an advanced theory of thin-walled girders with application ...an advanced theory of thin-walled...

Marine Structures 22 (2009) 387–437

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Marine Structuresjournal homepage: www.elsevier .com/locate/

marstruc

An advanced theory of thin-walled girders with applicationto ship vibrations

I. Senjanovic*, S. Tomasevic, N. VladimirUniversity of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, I. Lucica 5, 10000 Zagreb, Croatia

a r t i c l e i n f o

Article history:Received 8 July 2008Received in revised form 7 January 2009Accepted 24 March 2009

Keywords:Thin-walled girder theoryBeam modelCoupled vibrationsContainer shipFEM

* Corresponding author. Tel.: þ385 16168142; faE-mail address: [email protected] (I. Senjan

0951-8339/$ – see front matter � 2009 Elsevier Ltdoi:10.1016/j.marstruc.2009.03.004

a b s t r a c t

The paper presents an outline of the advanced theory of thin-walled girders. The improvement includes shear influence ontorsion as an extension of shear influence on bending. The analogybetween bending and torsion is recognized and pointed outthroughout the paper. Complete differential equations of coupledflexural and torsional vibrations for a prismatic girder are derived.In addition, the 8 d.o.f. beam finite element, utilizing the energyapproach, is constituted with stiffness and mass matrices, and loadvectors. The paper describes determining of geometrical propertiesof multi-cell open cross-sections by employing the strip elementmethod. Numerical procedures for vibration analyses are outlined.Furthermore, dry natural vibrations of a VLCS (Very LargeContainer Ship) are analysed by 1D FEM model as a prerogative forhydroelastic analyses of these relatively flexible vessels. Influenceof transverse bulkheads is taken into account by increasingtorsional stiffness of the ship hull proportionally to their defor-mation energies. Validation of 1D FEM model is checked bycorrelation analysis with the vibration response of the fine 3D FEMmodel.

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Nomenclature

A cross-section areaAi integration constantsAs shear areaak, bk, ck, dk, ek, fk, coefficients of shape functionsBw warping bimomentb one half of bulkhead breadthC energy coefficientc eccentricityE Young’s modulusEtot total energyf normal stress flowG shear modulusg shear stress flowH ship heightIb moment of inertia of cross-sectionIs shear inertia modulusIt torsional modulusIw warping modulusi, j, k indexesJb moment of inertia of distributed massJ0t ; Jt polar moment of inertia of distributed mass about centre of gravity and shear centre

Jw bimoment of inertia of distributed mass about warping centreL, l lengthM bending momentm distributed massn mode numberQ shear forceq distributed loadT torqueTt torsional torqueTw warping torquet timeU strain energyw deflectionwb bending deflectionws shear deflectionu, v membrane displacementsx, y, z coordinateszC, zS, zG coordinate of centroid, shear centre and gravity centre[C] damping matrix[D] elasticity matrix[K], [k] stiffness matrices[L] deformation matrix[M], [m] mass matrices[T] transformation matrix{F}, {f} force vectors

I. Senjanovic et al. / Marine Structures 22 (2009) 387–437388

{P}, {R} force vectors{U}, {V} displacement vectors{D}, {d} displacement vectorsa, h vibration coefficientsb, g stiffness ratiosz damping coefficientw variation of twist anglem distributed torquen Poisson’s coefficientx non-dimensional coordinates normal stresss shear stress4 rotation anglej twist anglejt pure twist anglejs shear twist angleu natural frequency[f] matrix of shape functions

I. Senjanovic et al. / Marine Structures 22 (2009) 387–437 389

1. Introduction

Increased sea transport requires building of ultra large container ships which are quite flexible [1].Therefore, their strength has to be checked by hydroelastic analysis [2]. The methodology of hydroe-lastic analysis is described in [3]. It includes the definition of the structural model, ship and cargo massdistributions, and geometrical model of ship wetted surface. Hydroelastic analysis is based on themodal superposition method. First, dry natural vibrations of ship hull are calculated. Then, modalhydrostatic stiffness, modal added mass, modal damping and modal wave load are determined. Finally,the calculation of wet natural vibrations is performed and transfer functions for determining shipstructural response to wave excitation are obtained [4,5]. The approach is checked by the model test ofthe flexible barge [6,7].

The intention of this paper is to present an advanced numerical procedure based on the beam andthin-walled girder theories for reliable calculation of dry natural vibrations of container ships, as animportant step in their hydroelastic analysis. A ship hull, as an elastic nonprismatic thin-walled girder,performs longitudinal, vertical, horizontal and torsional vibrations. Since the cross-sectional centre ofgravity and centroid, as well as the shear centre positions are not identical, coupled longitudinal andvertical, and horizontal and torsional vibrations occur, respectively.

The distance between the centre of gravity and centroid for longitudinal and vertical vibrations, aswell as distance between the former and shear centre for horizontal and torsional vibrations arenegligible for conventional ships. Therefore, in the above cases ship hull vibrations are usually analysedseparately. However, the shear centre in ships with large hatch openings is located outside the cross-section, i.e. below the keel, and therefore the coupling of horizontal and torsional vibrations isextremely high.

The above problem is rather complicated due to geometrical discontinuity of the hull cross-section.The accuracy of the solution depends on the reliability of stiffness parameters determination, i.e. ofbending, shear, torsional and warping moduli. The finite element method is a powerful tool to solve theabove problem in a successful way.

One of the first solutions for coupled horizontal and torsional hull vibrations, dealing with the finiteelement technique, is given in [8,9]. Generalised and improved solutions are presented in [10,11]. In allthese references, the determination of hull stiffness is based on the classical thin-walled girder theory,which does not give a satisfactory value for the warping modulus of the open cross-section [12,13].


Apart from that, the fixed values of stiffness moduli are determined, so that the application of the beamtheory for hull vibration analysis is limited to a few lowest natural modes only. Otherwise, if the modedependent stiffness parameters are used the application of the beam theory can be extended up to thetenth natural mode [14–16].

Based on the above described state-of-the-art and inspired motivation, this paper brings somenovelties and improvements to the thin-walled girder theory. Shear influence on torsion, as anextension of its influence on bending, is taken into account. Analogy between bending and torsion isused as a sign-post. All the relevant stiffness and mass parameters are included in the differentialequations of coupled flexural and torsional vibrations. A more complete beam finite element with eightdegrees of freedom is developed. The application of the advanced theory is illustrated in the case ofa very large container vessel. The influence of transverse bulkheads is incorporated in the hull stiffness[17,18]. The validity of the improved theory is checked by correlation of the 1D FEM vibration responsewith that obtained by 3D FEM analysis.

2. Differential equations of beam vibrations

Referring to the flexural beam theory [19,20], the total beam deflection, w, consists of the bendingdeflection, wb, and the shear deflection, ws, while the angle of cross-section rotation depends only onthe former, Fig. 1

Fig. 1. Beam bending and torsion.


w ¼ wb þws; 4 ¼ vwb: (1)
vx
The cross-sectional forces are the bending moment and the shear force

M ¼ �EIbv4

vx; (2)

Q ¼ GAsvws

vx; (3)

where E and G are Young’s and shear modulus, respectively, while Ib and As are the moment of inertia ofcross-section and shear area, respectively.

The inertia load consists of the distributed transverse load, qi, and the bending moment, mi, and inthe case of coupled horizontal and torsional vibration is specified as

qi ¼ �m

v2wvt2 þ c

v2j

vt2

!; (4)

mi ¼ �Jbv24

vt2 ; (5)

where m is the distributed ship and added mass, Jb is the moment of inertia of ship mass about z-axis,and c is the distance between the centre of gravity and the shear centre, c ¼ zG � zS, Fig. 2.

Concerning torsion, the total twist angle, j, consists of the pure twist angle, jt, and the shearcontribution, js, while the second beam displacement, which causes warping of cross-section, isvariation of the pure twist angle, i.e. Fig. 1 [18]

j ¼ jt þ js; w ¼ vjt

vx: (6)

Fig. 2. Cross-section of a thin-walled girder.


The cross-sectional forces include the pure torsional torque, Tt, warping bimoment, Bw, and addi-tional torque due to restrained warping, Tw

Tt ¼ GItw; (7)

Bw ¼ �EIwvw

vx; (8)

Tw ¼ GIsvjs

vx; (9)

where It, Iw and Is are the torsional modulus, warping modulus and shear inertia modulus, respectively.The inertia load consists of the distributed torque, mti, and the bimoment, bi, presented in the

following form:

mti ¼ �Jtv2j

vt2 �mcv2wvt2 ; (10)

bi ¼ �Jwv2w

vt2 ; (11)

where Jt is the polar moment of inertia of ship and added mass about the shear centre, and Jw is thebimoment of inertia of ship mass about the warping centre, Fig. 2.

