Marine Structures 22 (2009) Contents lists available at ScienceDirect. Marine Structures. journal homepage: marstruc

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1 Marine Structures (009) Contents lists available at ScienceDirect Marine Structures journal homepage: marstruc An advanced theory of thin-walled girders with application to ship vibrations I. Senjanović *, S. Tomašević, N. Vladimir University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, I. Lučića 5, Zagreb, Croatia article info abstract Article history: Received 8 July 008 Received in revised form 7 January 009 Accepted 4 March 009 Keywords: Thin-walled girder theory Beam model Coupled vibrations Container ship FEM The paper presents an outline of the advanced theory of thinwalled girders. The improvement includes shear influence on torsion as an extension of shear influence on bending. The analogy between bending and torsion is recognized and pointed out throughout the paper. Complete differential equations of coupled flexural and torsional vibrations for a prismatic girder are derived. In addition, the 8 d.o.f. beam finite element, utilizing the energy approach, is constituted with stiffness and mass matrices, and load vectors. The paper describes determining of geometrical properties of multi-cell open cross-sections by employing the strip element method. Numerical procedures for vibration analyses are outlined. Furthermore, dry natural vibrations of a VLCS (Very Large Container Ship) are analysed by 1D FEM model as a prerogative for hydroelastic analyses of these relatively flexible vessels. Influence of transverse bulkheads is taken into account by increasing torsional stiffness of the ship hull proportionally to their deformation energies. Validation of 1D FEM model is checked by correlation analysis with the vibration response of the fine 3D FEM model. Ó 009 Elsevier Ltd. All rights reserved. * Corresponding author. Tel.: þ ; fax: þ address: ivo.senjanovic@fsb.hr (I. Senjanović) /$ see front matter Ó 009 Elsevier Ltd. All rights reserved. doi: /j.marstruc

2 388 I. Senjanović et al. / Marine Structures (009) Nomenclature A cross-section area A i integration constants A s shear area a k,b k,c k,d k,e k,f k, coefficients of shape functions B w warping bimoment b one half of bulkhead breadth C energy coefficient c eccentricity E Young s modulus E tot total energy f normal stress flow G shear modulus g shear stress flow H ship height I b moment of inertia of cross-section I s shear inertia modulus I t torsional modulus I w warping modulus i, j, k indexes J b moment of inertia of distributed mass Jt 0; J t polar moment of inertia of distributed mass about centre of gravity and shear centre J w bimoment of inertia of distributed mass about warping centre L, l length M bending moment m distributed mass n mode number Q shear force q distributed load T torque T t torsional torque T w warping torque t time U strain energy w deflection w b bending deflection w s shear deflection u, v membrane displacements x, y, z coordinates z C, z S, z G 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

3 I. Senjanović et al. / Marine Structures (009) {P}, {R} force vectors {U}, {V} displacement vectors {D}, {d} displacement vectors a, h vibration coefficients b, g stiffness ratios z damping coefficient w variation of twist angle m distributed torque n Poisson s coefficient x non-dimensional coordinate s normal stress s shear stress 4 rotation angle j twist angle j t pure twist angle j s shear twist angle u natural frequency [f] matrix of shape functions 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 []. The methodology of hydroelastic analysis is described in [3]. It includes the definition of the structural model, ship and cargo mass distributions, and geometrical model of ship wetted surface. Hydroelastic analysis is based on the modal superposition method. First, dry natural vibrations of ship hull are calculated. Then, modal hydrostatic 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 ship structural response to wave excitation are obtained [4,5]. The approach is checked by the model test of the flexible barge [6,7]. The intention of this paper is to present an advanced numerical procedure based on the beam and thin-walled girder theories for reliable calculation of dry natural vibrations of container ships, as an important 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 of gravity and centroid, as well as the shear centre positions are not identical, coupled longitudinal and vertical, and horizontal and torsional vibrations occur, respectively. The distance between the centre of gravity and centroid for longitudinal and vertical vibrations, as well as distance between the former and shear centre for horizontal and torsional vibrations are negligible for conventional ships. Therefore, in the above cases ship hull vibrations are usually analysed separately. However, the shear centre in ships with large hatch openings is located outside the crosssection, i.e. below the keel, and therefore the coupling of horizontal and torsional vibrations is extremely 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. of bending, shear, torsional and warping moduli. The finite element method is a powerful tool to solve the above problem in a successful way. One of the first solutions for coupled horizontal and torsional hull vibrations, dealing with the finite element technique, is given in [8,9]. Generalised and improved solutions are presented in [10,11]. In all these 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 [1,13].

4 390 I. Senjanović et al. / Marine Structures (009) Apart from that, the fixed values of stiffness moduli are determined, so that the application of the beam theory for hull vibration analysis is limited to a few lowest natural modes only. Otherwise, if the mode dependent stiffness parameters are used the application of the beam theory can be extended up to the tenth natural mode [14 16]. Based on the above described state-of-the-art and inspired motivation, this paper brings some novelties and improvements to the thin-walled girder theory. Shear influence on torsion, as an extension of its influence on bending, is taken into account. Analogy between bending and torsion is used as a sign-post. All the relevant stiffness and mass parameters are included in the differential equations of coupled flexural and torsional vibrations. A more complete beam finite element with eight degrees of freedom is developed. The application of the advanced theory is illustrated in the case of a 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 response with that obtained by 3D FEM analysis.. Differential equations of beam vibrations Referring to the flexural beam theory [19,0], the total beam deflection, w, consists of the bending deflection, w b, and the shear deflection, w s, while the angle of cross-section rotation depends only on the former, Fig. 1 Fig. 1. Beam bending and torsion.

