Research Article Torsion Dynamic Behaviour of the Ship

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 793412 7 pageshttpdxdoiorg1011552013793412

Research ArticleTorsion Dynamic Behaviour of the Ship Hull Made out ofLayered Composites

Ionel Chirica

ldquoDunarea de Josrdquo University of Galati Domneasca Street No 47 800008 Galati Romania

Correspondence should be addressed to Ionel Chirica ionelchiricaugalro

Received 23 February 2013 Accepted 6 May 2013

Academic Editor Irene Penesis

Copyright copy 2013 Ionel ChiricaThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper addresses the dynamic torsion behaviour of the ship hull with very large open decks A short overview of the pastresearch is presented and several key findings and behavioral characteristics are discussed In the paper the author is focusing onthe dynamic part of a newmacroelementmodel used for torsion dynamic analysis of the ship hull made of composite materialsThenumerical analysis (using twomethods one of them being a new proposedmethod) and experiments are developed on a simplifiedtypical hull of a container ship The torsion analysis is performed on a scale model of a container ship made of layered compositeplates and the first 5 natural frequencies are determinedThe results obtained with the proposed numerical method (software codeTORS made by the author) are compared with the results obtained with FE analysis and with the experiments done on the physicalmodel

1 Introduction

In recent years the improved design fabrication andmechanical performance of low-cost composites have led toan increase in the use of composites for large patrol boatshovercraft mine hunters and corvettes Currently there areall-composite naval ships up to 80ndash90m long and this trendcontinues It is predicted that from 2020 the hulls for mid-sized warships such as frigates that are typically 120ndash160mlong may be built of composite materials [1]

Nowadays laminated composite panels are gaining pop-ularity in maritime structural applications such as ship hullsdecks and ship and offshore superstructuresThese panels arebecoming increasingly used in structural marine applicationsdue to their high specific stiffness and high specific strength[1ndash4]

Investigations into the vibration and the elastic behaviiorof thin-walled structures especially thin-walled beams withopen and closed cross-sections have been carried out sincethe early works [5 6]

Monographs written by Kollbrunner and Basler [7]Gjelsvik [8] and a paper written by Kawai and Fujitani [9]are useful references for developing new methods of thethin-walled beam theory and its applications The intensiveresearch works have been made to develop finite element

models that can accurately represent the complex flexural-torsional warping deformable response of these structures

Yonghao et al in [10] present the theory and method ofanalyzing horizontal-torsion-coupled dynamic behavior of aship hull with large hatch opening by the transfer matrixmethod considering the ship hull as a simplified free-freenonuniform thin-walled beam

Senjanovic and Grubisic in [11] and Senjanovic et al [12]have developed a flexural and torsion theory of thin-walledgirders by introducing the concept of effective stiffness andmass parameters as modal quantities

In [13] N I Kim and M Y Kim have proposed a generaltheory for the shear deformable thin-walled beam withnonsymmetric openclosed cross-sections by developing animproved shear deformable beam theory based on Vlasovrsquosassumption and applying Hellinger-Reissner principle

In [14] Ascione et al have developed a one-dimensionalkinematical model capable of assessing the dynamic behaviorof fibre-reinforced polymers (FRPs) thin-walled beams withopen cross-section The proposed model accounts for theeffects of shear deformability

Voros in [15] has proposed a finite element model withseven degrees of freedoms per node He has also analyzed thefree vibration and mode shapes of straight beams

2 Mathematical Problems in Engineering

The thin-walled beam methodology based code wasdeveloped by Chirica et al [16 17] Chirica and Beznea in[18] for static torsion behavior analysis of ship hull with openand closed cross-sections

The methodology is extended in the present paper toapproach the macroelement model for torsion dynamicbehavior of the ship hull made of orthotropic layered com-posite material

The author considers the methodology presented in thepaper to be appropriate and efficient in analyzing torsionvibration problem of the thin-walled laminated compositebeam with a special application in ship design activityThe proposed macroelement model offers a facile mode toprepare the input data for the ship hull geometry describing

2 Torsion Dynamic Model ofthe Macroelement

21 General Hypothesis The methodology presented in thepaper is based on a macro element model aimed to analyzethe torsion ship hull dynamic treated as thin-walled beamThis methodology proposes an extended model for dynamicanalysis using the macro element model treated in [16ndash19] Inthe methodology the outline of the section is considered asa polygonal one For a straight line portion of cross-sectionoutline is corresponding a longitudinal strip plate (Figure 1)Due to the torsion loading of the thin-walled beam in thestrip plate the stretching-compression bending and shearingoccur The strip plate is treated as an Euler-Bernoulli beam[16ndash19] The stiffness matrix and inertia matrix of the macro-element are obtained by assembling the stiffness and inertiamatrices of the strips [16]

For macroelement analysis two coordinate systems areused

(i) global system119877119883119884119885 having axis119877119883 along the torsioncenters line (119877) of the cross-sections