Considering the equilibrium of a differential element, one can write for flexural vibrations

vMvx¼ Q þ mi; (12)

vQvx¼ �qi � q; (13)

and for torsional vibrations [17]

vBw

vx¼ Tw þ bi; (14)

vTt

vxþ vTw

vx¼ �mti � m: (15)

The above equations can be reduced to two coupled partial differential equations as follows.Substituting Eqs. (2) and (3) into (12) yields

vws

vx¼ � EIb

GAs

v24

vx2þ Jb

GAs

v24

vt2: (16)

By inserting Eqs. (3) and (4) into (13) leads to

EIbv44

vx4þm

v24

vt2��

Jb þmEIbGAs

�v44

vx2vt2þmJb

GAs

v44

vt4þmc

v3j

vxvt2¼ vq

vx: (17)

In similar way, substituting Eqs. (8) and (9) into (14) yields


vjs

vx¼ �EIw

GIs

v2w

vx2 þJwGIs

v2w

vt2 : (18)

By inserting Eqs. (7), (9) and (10) into (15) one finds

EIwv4w

vx4 � GItv2w

vx2 þ Jtv2w

vt2 ��

Jw þ JtEIwGIs

�v4w

vx2vt2 þJwGIs

v4w

vt4 þmcv3w

vxvt2 ¼vm

vx: (19)

Furthermore, j in (17) can be split into jt þ js and the later term can be expressed with (18). Similarsubstitution can be done for w ¼ wb þws in (19), where ws is given with (16). Thus, taking into accountthat 4 ¼ vwb=vx and w ¼ vjt=vx, Eqs. (17) and (19) after integration per x read

EIbv4wb

vx4 þmv2wb

vt2 ��

Jb þmEIbGAs

�v4wb

vx2vt2 þmJbGAs

v4wb

vt4 þmc

v2jt

vt2 �EIwGIs

v4jt

vx2vt2 þJwGIs

v4jt

vt4

!¼ q

(20)

EIwv4jt

vx4 � GItv2jt

vx2 þ Jtv2jt

vt2 ��

Jw þ JtEIwGIs

�v4jt

vx2vt2 þJwGIs

v4jt

vt4

þmc

v2wb

vt2� EIb

GAs

v4wb

vx2vt2þ Jb

GAs

v4wb

vt4

!¼ m: ð21Þ

After solving Eqs. (20) and (21) the total deflection and twist angle are obtained by employing (16)and (18)

w ¼ wb þws ¼ wb �EIbGAs

v2wb

vx2 þJb

GAs

v2wb

vt2 þ f ðtÞ; (22)

j ¼ jt þ js ¼ jt �EIwGIs

v2jt

vx2 þJwGIs

v2jt

vt2 þ gðtÞ; (23)

where f(t) and g(t) are integration functions, which depend on initial conditions.The main purpose of developing differential equations of vibrations (20) and (21) is to get

insight into their constitution, position and role of the stiffness and mass parameters, andcoupling, which is realized through the inertia terms. If the pure torque Tt is excluded from theabove theoretical consideration, it is obvious that the complete analogy between bending andtorsion exists, [17].

Application of Eqs. (20) and (21) is limited to prismatic girders. It is illustrated in case of uncoupledtorsional natural vibrations of a uniform beam in Appendix A. For more complex problems, like shiphull, the finite element method, as a powerful tool, is on disposal.

The shape functions of beam finite element for vibration analysis have to satisfy the followingconsistency relations for harmonic vibrations obtained from Eqs. (22) and (23), [20]

w ¼ wb þws ¼�

1� u2 JbGAs

�wb �

EIbGAs

d2wb

dx2 ; (24)

j ¼ jt þ js ¼�

1� u2 JwGIs

�jt �

EIwGIs

d2jt

dx2 : (25)


3. Beam finite element

The properties of a finite element for the coupled horizontal and torsional vibration analysis can bederived from the total element energy. The total energy consists of the strain energy, the kinetic energy,the work of the external lateral load, q, and the torque, m, and the work of the boundary forces. Thus,according to [9,20],

Etot ¼12

Z l

0

"EIb

v2wb

vx2

!2

þGAs

�vws

vx

�2

þEIw

v2jt

vx2

!2

þGIs

�vjs

vx

�2

þGIt

�vjt

vx

�2#

dx

þ12

Z l

0

"m�

vwvt

�2

þJb

v2wb

vxvt

!2

þ2mcvwvt

vj

vtþ Jw

v2jt

vxvt

!2

þJt

�vj

vt

�2#

dx

�R l

0 ðqwþ mjÞdxþ ðQw�M4þ Tjþ BwwÞl0;

(26)

where l is the element length.Since the beam has four displacements, w;4;j;w, a two-node finite element has eight degrees of

freedom, i.e. four nodal shear-bending and torsion-warping displacements respectively, Fig. 3,

fUg ¼

8>><>>:

wð0Þ4ð0ÞwðlÞ4ðlÞ

9>>=>>;; fVg ¼

8>><>>:

jð0Þwð0ÞjðlÞwðlÞ

9>>=>>;: (27)

Fig. 3. Beam finite element.


Therefore, the basic beam displacements, wb and jt, can be presented as the third-orderpolynomials

wb ¼ CakD

nxko; jt ¼ CdkD

nxko; k ¼ 0;1;2;3;

x ¼ xl; C.D ¼ f.gT :

(28)

Furthermore, satisfying alternately the unit value for one of the nodal displacement {U} and zerovalues for the remaining displacements, and doing the same for {V}, it follows that:

wb ¼ CwbiDfUg; ws ¼ CwsiDfUg; w ¼ CwiDfUg;jt ¼ CjtiDfVg; js ¼ CjsiDfVg; j ¼ CjiDfVg; i ¼ 1;2; 3;4;

(29)

where wbi, wsi, wi and jti, jsi, ji are the shape functions specified below by employing relations (24)and (25)

wbi ¼ CaikD

nxko; wsi ¼ CbikD

nxko; wi ¼ CcikD

nxko

jti ¼ CdikD

nxko; jsi ¼ CeikD

nxko; ji ¼ CfikD

nxko (30)

½aik� ¼1

aðaþ 12bÞ

2664

aþ 6b 0 �3a 2a�4bðaþ 3bÞl aðaþ 12bÞl �2aðaþ 3bÞl a2l

6b 0 3a �2a�2bða� 6bÞl 0 �aða� 6bÞl a2l

3775 (31)

bi0 ¼ �ð1� aÞai0 � 2bai2bi1 ¼ �ð1� aÞai1 � 6bai3bi2 ¼ �ð1� aÞai2bi3 ¼ �ð1� aÞai3;

(32)

½cik� ¼ ½aik� þ ½bik�; i ¼ 1;2;3;4; k ¼ 0;1;2;3 (33)

a ¼ 1� u2 JbGAs

; b ¼ EIbGAsl2

(34)

Constitution of torsional matrices ½dik�, ½eik� and ½fik� is the same as ½aik�, ½bik� and ½cik�, but parametersa and b have to be exchanged with

h ¼ 1� u2 JwGIs

; g ¼ EIwGIsl2

(35)

according to (25).By substituting Eqs. (29) into (26) one obtains

Etot ¼12

�UV

�T� kbs 00 kws þ kt

��UV

�þ 1

2

�_U_V

�T�msb mstmts mtw

��_U_V

��

qm

�T�UV

�

��

PR

�T�UV

�; (36)

where, assuming constant values of the element properties,


½k�bs ¼ ½EIbR l

0ðd2wbi=dx2Þðd2wbj=dx2Þdxþ GAs

R l0ðdwsi=dxÞðdwsj=dxÞdx� – bending-shear stiffness

matrix,

½k�ws ¼ ½EIwR l

0ðd2jti=dx2Þðd2jtj=dx2Þdxþ GIs

R l0ðdjsi=dxÞðdjsj=dxÞdx� – warping-shear stiffness matrix,

½k�t ¼ ½GItR l

0ðdjti=dxÞðdjtj=dxÞdx� – torsion stiffness matrix,

½m�sb ¼ ½mR l

0 wiwjdxþ JbR l

0ðdwbi=dxÞðdwbj=dxÞdx� – shear-bending mass matrix,

½m�tw ¼ ½JtR l

0 jijjdxþ JwR l

0ðdjti=dxÞðdjtj=dxÞdx� – torsion-warping mass matrix,

½m�st ¼ ½mcR l

0 wijjdx ; ½m�ts ¼ ½m�Tst

i– shear-torsion mass matrix,

fqg ¼n R l

0 qwjdxo� shear load vector;

fmg ¼nR l

0 mjjdxo� torsion load vector;

i; j ¼ 1; 2;3;4:

(37)

The vectors {P} and {R} represent the shear-bending and torsion-warping nodal forces, respectively,

fPg ¼

8>><>>:�Qð0ÞMð0ÞQðlÞ�MðlÞ

9>>=>>;; fRg ¼

8>><>>:�Tð0ÞBwð0ÞTðlÞ�BwðlÞ

9>>=>>;: (38)

The above matrices are specified in Appendix B, as well as the load vectors for linearly distributedloads along the element, i.e.

q ¼ q0 þ q1x; m ¼ m0 þ m1x: (39)

Also, shape functions of sectional forces are listed in Appendix C.The total element energy has to be at its minimum. Satisfying the relevant conditions

vEtot

vfUg ¼ f0g;vEtot

vfVg ¼ f0g (40)

and employing Lagrange equations of motion, the finite element equation yields

ff g ¼ ½k�fdg þ ½m�n

€do� ff gqm; (41)

where

ff g ¼�

PR

�; ff gqm¼

�qm

�; fdg ¼

�UV

�

½k� ¼�

kbs 00 kws þ kt

�; ½m� ¼

�msb mstmts mtw

�:

(42)

It is obvious that coupling between the bending and torsion occurs through the mass matrix only,i.e. by the coupling matrices [m]st and [m]ts.