5 I. Senjanović et al. / Marine Structures (009) w ¼ w b þ w s ; 4 ¼ vw b vx : The cross-sectional forces are the bending moment and the shear force M ¼ EI b v4 vx ; (1) () Q ¼ GA s vw s vx ; (3) where E and G are Young s and shear modulus, respectively, while I b and A s are the moment of inertia of cross-section and shear area, respectively. The inertia load consists of the distributed transverse load, q i, and the bending moment, m i, and in the case of coupled horizontal and torsional vibration is specified as q i ¼ m! v w vt þ j cv vt ; (4) m i ¼ J b v 4 vt ; (5) where m is the distributed ship and added mass, J b 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 z G z S, Fig.. Concerning torsion, the total twist angle, j, consists of the pure twist angle, j t, and the shear contribution, j s, while the second beam displacement, which causes warping of cross-section, is variation of the pure twist angle, i.e. Fig. 1 [18] j ¼ j t þ j s ; w ¼ vj t vx : (6) Fig.. Cross-section of a thin-walled girder.

6 39 I. Senjanović et al. / Marine Structures (009) The cross-sectional forces include the pure torsional torque, T t, warping bimoment, B w, and additional torque due to restrained warping, T w T t ¼ GI t w; (7) B w ¼ EI w vw vx ; (8) T w ¼ GI s vj s vx ; where I t, I w and I s are the torsional modulus, warping modulus and shear inertia modulus, respectively. The inertia load consists of the distributed torque, m ti, and the bimoment, b i, presented in the following form: (9) m ti ¼ J t v j vt mcv w vt ; (10) b i ¼ J w v w vt ; where J t is the polar moment of inertia of ship and added mass about the shear centre, and J w is the bimoment of inertia of ship mass about the warping centre, Fig.. Considering the equilibrium of a differential element, one can write for flexural vibrations (11) vm vx ¼ Q þ m i; (1) vq vx ¼ q i q; and for torsional vibrations [17] vb w vx ¼ T w þ b i ; (13) (14) vt t vx þ vt w vx ¼ m ti m: The above equations can be reduced to two coupled partial differential equations as follows. Substituting Eqs. () and (3) into (1) yields vw s vx ¼ EI b GA s v 4 vx þ J b GA s v 4 vt : By inserting Eqs. (3) and (4) into (13) leads to v 4 4 EI b vx 4 þ 4 mv vt J b þ m EI b v4 4 GA s vx vt þ mj b v 4 4 GA s vt 4 þ mc v3 j vxvt ¼ vq vx : (17) In similar way, substituting Eqs. (8) and (9) into (14) yields (15) (16)

7 I. Senjanović et al. / Marine Structures (009) vj s vx ¼ EI w GI s v w vx þ J w GI s v w vt : (18) By inserting Eqs. (7), (9) and (10) into (15) one finds v 4 w EI w vx 4 GI v w t vx þ J v w t vt EI w v 4 w J w þ J t GI s vx vt þ J w v 4 w GI s vt 4 þ mc v3 w vxvt ¼ vm vx : (19) Furthermore, j in (17) can be split into j t þ j s and the later term can be expressed with (18). Similar substitution can be done for w ¼ w b þ w s in (19), where w s is given with (16). Thus, taking into account that 4 ¼ vw b =vx and w ¼ vj t =vx, Eqs. (17) and (19) after integration per x read EI b v 4 w b vx 4 þ w b mv vt J b þ m EI b v 4 w b GA s vx vt þ mj b v 4 w b GA s vt 4 þ mc v j t vt EI w GI s v 4 j t vx vt þ J w v 4 j t GI s vt 4! ¼ q (0) v 4 j EI t w vx 4 GI v j t t vx þ J v j t t vt EI w v4 j J w þ J t t GI s vx vt þ J w v 4 j t GI s vt 4 þ mc v w b vt EI b v 4 w b GA s vx vt þ J b v 4 w b GA s vt 4! ¼ m: ð1þ After solving Eqs. (0) and (1) the total deflection and twist angle are obtained by employing (16) and (18) w ¼ w b þ w s ¼ w b EI b v w b GA s vx þ J b v w b þ f ðtþ; () GA s vt j ¼ j t þ j s ¼ j t EI w v j t GI s vx þ J w v j t þ gðtþ; (3) GI s vt where f(t) and g(t) are integration functions, which depend on initial conditions. The main purpose of developing differential equations of vibrations (0) and (1) is to get insight into their constitution, position and role of the stiffness and mass parameters, and coupling, which is realized through the inertia terms. If the pure torque T t is excluded from the above theoretical consideration, it is obvious that the complete analogy between bending and torsion exists, [17]. Application of Eqs. (0) and (1) is limited to prismatic girders. It is illustrated in case of uncoupled torsional natural vibrations of a uniform beam in Appendix A. For more complex problems, like ship hull, 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 following consistency relations for harmonic vibrations obtained from Eqs. () and (3), [0] w ¼ w b þ w s ¼ 1 u J b w GA b EI b d w b s GA s dx ; (4) j ¼ j t þ j s ¼ 1 u J w j GI t EI w d j t s GI s dx : (5)

8 394 I. Senjanović et al. / Marine Structures (009) Beam finite element The properties of a finite element for the coupled horizontal and torsional vibration analysis can be derived 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 [9,0], Z " l # E tot ¼ 1 þ 1 0 Z l 0! v w EI b vws b vx þga s þei w vx "! vw v w m þj b vt b þmc vw vxvt vt v j t vx! vjs vjt þgi s þgi t vx vx! v # j t vj þj t dx vxvt vt vj vt þ J w R l 0 ðqw þ mjþdx þðqw M4 þ Tj þ B wwþ l 0 ; 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, wð0þ >< >= 4ð0Þ fug ¼ wðlþ >: 4ðlÞ ; fvg ¼ >; jð0þ >< >= wð0þ jðlþ >: wðlþ dx (6) : (7) >; Fig. 3. Beam finite element.