(ii) local system attached to each strip plate 119896 (119862119896119909119896119910119896119911119896)having the axis 119862119896119909119896 parallel with 119877119883 along theweight centers of the cross-sections of the strip plate

The macro element is considered to be made of anorthotropic composite material

The methodology presented in this paper is treatingin different ways and different hypothesis depending onthe type of cross-section open and closed cross-sectionsin the papers developed in [16ndash19] where the appropriatemathematicalmodel explaining how the finite stripmethod isapplied for the static analysis of a composite hull is describedTherefore in this paper only the dynamic part of the modelis presented

22 Composite Characteristics of the Thin Wall In the nextchapter the thin-walled beam theory based methodology ispresented where the ship hull is considered as being madeout of isotropic material

In this paper the methodology is applied to the thin-walled beam structure of the ship hull made out oforthotropic layered composite plates

Z

Y

X

xk

ykzk

R

Ck

Figure 1 The macro-element model and the longitudinal strip

The equivalent stiffness coefficients for the tension-compression bending shearing and torsion loading of thestrip 119896 having 119899119904 layers have been determined by Chiricaet al in [16ndash18] according to the condition to have thesame stiffnesses for the structure made out in two materialcases isotropic material and orthotropic layered materialThis condition is resulting from the equivalent equations ofthe axial forces 119873119894119896 shear forces 119865119894119896 and bending moments119872119894119896 acting in the weight centers (119862119894119896) of all layers (119894 = 1 119899119904)

belonging to the 119896th longitudinal strip (see Figure 2)By considering the same geometry for the section (ie

by keeping the same dimensions ℎ119896 and 120575119896) the equivalentstiffness coefficients are resulting as follows [16ndash18]

(i) for tension-compression

(119864119860)119896 = 2(

119899119904

sum

119894=1

119864119894119896119892119894119896)ℎ119896 (1)

(ii) for bending

(119864119868)119896 = 2

119899119904

sum

119894=1

119864119894119896119868119894119896 =1

6(

119899119904

sum

119894=1

119864119894119896119892119894119896)ℎ3

119896 (2)

(iii) for torsion

(119866119868119879)119896= 8ℎ119896

119899119904

sum

119894=1

1198661198941198961199112

119894119896119892119894119896 (3)

Also the equivalent Youngrsquos (119864119896) and shear (119866119896) moduliare obtained as

119864119896 =2

120575119896

119899119904

sum

119894=1

119864119894119896119892119894119896

119866119896 =24

1205753

119896

119899119904

sum

119894=1

1198661198941198961199112

119894119896119892119894119896

(4)

In (1)ndash(4)119864119894119896 and119866119894119896 are theYoungrsquos and shearmodulusrespectively of the 119894th layer belonging to the 119896th strip

The other parameters used in (1)ndash(4) and Figure 2 are asfollows

(i) 119892119894119896 is the thickness of the 119894th layer belonging to the119896th longitudinal strip

Mathematical Problems in Engineering 3

yk

1i

Fik

Mik Cikhk

Nik

xk

gikzik

zk

120575k

Ck

ns

Figure 2 Forces acting on the 119894th layer of the 119896th longitudinal strip

yk

i 120579i ui

120593i 120593998400i

uijj 120579j uj

120593j 120593998400j

120585 = 05

L

xk

i j

Figure 3 Kinematic parameters of the 119896th longitudinal strip beam

(ii) ℎ119896 is the length of the 119896th longitudinal strip

(iii) 120575119896 is the total thickness of the 119896th longitudinal strip

(iv) 119911119894119896 is the 119911 coordinate (measured along the thickness)of the 119894th layer belonging to the 119896th longitudinal strip

23 Specific Matrices of the Macroelement Initially to applythe Vlasovrsquos theory the macroelement is considered to bemade of isotropic material with the characteristics deter-mined in the previous chapter

For the 119896th striprsquos displacements 120593(120585) and V119896(120585) (seeFigure 3) polynomial functions (third order) are chosen (see[16ndash18])

120593 (120585) = 1198671 (120585) 120593119894 + 1198711198673 (120585) 1205931015840

119894+ 1198672 (120585) 120593119895 + 1198711198674 (120585) 120593

1015840

119895

V119896 (120585) = 1198671 (120585) V119896

119894+ 1198711198673 (120585) 120579

119896

119894+ 1198672 (120585) V

119896

119895+ 1198711198674 (120585) 120579

119896

119895

(5)

The polynomial functions used in (5) have the form

1198671 (120585) = 1 minus 31205852+ 21205853 1198672 (120585) = 3120585

2minus 21205853

1198673 (120585) = 120585 minus 21205852+ 1205853 1198674 (120585) = 120585

3minus 1205852

(6)

where 120585 = 119909119871

In (5) 119871 is the strip length (equal to the macro-elementlength) and

120579119896

119894=119889V119896119894

119889119909 120579

119896

119895=

119889V119896119895

119889119909 (7)