4. Finite element transformation

In the finite element equation (41), first the element properties related to bending and then thoserelated to torsion appear. To make an ordinary finite element assembling possible, it is necessary totransform Eq. (41) in such a way that first all properties related to the first node are specified and


then those belonging to the second one. Thus, the rearranged nodal force and displacement vectorsread

n~fo¼

8>>>>>>>>>><>>>>>>>>>>:

�Qð0ÞMð0Þ�Tð0ÞBwð0ÞQðlÞ�MðlÞ

TðlÞ�BwðlÞ

9>>>>>>>>>>=>>>>>>>>>>;;n

~do¼

8>>>>>>>>>><>>>>>>>>>>:

wð0Þ4ð0Þjð0Þ4ð0ÞwðlÞ4ðlÞjðlÞ4ðlÞ

9>>>>>>>>>>=>>>>>>>>>>;: (43)

The same transformation has to be done for the load vector ff gqm resulting in f~f gqm. The abovevector transformation implies the row and column exchange in the matrices according to the followingset form:

1 2 5 6 3 4 7 8

1 11 12 15 16 13 14 17 18

2 21 22 25 26 23 24 27 28

5 51 52 55 56 53 54 57 58

6 61 62 65 66 63 64 67 68

3 31 32 35 36 33 34 37 38

4 41 42 45 46 43 44 47 48

7 71 72 75 76 73 74 77 78

8 81 82 85 86 83 84 87 88

The element deflection refers to the shear centre as the origin of a local coordinate system. Since thevertical position of the shear centre varies along the ship’s hull, it is necessary to prescribe the elementdeflection for a common line, in order to be able to assemble the elements. Thus, choosing the x-axis(base line) of the global coordinate system as the referent line, the following relation between theformer and the latter nodal deflections exists:

wð0Þ ¼ wð0Þ þ zSjð0ÞwðlÞ ¼ wðlÞ þ zSjðlÞ; (45)

where zS is the coordinate of the shear centre, Fig. 2. Other displacements are the same in bothcoordinate systems. Twist angle j does not have influence on the cross-section rotation angle 4. Thelocal displacement vector can be expressed as

n~do¼h~Tin

~do; (46)

where ½~T � is the transformation matrix

h~Ti¼�½T� ½0�½0� ½T�

�; ½T� ¼

2664

1 0 zS 00 1 0 00 0 1 00 0 0 1

3775: (47)

Since the total element energy is not changed by the above transformations, a new elementequation can be derived taking (46) into account. Thus, one obtains in the global coordinate system

n~fo¼h

~kin

~doþh

~min€~d

o�n

~fo

qm; (48)

where

(44)


n~fo¼�~TTn~f

o �~k ¼ �~TT�~k�~T �~m¼�~TT�

~m�

~T n

~fo

qm¼�~TTn~f

oqm: (49)

The first of the above expressions transforms the nodal torques into the form

�Tð0Þ ¼ �Tð0Þ � zSQð0ÞTðlÞ ¼ TðlÞ þ zSQðlÞ: (50)

5. Numerical procedure for vibration analysis

A ship’s hull is modelled by a set of beam finite elements. Their assemblage in the global coordinatesystem, performed in the standard way, results in the matrix equation of motion, which may beextended by the damping forces

½K�fDg þ ½C�n

_Doþ ½M�

n€Do¼ fFðtÞg; (51)

where [K], [C] and [M] are the stiffness, damping and mass matrices, respectively; fDg; f _D and f€Dgo

are the displacement, velocity and acceleration vectors, respectively; and {F(t)} is the load vector.In case of natural vibration {F(t)}¼ {0} and the influence of damping is rather low for ship struc-

tures, so that the damping forces may be ignored. Assuming

fDg ¼ ffgeiut ; (52)

where ffg and u are the mode vector and natural frequency respectively, Eq. (51) leads to theeigenvalue problem

½K� � u2½M��ffg ¼ f0g; (53)

which may be solved by employing different numerical methods [21]. The basic one is the determinantsearch method in which u is found from the condition��½K� � u2½M�

�� ¼ 0 (54)

by an iteration procedure. Afterwards, ffg follows from (53) assuming unit value for one element inffg.

The forced vibration analysis may be performed by direct integration of Eq. (51), as well as by themodal superposition method. In the latter case the displacement vector is presented in the form

fDg ¼ ½f�fXg; (55)

where ½f� ¼ ½ffg� is the undamped mode matrix and {X} is the generalised displacement vector.Substituting (55) into (51), the modal equation yields

½k�fXg þ ½c�

_X�þ ½m�

n€Xo¼ ff ðtÞg; (56)

where

½k� ¼ ½f�T ½K�½f� � modal stiffness matrix

½c� ¼ ½f�T ½C�½f� � modal damping matrix

½m� ¼ ½f�T ½M�½f� � modal mass matrix

ff ðtÞg ¼ ½f�TfFðtÞg � modal load vector:

(57)

The matrices [k] and [m] are diagonal, while [c] becomes diagonal only in a special case, for instanceif [C]¼ a0 [M]þ b0 [K], where a0 and b0 are coefficients [20].


Solving (56) for undamped natural vibration, [k]¼ [u2m] is obtained, and by its backward substi-tution into (56) the final form of the modal equation yieldsh

u2ifXg þ 2½u�½z�

_X�þn

€Xo¼ f4ðtÞg; (58)

where

½u� ¼" ffiffiffiffiffiffiffi

kii

mii

s #�� natural frequency matrix

½z� ¼"

cij

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkiimiiÞ

p#�� relative damping matrix

f4ðtÞg ¼�

fiðtÞmii

�� relative load vector:

(59)

If [z] is diagonal, the matrix Eq. (58) is split into a set of uncoupled modal equations.The ship vibration is caused by the engine and propeller excitation forces, which are of periodical

nature and therefore can be split into harmonics. Thus, the ship’s hull response is obtained solvingeither (51) or (56). In both cases, the system of differential equations is transformed into a system ofalgebraic equations.

If hull vibration is induced by waves, the time integration of (51) or (56) has to be performed. Severalnumerical methods are available for this purpose, as for instance the Houbolt, the Newmark and theWilson q method [21], as well as the harmonic acceleration method [22,23].

It is important to point out that all stiffness and mass matrices of the beam finite element (andconsequently those of the assembly) are frequency dependent quantities, due to coefficients a and h inthe formulation of the shape functions, Eqs. (34) and (35). Therefore, for solving the eigenvalueproblem (53) an iteration procedure has to be applied. As a result of frequency dependent matrices, theeigenvectors are not orthogonal. If they are used in the modal superposition method for shiphydroelastic analysis, full modal stiffness and mass matrices (as those of added mass and hydrody-namic damping) are generated. Since the inertia terms are much smaller than the deformation ones inEqs. (24) and (25), the off-diagonal elements in modal stiffness and mass matrices are very smallcompared to the diagonal elements and can be neglected.

It is obvious that the usage of the physically consistent non-orthogonal natural modes in the modalsuperposition method is not practical, especially not in the case of time integration. Therefore, it ispreferable to use mathematical orthogonal modes for that purpose. They are created by the staticdisplacement relations yielding from Eqs. (24) and (25) with u ¼ 0, that leads to a ¼ h ¼ 1. In thatcase all finite element matrices, defined with Eqs. (37) and in Appendix B, can be transformed intoexplicit form, Appendix D. The resulting discrepancies of these two different formulations are analysedanalytically in Appendix E and numerically within the following illustrative example.

6. Particulars of container ship

The application of the improved theory and numerical procedure is illustrated in case of an 11400TEU VLCS (Very Large Container Ship), Fig. 4. The main vessel particulars are the following:

Length overall, Loa¼ 363.44 mLength between perpendiculars, Lpp¼ 348 mBreadth, B¼ 45.6 mDepth, H¼ 29.74 mDraught, T¼ 15.5 mDisplacement, full load, Df¼ 171445 tDisplacement, ballast, Db¼ 74977 tDisplacement, lightweight, Dl¼ 37151 tEngine power, P¼ 72240 kWShip speed, v¼ 24.7 kn

Fig.

4.11

400

TEU

con

tain

ersh

ip.



The midship section, which shows a double skin structure with the web frames and longitudinals,is presented in Fig. 5. The ship hull stiffness properties are calculated by the program STIFF [24], basedon the theory of thin-walled girders [14,25], Appendix F. Their distributions along the ship are shownin Fig. 6. It is evident that geometrical properties rapidly change values in the engine and super-structure area, due to closed ship cross-section. This is especially pronounced in the case of torsionalmodulus, which takes quite low values for the open cross-section and rather high values for theclosed one, Fig. 6e. Novelty is the distribution of the shear inertia modulus Is, Fig. 6f. Bulkheadinfluence is taken into account by increasing the value of the torsional modulus I�t ¼ 1:9It as it iselaborated in Appendix G.

Dry vibration analysis is performed for the lightweight loading condition. The containersare excluded from the analysis since they influence only the mass properties while thestructural model is of primary interest. Ship mass distribution and its properties are shownin Fig. 7. The mass bimoment of inertia Jw is neglected due to its very small influence onvibrations.

7. 1D vibration analysis

Dry natural vibrations are calculated by the modified and improved program DYANA within[4]. The ship hull is divided into 50 beam finite elements. Finite elements of closed cross-section(6 d.o.f., w excluded) are used in the ship bow, ship aft and in the engine room area. First,natural frequencies of horizontal vibrations are calculated by the finite element with consistentfrequency dependent and independent properties, as well as for the case of no mass rotationTable 1. Relations between the results are similar to those noticed in the analytical evaluation ofthe approximate solution in Appendix E. Negligible small discrepancies between naturalfrequencies of the mathematical and physical modes are evident. Influence of mass rotation onfrequencies is within 4%. So, the mathematical natural modes determined by finite elements withfrequency independent shape functions are very reliable to be used in the ship hydroelasticanalysis.

