improvement in resistance due to thecentral stiffener is more remarkable than that of the lateralstiffener In addition a smaller distance between the lateral

stiffener and the indentation position can lead to a lowercritical penetration depth of the ship side panel Moreoverthe resistance response influenced by the tripping of thestiffener web is small Furthermore the similarity of theexperimental and analytical resistance-penetration curvesdemonstrates the reliability and accuracy of the proposedsimplified analytical method

Nomenclature

b0 Half-width of the rectangular platedwls Penetration depth reduced by the lateral stiffenerE Elastic modulusFe_cs Elastic resistance of the central stiffenerFls Resistance of the lateral stiffenerFp Resistance of the unstiffened plateFp_cs Plastic resistance of the central stiffenerFsp Resistance of the stiffened platehs Stiffener heightls Half-length of the stiffenern Work hardening exponent of the plate materialRb Radius of the spherical punchtp Plate thicknessts Stiffener thicknesswc Vertical distance from the point C to initial shapewcsmax Maximum deflection of the central stiffenerwls Deflection of the lateral stiffenerwlsmax Maximum transverse deflection of the lateral

stiffenerwp f Critical penetration depth of the platewsp f Critical penetration depth of the stiffened plateφc Indenter wrapping angle at the outermost contact

pointc Rotation angle of the central stiffenerεcs Tension strain of the central stiffenerεls Tension strain of the lateral stiffenerxc Horizontal distance from the point C to plate edgexls Initial horizontal distance between the stiffener and

plate edgeσ0p Flow stress of the plateσ0s Flow stress of the stiffener

Critical penetrationdepth of the plate

Influence of thestiffener

(25)

wp_f

(35)

dwls (φc)

if |wcsmax (φc) + dwls (φc) ndash wp_f| lt 02

wsp_f = wcsmax (φc)2FB dwls = 1019mm wsp_f = 1002mm

3FB dwls = 649mm wsp_f = 1036mm

Critical penetration depthof the stiffened plate

Lateral stiffener

(3) (24)(16)

Fp ndash wcsmax Fcs ndash wcsmax

Fls ndash wcsmax

Fsp ndash wcsmax

Fe_cs ndash wcsmax Fp_cs ndash wcsmax

(5)

Elastic stage Plastic stage

Central stiffenerPlate

Deformation resistanceof the stiffened plate

Resistance Penetration

Ref [15](3)

Figure 15 Calculation of the resistance-penetration relation for the stiffened plate

10 Mathematical Problems in Engineering

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is study was supported by the Foundation of WuhanPolytechnic University (grant no 2020Y09) and NationalNatural Science Foundation of China (grant no 51579110)

References

[1] L Hong Simplifed analysis and design of ships subjected tocollision and grounding PhD thesis Norwegian University ofScience and Technology Trondheim Norway 2009

[2] H S Alsos and J Amdahl ldquoOn the resistance to penetration ofstiffened plates part Imdashexperimentsrdquo International Journal ofImpact Engineering vol 36 no 6 pp 799ndash807 2009

[3] D Morin B L Kaarstad B Skajaa O S Hopperstad andM Langseth ldquoTesting and modelling of stiffened aluminiumpanels subjected to quasi-static and low-velocity impactloadingrdquo International Journal of Impact Engineeringvol 110 pp 97ndash111 2017

[4] M Kotildergesaar J Romanoff H Remes and P PalokangasldquoExperimental and numerical penetration response of laser-welded stiffened panelsrdquo International Journal of ImpactEngineering vol 114 pp 78ndash92 2018

[5] L Yang and D-Y Wang ldquoExperimental and numerical in-vestigation on the response of stiffened panels subjected tolateral impact considering material failure criteriardquo OceanEngineering vol 184 pp 193ndash205 2019

[6] M A G Calle and M Alves ldquoA review-analysis on materialfailure modeling in ship collisionrdquo Ocean Engineeringvol 106 pp 20ndash38 2015

[7] M Kotildergesaar ldquoe effect of low stress triaxialities and de-formation paths on ductile fracture simulations of large shellstructuresrdquo Marine Structures vol 63 pp 45ndash64 2019

[8] B C Cerik K Lee S-J Park and J Choung ldquoSimulation ofship collision and grounding damage using Hosford-Cou-lomb fracture model for shell elementsrdquo Ocean Engineeringvol 173 pp 415ndash432 2019

[9] H Zhiqiang A Joslashrgen and H Lin ldquoVerification of a sim-plified analytical method for predictions of ship groundingsover large contact surfaces by numerical simulationsrdquoMarineStructures vol 24 no 4 pp 436ndash458 2011

[10] N Jones Structual Impact Cambridge University PressCambridge UK 2nd edition 2012

[11] B Liu ldquoAnalytical method to assess double-hull ship struc-tures subjected to bulbous bow collisionrdquo Ocean Engineeringvol 142 pp 27ndash38 2017

[12] B C Simonsen and L P Lauridsen ldquoEnergy absorption andductile failure in metal sheets under lateral indentation by asphererdquo International Journal of Impact Engineering vol 24no 10 pp 1017ndash1039 2000

[13] Y-W Lee J C Woertz and T Wierzbicki ldquoFracture pre-diction of thin plates under hemi-spherical punch with cal-ibration and experimental verificationrdquo International Journalof Mechanical Sciences vol 46 no 5 pp 751ndash781 2004

[14] Y Gong Y Gao D Xie and J Liu ldquoDeflection and fracture ofa clamped plate under lateral indentation by a sphereflectionand fracture of a clamped plate under lateral indentation by asphererdquo Ocean Engineering vol 103 pp 21ndash30 2015

[15] M Zhang Q Sun J Liu Z Hu and S Zhang ldquoA study of therupture behavior of a ship side plate laterally punched by afull-shape bulbous bow indenterrdquo Ocean Engineeringvol 182 pp 48ndash60 2019

[16] S-R Cho and H-S Lee ldquoExperimental and analytical in-vestigations on the response of stiffened plates subjected tolateral collisionsrdquoMarine Structures vol 22 no 1 pp 84ndash952009

[17] B Liu R Villavicencio and C Guedes Soares ldquoSimplifiedanalytical method to evaluate tanker side panels during minorcollision incidentsrdquo International Journal of Impact Engi-neering vol 78 pp 20ndash33 2015

[18] B Sun Z Hu and G Wang ldquoAn analytical method forpredicting the ship side structure response in raked bowcollisionsrdquo Marine Structures vol 41 pp 288ndash311 2015

[19] M Zhang J Liu Z Hu and Y Zhao ldquoOn resistance of arectangular thin plate under lateral indentation by a wedgeindenterrdquo Ships and Offshore Structures vol 13 no 6pp 617ndash629 2018

[20] Z Yu J Amdahl and Y Sha ldquoLarge inelastic deformationresistance of stiffened panels subjected to lateral loadingrdquoMarine Structures vol 59 pp 342ndash367 2018

[21] Z G Gao S Yang and Z Q Hu ldquoe resistance of ship webgirders in collision and groundingrdquo Mathematical Problemsin Engineering vol 2014 Article ID 720542 13 pages 2014

Mathematical Problems in Engineering 11