The other parameters used in (5) are shown in Figure 3For horizontal bending and torsion of the 119896th longitudi-

nal strip beam the well-known stiffness and inertia matricesare obtained

k119896V =119864119868119896

1198713(

12 6119871 minus12 6119871

41198712

minus6119871 21198712

symm 12 minus6119871

41198712

)

k119896120593=119866119868119879119896

119871

((((((

(

6

5

119871

10minus6

5

119871

10

21198712

15minus119871

10minus1198712

10

symm6

5minus119871

10

21198712

15

))))))

)

m119896V =119898119871

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

m119896120593=120588119868119879119896119871

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

(8)

where

(i) 119898 is the distributed mass per unit length

(ii) 120588 is the material density

(iii) 119871 is the length of macroelement

(iv) 119868119879119896 is the conventional polar moment of the inertia ofthe 119896th strip plate

(v) 119868119896 is the moment of inertia regarding the lateralbending

(vi) 119864 and 119866 are the Youngrsquos and shear moduli respec-tively of the material

The equation between stretching-compression and bend-ing degrees of freedom (u119896 k119896) of the 119896th strip and degreesof freedom of the macroelement 120593 is obtained as the relationbetween master and slave nodes

u119896 = C119896119906120593120593 k

119896= C119896V120593120593 (9)


where C119896119906120593

and C119896V120593 are ldquoformrdquo matrices of the strip 119896presented in [16ndash18] The displacements vectors are

u119896 = (

119906119896

119894

119906119896

119894119895

119906119896

119895

) 120593 =((

(

120593119894

1205931015840

119894

120593119895

1205931015840

119895

))

)

(10)

Based on the coupling matrices for each strip beam thestripsrsquo stiffness and inertia matrices for axial and bendingloading are assembling to obtain the stiffness and inertiamatrices of the macroelement The assembling process isperformed by the transforming process using the orthogonalprocess [20]

The stiffness matrix of the macroelement is obtained by

K =119864 119868120596

1198713K119864119861 +

119866 119868119879

119871K119878119881 (11)

In (11) the matrices K119864119861 and K119878119881 are stiffness matrixfor warping torsion and stiffness matrix for free torsiondeveloped in [16ndash19]

The inertia matrix of the macroelement is obtained by

M =120588119868120596

119871M119864119861 + 120588119868119879119871M119878119881 (12)

In (12) the matrices M119864119861 and M119878119881 have the forms

M119864119861 =1

30(

36 3119871 minus36 6119871

41198712

minus31198711198712

2


1198712

)

M119878119881 =1

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

(13)

After the coupling of all macro-elements the inertiamatrix (M) and stiffness matrix (K) of the ship hull areobtained The equation of natural vibrations of the ship hullhas the form

MΔ + KΔ = 0 (14)

where Δ and Δ are the accelerations and displacementsvectors respectively

Assuming that Δ = Δ0119890119894119901119905 where Δ0 and 119901 are the mode

vector and natural frequency respectively (14) results in theeigenvalue problem

(K minus 1199012M)Δ0 = 0 (15)

3 Numerical Analysis FEM CodeTORS versus FE Analysis

Based on the methodology described in the previous chaptera macroelement-based code was developed The code TORSdeveloped by the author for static analysis in [16ndash18] and nowextended for dynamic numerical analysis may be applied toship hull behavior loaded to torsion For the present studya simplified model of a container ship made of compositematerials was analyzedThis type of ship has a very large deckopening for an efficientmaneuvering of the containers duringloading-unloading process Therefore in the middle part ofthe ship the deck is reduced to two longitudinal strips placedon both sides (see Figure 8)

The main characteristics of the model are length 119871119898 =24m breadth 119861119898 = 04m and depth 119863119898 = 02m Thelength of the open area is 12m and the lengths of the closedareas are 06m The material is E-glasspolyester having thecharacteristics determined by experimental tests in [21 22]

119864119909 = 46GPa 119864119910 = 46GPa 119864119911 = 13GPa 119866119909119910 = 5GPa119866119909119911 = 5GPa 119866119910119911 = 46GPa 120588 = 2045 kgm3120583119909119910

= 03 120583119910119911

= 042 120583119909119911

= 03

The ship hull model is divided into two parts the aft partand fore part with closed cross-sections and the middle parthaving open cross-sections (see Figure 8)

The thicknessesrsquo values of the hull shell are 2mm forside shell and 3mm for deck and bulkheads The model is asimplified ship hull of a container ship having the prismaticshell

To fulfill the thin-walled beam hypothesis (nondeforma-bility of the cross section) the model has 13 transversalbulkheads placed at every 02m (see Figure 8) The influenceof the bulkheads on the ship hull torsion behavior wasconsidered by introducing the supplementary inertias on thelateral displacement and torsion angle on the macro-elementlevel