The first five natural frequencies for vertical and horizontal vibrations are listed in Tables 2 and 3 forfurther analysis. The corresponding natural modes are of ordinary shapes. Natural frequencies ofcoupled horizontal and torsional vibrations are listed in Table 4. The first five natural modes presentedby the twist angle, j, and the hull deflection at the level of center of gravity, wG, are shown in Fig. 8.These functions are mutually dependent. However, they are normalized by their own maximum valuesdue to a simpler presentation. There always occurs coupling between frequency close symmetricflexural and torsional modes as well as anti-symmetric ones [26,27]. This is indicated in Table 4, wherethe 5th coupled mode is an extraordinary natural mode comprised of the 2nd torsional and the 5thflexural elementary modes.

Once the dry natural modes of ship hull are determined, it is possible to transfer the beam nodedisplacements to the ship wetted surface for the hydrodynamic calculation. The transformation(spreading) functions for vertical and coupled horizontal and torsional vibration yield respectively [3]

hv ¼ �dwv

dxðZ � zCÞiþwvk (60)

hht ¼�� dwhY þ dj

u�

iþ ½wh þ jðZ � zSÞ�j� jYk; (61)
dx dx
where w is hull deflection, j is twist angle, u ¼ uðx;Y ; ZÞ is the cross-section warping functionreduced to the wetted surface, zC and zS are coordinates of centroid and shear centre respectively,and Y and Z are coordinates of the point on the ship surface. The first two dry natural modes of theship wetted surface in case of vertical and coupled horizontal and torsional vibrations are shown inFigs. 9–12 respectively. In the latter case, the orthogonal view on the vertical and horizontal planes isused.

Fig.

5.M

idsh

ipse

ctio

n.


Fig. 6. Longitudinal distribution of ship cross-section geometrical properties.


8. 3D vibration analysis

For this purpose, a fine 3D FEM model of the container vessel is created by NASTRAN program [28].The model includes 33072 nodes, 84076 finite elements (38288 shells and 45788 beams), and 187290d.o.f. The ship mass distribution (steel and equipment) is given by adjusted mass density for the finiteelement blocks. Only the main engine mass is specified by the lumped masses. The Lanczos method isused for the solution of the eigenvalue problem.

In order to uncouple the horizontal vibrations from the torsional ones, for the purpose of a moredetailed correlation analysis, the FEM model is reinforced by a set of vertical rigid massless beams inthe longitudinal symmetry plane. The rotation of all beam nodes around axial axis is fixed.

Natural frequencies of vertical, horizontal and coupled horizontal and torsional vibrations are listedin Tables 2–4, respectively. The first five natural modes of the coupled vibrations, which are of primaryinterest, are shown in Figs. 13–17. The lateral and bird views are used in order to achieve easierrecognition of the elementary modes in each coupled mode. We can see that in all vibration modes

Fig. 6. (continued).


warping of the front (collision) bulkhead and the aft (transom) bulkhead is rather low. Thus, theassumption of restrained warping of hull peaks, introduced in 1D FEM model, is quite realistic.

9. Correlation analysis

It is possible to compare now the first two vibration modes determined by 1D and 3D FEM analyses,Figs. 11–14, respectively. The same mode shapes are obvious. Natural frequencies of vertical, horizontaland coupled horizontal and torsional vibrations determined by 1D and 3D FEM models are correlatedin Tables 2–4, respectively, with indicated discrepancies. The general opinion concerning the accuracyof 1D vibration analysis is that the first five eigenpairs are acceptable from the engineering viewpoint,with frequencies tolerance discrepancy of up to 5%. In the considered case, the agreement of naturalfrequencies for vertical vibrations is quite good. Somewhat large discrepancies appear for the fifthfrequency of horizontal vibration (8.78%). This discrepancy is magnified in the fifth frequency of theextraordinary coupled horizontal and torsional vibrations (16.81%), comprised of the 5th elementaryhorizontal mode and the 2nd torsional mode, Fig. 17.

It is necessary to point out that 1D eigenpairs are determined for the fixed ship stiffness and massproperties. However, if the mode dependent effective values of parameters are taken into account,discrepancies between 1D and 3D results are considerably reduced, and the validity of 1D vibration analysescanbe extended up to the 10th mode. The effective stiffness parameters are determined by utilizing the stripelement method and the energy approach [29]. The main idea is to specify the harmonic load distributionand the harmonic mode shapes and split the strain energy into normal and shear parts, and then to estimatethe effective bending modulus and shear area based on the separated energies, Appendix H.

In order to illustrate the above fact, 1D vibration calculation is repeated, taking the mode dependentmoment of inertia of cross-section, IZ5 ¼ i5IZ , and shear area, Asy5 ¼ a5Asy into account. The roughlyestimated values i5 ¼ 0:2 and a5 ¼ 1:2 are taken from the channel girder analysis, Fig. 18 [14]. Sincedeflection of a girder for higher modes is mainly due to shear, values of an are increased to some extent,while those of in asymptotically approach zero. The actual natural frequencies are compared in Table 5.It is obvious that the discrepancies between 1D and 3D results are now considerably reduced.

Fig. 7. Longitudinal distribution of ship mass properties, lightweight.

Table 1Natural frequencies of horizontal hull vibrations, ui (Hz).

Mode no. Physical mode, a Mathematical mode, b Jbz¼ 0, c Discrepancy %

b/a� 1 c/a� 1

1 1.552 1.552 1.584 0 2.062 2.740 2.740 2.801 0 2.233 4.022 4.021 4.070 �0.02 1.194 5.697 5.694 5.743 �0.05 0.085 7.398 7.392 7.483 �0.08 1.156 8.451 8.444 8.766 �0.08 3.737 9.569 9.557 9.802 �0.13 2.438 11.370 11.359 11.469 �0.10 0.879 13.352 13.329 13.115 �0.17 �1.8110 14.490 14.453 14.311 �0.26 �2.56


Table 3Natural frequencies of horizontal hull vibrations, ui (Hz).

Mode no. 1D FEM 3D FEM Discrepancy %

1 1.552 1.625 �4.492 2.740 2.787 �1.693 4.021 4.018 0.104 5.694 5.505 3.495 7.392 6.798 8.78

Table 4Natural frequencies of coupled horizontal and torsional hull vibrations, ui (Hz).

Mode no. Coupled modes 1D FEM 3D FEM Discrepancy %

1 T1 0.639 0.638 0.162 T2þH1 1.056 1.076 �1.863 T3þH2 1.745 1.749 �0.234 T4þH3 2.233 2.429 �8.075 T2þH5 3.072 2.630 16.816 T5þH4 3.350 3.519 �4.80

Table 2Natural frequencies of vertical hull vibrations, ui (Hz).

Mode no. 1D FEM 3D FEM Discrepancy %

1 1.149 1.159 �0.862 2.318 2.327 �0.393 3.695 3.654 1.124 5.457 5.409 0.895 6.913 6.605 4.66


The 5th natural mode determined by 1D FEM model with the mode dependent stiffnesses is shownin Fig. 19. Changes of functions wG and j are noticeable especially in the area of engine room. The newmode shape is now closer to the 3D one, Fig. 17, than that shown in Fig. 8e.

Zoom of the deformed ship aftbody in case of the 5th mode vibrations, is shown in Fig. 20. Warpingof the transverse bulkheads is quite pronounced in this mode. The engine room structure is rather stiffconcerning bending, while there is some warping release of its bulkheads. The warping magnitude inthe beam model is represented by variation of twist angle w. Its given value is zero within the engineroom as well as hull peaks, due to closed cross-sections, Fig. 19. It is quite difficult to simulate the localwarping effect in the engine room by 1D FEM model.

In 1D vibration analysis we distinguish the so-called mathematical natural modes and the actualphysical modes, determined by fixed and mode dependent vibration properties, respectively. Theformer modes are mutually orthogonal and therefore suitable to be used as the coordinate functions fordetermining structural forced response by the modal superposition method. As result of that property,the diagonal modal stiffness and mass matrices are generated. Consequently, the modal equations areweakly coupled only by the modal damping. Thus, the equations can be easily solved by an iterationprocedure.

The application of the actual physical vibration modes, i.e. those determined with the effectivevalues of vibration parameters, in the modal superposition method is not suitable due to coupling ofmodal equations as a result of mode non-orthogonality, Appendix H. This is anyway out of interest incase of ship hydroelasticity analysis, since the ship structure response to waves occurs in the lowerfrequency domain and is successfully described by the first few natural modes only. In the frequencydomain the mathematical modes are rather close to the actual ones, and it does not matter which areused in the modal superposition method. The calculated ship response will be the same in both cases.

Fig. 8. Natural modes of coupled horizontal and torsional vibrations, 1D model.


10. Conclusion

Very large container ships are quite flexible and their design and construction are at the margin ofthe present classification rules. Therefore, great effort is done nowadays to investigate the shiphydroelasticity phenomenon. Dry natural vibration analysis, utilizing either 1D or 3D FEM model, is thefirst step in solving this challenging problem. For the research purpose and preliminary design stage,the beam analysis combined with thin-walled girder theory is more suitable and convenient. It makesit possible to investigate the influence of all stiffness and mass parameters on ship dynamic behaviourand at the same time reduces the amount of work needed to carry out the analysis.

In order to increase the reliability of thin-walled girder theory some improvement of the classicaltheory is proposed. Shear influence on torsion is included in the theoretical consideration byfollowing analogy between bending and torsion. Shear influence is more pronounced in torsion thanin bending. The beam finite element for coupling flexural and torsional vibrations is developed withcomplete stiffness and mass matrices. Two approaches are used, the consistent frequency dependentshape functions, and frequency independent ones, which follows from static beam theory. It isshown that discrepancies between the vibration results are negligible. Also, the influence of

Fig. 9. The first natural mode of vertical vibrations, u1¼1.149 Hz, 1D model.


transverse bulkheads is incorporated in the hull torsional stiffness, based on detailed determinationof the bulkhead strain energy as an orthotropic plate with rigid stool.