9 I. Senjanović et al. / Marine Structures (009) Therefore, the basic beam displacements, w b and j t, can be presented as the third-order polynomials w b ¼ Ca k Dn x k o ; j t ¼ Cd k Dn x k o ; k ¼ 0; 1; ; 3; x ¼ x l ; C.D ¼f.gT : (8) Furthermore, satisfying alternately the unit value for one of the nodal displacement {U} and zero values for the remaining displacements, and doing the same for {V}, it follows that: w b ¼ Cw bi DfUg; w s ¼ Cw si DfUg; w ¼ Cw i DfUg; j t ¼ Cj ti DfVg; j s ¼ Cj si DfVg; j ¼ Cj i DfVg; i ¼ 1; ; 3; 4; (9) where w bi, w si, w i and j ti, j si, j i are the shape functions specified below by employing relations (4) and (5) w bi ¼ Ca ik Dn o x k ; w si ¼ Cb ik Dn o n o x k ; w i ¼ Cc ik D x k j ti ¼ Cd ik Dn o x k ; j si ¼ Ce ik Dn o n x k ; j o (30) i ¼ Cf ik D x k 3 a þ 6b 0 3a a 1 ½a ik Š¼ 6 4bða þ 3bÞl aða þ 1bÞl aða þ 3bÞl a l 7 aða þ 1bÞ4 6b 0 3a a 5 (31) bða 6bÞl 0 aða 6bÞl a l b i0 ¼ ð1 aþa i0 ba i b i1 ¼ ð1 aþa i1 6ba i3 b i ¼ ð1 aþa i b i3 ¼ ð1 aþa i3 ; (3) ½c ik Š¼½a ik Šþ½b ik Š; i ¼ 1; ; 3; 4; k ¼ 0; 1; ; 3 (33) a ¼ 1 u J b GA s ; b ¼ EI b GA s l (34) Constitution of torsional matrices ½d ik Š, ½e ik Š and ½f ik Š is the same as ½a ik Š, ½b ik Š and ½c ik Š, but parameters a and b have to be exchanged with h ¼ 1 u J w GI s ; g ¼ EI w GI s l (35) according to (5). By substituting Eqs. (9) into (6) one obtains E tot ¼ 1 T U kbs 0 V 0 k ws þ k t T P U ; R V U þ 1 _U_V T msb V m st _U_V m ts m tw q m T U V (36) where, assuming constant values of the element properties,

10 396 I. Senjanović et al. / Marine Structures (009) R l ½kŠ bs ¼½EI b 0 ðd w bi =dx Þðd w bj =dx R l Þdx þ GA s 0 ðdw si=dxþðdw sj =dxþdxš bending-shear stiffness matrix, ½kŠ ws ¼½EI w R l 0 ðd j ti =dx Þðd j tj =dx Þdx þ GI s R l 0 ðdj si=dxþðdj sj =dxþdxš warping-shear stiffness matrix, R l ½kŠ t ¼½GI t 0 ðdj ti=dxþðdj tj =dxþdxš torsion stiffness matrix, ½mŠ sb ¼½m R l 0 w R l iw j dx þ J b 0 ðdw bi=dxþðdw bj =dxþdxš shear-bending mass matrix, R l ½mŠ tw ¼½J t j R l 0 ij j dx þ J w 0 ðdj ti=dxþðdj tj =dxþdxš torsion-warping mass matrix, ½mŠ st ¼½mc R i l 0 w ij j dx ; ½mŠ ts ¼½mŠ T st shear-torsion mass matrix, n R l fqg ¼ 0 jdxo qw n R l fmg ¼ 0 mj jdxo i; j ¼ 1; ; 3; 4: shear load vector; torsion load vector; (37) The vectors {P} and {R} represent the shear-bending and torsion-warping nodal forces, respectively, 8 9 Qð0Þ >< >= Mð0Þ fpg ¼ QðlÞ >: MðlÞ ; frg ¼ >; 8 >< Tð0Þ B w ð0þ TðlÞ >: B w ðlþ 9 >= : (38) >; The above matrices are specified in Appendix B, as well as the load vectors for linearly distributed loads along the element, i.e. q ¼ q 0 þ q 1 x; m ¼ m 0 þ m 1 x: (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 ve tot vfug ¼ f0g; ve tot vfvg ¼ f0g (40) and employing Lagrange equations of motion, the finite element equation yields ff g ¼½kŠfdg þ½mšn d o ff g qm ; (41) where P q U ff g¼ ; ff g R qm ¼ ; fdg ¼ m V kbs 0 msb m ½kŠ ¼ ; ½mŠ ¼ st : 0 k ws þ k t m ts m tw (4) 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 those related to torsion appear. To make an ordinary finite element assembling possible, it is necessary to transform Eq. (41) in such a way that first all properties related to the first node are specified and

11 I. Senjanović et al. / Marine Structures (009) then those belonging to the second one. Thus, the rearranged nodal force and displacement vectors read 8 9 Qð0Þ Mð0Þ Tð0Þ n o >< >= ~f B ¼ w ð0þ ; QðlÞ MðlÞ TðlÞ >: >; B w ðlþ 8 9 wð0þ 4ð0Þ n ~ d o jð0þ >< >= 4ð0Þ ¼ : (43) wðlþ 4ðlÞ jðlþ >: >; 4ðlÞ The same transformation has to be done for the load vector ff g qm resulting in f ~ f g qm. The above vector transformation implies the row and column exchange in the matrices according to the following set form: (44) The element deflection refers to the shear centre as the origin of a local coordinate system. Since the vertical position of the shear centre varies along the ship s hull, it is necessary to prescribe the element deflection 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 the former and the latter nodal deflections exists: wð0þ ¼wð0Þþz S jð0þ wðlþ ¼wðlÞþz S jðlþ; (45) where z S is the coordinate of the shear centre, Fig.. Other displacements are the same in both coordinate systems. Twist angle j does not have influence on the cross-section rotation angle 4. The local displacement vector can be expressed as n ~ d o h in ¼ ~T ~ d o ; (46) where ½ TŠ ~ is the transformation matrix 3 h i 1 0 z S 0 ~T ½TŠ ½0Š ¼ ; ½TŠ ¼ ½0Š ½TŠ : (47) Since the total element energy is not changed by the above transformations, a new element equation can be derived taking (46) into account. Thus, one obtains in the global coordinate system where n o ~f ¼ h in ~k ~ d o h in ~ þ ~m d o n o ~f qm ; (48)