The first numerical model is the 3Dmodel using macroe-lements from TORS code

In the beginning a convergence analysis was performedFinally the optimum number of strip elements and the opti-mum lengths of the macroelements were determined

Therefore model mesh used for TORS calculus isconcerning 12 macro-elements 6 macro-elements in theclosed parts (3 macro-elements for each end part) of theship model and 6 macro-elements in the open part (middlepart) Each macro-element representing a piece of shipmodel was modeled with longitudinal plane strip elementsThe closed section type macro-elements concern 14 stripelements (Figure 4) The open section type macro-elementsare modeled with 12 longitudinal strip elements as it is seenin Figure 5 The total number of strip elements (having thesame length of 02m) is 228 The outline of the cross sectionof the model was approached with a polygonal lineThe bilgewas modelled with 3 longitudinal strip elements (S2 S3 andS4 in Figures 4 and 5)

The second numerical model is a 3-D model with 4-nodeshell composite finite elements


X

Y

Z

Oo(R)

S1S2

S3S4

S5S6 S7

Figure 4 The mesh concerning longitudinal strips for closed sec-tion

X

Y

Z

Oo(R)

S1S2S3S4

S5

S6

Figure 5Themesh concerning longitudinal strips for open section

In the beginning a convergence analysis was performed[18] Finally the optimum dimension of the quadrilateralelement side (002m) was determined and a number of 18110shell elements were used in the mesh model

The results obtained with the code TORS were comparedwith the results obtained by FE analysis (see Table 1) Due tothe fact that the real ship hasmuch stiffened structure in bothends in the both modeling types (TORS and FE analysis) themodel is considered as clamped at the ends

The graphical representations of the first 3 vibrationmodes in air presented in the Figure 6 give a good infor-mation about the differences in torsion stiffness betweenclosed and opened section (for the ship hullmodel the torsionstiffnesses ratio is about 104)

On this count the numerical analysis of the simplifiedmodel (named as ldquofullrdquo model) was further done by con-sidering only the open part of the model (named as ldquoopenrdquomodel) clamped in the joining sections between opened andclosed parts The results obtained after numerical calculuswith TORS are presented in Figure 7 as a graphical zoom ofthe first torsion modal deformed shape (variation of torsionangle) in the area between sections placed at 04m and 06mThe curve 1 represents variation of torsion angle for ldquofullrdquomodel The curve 2 represents variation of torsion angle forldquoopenrdquomodel As it is seen the both curves are approaching inthe open area The results obtained for the first 5 frequenciesof both models (model 1 is considered as ldquofullrdquo model andmodel 2 is considered as ldquoopenrdquo model) are presented in theTable 1

Table 1 Natural frequencies (Hz)

Modeno

TORS FE TORS FE ExperimentModel 1 Model 1 Model 2 Model 2

1 309 301 300 298 2902 329 319 323 313 3073 353 342 347 337 3324 368 373 361 367 3595 416 411 403 406 395

120593x

(rad

)

06

04

02

0

minus02

minus04

minus060 06 12 18 24

x (m)

Mode 1

Mode 2 Mode 3

Figure 6The first three vibrationmodes in air obtained by numer-ical calculus with TORS

2

04 06

1

Figure 7 Graphical zoom for vibration modes

4 Experimental Tests

It is often difficult to achieve frequencies and mode shapeswith complex simulation models used in a model testNumerical model tested with TORS was verified with ademonstrator of a simple ship hull model made out of layeredcomposites

The test rig for torsional vibrations estimation of the shiphull model is shown in Figure 8 The torque is obtained witha dynamic couple of two forces acting on a frame placed inthe midship section of the ship model


(a) (b)

Figure 8 Test rig

The natural frequencies of the model were determinedwith an equipment concerning a vibrometer and an accel-erometer The second measuring equipment used for exper-iments consists of gauges measurements Both equipmentshave generated the results in a good agreement

Impact excitation method is used in this study to predictthe natural frequencies of the model The impact excitationswere obtained with a special hammer

By striking the model at any point on the special frameplaced on the middle section of the model and displayingthe response signals from the accelerometer in the frequencydomain the fundamental natural frequencies were measuredat resonant peaks To determine the torsion-horizontal bend-ing mode shape corresponding to the natural frequency ofthe model a large number of repeated tests were done (seeFigure 8) To obtain the best results for torsion coupled withhorizontal bending vibrations the accelerometer was placedin various positions on deck plate into side part of themodelThe vertical-bending natural frequencies were filteredaccording to the calculations done for this type of vibrations

The aim of the work was to determine the frequencies ofthe ship hull (global vibrations) To avoid the detecting of thelocal vibrations of the panels (not interested in this work) theaccelerometer was placed only in the correspondence of thetransversal bulkheads or very close to the special frame