The developed procedure is illustrated in case of a very large container ship and the accuracy of theresults is verified by the correlation analysis with the results from 3D FEM analysis. By taking allrelevant ship stiffness and mass parameters into account, quite good agreement of the results isachieved in the low frequency domain. The application of the beam model for ship hull vibrations islimited to the first five natural modes. However, this is sufficient for the ship hydroelasticity analysissince the most energy of wave spectrum is concentrated in the low frequency domain, i.e. belownatural frequency of the first elastic mode of the ship hull.

The 3D FEM analysis shows that the assumption on restrained warping in the ship peaks isrealistic. However, in spite of the fact that the engine room is a closed cross-section substructure,some warping release is noticed in case of extraordinary and higher natural modes. The presentsolutions in the literature, as for instance that shown in [11], which is correlated to a simplified one in[30], are not very promising. Therefore, this problem requires further investigation and incorporationin the beam model. The basic idea is to consider engine room structure as an open cross-sectionsegment with week deck and platforms and determine effective torsional, warping and shear moduliutilizing the energy approach.

Fig. 10. The second natural mode of vertical vibrations, u2¼ 2.318 Hz, 1D model.

Fig. 11. The first natural mode of coupled horizontal and torsional vibrations, u1¼0.639 Hz, 1D model.

I.Senjanovicet

al./M

arineStructures

22(2009)

387–437409

Fig. 12. The second natural mode of coupled horizontal and torsional vibrations, u2¼1.056 Hz, 1D model.

I.Senjanovicet

al./M

arineStructures

22(2009)

387–437410

Fig. 13. The first natural mode of coupled horizontal and torsional vibrations, u1¼0.638 Hz.

Fig. 14. The second natural mode of coupled horizontal and torsional vibrations, u2¼1.076 Hz.

Fig. 15. The third natural mode of coupled horizontal and torsional vibrations, u3¼1.749 Hz.


The beam model of ship hull presented in this paper is reliable enough to be directly applied in shiphydroelastic analyses. The most discrepancies between 1D and 3D model are reduced by taking shearinfluence on torsion as an additional stiffness parameter into account, as well as by incorporatingtransverse bulkhead stiffness contribution in a very efficient way.

Fig. 16. The fourth natural mode of coupled horizontal and torsional vibrations, u4¼ 2.429 Hz.

Fig. 17. The fifth natural mode of coupled horizontal and torsional vibrations, u5¼ 2.630 Hz.


Further improvement of the beam model, insisting on the consistence of compatibility conditions atjoints of open and closed cross-section hull segments will have very small influence on global sectionalforces of ship hull. In any case, beam model cannot get proper stress concentration in the deck cornersfor fatigue analysis. Therefore, it is necessary to employ the substructure technique, imposing sectionalforces at substructure boundaries.

Fig. 18. Efficiency factors of horizontal bending modulus and shear area of channel girder in, an.

Table 5The 5th natural frequencies, u5 (Hz), IZ5¼ 0.2IZ, ASy5¼1.2ASy.

1D FEM 3D FEM Discrepancy %

Horizontal 7.018 6.798 3.24Coupled 2.747 2.630 4.45


Acknowledgment

This research was supported by the Ministry of Science, Education and Sport of the Republic ofCroatia (Project No. 120-1201703-1704). The authors would like to express their gratitude to Ms. EstelleStumpf, research engineer at Bureau Veritas, Marine Division – Research Department, for generating3D FEM model of the container ship.

Appendix A. Torsional vibrations of prismatic beam

In order to analyse influence of stiffness and mass parameters on torsional vibrations, the naturalvibrations of a prismatic beam are considered. The governing differential equation of motion isdeduced from (21) by neglecting distance between center of gravity and shear center, c¼ 0, andwarping bimoment of inertia, Jw¼ 0, due to reason of simplicity.

EIwv4jt

vx4 � GItv2jt

vx2 þ J0t

v2jt

vt2 �EIwGIs

v4jt

vx2vt2

!¼ 0: (A1)

Since natural vibrations are harmonic Eq. (A1) leads to the ordinary differential equation

EIwd4jt

dx4 � GIt

�1� u2J0

tEIw

GItGIs

�d2jt

dx2 � u2J0t jt ¼ 0; (A2)

where jt and u are natural mode and frequency, respectively.Solution of (A2) is assumed in the exponential form

jt ¼ eax; (A3)

and by substituting it in (A2) one finds the biquadratic characteristic equation

Fig. 19. The 5th natural mode of 1D FEM model, effective parameters IZ5¼ 0.2IZ and ASy5¼1.2ASy.

Fig.

20.

Bird

view

ofsh

ipaf

tbod

y,th

e5t

hn

atur

alm

ode.


Fig. F1. Shear stress flow due to vertical bending.


a4 þ ba2 þ c ¼ 0; (A4)

where

b ¼ GItEIw

�u2J0

tEIw

GItGIs� 1

�; c ¼ �u2J0

tEIw

: (A5)

Solutions of (A4) read

aj ¼ �h; �ic; (A6)

where

Fig. F2. Shear stress flow due to horizontal bending.


Fig. F3. Shear stress flow due to torsion.


h ¼ 1ffiffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4c

p� b

q;c ¼ 1ffiffiffi

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4c

pþ b

q: (A7)

Thus, solution of (A2) takes the form

jt ¼ A1shhxþ A2chhxþ A3sincxþ A4coscx: (A8)


Let us consider vibrations of a free beam of length 2l, with restrained warping, u ¼ uw; at its ends.The relevant boundary conditions read

x ¼ �l : T ¼ 0; u ¼ 0 (A9)

that leads to

x ¼ �l :djt

dx¼ 0;

d3jt

dx3 ¼ 0: (A10)

In the case of symmetric modes A1 ¼ A3 ¼ 0; while for anti-symmetric modes A2 ¼ A4 ¼ 0. Thecorresponding eigenvalue problems yield�

hshhl �csinclh3shhl c3sincl

��A2A4

�¼ f0g; (A11)

Fig. F4. Shear stress flow due to restrained warping.

Fig. F5. Normal stress flow due to restrained warping.


Fig. G1. Bulkhead deflection due to warping of cross-section.


�h ch hl c cos clh3 ch hl �c3 cos cl

��A1A3

�¼ f0g: (A12)

For nontrivial solutions determinants of (A11) and (A12) have to be zero. That leads to the frequencyequations

hc

h2 þ c2�

sh hl sin cl ¼ 0 (A13)

hc

h2 þ c2�

chhl cos cl ¼ 0 (A14)

with the same eigenvalue formula for symmetric (n¼ 0, 2.) and anti-symmetric (n¼ 1, 3.) modes

cl ¼ np2; n ¼ 0;1;2. (A15)

Substituting (A15) into (A7) for c, one finds the following expression for natural frequencies oftorsional vibrations

Fig. G2. Longitudinal section of container ship hold.

Fig. G3. Watertight bulkhead.


Fig. G4. Support bulkhead.



un ¼np2l

ffiffiffiffiffiffiffiGIt

J0t

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

np2l

�2EIwGIt

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

np2l

�2EIwGIs

r ; n ¼ 0; 1;2. (A16)

The first term in the above formula represents natural frequencies of a free beam with free warping.

~un ¼np2l

ffiffiffiffiffiffiffiGIt

J0t

s: (A17)

Fig. G5. Bird view on deformed bulkheads.


Warping stiffness EIw in the nominator of (A16) increases natural frequencies as a result ofrestrained warping. Shear stiffness GIs in the denominator reduces natural frequencies since additionaltwist angle due to shear influence acts as release.

Integration constants A2 and A4, and A1 and A3 are determined from (A11) and (A12), respectively.The symmetric and anti-symmetric modes according to (A8) yield

jtn ¼ cn sin cnl ch hnxþ hn sh hnl cos cnx; n ¼ 0;2. (A18)

jtn ¼ cn cos cnl sh hnx� hn ch hnl sin cnx; n ¼ 1;3. (A19)

Referring to (23), the total twist angle consisted of torsion and shear contribution takes thefollowing form

jn ¼�

1� EIwGIs

h2n

�cn sin cnl ch hnxþ

�1þ EIw

GIsc2

n

�hn sh hnl cos cnx; n ¼ 0;2. (A20)

jn ¼�

1� EIwGIs

h2n

�cn cos cnl sh hnx�

�1� EIw

GIsc2

n

�hn ch hnl sin cnx; n ¼ 1;3. (A21)

Appendix B. Consistent finite element properties, according to Eq. (37) (Frequency dependentformulation)

Bending-shear stiffness matrix:

½k�bs¼ ½k�bþ½k�s (B1)

kbij ¼

4EIbl3

�ai2aj2 þ

32

�ai2aj3 þ ai3aj2

�þ 3ai3aj3

�(B2)

ksij ¼

GAs

l

�bi1bj1 þ bi1bj2 þ bi2bj1 þ bi1bj3 þ bi3bj1 þ

43

bi2bj2 þ32

�bi2bj3 þ bi3bj2

�þ 9

5bi3bj3

�(B3)

The warping-shear stiffness matrix ½k�ws is of the same constitution as ½k�bs, but Ib, As, aik and bik haveto be replaced with Iw, Is, dik and eik, respectively, Eqs. (35). Torsional stiffness matrix ½k�t is of the sametype as ½k�s, but As and bik have to be replaced with It and aik, respectively.