12 398 I. Senjanović et al. / Marine Structures (009) n ~f o ¼ ~ T T n ~ f o ~k ¼ ~T T ~k ~T ~m ¼ ~T T ~m ~T n ~f o The first of the above expressions transforms the nodal torques into the form qm ¼ ~ T T n ~ f o qm : (49) Tð0Þ ¼ Tð0Þ z S Qð0Þ TðlÞ ¼TðlÞþz S QðlÞ: (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 coordinate system, performed in the standard way, results in the matrix equation of motion, which may be extended by the damping forces ½KŠfDg þ½cšn D o _ þ½mšn D o ¼ ffðtþg; (51) where [K], [C] and [M] are the stiffness, damping and mass matrices, respectively; fdg; f D _ o and f Dg 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 structures, so that the damping forces may be ignored. Assuming fdg ¼ ffge iut ; (5) where ffg and u are the mode vector and natural frequency respectively, Eq. (51) leads to the eigenvalue problem ½KŠ u ½MŠ ffg ¼ f0g; (53) which may be solved by employing different numerical methods [1]. The basic one is the determinant search method in which u is found from the condition ½KŠ u ½MŠ ¼ 0 (54) by an iteration procedure. Afterwards, ffg follows from (53) assuming unit value for one element in ffg. The forced vibration analysis may be performed by direct integration of Eq. (51), as well as by the modal superposition method. In the latter case the displacement vector is presented in the form fdg ¼ ½fŠfXg; (55) where ½fŠ ¼ ½ffgŠ is the undamped mode matrix and {X} is the generalised displacement vector. Substituting (55) into (51), the modal equation yields where ½kŠfXg þ½cš _ X n o þ½mš X ¼ ff ðtþg; (56) ½kŠ ¼½fŠ T ½KŠ½fŠ modal stiffness matrix ½cŠ ¼½fŠ T ½CŠ½fŠ modal damping matrix ½mŠ ¼½fŠ T ½MŠ½fŠ modal mass matrix ff ðtþg ¼ ½fŠ T ffðtþg modal load vector: (57) The matrices [k] and [m] are diagonal, while [c] becomes diagonal only in a special case, for instance if [C] ¼ a 0 [M] þ b 0 [K], where a 0 and b 0 are coefficients [0].

13 I. Senjanović et al. / Marine Structures (009) Solving (56) for undamped natural vibration, [k] ¼ [u m] is obtained, and by its backward substitution into (56) the final form of the modal equation yields h u i fxg þ ½uŠ½zŠ _ X n o þ X ¼ f4ðtþg; (58) where " s ffiffiffiffiffiffiffi# k ½uŠ ¼ ii natural frequency matrix m " ii # c ij ½zŠ ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relative damping matrix ðk ii m ii Þ fi ðtþ f4ðtþg ¼ relative load vector: m ii (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 solving either (51) or (56). In both cases, the system of differential equations is transformed into a system of algebraic equations. If hull vibration is induced by waves, the time integration of (51) or (56) has to be performed. Several numerical methods are available for this purpose, as for instance the Houbolt, the Newmark and the Wilson q method [1], as well as the harmonic acceleration method [,3]. It is important to point out that all stiffness and mass matrices of the beam finite element (and consequently those of the assembly) are frequency dependent quantities, due to coefficients a and h in the formulation of the shape functions, Eqs. (34) and (35). Therefore, for solving the eigenvalue problem (53) an iteration procedure has to be applied. As a result of frequency dependent matrices, the eigenvectors are not orthogonal. If they are used in the modal superposition method for ship hydroelastic analysis, full modal stiffness and mass matrices (as those of added mass and hydrodynamic damping) are generated. Since the inertia terms are much smaller than the deformation ones in Eqs. (4) and (5), the off-diagonal elements in modal stiffness and mass matrices are very small compared to the diagonal elements and can be neglected. It is obvious that the usage of the physically consistent non-orthogonal natural modes in the modal superposition method is not practical, especially not in the case of time integration. Therefore, it is preferable to use mathematical orthogonal modes for that purpose. They are created by the static displacement relations yielding from Eqs. (4) and (5) with u ¼ 0, that leads to a ¼ h ¼ 1. In that case all finite element matrices, defined with Eqs. (37) and in Appendix B, can be transformed into explicit form, Appendix D. The resulting discrepancies of these two different formulations are analysed analytically 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 TEU VLCS (Very Large Container Ship), Fig. 4. The main vessel particulars are the following: Length overall, L oa ¼ m Length between perpendiculars, L pp ¼ 348 m Breadth, B ¼ 45.6 m Depth, H ¼ 9.74 m Draught, T ¼ 15.5 m Displacement, full load, D f ¼ t Displacement, ballast, D b ¼ t Displacement, lightweight, D l ¼ t Engine power, P ¼ 740 kw Ship speed, v ¼ 4.7 kn

14 400 I. Senjanović et al. / Marine Structures (009) Fig TEU container ship.