Data collection was performed according to a test pro-gram and procedure

(i) the determining of the appropriate measuring points(positions of the accelerometers and strain gauges)chosen so that to avoid asmuch as possible of the localvibrations

(ii) the determining of the appropriate impact points(positions of the hammer) on the frame so that toperform as much as possible of only global vibrationsexciting (and not the frame vibrations)

(iii) collecting data with accelerometers and strain gaugesduring impact excitations

(iv) data processing(v) post-test verification The model test was verified

by post-test analysis This analysis was feeding the

measured natural frequencies back into the numericalmodel of the ship hull model by comparing thenumerical and experimental frequencies The pur-pose of this check is to verify that the global condi-tions imposed in the numerical simulation (boundaryconditions) of the model provide a good agree-ment with the global conditions of the experimentalmodel

(vi) the boundary conditions were reconsidered into thenumerical analysis and the experimental testingswere remade

The fundamental natural frequencies of the model can bepredicted from the frequency domain of the accelerometerresponse The first 5 frequencies obtained from experimentsgive the values closed to the natural frequencies obtainedfromnumerical analysis (TORS and FE analysis) as are shownin the Table 1

5 Conclusions

The proposed methodology is a developed and extendedtheory from static to dynamic problems presented in [16ndash18]

The methodology was found to be appropriate andefficient in analyzing torsion vibration problem of a thin-walled laminated composite beam with a special applicationin ship design activity

The facile mode to prepare the input data for definingthe cross sections and finally the entire ship hull by usingmacroelement model and code TORS recommends themethodology as a good tool for torsion analysis of shiphull made of composite materials Additionally the CPUtime machine with TORS is much lesser than the CPU timeobtained with FE analysis [18]

Comparison between the results obtained in numericaland experimental analysis shows good agreements for theresonance frequencies in most cases

Also the close comparation between FEA analysis andTORS calculus provides close values for the natural frequen-cies and modal shapes and reveals macro-element model-based methodology as a good and efficient tool


Certain remarks can be developed

(i) The analysed model is a prismatic one to avoid thedistortions in results due to sudden variation of shipforms on the ends

(ii) As it is seen in the graphical representations inFigure 6 due to very huge torsion stiffness differencebetween closed and opened sections the ship hullends may be considered as clamped

(iii) According to the torsion deformations modes theapproach calculations can be done only for the openpart of the ship considering the middle part as beingclamped in the joint sections with closed parts Thisapproach calculus can be done with fair results for thedesign stage of the ship hullswith large deck openings

(iv) Due to the variation of the cross section shape ofthe model a coupled torsion vibrations with lateralbending vibrations that cannot be separated occurredTherefore the frequencies for each of both vibrationtypes cannot be separated

(v) The results obtained by the three methods (macroele-mentmodelmdashTORS software FE analysis and exper-imental tests) are in a fair agreement

Acknowledgment

The work has been performed in the scope of the NationalProject 168 EU-UEFISCDI (2012ndash2014)

References

[1] A P Mouritz E Gellert P Burchill and K Challis ldquoReview ofadvanced composite structures for naval ships and submarinesrdquoComposite Structures vol 53 no 1 pp 21ndash24 2001

[2] I Chirica and E F Beznea ldquoBuckling analysis of the compositeplates with delaminationsrdquo Computational Materials Sciencevol 50 no 5 pp 1587ndash1591 2011

[3] I Chirica D Boazu and E F Beznea ldquoResponse of ship hulllaminated plates to close proximity blast loadsrdquo ComputationalMaterials Science vol 52 no 1 pp 197ndash203 2012

[4] E F Beznea and I Chirica ldquoBuckling behaviour of imperfectcomposite platerdquo Revista de Materiale Plastice vol 48 no 3 pp231ndash239 2011

[5] V Z Vlasov Thin-Walled Elastic Beams Program of ScientificTranslations Jerusalem Israel 2nd edition 1961

[6] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 1961

[7] C F Kollbrunner and K Basler Torsion in Structures SpringerBerlin Germany 1969

[8] A GjelsvikTheTheory ofThinWalled Bars JohnWiley amp SonsNew York NY USA 1981

[9] T Kawai and Y Fujitani Some Considerations on the ModernBeamTheorymdashDevelopment of the Practical Methods vol 32 ofReport of the Institute of Industrial Science The University ofTokyo 1986

[10] P Yonghao P Yougang and Y Xueyong ldquoThe horizontal-torsion-coupled model analysis and dynamic response calcu-lation of the ship hull with a large hatch openingrdquo Journal ofTianjin University pp 51ndash58 1993

[11] I Senjanovic and R Grubisic ldquoCoupled horizontal and tor-sional vibration of a ship hull with large hatch openingsrdquoComputers and Structures vol 41 no 2 pp 213ndash226 1991