Shear-bending mass matrix

½m�sb¼ ½m�sþ½m�b (B4)

msij ¼ ml

"ci0cj0 þ

12

�ci1cj0 þ ci0cj1

�þ 1

3

�ci2cj0 þ ci1cj1 þ ci0cj2

�þ1

4

�ci0cj3 þ ci1cj2 þ ci2cj1 þ ci3cj0

�þ 1

5

�ci1cj3 þ ci2cj2 þ ci3cj1

�þ1

6

�ci2cj3 þ ci3cj2

�þ 1

7ci3cj3

# (B5)

mbij ¼

Jbl

�ai1aj1 þ ai1aj2 þ ai2aj1 þ ai1aj3 þ ai3aj1 þ

43

ai2aj2 þ32

�ai2aj3 þ ai3aj2

�þ 9

5ai3aj3

�(B6)


The torsion-warping mass matrix ½m�tw is of the same constitution as ½m�sb, but m, Jb, cik, and aik haveto be replaced with Jt, Jw, fik and dik respectively, Eq. (35).

Shear-torsion mass matrix

mstij ¼ mcl

"ci0fj0 þ

12

ci1fj0 þ ci0fj1

�þ 1

3

ci2fj0 þ ci1fj1 þ ci0fj2

�

þ14

ci0fj3 þ ci1fj2 þ ci2fj1 þ ci3fj0

�þ 1

5

ci1fj3 þ ci2fj2 þ ci3fj1

�þ1

6

ci2fj3 þ ci3fj2

�þ 1

7ci3fj3

# (B7)

Shear load vector

qi ¼ q0l�

ci0 þ12

ci1 þ13

ci2 þ14

ci3

�þ q1l

�12

ci0 þ13

ci1 þ14

ci2 þ15

ci3

�(B8)

Torsion load vector mi is of same form as (B8), expressed with m0, m1 and fik, Eqs. (35) and (39),instead of q0, q1 and eik, respectively. In all above formulae indeces i, j take values 1, 2, 3 and 4.

Appendix C. Consistent shape functions of finite element sectional forces

Bending moment, Eq. (2):

Mi ¼ �EIbd2wbi

dx2 ¼ �EIbD

að2Þik

Enxko; k ¼ 0; 1;2; 3: (C1)

Shear force, Eq. (3):

Qi ¼ GAsdwsi

dx¼ GAs

Dbð1Þik

Enxko; k ¼ 0;1;2;3: (C2)

Pure torque, Eq. (7):

Tti ¼ GItdjti

dx¼ GIt

Ddð1Þik

Enxko; k ¼ 0;1;2;3: (C3)

Warping torque, Eq. (9):

Twi ¼ GIsdjsi

dx¼ GIs

Deð1Þik

Enxko; k ¼ 0;1;2;3: (C4)

Bimoment, Eq. (8):

Bwi ¼ �EIwd2jti

dx2 ¼ �EIwD

dð2Þik

Enxko; k ¼ 0;1;2;3: (C5)

Vectors of the shape functions, formulae (29) and (32):Dað2Þik

E¼ 2

l2Cai2;3ai3; 0;0DDbð1Þik

E¼ 1

l Cbi1;2bi2; 3bi3;0DDdð1Þik

E¼ 1

l Cdi1;2di2;3di3;0DDeð1Þik

E¼ 1

l Cei1;2ei2;3ei3;0DDdð2Þik

E¼ 2

l2Cdi2;3di3;0;0D

(C6)


Appendix D. Simplified finite element properties, from Appendix B (Frequency independentformulation)

Stiffness matrices:

½k�bs¼2EIb

ð1þ 12bÞl3

2664

6 3l �6 3l2ð1þ 3bÞl2 �3l ð1� 6bÞl2

6 �3lSym: 2ð1þ 3bÞl2

3775 (D1)

½k�ws¼2EIw

ð1þ 12gÞl3

2664

6 3l �6 3l2ð1þ 3gÞl2 �3l ð1� 6gÞl2

6 �3lSym: 2ð1þ 3gÞl2

3775 (D2)

½k�t¼GIt

30ð1þ12gÞ2l

2664

36 3ð1�60gÞl �36 3ð1�60gÞl4�1þ15gþ360g2

�l2 �3ð1�60gÞl �

�1þ60g�720g2

�l2

36 �3ð1�60gÞlSym: 4

�1þ15gþ360g2

�l2

3775 (D3)

Mass matrices:

½m�sb¼ ½m�sþ½m�b (D4)

½m�s¼ml

420ð1þ12bÞ2

�

266666666664

156þ3528bþ20160b2

22þ462bþ2520b2�

l 54þ1512bþ10080b2 �

13þ378bþ2520b2�

l4þ84bþ504b2

�l2

13þ378bþ2520b2

�l �

3þ84bþ504b2

�l2

156þ3528bþ20160b2�

22þ462bþ2520b2�

l

Sym:

4þ84bþ504b2�

l2

377777777775

(D5)


½m�b¼Jb

30ð1þ12bÞ2l

266664

36 ð3�180bÞl �36 ð3�180bÞl4þ60bþ1440b2

�l2 ð�3þ180bÞl �

1þ60b�720b2

�l2

36 ð�3þ180bÞlSym:

4þ60bþ1440b2

�l2

377775 (D6)

½m�tw¼ ½m�tþ½m�w (D7)

½m�t¼Jt l

420ð1þ12gÞ2

�

2666664

156þ3528gþ20160g2�22þ462gþ2520g2

�l 54þ1512gþ10080g2 �

�13þ378gþ2520g2

�l�

4þ84gþ504g2�l2

�13þ378gþ2520g2

�l �

�3þ84gþ504g2

�l2

156þ3528gþ20160g2 ��22þ462gþ2520g2�l

Sym:�4þ84gþ504g2

�l2

3777775(D8)

½m�w¼Jw

30ð1þ12gÞ2l

2664

36 ð3�180gÞl �36 ð3�180gÞl�4þ60gþ1440g2

�l2 ð�3þ180gÞl �

�1þ60g�720g2

�l2

36 ð�3þ180gÞlSym:

�4þ60gþ1440g2�l2

3775 (D9)

½m�st¼mlc

420ð1þ12bÞð1þ12gÞ

�

2664

156þ1764bþ1764gþ20160bg ð22þ252bþ210gþ2520bgÞl 54þ756bþ756gþ10080bg �ð13þ168bþ210gþ2520bgÞlð22þ210bþ252gþ2520bgÞl ð4þ42bþ42gþ504bgÞl2 ð13þ210bþ168gþ2520bgÞl �ð3þ42bþ42gþ504bgÞl254þ756bþ756gþ10080bg ð13þ168bþ210gþ2520bgÞl 156þ1764bþ1764gþ20160bg �ð22þ252bþ210gþ2520bgÞl�ð13þ210bþ168gþ2520bgÞl �ð3þ42bþ42gþ504bgÞl2 �ð22þ210bþ252gþ2520bgÞl ð4þ42bþ42gþ504bgÞl2

3775 (D10)

½m�ts¼ ½m�Tst (D11)

Load vectors:

fqg ¼ q0l12

8>><>>:

6l6�l

9>>=>>;þ

q1l60ð1þ 12bÞ

8>><>>:

9þ 120bð2þ 30bÞl21þ 240b�ð3þ 30bÞl

9>>=>>; (D12)

fmg ¼ m0l12

8>><>>:

6l6�l

9>>=>>;þ

m1l60ð1þ 12gÞ

8>><>>:

9þ 120gð2þ 30gÞl21þ 240g�ð3þ 30gÞl

9>>=>>; (D13)

Stiffness ratios:


b ¼ EIbGAsl2

; g ¼ EIwGIsl2

: (D14)

Appendix E. Estimation of inaccuracy due to application of inconsistent finite elementformulation

The beam finite element properties are determined by the energy approach, employing both thefrequency dependent and independent shape functions. The former formulation is based on theconsistent vibration beam theory, while the latter follows the static beam theory. Accuracy of thesimplified formulation can be evaluated analysing natural vibrations of prismatic beam by the energymethod. The governing equation for flexural natural frequencies is deduced from the energy balance,Eq. (26), in the form of Rayleigh quotient

u2 ¼EIR L

0

d2wbdx2

�2dxþ GAs

R L0

dwsdx

�2dx

mR L

0 w2dxþ JR L

0

dwbdx

�2dx

: (E1)

For the reason of simplicity let us use the sinusoidal natural modes into account

wbn ¼ sinnpx

L: (E2)

In that case the shear and total deflection according to (24) read

wsn ¼�� u2 J

GAsþnp

L

�2 EIGAs

�sin

npxL

(E3)

wn ¼�

1� u2 JGAsþnp

L

�2 EIGAs

�sin

npxL: (E4)

By substituting (E2)–(E4) into (E1), and taking into account

Z L

0sin2npx

Ldx ¼

Z L

0cos2npx

Ldx ¼ L

2(E5)

yields

u2 ¼np

L

�2EIm

�npL

�2þGAsEI

h�npL

�2 EIGAs� u2

nJ

GAs

i2

h1þ

�npL

�2 EIGAs� u2

nJ

GAs

i2þ�

npL

�2 Jm

: (E6)

Since both the modal stiffness (nominator) and modal mass (denominator) depend on theunknown natural frequency un (as in the finite element formulation), an iteration procedure has to beused to solve the implicit Eq. (E6). The convergence is very fast due to small influence of u2

n on the righthand side of (E6).