15 I. Senjanović et al. / Marine Structures (009) 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 [4], based on the theory of thin-walled girders [14,5], Appendix F. Their distributions along the ship are shown in Fig. 6. It is evident that geometrical properties rapidly change values in the engine and superstructure area, due to closed ship cross-section. This is especially pronounced in the case of torsional modulus, which takes quite low values for the open cross-section and rather high values for the closed one, Fig. 6e. Novelty is the distribution of the shear inertia modulus I s, Fig. 6f. Bulkhead influence is taken into account by increasing the value of the torsional modulus It ¼ 1:9I t as it is elaborated in Appendix G. Dry vibration analysis is performed for the lightweight loading condition. The containers are excluded from the analysis since they influence only the mass properties while the structural model is of primary interest. Ship mass distribution and its properties are shown in Fig. 7. The mass bimoment of inertia J w is neglected due to its very small influence on vibrations. 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 consistent frequency dependent and independent properties, as well as for the case of no mass rotation Table 1. Relations between the results are similar to those noticed in the analytical evaluation of the approximate solution in Appendix E. Negligible small discrepancies between natural frequencies of the mathematical and physical modes are evident. Influence of mass rotation on frequencies is within 4%. So, the mathematical natural modes determined by finite elements with frequency independent shape functions are very reliable to be used in the ship hydroelastic analysis. The first five natural frequencies for vertical and horizontal vibrations are listed in Tables and 3 for further analysis. The corresponding natural modes are of ordinary shapes. Natural frequencies of coupled horizontal and torsional vibrations are listed in Table 4. The first five natural modes presented by the twist angle, j, and the hull deflection at the level of center of gravity, w G, are shown in Fig. 8. These functions are mutually dependent. However, they are normalized by their own maximum values due to a simpler presentation. There always occurs coupling between frequency close symmetric flexural and torsional modes as well as anti-symmetric ones [6,7]. This is indicated in Table 4, where the 5th coupled mode is an extraordinary natural mode comprised of the nd torsional and the 5th flexural elementary modes. Once the dry natural modes of ship hull are determined, it is possible to transfer the beam node displacements to the ship wetted surface for the hydrodynamic calculation. The transformation (spreading) functions for vertical and coupled horizontal and torsional vibration yield respectively [3] h v ¼ dw v dx ðz z CÞi þ w v k (60) h ht ¼ dw h dx Y þ dj dx u i þ½w h þ jðz z S ÞŠj jyk; (61) where w is hull deflection, j is twist angle, u ¼ uðx; Y; ZÞ is the cross-section warping function reduced to the wetted surface, z C and z S 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 the ship wetted surface in case of vertical and coupled horizontal and torsional vibrations are shown in Figs. 9 1 respectively. In the latter case, the orthogonal view on the vertical and horizontal planes is used.

16 40 I. Senjanović et al. / Marine Structures (009) Fig. 5. Midship section.

17 I. Senjanović et al. / Marine Structures (009) 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 [8]. The model includes 3307 nodes, finite elements (3888 shells and beams), and d.o.f. The ship mass distribution (steel and equipment) is given by adjusted mass density for the finite element blocks. Only the main engine mass is specified by the lumped masses. The Lanczos method is used for the solution of the eigenvalue problem. In order to uncouple the horizontal vibrations from the torsional ones, for the purpose of a more detailed correlation analysis, the FEM model is reinforced by a set of vertical rigid massless beams in the 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 listed in Tables 4, respectively. The first five natural modes of the coupled vibrations, which are of primary interest, are shown in Figs The lateral and bird views are used in order to achieve easier recognition of the elementary modes in each coupled mode. We can see that in all vibration modes

18 404 I. Senjanović et al. / Marine Structures (009) Fig. 6. (continued). warping of the front (collision) bulkhead and the aft (transom) bulkhead is rather low. Thus, the assumption 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 , respectively. The same mode shapes are obvious. Natural frequencies of vertical, horizontal and coupled horizontal and torsional vibrations determined by 1D and 3D FEM models are correlated in Tables 4, respectively, with indicated discrepancies. The general opinion concerning the accuracy of 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 natural frequencies for vertical vibrations is quite good. Somewhat large discrepancies appear for the fifth frequency of horizontal vibration (8.78%). This discrepancy is magnified in the fifth frequency of the extraordinary coupled horizontal and torsional vibrations (16.81%), comprised of the 5th elementary horizontal mode and the nd torsional mode, Fig. 17. It is necessary to point out that 1D eigenpairs are determined for the fixed ship stiffness and mass properties. 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 analyses can be extended up to the 10th mode. The effective stiffness parameters are determined by utilizing the strip element method and the energy approach [9]. The main idea is to specify the harmonic load distribution and the harmonic mode shapes and split the strain energy into normal and shear parts, and then to estimate the 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 dependent moment of inertia of cross-section, I Z5 ¼ i 5 I Z, and shear area, A sy5 ¼ a 5 A sy into account. The roughly estimated values i 5 ¼ 0: and a 5 ¼ 1: are taken from the channel girder analysis, Fig. 18 [14]. Since deflection of a girder for higher modes is mainly due to shear, values of a n are increased to some extent, while those of i n 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.

19 I. Senjanović et al. / Marine Structures (009) Fig. 7. Longitudinal distribution of ship mass properties, lightweight. Table 1 Natural frequencies of horizontal hull vibrations, u i (Hz). Mode no. Physical mode, a Mathematical mode, b J bz ¼ 0, c Discrepancy % b/a 1 c/a

20 406 I. Senjanović et al. / Marine Structures (009) Table Natural frequencies of vertical hull vibrations, u i (Hz). Mode no. 1D FEM 3D FEM Discrepancy % Table 3 Natural frequencies of horizontal hull vibrations, u i (Hz). Mode no. 1D FEM 3D FEM Discrepancy % Table 4 Natural frequencies of coupled horizontal and torsional hull vibrations, u i (Hz). Mode no. Coupled modes 1D FEM 3D FEM Discrepancy % 1 T Tþ H T3þ H T4þ H Tþ H T5þ H The 5th natural mode determined by 1D FEM model with the mode dependent stiffnesses is shown in Fig. 19. Changes of functions w G and j are noticeable especially in the area of engine room. The new mode 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. 0. Warping of the transverse bulkheads is quite pronounced in this mode. The engine room structure is rather stiff concerning bending, while there is some warping release of its bulkheads. The warping magnitude in the beam model is represented by variation of twist angle w. Its given value is zero within the engine room as well as hull peaks, due to closed cross-sections, Fig. 19. It is quite difficult to simulate the local warping effect in the engine room by 1D FEM model. In 1D vibration analysis we distinguish the so-called mathematical natural modes and the actual physical modes, determined by fixed and mode dependent vibration properties, respectively. The former modes are mutually orthogonal and therefore suitable to be used as the coordinate functions for determining 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 are weakly coupled only by the modal damping. Thus, the equations can be easily solved by an iteration procedure. The application of the actual physical vibration modes, i.e. those determined with the effective values of vibration parameters, in the modal superposition method is not suitable due to coupling of modal equations as a result of mode non-orthogonality, Appendix H. This is anyway out of interest in case of ship hydroelasticity analysis, since the ship structure response to waves occurs in the lower frequency domain and is successfully described by the first few natural modes only. In the frequency domain the mathematical modes are rather close to the actual ones, and it does not matter which are used in the modal superposition method. The calculated ship response will be the same in both cases.