[12] I Senjanovic S Tomasevic and R Grubisic ldquoCoupled horizon-tal and torsional vibrations of container shipsrdquo Brodogradniavol 58 no 4 pp 365ndash379 2007

[13] N I Kim and M Y Kim ldquoExact dynamicstatic stiffnessmatrices of non-symmetric thin-walled beams consideringcoupled shear deformation effectsrdquoThin-Walled Structures vol43 no 5 pp 701ndash734 2005

[14] L Ascione L Feo and G Mancusi ldquoOn the statical behaviourof fibre-reinforced polymer thin-walled beamsrdquo Composites Bvol 31 no 8 pp 643ndash654 2000

[15] G M Voros ldquoFree vibration of thin-walled beamsrdquo Journal ofComputational and Applied Mechanics vol 9 no 2 pp 1ndash142008

[16] I Chirica S D Musat R Chirica and E F Beznea ldquoTorsionalbehaviour of the ship hull composite modelrdquo ComputationalMaterials Science vol 50 no 4 pp 1381ndash1386 2011

[17] I Chirica D Boazu E F Beznea and A Chirica ldquoShip hullcomposite plates analysis under blast loadsrdquo in Proceedings ofthe of the 3rd International Conference on Advances in MarineStructures (MARSTRUCT rsquo11) pp 343ndash350 Taylor amp FrancisHamburg Germany 2011

[18] I Chirica and E F Beznea ldquoA numerical model for torsionanalysis of composite ship hullsrdquo Mathematical Problems inEngineering vol 2012 Article ID 212346 17 pages 2012

[19] I Chirica E F Beznea and D Boazu ldquoTorsion dynamicanalysis of a ship hull composite modelrdquo in Proceedings ofthe of the 14th International Congress of International MaritimeAssociation of the Mediterranean vol 1 pp 331ndash337 Taylor ampFrancis Genova Italy 2011

[20] S K Ider and F M L Amirouche ldquoCoordinate reduction in thedynamics of constrained multibody systemsmdasha new approachrdquoJournal of Applied Mechanics vol 55 no 4 pp 899ndash904 1988

[21] E F Beznea Studies and researches on the buckling behaviour ofthe composite panels [Doctoral thesis] UniversityDunarea de Josof Galati 2008

[22] E F Beznea Composite StructuresmdashLaboratory ApplicationsGalati University Press 2012 (Romanian)

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Stochastic AnalysisInternational Journal of


The thin-walled beam methodology based code wasdeveloped by Chirica et al [16 17] Chirica and Beznea in[18] for static torsion behavior analysis of ship hull with openand closed cross-sections

The methodology is extended in the present paper toapproach the macroelement model for torsion dynamicbehavior of the ship hull made of orthotropic layered com-posite material

The author considers the methodology presented in thepaper to be appropriate and efficient in analyzing torsionvibration problem of the thin-walled laminated compositebeam with a special application in ship design activityThe proposed macroelement model offers a facile mode toprepare the input data for the ship hull geometry describing

2 Torsion Dynamic Model ofthe Macroelement

21 General Hypothesis The methodology presented in thepaper is based on a macro element model aimed to analyzethe torsion ship hull dynamic treated as thin-walled beamThis methodology proposes an extended model for dynamicanalysis using the macro element model treated in [16ndash19] Inthe methodology the outline of the section is considered asa polygonal one For a straight line portion of cross-sectionoutline is corresponding a longitudinal strip plate (Figure 1)Due to the torsion loading of the thin-walled beam in thestrip plate the stretching-compression bending and shearingoccur The strip plate is treated as an Euler-Bernoulli beam[16ndash19] The stiffness matrix and inertia matrix of the macro-element are obtained by assembling the stiffness and inertiamatrices of the strips [16]

For macroelement analysis two coordinate systems areused

(i) global system119877119883119884119885 having axis119877119883 along the torsioncenters line (119877) of the cross-sections

(ii) local system attached to each strip plate 119896 (119862119896119909119896119910119896119911119896)having the axis 119862119896119909119896 parallel with 119877119883 along theweight centers of the cross-sections of the strip plate

The macro element is considered to be made of anorthotropic composite material

The methodology presented in this paper is treatingin different ways and different hypothesis depending onthe type of cross-section open and closed cross-sectionsin the papers developed in [16ndash19] where the appropriatemathematicalmodel explaining how the finite stripmethod isapplied for the static analysis of a composite hull is describedTherefore in this paper only the dynamic part of the modelis presented

22 Composite Characteristics of the Thin Wall In the nextchapter the thin-walled beam theory based methodology ispresented where the ship hull is considered as being madeout of isotropic material

In this paper the methodology is applied to the thin-walled beam structure of the ship hull made out oforthotropic layered composite plates

Z

Y

X

xk

ykzk

R

Ck

Figure 1 The macro-element model and the longitudinal strip

The equivalent stiffness coefficients for the tension-compression bending shearing and torsion loading of thestrip 119896 having 119899119904 layers have been determined by Chiricaet al in [16ndash18] according to the condition to have thesame stiffnesses for the structure made out in two materialcases isotropic material and orthotropic layered materialThis condition is resulting from the equivalent equations ofthe axial forces 119873119894119896 shear forces 119865119894119896 and bending moments119872119894119896 acting in the weight centers (119862119894119896) of all layers (119894 = 1 119899119904)

belonging to the 119896th longitudinal strip (see Figure 2)By considering the same geometry for the section (ie

by keeping the same dimensions ℎ119896 and 120575119896) the equivalentstiffness coefficients are resulting as follows [16ndash18]

(i) for tension-compression

(119864119860)119896 = 2(

119899119904

sum

119894=1

119864119894119896119892119894119896)ℎ119896 (1)

(ii) for bending

(119864119868)119896 = 2

119899119904

sum

119894=1

119864119894119896119868119894119896 =1

6(

119899119904

sum

119894=1

119864119894119896119892119894119896)ℎ3

119896 (2)

(iii) for torsion

(119866119868119879)119896= 8ℎ119896

119899119904

sum

119894=1

1198661198941198961199112

119894119896119892119894119896 (3)

Also the equivalent Youngrsquos (119864119896) and shear (119866119896) moduliare obtained as

119864119896 =2

120575119896

119899119904

sum

119894=1

119864119894119896119892119894119896

119866119896 =24

1205753

119896

119899119904

sum

119894=1

1198661198941198961199112

119894119896119892119894119896

(4)

In (1)ndash(4)119864119894119896 and119866119894119896 are theYoungrsquos and shearmodulusrespectively of the 119894th layer belonging to the 119896th strip

The other parameters used in (1)ndash(4) and Figure 2 are asfollows

(i) 119892119894119896 is the thickness of the 119894th layer belonging to the119896th longitudinal strip


yk

1i

Fik

Mik Cikhk

Nik

xk

gikzik

zk

120575k

Ck

ns


yk

i 120579i ui

120593i 120593998400i

uijj 120579j uj

120593j 120593998400j

120585 = 05

L

xk

i j







120593 (120585) = 1198671 (120585) 120593119894 + 1198711198673 (120585) 1205931015840

119894+ 1198672 (120585) 120593119895 + 1198711198674 (120585) 120593

1015840

119895

V119896 (120585) = 1198671 (120585) V119896

119894+ 1198711198673 (120585) 120579

119896

119894+ 1198672 (120585) V

119896

119895+ 1198711198674 (120585) 120579

119896

119895

(5)


1198671 (120585) = 1 minus 31205852+ 21205853 1198672 (120585) = 3120585

2minus 21205853

1198673 (120585) = 120585 minus 21205852+ 1205853 1198674 (120585) = 120585

3minus 1205852

(6)

where 120585 = 119909119871


120579119896

119894=119889V119896119894

119889119909 120579

119896

119895=

119889V119896119895

119889119909 (7)



k119896V =119864119868119896

1198713(

12 6119871 minus12 6119871

41198712

minus6119871 21198712


41198712

)

k119896120593=119866119868119879119896

119871

((((((

(

6

5

119871

10minus6

5

119871

10

21198712

15minus119871

10minus1198712

10

symm6

5minus119871

10

21198712

15

))))))

)

m119896V =119898119871

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

m119896120593=120588119868119879119896119871

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

(8)

where








u119896 = C119896119906120593120593 k

119896= C119896V120593120593 (9)


where C119896119906120593


u119896 = (

119906119896

119894

119906119896

119894119895

119906119896

119895

) 120593 =((

(

120593119894

1205931015840

119894

120593119895

1205931015840

119895

))

)

(10)



K =119864 119868120596

1198713K119864119861 +

119866 119868119879

119871K119878119881 (11)



M =120588119868120596

119871M119864119861 + 120588119868119879119871M119878119881 (12)


M119864119861 =1

30(

36 3119871 minus36 6119871

41198712

minus31198711198712

2


1198712

)

M119878119881 =1

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

(13)


MΔ + KΔ = 0 (14)




(K minus 1199012M)Δ0 = 0 (15)





= 03 120583119910119911

= 042 120583119909119911

= 03









X

Y

Z

Oo(R)

S1S2

S3S4

S5S6 S7


X

Y

Z

Oo(R)

S1S2S3S4

S5

S6







Modeno


1 309 301 300 298 2902 329 319 323 313 3073 353 342 347 337 3324 368 373 361 367 3595 416 411 403 406 395

120593x

(rad

)

06

04

02

0

minus02

minus04

minus060 06 12 18 24

x (m)

Mode 1

Mode 2 Mode 3


2

04 06

1






(a) (b)