However, Eq. (E6) can also be solved in a close form if it is written as a polynomial. It is obvious thatwe are faced with the polynomial of 3rd order per u2

n, which can be written as product of twopolynomials

P3

u2

n

�¼ P1

u2

n

�P2

u2

n

�¼ 0; (E7)

where

Table E1Natural frequencies of prismatic pontoon, ui (Hz).

Mode no. Exact (E8), a Approximate (E9), b J¼ 0 (E10), c Discrepancy %

b/a� 1 c/a� 1

1 0.7540 0.7541 0.7588 0.00 0.632 2.1418 2.1422 2.1560 0.02 0.653 3.5739 3.5745 3.5900 0.02 0.454 4.9789 4.9796 4.9940 0.01 0.305 6.3610 6.3617 6.3745 0.01 0.216 7.7278 7.7284 7.7397 0.01 0.157 9.0844 9.0850 9.0951 0.01 0.128 10.4342 10.4347 10.4437 0.01 0.099 11.7791 11.7796 11.7877 0.00 0.0710 13.1205 13.1209 13.1284 0.00 0.06


P1

u2

n

�¼ J

GAsu2

n ��

1þnp

L

�2 EIGAs

�(E8)

P2

u2

n

�¼ J

GAsu4

n ��

1þnp

L

�2�

EIGAsþ J

m

��u2

n þnp

L

�4EIm: (E9)

The solution of P1ðu2nÞ ¼ 0 doesn’t have physical meaning, as well as the first eigenvalue of

P2ðu2nÞ ¼ 0. Its second eigenvalue represents the actual natural frequencies

u2n ¼

GAs

2J

8<:�

1þnp

L

�2�

EIGAsþ J

m

��(�

1þnp

L

�2�

EIGAsþ J

m

��2

�4np

L

�4 EIGAs

Jm

)12

9=;: (E10)

It is interesting to point out that polynomial P2ðu2nÞ is identical to the frequency equation of the

uncoupled flexural vibrations, which yields from Eq. (20).In case of static displacement relations, Eqs. (E3) and (E4) with u ¼ 0, Eq. (E6) is reduced to the

approximate formula for determining natural frequencies

u2n ¼

npL

�4EIm

1þ�

npL

�2 EIGAsh

1þ�

npL

�2 EIGAs

i2þ�

npL

�2 Jm

: (E11)

Furthermore, if mass rotation is neglected, i.e. J ¼ 0, one finds from (E11)

u2n ¼

ðnpL

�4EIm

1þ�

npL

�2 EIGAs

: (E12)

In order to estimate reliability of the approximate solution (E11) and influence of mass rotation onnatural frequencies, Eq. (E12), let us analyse vibrations of a pontoon with the cross-section parametersof a real ship specified in Section 6: L¼ 348 m, A¼ 7.049 m2, Asy¼ 1.056 m2, Ibz¼ 2428 m4, m¼ 100 t/m,E¼ 2.1$108 kN/m2, n¼ 0.3, E=G ¼ 2ð1þ nÞ; ðJ=mÞ ¼ ðIbz=AÞ. The calculated values of naturalfrequencies in the above considered cases are listed in Table E1.

Discrepancies between the approximate and the exact solution are very small and negligible. Thesolution for J ¼ 0 shows some differences at the first modes, but for the higher modes it converges tothe exact solution.


Appendix F. Ship cross-section properties

Geometrical properties of a thin-walled girder include cross-section area A, moment of inertia ofcross-section Ib, shear area As, torsional modulus It, warping modulus Iw and shear inertia modulus Is.These parameters are determined analytically for a simple cross-section as pure geometrical properties[12,13,17,18].

However, determination of cross-section properties for an open multi-cell cross-section, as in caseof ship structures, is quite a difficult task. Therefore, the strip element method is applied for solving thisstatically indetermined problem [29]. Firstly, axial node displacements are calculated due to bendingcaused by shear force, and due to torsion caused by variation of twist angle. Then, shear stress inbending sb, shear stress due to pure torsion st, shear and normal stresses due to restrained warping sw

and sw, respectively, are determined. Based on the equivalence of strain energies induced by sectionalforces and calculated stresses, it is possible to specify cross-section properties in the same formulationas presented below. Furthermore, those formulae can be expressed by stress flows, i.e. stresses due tounit sectional forces [14,25].

Shear area:

As ¼Q2R

As2

bdA¼ 1R

Ag2

b dA; gb ¼

sb

Q: (F1)

Torsional modulus:

It ¼T2

tRA

s2t dA

¼ 1RA

g2t dA

; gt ¼st

Tt: (F2)

Shear inertia modulus:

Is ¼T2

wRA

s2wdA

¼ 1RA

g2wdA

; gw ¼sw

Tw: (F3)

Warping modulus:

Iw ¼B2

wRA

s2wdA

¼ 1RA

f 2wdA

; fw ¼sw

Bw: (F4)

The above quantities are not pure geometrical cross-section properties any more, since they alsodepend on Poisson’s ratio as a physical parameter.

For a ship in lightweight loading condition the mass parameters can be determined approximatelybased on the given mass distribution per unit length, m, and calculated cross-section parameters, i.e.

Jb ¼mA

Ib; J0t ¼

mA

Iby þ Ibz

�; Jw ¼

mA

Iw (F5)

Stress flows at midship section for the considered ship, determined by program STIFF [24], areshown in Figs. F1–F5. The midship section properties are the following:

Cross-section area, A¼ 7.049 m2

Vertical shear area, Asz¼ 1.604 m2

Horizontal shear area, Asy¼ 1.056 m2

Vertical position of centroid, zC¼ 13.36 mVertical position of shear centre, zS¼�15.61 mVertical moment of inertia, Iby¼ 1040 m4

Horizontal moment of inertia, Ibz¼ 2428 m4


Torsional modulus, It¼ 20.20 m4

Shear inertia modulus, Is¼ 927 m4

Warping modulus, Iw¼ 321500 m6.

Appendix G. Contribution of transverse bulkheads to hull stiffness

This problem for container ships is extensively analysed in [31], where torsional modulus of shipcross-section is increased proportionally to the bulkhead strain energy. The bulkhead is considered asan orthotropic plate with very strong stool [32]. The bulkhead strain energy is determined for the givenwarping of cross-section as a boundary condition. The warping causes bulkhead screwing and bending.Here, only the review of the final results is presented.

The bulkhead deflection (axial displacement) is given by the following formula, Fig. G1:

uðy; zÞ ¼ �y��

z� d�þ�

1�y

b

�2�

z2

H

2� z

H

��j0; (G1)

where H is the ship height, b is one half of bulkhead breadth, d is the distance of warping centre fromdouble bottom neutral line, y and z are transverse and vertical coordinates, respectively, and j0 is thevariation of twist angle.

The bulkhead grillage strain energy includes vertical and horizontal bending with contraction, andtorsion [30].

Ug ¼1

1� n2

"116H3

35biy þ

32b3

105Hiz þ

8Hb75

n�iy þ iz

�þ 143Hb

75ð1� nÞit

#Ej02; (G2)

where iy, iz and it are the average moments of inertia of cross-section and torsional modulus per unitbreadth, respectively.

The stool strain energy is comprised of the bending, shear and torsional contributions

Us ¼"

12h2Isb

bþ 72ð1þ nÞh

2

b3

I2sbAsþ 9bIst

10ð1þ nÞ

#Ej02; (G3)

where Isb, As and Ist are the moment of inertia of cross-section, shear area and torsional modulus,respectively. Quantity h is the stool distance from the inner bottom, Fig. G2.

The equivalent torsional modulus yields, Fig. G2

I�t ¼�

1þ al1þ he

4ð1þ nÞCItl0

�It ; (G4)

Table G1Stiffness parameters of watertight bulkhead.

Girder Momentof inertia,I [m4]

Torsionalmodulus,It [m4]

Girderspacing,c [m]

Moment of inertiaper unit breadth,i [m3]

Torsional modulusper unit breadth,it [m3]

Horizontal 0.02160 0.00905 5.184 0.004164 0.002843Vertical 0.03094 0.02333 5.04 0.006139

Table G2Stiffness parameters of support bulkhead.

Girder Momentof inertia,I [m4]

Torsional modulus,It [m4]

Girder spacing,c [m]

Moment of inertiaper unit breadth,i [m3]

Torsional modulusper unit breadth,it [m3]

Horizontal 0.00972 0.00486 5.184 0.001875 0.002293Vertical 0.02017 0.02827 5.04 0.004002

Table G3Stool stiffness parameters.

Shear area Moment of inertia Torsional modulus

As [m2] Is [m4] Its [m4]0.045 0.12236 0.433

Table G4Bulkhead strain energy, U/[Ej’2]

Measure Watertight bulkhead Support bulkhead Energy coefficient

Grillage Stool Grillage Stool C, Eq. (D5)

(1) (2) (3) (4) (5) (6)¼ [(2)þ(3)þ(4)þ(5)]/2U/[Ej’2] 22.856 69.708 11.341 69.708 86.807


where a is the web height of bulkhead girders (frame spacing), l0 is the bulkhead spacing, l1 ¼ l0 � a isthe net length, he is the efficiency factor, and C is the energy coefficient

C ¼ Ug þ Us

Ej02: (G5)

The second term in (G4) is the main contribution of the bulkhead as the closed cross-sectionsegment of ship hull, and the third one comprises the bulkhead strain energy.

Large container ships are designed with alternate watertight and support bulkheads, Fig. G2. Thesebulkheads for the considered ship are shown in Figs. G3 and G4. The stiffness parameters of thebulkhead girders are listed in Tables G1 and G2, while the stool parameters are given in Table G3. Thebulkhead dimensions are the following:

H ¼ 29:74 m; b ¼ 20:45 m; l0 ¼ 14:44 m; a ¼ 1:81 m:

The bulkhead strain energy, determined according to (G2) and (G3), is summarized in Table G4,where also the energy coefficient is calculated as the average value of the watertight and supportbulkhead strain energies. Most of the hull induced energy is absorbed by the stool. The efficiencyfactor, he, takes into account the release of the bulkhead support, Fig. G5. It can be expressed as the ratioof the stool boundary rotation angle and the global deck rotation angle. In the considered case,he ¼ 0:55. Thus, the equivalent torsional modulus for midship section yields I�t ¼ 1:9It . This value isapplied for all ship cross-sections as the first approximation.

Appendix H. Formulation of effective stiffness of thin-walled girders

It is well known that Timoshenko’s beam theory for flexural vibrations with the shear area includedis valid for the first couple natural modes. In literature there is a large number of references dealing

Fig. H1. Membrane strip element.


with effective stiffness parameters in order to increase validity of beam model to higher modes as forinstance [33–37]. One way is to keep moment of inertia of cross-section constant and to vary sheararea, and another is to vary both parameters, [38]. Since shear deflection is increasing and pure bendingdeflection is decreasing for higher modes, the former treatment is artificial one, while the latter isphysically consistent and therefore preferable.

Energy approach for determining effective stiffness of thin-walled girders for flexural and torsionalvibrations is described in details in [14]. Here, only the basic idea and main formulae for flexuralvibrations are presented informatively.

One half of ship cross-section is divided into strip elements stretching in axial direction, which canbe of the following types: membrane, plate, bar and beam. In order to avoid the distortion of cross-section, all transverse bulkheads within assumed prismatic hull are condensed in one bulkhead ofcommon thickness and modelled by ordinary membrane finite elements.

A membrane strip element, with geometric and physical characteristics, nodal forces anddisplacements, is shown in Fig. H1. Displacement field is described by harmonic functions, that is validfor simply supported edges

ff g ¼�

uv

�¼ ½f�fdg; (H1)

where

½f� ¼ ½f�C cos anxþ ½f�S sin anx; an ¼ npL ; n ¼ 1;2.

½f�C ¼"

f1 0 f2 0

0 0 0 0

#; ½f�S¼

"0 0 0 0

0 f1 0 f2

#

fdg ¼

8>>>><>>>>:

u1

v1

u2

v2

9>>>>=>>>>;:

(H2)

Functions f1 and f2 are the linear strip shape functions

f1 ¼ 1� h; f2 ¼ h; h ¼ y=b: (H3)

Furthermore, membrane deformation field reads

feg ¼ ½L�fdg ¼ ½L�½f�fdg; (H4)

where ½L� is the membrane differential operator

½L� ¼

26666664

v

vx0

0v

vyv

vyv

vx

37777775

(H5)

leading to

½L� ¼ ½L�C cos anxþ ½L�S sin anx

½L�C ¼

264

0 0 0 0

0 0 0 0

f01 anf1 f02 anf2

375; ½L�S¼

264�anf1 0 �anf2 0

0 f01 0 f02

0 0 0 0

375: (H6)


The membrane elasticity matrix reads

2 3
½D� ¼ h
1� n264

E nE 0

nE E 0

0 0�1� n2

�G

75: (H7)

According to the definition of stiffness matrix in the finite element method, and by employing (H3),one finds separated normal and shear stiffness matrices

½K� ¼Z L

0

Z b

0½L�T ½D�½L�dxdy ¼ ½K�Eþ½K�G; (H8)

where

½K�E ¼ELh

2�1� n2

�b

2666666664

b2n

3nbn

2b2

n6

�nbn

2

1nbn

2�1

b2n

3�nbn

2Sym: 1

3777777775

½K�G¼GLh2b

266666666664

1 �bn

2�1 �bn

2b2

n

3bn

2b2

n

6

1bn

2

Sym:b2

n3

377777777775

bn ¼ anb ¼ npbL:

(H9)

By following the standard finite element procedure for assembling of finite elements, the systemequation for each the nth mode yields

�½K�Enþ½K�Gn

�fdgn¼

L2fqgn; (H10)

where qn is the amplitude of the assumed vertical and horizontal load for vertical and horizontalbending, respectively.

Shear force Q and bending moment M are obtained by single and double integration of harmonicload q along prismatic hull, respectively. Thus, their amplitudes for complete cross-section read

Qn ¼2an

Xk

i¼1

qin; Mn ¼

2a2

n

Xk

i¼1

qin; (H11)

where k is number of loaded nodes at one half of cross-section.

Table H1The 5th natural frequencies with effective stiffness, u5 (Hz).

Vibration type 1D FEM 3D FEM Discrepancy %

a�5 ¼ 0:804 a�5 ¼ 0:3 a�5 ¼ 0:804 a�5 ¼ 0:3

Horizontal 6.864 4.665 6.798 0.97 �31.38Coupled 3.029 2.687 2.630 15.17 2.17


Furthermore, the bending and shear strain energy for the uniform hull due to harmonic M and Qtake values

UMn ¼1

2EIn

Z L

0M2dx ¼ LM2

n4EIn

UQn ¼1

2GASn

Z L

0Q2dx ¼ LQ2

n

4GASn:

(H12)

On the other side, the bending and shear strain energy for complete hull are also obtained bymultiplying Eq. (H10) with CdDn. Finally, equalizing those expressions with (H12), one finds formulae foreffective moment of inertia of cross-section and shear area, respectively

In ¼LM2

n

4ECdDn½K�Enfdgn

ASn ¼LQ2

n

4GCdDn½K�Gnfdgn:

(H13)

The effective coefficients are in ¼ In=I0 and an ¼ ASn=AS0. In similar way, the torsional effectivestiffness parameters can also be determined, [14].

If one prefers to keep the value of moment of inertia of cross-section constant and to vary shear areaonly, than complete strain energy has to be used for shear area correction. By employing the followingrelation yielding from Eqs. (H11)

Mn ¼Qn

an; (H14)

and after some manipulations one finds

a�n ¼an

1þ 12ð1þ nÞa2

n

As

Ian

in

: (H15)

In the considered numerical example, the horizontal stiffness parameters for midship section read:Asy ¼ 1:056 m2, Ibz ¼ 2428 m4, Appendix F. This leads to a�5 ¼ 0:804 and the modified naturalfrequencies are listed in Table G1. Discrepancy between 1D and 3D natural frequencies is considerablyreduced in the case of horizontal vibrations, while that of the coupled vibrations is still high. By takinga�5 ¼ 0:3 into account, the effect is opposite, Table H1. It is obvious that unrealistic ratio of the bendingand shear deflection, combined with torsional properties in coupled vibrations results with incorrectfrequency values. So, the conclusion is that effective values of both bending and shear stiffnesses haveto be taken into account in order to simultaneously minimize the discrepancies of flexural and coupledflexural and torsional frequencies, as it is done in Section 9, Table 4. In addition, let us consider theproperties of natural vibration modes calculated with constant and mode dependent stiffnessparameters. In the former case modes are orthogonal, i.e.

Z L

0wiwjdx ¼ 0; if isj (H16)

In the later case, each natural mode determined with own effective parameters can be expressed ina series of the above orthogonal modes. Since the considered modes are not orthogonal

Z L

0wi

iwjjdx ¼

Z L

0

XNk¼1

aikwk

XNl¼1

ajlwldx ¼

XNk¼1

aikaj

k

Z L

0w2

kdxs0: (H17)


References

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dynamic transient response. Computers & Structures 1988;29(2):227–40.[24] Fan Y, Senjanovic I. STIFF, User’s Manual. Zagreb: Faculty of Mechanical Engineering and Naval Architecture; 1990.[25] Senjanovic I, Fan Y. A finite element formulation of initial ship cross-section properties. Brodogradnja 1993;41(1):27–36.[26] Senjanovic I, Catipovic I, Tomasevic S. Coupled horizontal and torsional vibrations of a flexible barge. Engineering

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vibration analysis. Journal of Ship Research 1989;33(4):298–309.[36] Senjanovic I, Fan Y. The bending and shear coefficients of thin-walled girders. Thin-Walled Structures 1990;10:31–57.[37] Senjanovic I, Fan Y. On torsional and warping stiffness of thin-walled girders. Thin-Walled Structures 1991;11:233–76.[38] Jensen JJ. On the shear coefficient in Timoshenko’s beam theory. Journal of Sound and Vibration 1983;87(4):621–35.

Ivo Senjanovic, D.Sc. Professor of Naval Architecture at the Faculty of Mechanical Engineering and Naval Architecture, Universityof Zagreb. Teaching several courses on strength and vibration of ship and offshore structures, submarines etc. Investigation fieldsare ship stability, launching, numerical methods, numerical simulations, non-linear dynamics, ship strength and vibration, shelltheory and design of pressure vessels. Published books on shell theory, ship vibrations and finite element method, and ca. 200scientific and professional papers. E-mail: [email protected]; http://mahazu.hazu.hr/Akademici/ISenjanovic.html.

Stipe Tomasevic, Ph.D. Lecturer at the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb. Supportin education on the finite element method and software application. Investigation of ship strength, vibration, fatigue andhydroelasticity. E-mail: [email protected].

Nikola Vladimir, Dipl. Ing., Ph.D. student at the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb.E-mail: [email protected]


http://mahazu.hazu.hr/Akademici/ISenjanovic.html

mailto:[email protected].


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