21 I. Senjanović et al. / Marine Structures (009) 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 of the present classification rules. Therefore, great effort is done nowadays to investigate the ship hydroelasticity phenomenon. Dry natural vibration analysis, utilizing either 1D or 3D FEM model, is the first 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 makes it possible to investigate the influence of all stiffness and mass parameters on ship dynamic behaviour and 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 classical theory is proposed. Shear influence on torsion is included in the theoretical consideration by following analogy between bending and torsion. Shear influence is more pronounced in torsion than in bending. The beam finite element for coupling flexural and torsional vibrations is developed with complete stiffness and mass matrices. Two approaches are used, the consistent frequency dependent shape functions, and frequency independent ones, which follows from static beam theory. It is shown that discrepancies between the vibration results are negligible. Also, the influence of

22 408 I. Senjanović et al. / Marine Structures (009) Fig. 9. The first natural mode of vertical vibrations, u 1 ¼1.149 Hz, 1D model. transverse bulkheads is incorporated in the hull torsional stiffness, based on detailed determination of 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 the results is verified by the correlation analysis with the results from 3D FEM analysis. By taking all relevant ship stiffness and mass parameters into account, quite good agreement of the results is achieved in the low frequency domain. The application of the beam model for ship hull vibrations is limited to the first five natural modes. However, this is sufficient for the ship hydroelasticity analysis since the most energy of wave spectrum is concentrated in the low frequency domain, i.e. below natural 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 is realistic. 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 present solutions 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 incorporation in the beam model. The basic idea is to consider engine room structure as an open cross-section segment with week deck and platforms and determine effective torsional, warping and shear moduli utilizing the energy approach. Fig. 10. The second natural mode of vertical vibrations, u ¼.318 Hz, 1D model.

23 I. Senjanovic et al. / Marine Structures (009) Fig. 11. The first natural mode of coupled horizontal and torsional vibrations, u1 ¼ Hz, 1D model. 409

24 410 I. Senjanović et al. / Marine Structures (009) Fig. 1. The second natural mode of coupled horizontal and torsional vibrations, u ¼ Hz, 1D model.

25 Fig. 13. The first natural mode of coupled horizontal and torsional vibrations, u1 ¼ Hz. Fig. 14. The second natural mode of coupled horizontal and torsional vibrations, u ¼ Hz.

26 41 I. Senjanovic et al. / Marine Structures (009) Fig. 15. The third natural mode of coupled horizontal and torsional vibrations, u3 ¼ Hz. The beam model of ship hull presented in this paper is reliable enough to be directly applied in ship hydroelastic analyses. The most discrepancies between 1D and 3D model are reduced by taking shear influence on torsion as an additional stiffness parameter into account, as well as by incorporating transverse bulkhead stiffness contribution in a very efficient way. Fig. 16. The fourth natural mode of coupled horizontal and torsional vibrations, u4 ¼.49 Hz.

27 I. Senjanović et al. / Marine Structures (009) Fig. 17. The fifth natural mode of coupled horizontal and torsional vibrations, u 5 ¼.630 Hz. Further improvement of the beam model, insisting on the consistence of compatibility conditions at joints of open and closed cross-section hull segments will have very small influence on global sectional forces of ship hull. In any case, beam model cannot get proper stress concentration in the deck corners for fatigue analysis. Therefore, it is necessary to employ the substructure technique, imposing sectional forces at substructure boundaries. Fig. 18. Efficiency factors of horizontal bending modulus and shear area of channel girder i n, a n.

28 414 I. Senjanović et al. / Marine Structures (009) Table 5 The 5th natural frequencies, u 5 (Hz), I Z5 ¼ 0.I Z, A Sy5 ¼ 1.A Sy. 1D FEM 3D FEM Discrepancy % Horizontal Coupled Acknowledgment This research was supported by the Ministry of Science, Education and Sport of the Republic of Croatia (Project No ). The authors would like to express their gratitude to Ms. Estelle Stumpf, research engineer at Bureau Veritas, Marine Division Research Department, for generating 3D 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 natural vibrations of a prismatic beam are considered. The governing differential equation of motion is deduced from (1) by neglecting distance between center of gravity and shear center, c ¼ 0, and warping bimoment of inertia, J w ¼ 0, due to reason of simplicity. v 4 j EI t w vx 4 GI v j t t vx þ J0 t v j t vt EI w GI s v 4 j t vx vt! ¼ 0: (A1) Since natural vibrations are harmonic Eq. (A1) leads to the ordinary differential equation d 4 j EI t w dx 4 GI t 1 u Jt 0 EI w d j t GI t GI s dx u Jt 0j t ¼ 0; where j t and u are natural mode and frequency, respectively. Solution of (A) is assumed in the exponential form j t ¼ e ax ; (A) (A3) and by substituting it in (A) one finds the biquadratic characteristic equation Fig. 19. The 5th natural mode of 1D FEM model, effective parameters I Z5 ¼ 0.I Z and A Sy5 ¼ 1.A Sy.