Figure 8 Test rig














5 Conclusions













Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










yk

1i

Fik

Mik Cikhk

Nik

xk

gikzik

zk

120575k

Ck

ns


yk

i 120579i ui

120593i 120593998400i

uijj 120579j uj

120593j 120593998400j

120585 = 05

L

xk

i j







120593 (120585) = 1198671 (120585) 120593119894 + 1198711198673 (120585) 1205931015840

119894+ 1198672 (120585) 120593119895 + 1198711198674 (120585) 120593

1015840

119895

V119896 (120585) = 1198671 (120585) V119896

119894+ 1198711198673 (120585) 120579

119896

119894+ 1198672 (120585) V

119896

119895+ 1198711198674 (120585) 120579

119896

119895

(5)


1198671 (120585) = 1 minus 31205852+ 21205853 1198672 (120585) = 3120585

2minus 21205853

1198673 (120585) = 120585 minus 21205852+ 1205853 1198674 (120585) = 120585

3minus 1205852

(6)

where 120585 = 119909119871


120579119896

119894=119889V119896119894

119889119909 120579

119896

119895=

119889V119896119895

119889119909 (7)



k119896V =119864119868119896

1198713(

12 6119871 minus12 6119871

41198712

minus6119871 21198712


41198712

)

k119896120593=119866119868119879119896

119871

((((((

(

6

5

119871

10minus6

5

119871

10

21198712

15minus119871

10minus1198712

10

symm6

5minus119871

10

21198712

15

))))))

)

m119896V =119898119871

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

m119896120593=120588119868119879119896119871

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

(8)

where








u119896 = C119896119906120593120593 k

119896= C119896V120593120593 (9)


where C119896119906120593


u119896 = (

119906119896

119894

119906119896

119894119895

119906119896

119895

) 120593 =((

(

120593119894

1205931015840

119894

120593119895

1205931015840

119895

))

)

(10)



K =119864 119868120596

1198713K119864119861 +

119866 119868119879

119871K119878119881 (11)



M =120588119868120596

119871M119864119861 + 120588119868119879119871M119878119881 (12)


M119864119861 =1

30(

36 3119871 minus36 6119871

41198712

minus31198711198712

2


1198712

)

M119878119881 =1

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

(13)


MΔ + KΔ = 0 (14)




(K minus 1199012M)Δ0 = 0 (15)





= 03 120583119910119911

= 042 120583119909119911

= 03









X

Y

Z

Oo(R)

S1S2

S3S4

S5S6 S7


X

Y

Z

Oo(R)

S1S2S3S4

S5

S6







Modeno


1 309 301 300 298 2902 329 319 323 313 3073 353 342 347 337 3324 368 373 361 367 3595 416 411 403 406 395

120593x

(rad

)

06

04

02

0

minus02

minus04

minus060 06 12 18 24

x (m)

Mode 1

Mode 2 Mode 3


2

04 06

1






(a) (b)

Figure 8 Test rig














5 Conclusions













Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where C119896119906120593


u119896 = (

119906119896

119894

119906119896

119894119895

119906119896

119895

) 120593 =((

(

120593119894

1205931015840

119894

120593119895

1205931015840

119895

))

)

(10)



K =119864 119868120596

1198713K119864119861 +

119866 119868119879

119871K119878119881 (11)



M =120588119868120596

119871M119864119861 + 120588119868119879119871M119878119881 (12)


M119864119861 =1

30(

36 3119871 minus36 6119871

41198712

minus31198711198712

2


1198712

)

M119878119881 =1

420(

156 22119871 54 minus13119871

41198712

13119871 minus31198712


41198712

)

(13)


MΔ + KΔ = 0 (14)




(K minus 1199012M)Δ0 = 0 (15)





= 03 120583119910119911

= 042 120583119909119911

= 03









X

Y

Z

Oo(R)

S1S2

S3S4

S5S6 S7


X

Y

Z

Oo(R)

S1S2S3S4

S5

S6







Modeno


1 309 301 300 298 2902 329 319 323 313 3073 353 342 347 337 3324 368 373 361 367 3595 416 411 403 406 395

120593x

(rad

)

06

04

02

0

minus02

minus04

minus060 06 12 18 24

x (m)

Mode 1

Mode 2 Mode 3


2

04 06

1






(a) (b)

Figure 8 Test rig














5 Conclusions













Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X

Y

Z

Oo(R)

S1S2

S3S4

S5S6 S7


X

Y

Z

Oo(R)

S1S2S3S4

S5

S6







Modeno


1 309 301 300 298 2902 329 319 323 313 3073 353 342 347 337 3324 368 373 361 367 3595 416 411 403 406 395

120593x

(rad

)

06

04

02

0

minus02

minus04

minus060 06 12 18 24

x (m)

Mode 1

Mode 2 Mode 3


2

04 06

1






(a) (b)

Figure 8 Test rig














5 Conclusions













Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

Figure 8 Test rig














5 Conclusions













Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Torsion Dynamic Behaviour of the Ship