29 I. Senjanović et al. / Marine Structures (009) Fig. 0. Bird view of ship aftbody, the 5th natural mode.

30 416 I. Senjanović et al. / Marine Structures (009) Fig. F1. Shear stress flow due to vertical bending. where a 4 þ ba þ c ¼ 0; b ¼ GI t u Jt 0 EI w EI w 1 GI t GI s ; c ¼ u J 0 t EI w : (A4) (A5) Solutions of (A4) read a j ¼h; ic; (A6) where

31 I. Senjanović et al. / Marine Structures (009) Fig. F. Shear stress flow due to horizontal bending.

32 418 I. Senjanović et al. / Marine Structures (009) Fig. F3. Shear stress flow due to torsion. h ¼ 1 q p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi b 4c b; c ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi b 4c þ b: (A7) Thus, solution of (A) takes the form j t ¼ A 1 shhx þ A chhx þ A 3 sincx þ A 4 coscx: (A8)

33 I. Senjanović et al. / Marine Structures (009) Let us consider vibrations of a free beam of length l, with restrained warping, u ¼ uw; at its ends. The relevant boundary conditions read x ¼l : T ¼ 0; u ¼ 0 (A9) that leads to x ¼l : dj t dx ¼ 0; d 3 j t ¼ 0: (A10) dx3 In the case of symmetric modes A 1 ¼ A 3 ¼ 0; while for anti-symmetric modes A ¼ A 4 ¼ 0. The corresponding eigenvalue problems yield hshhl h 3 shhl csincl c 3 sincl A A 4 ¼f0g; (A11) Fig. F4. Shear stress flow due to restrained warping.

34 40 I. Senjanović et al. / Marine Structures (009) Fig. F5. Normal stress flow due to restrained warping.

35 I. Senjanović et al. / Marine Structures (009) Fig. G1. Bulkhead deflection due to warping of cross-section. h ch hl h 3 ch hl c cos cl c 3 cos cl A1 A 3 ¼f0g: (A1) For nontrivial solutions determinants of (A11) and (A1) have to be zero. That leads to the frequency equations hc h þ c sh hl sin cl ¼ 0 (A13) hc h þ c chhl cos cl ¼ 0 (A14) with the same eigenvalue formula for symmetric (n ¼ 0,.) and anti-symmetric (n ¼ 1, 3.) modes cl ¼ np ; n ¼ 0; 1;. (A15) Substituting (A15) into (A7) for c, one finds the following expression for natural frequencies of torsional vibrations Fig. G. Longitudinal section of container ship hold.

36 4 I. Senjanović et al. / Marine Structures (009) Fig. G3. Watertight bulkhead.

37 I. Senjanović et al. / Marine Structures (009) Fig. G4. Support bulkhead.

38 44 I. Senjanović et al. / Marine Structures (009) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n ¼ np sffiffiffiffiffiffi EIw GI 1 þ np t l GI t l Jt 0 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; n ¼ 0; 1;. (A16) EIw 1 þ np l GI s The first term in the above formula represents natural frequencies of a free beam with free warping. ~u n ¼ np sffiffiffiffiffiffi GI t l Jt 0 : (A17) Fig. G5. Bird view on deformed bulkheads.

39 I. Senjanović et al. / Marine Structures (009) Warping stiffness EI w in the nominator of (A16) increases natural frequencies as a result of restrained warping. Shear stiffness GI s in the denominator reduces natural frequencies since additional twist angle due to shear influence acts as release. Integration constants A and A 4, and A 1 and A 3 are determined from (A11) and (A1), respectively. The symmetric and anti-symmetric modes according to (A8) yield j tn ¼ c n sin c n lchh n x þ h n sh h n l cos c n x; n ¼ 0;. (A18) j tn ¼ c n cos c n lshh n x h n ch h n l sin c n x; n ¼ 1; 3. (A19) Referring to (3), the total twist angle consisted of torsion and shear contribution takes the following form j n ¼ j n ¼ 1 EI w h n GI s 1 EI w h n GI s c n sin c n lchh n x þ 1 þ EI w c n GI s c n cos c n lshh n x 1 EI w c n GI s h n sh h n l cos c n x; n ¼ 0;. (A0) h n ch h n l sin c n x; n ¼ 1; 3. (A1) Appendix B. Consistent finite element properties, according to Eq. (37) (Frequency dependent formulation) Bending-shear stiffness matrix: ½kŠ bs ¼½kŠ b þ½kš s (B1) k b ij ¼ 4EI b l 3 a i a j þ 3 ai a j3 þ a i3 a j þ 3ai3 a j3 (B) k s ij ¼ GA s b l i1 b j1 þ b i1 b j þ b i b j1 þ b i1 b j3 þ b i3 b j1 þ 4 3 b ib j þ 3 9 bi b j3 þ b i3 b j þ 5 b i3b j3 (B3) The warping-shear stiffness matrix ½kŠ ws is of the same constitution as ½kŠ bs, but I b, A s, a ik and b ik have to be replaced with I w, I s, d ik and e ik, respectively, Eqs. (35). Torsional stiffness matrix ½kŠ t is of the same type as ½kŠ s, but A s and b ik have to be replaced with I t and a ik, respectively. Shear-bending mass matrix ½mŠ sb ¼½mŠ s þ½mš b (B4) " m s ij ¼ ml c i0 c j0 þ 1 1 ci1 c j0 þ c i0 c j1 þ ci c 3 j0 þ c i1 c j1 þ c i0 c j þ 1 1 ci0 c 4 j3 þ c i1 c j þ c i c j1 þ c i3 c j0 þ ci1 c 5 j3 þ c i c j þ c i3 c j1 # þ 1 1 ci c 6 j3 þ c i3 c j þ 7 c i3c j3 m b ij ¼ J b a l i1 a j1 þ a i1 a j þ a i a j1 þ a i1 a j3 þ a i3 a j1 þ 4 3 a ia j þ 3 9 ai a j3 þ a i3 a j þ 5 a i3a j3 (B5) (B6)

40 46 I. Senjanović et al. / Marine Structures (009) The torsion-warping mass matrix ½mŠ tw is of the same constitution as ½mŠ sb, but m, J b, c ik, and a ik have to be replaced with J t, J w, f ik and d ik respectively, Eq. (35). Shear-torsion mass matrix m st ij " ¼ mcl c i0 f j0 þ 1 c i1 f j0 þ c i0 f j1 þ 1 c 3 i f j0 þ c i1 f j1 þ c i0 f j þ 1 c 4 i0 f j3 þ c i1 f j þ c i f j1 þ c i3 f j0 þ 1 c 5 i1 f j3 þ c i f j þ c i3 f j1 # þ 1 c 6 i f j3 þ c i3 f j þ 1 7 c i3f j3 (B7) Shear load vector q i ¼ q 0 l c i0 þ 1 c i1 þ 1 3 c i þ c i3 þ q 1 l c i0 þ 1 3 c i1 þ 1 4 c i þ 1 5 c i3 (B8) Torsion load vector m i is of same form as (B8), expressed with m 0, m 1 and f ik, Eqs. (35) and (39), instead of q 0, q 1 and e ik, respectively. In all above formulae indeces i, j take values 1,, 3 and 4. Appendix C. Consistent shape functions of finite element sectional forces Bending moment, Eq. (): d w M i ¼ EI bi b dx Shear force, Eq. (3): dw Q i ¼ GA si s dx ¼ GA s Pure torque, Eq. (7): T ti ¼ GI t dj ti dx D ¼ GI t Warping torque, Eq. (9): dj T wi ¼ GI si s dx ¼ GI s Bimoment, Eq. (8): ¼ EI b D D d ð1þ ik D b ð1þ ik e ð1þ ik a ðþ ik En x k o ; k ¼ 0; 1; ; 3: (C1) En x k o ; k ¼ 0; 1; ; 3: (C) En x k o ; k ¼ 0; 1; ; 3: (C3) En x k o ; k ¼ 0; 1; ; 3: (C4) B wi ¼ EI w d j ti dx D ¼ EI w d ðþ ik En x k o ; k ¼ 0; 1; ; 3: (C5) Vectors of the shape functions, formulae (9) and (3): D E a ðþ ¼ Ca ik l i ; 3a i3 ; 0; 0D D E b ð1þ ¼ 1 ik l Cb i1; b i ; 3b i3 ; 0D D E d ð1þ ¼ 1 ik l Cd i1; d i ; 3d i3 ; 0D D E e ð1þ ¼ 1 ik l Ce i1; e i ; 3e i3 ; 0D D E d ðþ ¼ Cd ik l i ; 3d i3 ; 0; 0D (C6)

41 I. Senjanović et al. / Marine Structures (009) Appendix D. Simplified finite element properties, from Appendix B (Frequency independent formulation) Stiffness matrices: EI ½kŠ bs ¼ b 6 ð1 þ 1bÞl l 6 3l ð1 þ 3bÞl 3l ð1 6bÞl 6 3l Sym: ð1 þ 3bÞl (D1) EI w ½kŠ ws ¼ 6 ð1 þ 1gÞl l 6 3l ð1 þ 3gÞl 3l ð1 6gÞl 6 3l Sym: ð1 þ 3gÞl (D) ð1 60gÞl 36 3ð1 60gÞl GI t ½kŠ t ¼ þ 15g þ 360g l 3ð1 60gÞl 1 þ 60g 70g l 7 30ð1 þ 1gÞ 4 l 36 3ð1 60gÞl Sym: 4 5 (D3) 1 þ 15g þ 360g l Mass matrices: ½mŠ sb ¼½mŠ s þ½mš b (D4) ml ½mŠ s ¼ 40ð1þ1bÞ 3 156þ358bþ0160b þ46bþ50b l 54þ151bþ10080b 13þ378bþ50b l 4þ84bþ504b l 13þ378bþ50b l 3þ84bþ504b l 156þ358bþ0160b þ46bþ50b l 6 4 Sym: 4þ84bþ504b l 7 5 (D5)

42 48 I. Senjanović et al. / Marine Structures (009) ð3 180bÞl 36 ð3 180bÞl J ½mŠ b ¼ b 4þ60bþ1440b l ð 3þ180bÞl 1þ60b 70b l 30ð1þ1bÞ 6 l4 36 ð 3þ180bÞl 7 Sym: 4þ60bþ1440b 5 l (D6) ½mŠ tw ¼½mŠ t þ½mš w (D7) J t l ½mŠ t ¼ 40ð1þ1gÞ 156þ358gþ0160g þ46gþ50g l 54þ151gþ10080g 3 13þ378gþ50g l 4þ84gþ504g l 13þ378gþ50g l 3þ84gþ504g l 6 156þ358gþ0160g 4 þ46gþ50g l Sym: 4þ84gþ504g 7 l 5 (D8) 3 36 ð3 180gÞl 36 ð3 180gÞl J w ½mŠ w ¼ 6 4þ60gþ1440g l ð 3þ180gÞl 1þ60g 70g l 7 30ð1þ1gÞ 4 l 36 ð 3þ180gÞl Sym: 4þ60gþ1440g 5 (D9) l mlc ½mŠ st ¼ 40ð1þ1bÞð1þ1gÞ 3 156þ1764bþ1764gþ0160bg ðþ5bþ10gþ50bgþl 54þ756bþ756gþ10080bg ð13þ168bþ10gþ50bgþl 6 ðþ10bþ5gþ50bgþl ð4þ4bþ4gþ504bgþl ð13þ10bþ168gþ50bgþl ð3þ4bþ4gþ504bgþl þ756bþ756gþ10080bg ð13þ168bþ10gþ50bgþl 156þ1764bþ1764gþ0160bg ðþ5bþ10gþ50bgþl5 (D10) ð13þ10bþ168gþ50bgþl ð3þ4bþ4gþ504bgþl ðþ10bþ5gþ50bgþl ð4þ4bþ4gþ504bgþl ½mŠ ts ¼½mŠ T st Load vectors: þ 10b fqg ¼ q >< >= 0l l q þ 1 l >< >= ð þ 30bÞl ð1 þ 1bÞ 1 þ 40b >: >; >: >; l ð3 þ 30bÞl (D11) (D1) fmg ¼ m 6 9 þ 10g >< >= 0l l m þ 1 l >< >= ð þ 30gÞl ð1 þ 1gÞ 1 þ 40g >: >; >: >; l ð3 þ 30gÞl (D13) Stiffness ratios: