An explicit formulation for restoring stiffness and its performance in ship hydroelasticity

ARTICLE IN PRESS Ocean Engineering 35 (2008) 1322– 1338

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

An explicit formulation for restoring stiffness and its performance in ship hydroelasticity Ivo Senjanovic´ , Marko Tomic´, Stipe Tomasˇevic´ Department of Naval Architecture and Ocean Engineering, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, I. Lucˇic´a 5, 10000 Zagreb, Croatia

a r t i c l e in f o

a b s t r a c t

Article history: Received 1 April 2008 Accepted 11 June 2008 Available online 17 June 2008

The ship hydroelasticity is an issue nowadays due to the fact that very large container ships, which are quite flexible, are being built. Hydroelastic analysis includes definition of geometrical, structural, hydrostatic, hydrodynamic and, finally, hydroelastic model. Within the hydrostatic model, the modal restoring stiffness, well known in literature as hydrostatic stiffness, is formulated. This problem is approached either from hydromechanical or structural point of view, and, perhaps just for this reason, there are still some open questions. This paper offers a unique formulation of the restoring stiffness. It deals with state-of-the-art, definition of the modal restoring stiffness, flexural and torsional stiffness of pseudo-cylinder, beam finite element formulation, asymmetry analysis of restoring stiffness matrix and incorporation of the restoring stiffness into ship hydroelastic analysis. The application of the developed procedure for determining modal restoring stiffness is illustrated in case of a flexible barge. At the end, some interesting aspects of this challenging task, such as the influence of structure deformation on restoring stiffness and asymmetry of its matrix, are further discussed. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Ship hydroelasticity Hydrostatic stiffness Flexible barge FEM

1. Introduction Ship hydroelasticity has been a known issue for many years (Bishop and Price, 1979). However, its significance became more pronounced in recent years due to the fact that very large ships, especially container vessels, are being built, with flexibility as their intrinsic characteristic (Mikkelsen, 2006). The methodology of ship hydroelastic analysis includes definition of structural model, ship structure and cargo mass distribution, geometry of ship wetted surface, calculation of dry natural vibrations, restoring stiffness, added mass, damping and wave loads (Senjanovic´ et al., 2008). In order to determine natural vibrations, one has an option of using either a 1D or a 3D FEM model (Hirdaris et al., 2006). The former is a simpler one and more convenient for preliminary design stage, while the latter is more suitable for final check of the ship strength. Within the context of linearity, the hydroelastic analysis can be performed by Rayleigh’s modal superposition method (Bishop and Price, 1979). Radiation–diffraction hydrodynamic problem is currently solved using either a 2D strip theory (Salvesen et al., 1970) or 3D panel methods (Noblesse, 1982; Malenica, 2003; Hirdaris et al., 2003). The hydrostatic model seems to be simpler than the hydrodynamic one. However, there is still no unique solution to this challenging problem. Also, there are some open questions  Corresponding author. Tel.: +385 1 6168 142; fax: +385 1 6156940.

E-mail address: [email protected] (I. Senjanovic´). 0029-8018/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2008.06.004

concerning the formulation of the restoring stiffness, which consists of hydrostatic part and gravity contribution. This paper tries to shed some light on the physical background of the modal restoring stiffness.

2. State-of-the-art It seems that the determination of the ship restoring stiffness is a more difficult problem than one would expect. Basically, there are two approaches to solve the problem, a hydromechanical one and a structural one. Within the former approach, there are three well-known formulations for the hydrostatic stiffness of a flexible structure. They are presented here in the index (tensor) notation. Price and Wu (1985): ZZ j i C ij ¼ rg h3 hk nk dS, (1) S

Newman (1994): ZZ i i j C ij ¼ rg ðh3 þ Zhl;l Þhk nk dS,

(2)

S

Riggs (1996): ZZ ZZZ i i j rS hi3;k hjk dV, C ij ¼ rg ðh3 þ Zhl;l Þhk nk dS þ g S

V

(3)

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 1322–1338

where Cij is the element of the modal restoring stiffness matrix; h the natural mode (rigid and elastic); X, Y, Z, the global coordinates; S the wetted surface; n the unit normal vector to the wetted surface; V the volume of the structure; r, rS the fluid and structure mass density; and g gravity constant. In the first formulation, Eq. (1), only the basic hydrostatic pressure term is considered. Eq. (2) extends the first one, giving the correct hydrostatic pressure coefficients. However, neither first formulation, nor the second one gives the complete restoring stiffness coefficients for the rigid body motion because the gravity part is missing. The necessity of the gravity contribution was commented in Newman (1994). This is realized in the third formulation, Eq. (3). All three formulations result in an asymmetric stiffness matrix. The formulation of Huang and Riggs (2000), as a more structural approach, results in the following definition of the complete restoring stiffness, also including the structure internal stresses and structure gravity as a special case: ZZ ZZ i j j i j C ij ¼  rg hk ðh3 þ Zhl;l Þnk dS þ rg Zhl hk;l nk dS ZZZ þ

S

S i j lm hk;l hk;m

s

dV.

(4a)

V

In that particular paper, the importance of internal stresses, as a way to complete restoring stiffness, is pointed out. Moreover, the authors claim that the restoring stiffness matrix formulated in such a way has the important symmetry property, as an obvious advantage in the hydroelasticity analysis. The fifth formulation of restoring stiffness, presented in Malenica (2003) and applied in Senjanovic´ et al. (2008), is based on the variational principle and can be presented in the index notation like the others: ZZ ZZ j i j i C ij ¼ rg h3 hk nk dS þ rg Zhl hk;l nk dS S

þ rg

ZZ

S j

j

i

Z½hl;l nk  hl;k nl hk dS

ZZZS rS hi3;k hjk dV. þg

(5)

V

By comparing (4b) and (5), it is obvious that the first integrals (pressure variation) are identical. The second integral (mode variation) of (5) is not taken into account in (4b). The third integral in (5) and the second integral in (4b) are related to the normal vector variation. Their first terms are identical, while there are some differences in the second terms. The volume integral in (5) (gravity force variation) is equal to that in (3), and to the rigid body contribution of the stress integral in (4b). Differences between (4b) and (5) are discussed in Section 12. Formulation (5) is subject to further investigation in this article. Matrix and vector notation are used since the authors believe that it makes the physical background of the governing equations more transparent.

3. Motivation and terminology General opinion is that theory starts where understanding stops. Both are based on knowledge, and intuition plays quite an important role in theoretical consideration. Therefore, searching for the solution of a difficult problem should always be done with as much insight into the physical side as possible. After the solution is achieved by rather complex mathematical procedures, physics should once again be involved to explain its meaning. Resulting from the above observation, the intention of this paper is to shed some light on the restoring stiffness problem and to avoid any doubt in the achieved solution. Therefore, it is the authors’ firm belief that such a consideration should start from the very beginning. For a ship floating in still water, the weight is in equilibrium with buoyancy. An additional external force obviously causes additional buoyancy: F ¼ B ¼ kh w,

S ZZZ rs ðhj rÞhiz dV. þg

(a)

where w is vertical displacement and kh is hydrostatic stiffness. The static equilibrium is used as the referent configuration for a dynamic analysis. An external moment M, acting on the ship in vertical, longitudinal or transverse plane is equalled by the hydrostatic moment Mh and gravity moment Mg: M ¼ M h  M g ¼ ðkh  kg Þc,

Molin (2003) derived the restoring coefficients by employing the vector differential and integral calculus (Kreyszig, 1993), in the following form: ZZ j j i j i i j C ij ¼ rg fðhz þ Z r  h Þh n þ Z½ðh rÞh  ðh rÞh ng dS

1323

(b)

where c is rotation angle, kh and kg are hydrostatic and gravity stiffness, respectively, resulting in the restoring stiffness, kr ¼ kh  kg : Thus, in the general case we are dealing with the restoring stiffness consisting of hydrostatic and gravity contributions. This terminology is used all throughout the paper.

(6)

V

It can be proved that formulations (5) and (6) are identical. The above formulae show the development of the restoring stiffness from the basic hydrostatic part, through an extension by the gravity contribution up to taking the stress distribution into account. In formulation (4a), the orientation of the normal vector is opposite to that in (5). Due to easier correlation of these two formulations, let us change the sign of the normal vector in (4a) and present it in the form similar to Eq. (5): ZZ ZZ j i j i j i C ij ¼ rg h3 hk nk dS þ rg Zðhl;l hk  hk;l hl Þnk dS S

ZZZ þ V

S

slm hik;l hjk;m dV.

(4b)

4. Definition of restoring stiffness 4.1. General Let us first consider a linear single-degree-of-freedom-system. It is well known that force is a product of stiffness K and displacement u: F ¼ Ku.

(7)

Thus, the stiffness can be defined as (Fig. 1) K¼

dF , du

(8)

where dF is a chosen variation, and du is the dependent variation (Fig. 1). Coming back to (7), the force can be presented

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 1322–1338

1324

Fig. 2. Variations of pressure force.

That is a scalar quantity and its variation, similarly to (11) and (13), yields

dF Hi ¼ rg

Fig. 1. Force-displacement relation in 1 DOF system.

ZZ

ðdZhi n þ Z dhi n þ Zhi dnÞ dS.

(15)

S

in another form: F ¼ dF x,

(9)

The variation of a particular quantity can be determined by applying the notion of the directional derivative hjr, where r is Hamilton differential operator:

where dF is relative stiffness and

x¼

u

(10)

du

is relative displacement. The above formulation may seem a nonsense. However, it finds application in both the finite element formulation and the modal formulation of the restoring stiffness. 4.2. Restoring force Restoring force includes gravity force and hydrostatic force. The former is constant, while the latter is variable. Hydrostatic force results from pressure integration over the wetted surface. For a panel of surface S, the force is a vector: ZZ F ¼ rg Zn dS, (11) S

where Z ¼ Z(x, y) is the surface coordinate measured from the waterline and n is the surface unit normal vector. The variation of hydrostatic force is the difference between its instantaneous value and the initial (reference) value (Fig. 2): ZZ ZZ dF ¼ rg Z~ n~ dS~ þ rg Zn dS. (12) S

S

Each of the above quantities can be presented in the form ðg  Þ ¼ ð  Þ þ dð  Þ, where d denotes the variation. By neglecting small terms of higher order, one gets ZZ dF ¼ rg ðdZn þ Z dnÞ dS, (13) s

where according to (9), dF is relative stiffness. The first variation in (13) increases the force amplitude (Fig. 2a), and the second one changes ratios between the force components while keeping its magnitude constant (Fig. 2b). 4.3. Modal hydrostatic forces Now, let us consider the modal force, which represents the work of the actual force on modal displacements, over the whole wetted surface S: ZZ FH Zhi n dS. (14) i ¼ rg S

q j q j q þ hy þ hz . qx qy qz

j

hj r ¼ hx

(16)

As a result, one finds

dZ ¼ ðhj rÞZ ¼ hj k;

dhi ¼ ðhj rÞhi ;

dn ¼ ðhj rÞn.

(17)

Variations dZ and dhi can be determined directly, while determining of dn is a rather difficult task. It is presented in Senjanovic´ et al. (2008) for the case of the parametric definition of wetted surface. If wetted surface is given in the explicit form, as it is conventionally done for ships by the offsets of waterline Y ¼ Yðx; zÞ, the procedure for determining dn is somewhat simpler as shown in Appendix A. Based on analogy with the actual force, Eq. (9), the modal force (14) can be represented as the product of its variation (15) and relative displacement. Since the variation depends on all modes, Eq. (17), the modal force is expressed by series FH i ¼

N X

xj dF Hij ¼

j¼1

N X

xj C Hij ,

(18)

j¼1

where xj are modal displacements (generalized coordinates) and CH ij is a typical element of the modal hydrostatic stiffness matrix. It consists of three parts, i.e., static pressure, surface mode and normal vector contributions, respectively: Hp Hh Hn CH ij ¼ C ij þ C ij þ C ij ,

(19)

where ¼ rg C Hp ij

ZZ

j

i

i

i

hz ðhx nx þ hy ny þ hz nz Þ dS,

(20)

S

C Hh ij

¼ rg

ZZ

" Z

S

!

j hx

i i qhix j qh j qh þ hy x þhz x nx qx qy qz

!

j

i i qhiy j qhy j qhy þ hy þhz ny qx qy qz

j

i i qhiz j qh j qh þ hy z þhz z nz dS, qx qy qz

þ

hx

þ

hx

! #

(21)

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 1322–1338

C Hn ij

ZZ

¼ rg

"

("

#

!

qhjy qhjz qhjy qhj i þ ny  z nz hx nx  qy qz qx qx

Z

S

þ  "

qhjx qhjx qhjz qhj i nx þ þ ny  z nz hy qy qx qz qy j

þ 

Furthermore, the vertical dry modes of the wetted surface, as determined by 1D analysis, yield

#

!

j

hvi ¼ ðZ  zN Þ

! # )

qhy qhix qhjx qhy i nx  ny þ þ nz hz dS. qz qz qx qy

1325

(22)

Based on the constitution of the above coefficients, it is evident that the hydrostatic stiffness matrix is neither diagonal nor symmetric.

dwi i þ wi k, dx

(29)

where wi is the cylinder deflection and zN is the coordinate of the neutral line. i Since in that case nx ¼ 0 and hy ¼ 0 for the cylinder shell, one finds the following expressions for the restoring coefficients: C Hp ij ¼ Ap Iðwi wj Þ;

C Hhn ¼ Ahn Iðwi w00j Þ; ij

0 0 Cm ij ¼ Am Iðwi wj Þ,

(30)

where 4.4. Modal gravity forces

Ap ¼ rg

Z

nz ds,

(31)

s

The modal gravity force is the work of body weight on modal displacements: ZZZ Fm rs hi k dV, (23) i ¼ g where V is the body volume and rs is differential mass density of the structure. Thus, variation of the modal gravity force associated with a particular mode yields ZZZ dF m rs dhi k dV. (24) i ¼ g V

By employing (17) and following (18), one can write for the total force (23): N X

N X

xj dF m ij ¼

j¼1

xj C m ij ,

ZZ

Iðwi wj Þ ¼ Iðw0i w0j Þ

Z

¼ 0

Ahn ¼ 2rg

m C ij ¼ C H ij þ C ij .

(27)

The above expressions for the hydrostatic and gravity coefficients are general and therefore valid not only for the elastic modes but also for the rigid body modes as well as for their coupling. The rigid body modes are special case of elastic modes, and corresponding modal restoring stiffness is condensed to the hydrostatic coefficients (Principles of Naval Architecture, 1988).

Let us consider a pseudo-cylinder with uniform ship-like crosssection. In that case, all surface integrals in the stiffness matrices can be split into product of two separated integrals, one per crosssection contour, s, and another per cylinder length, L: f ðy; zÞ ds

0

wi

d wj dx, dx2

dwi dwj dx. dx dx

(34)

Z

(35) Y WL

ZðZ  zN Þ dy,

gðxÞ dx.

(28)

(36)

0

(37)

(38)

where

Bh ¼ rg

Bn ¼ rg

ZZ

ZZ

ZZ

ðZ  zN Þ dy dz,

(39)

ZðZ  zN Þ2 dy dz,

(40)

Z dy dz,

(41) 2

L 0

2

L

L CH ij ¼ ½ðBp  Bn Þji wj þ Bh ki jj þ Bn wi jj 0 ,

5. Flexural restoring stiffness of pseudo-cylinder

s

Z

where YWL is the waterline offset, m is the mass per unit length and zm is the coordinate of gravity centre at particular crosssection. Since the static pressure also acts on the cylinder heads, it is necessary to determine their contribution to the restoring i stiffness. For the front head nx ¼ 1; ny ¼ 0; nz ¼ 0 and hy ¼ 0: Thus, formulae (20), (21) and (22) are reduced to the same expression for both heads, but with the respective x boundary values:

Bp ¼ rg

Fðx; y; zÞ dS ¼

L

Iðwi w00j Þ ¼

Am ¼ gðzm  zN Þm,

The complete restoring coefficients combine the hydrostatic and the gravity ones:

S

wi wj dx;

Ap ¼ 2rgY WL ,

4.5. Modal restoring stiffness

Z

L

0

Z

V

Z

(33)

(26)

are coefficients of the modal gravity matrix.

ZZ

ðZ  zN Þ dA,

A

(25)

j¼1

! ZZZ i i i j qhz j qhz j qhz þ hy þ hz ¼g rs hx dV qx qy qz

(32)

It is necessary to point out that coefficient Ah is cancelled out with one of two terms in An, forming finally coefficient Ahn. Since natural modes are orthogonal, matrix Iðwi wj Þ is diagonal. That is not valid for the integrals of the mode derivatives. In formula (33), rs is the structure density and A is the crosssection area. Taking the relation nz ds ¼ dy into account, formulae (31), (32) and (33) are presented in a simpler form:

where Cm ij

ZðZ  zN Þnz ds, s

Am ¼ rs g

V

Fm i ¼

Z

Ahn ¼ rg

and j ¼ dw=dx, k ¼ d w=dx2 . Here, the gravity contribution is zero. In (38), the integration notation is used, i.e., H H CH ij ¼ C ij ðLÞ  C ij ð0Þ.

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 1322–1338

1326

Analyzing the constitution of the above coefficients, the following relations are established:

6. Coupled flexural and torsional restoring stiffness of pseudo-cylinder Dry modes of the wetted surface in case of coupled horizontal and torsional vibrations, as determined by 1D analysis, are given in the following form:   dwi dci i hht ¼ Y þ u¯ i þ ½wi þ ðZ  zs Þci j  Y ci k, (42) dx dx where wi and ci are deflection and twist angle, respectively, u¯ ¼ uðX; Y; ZÞ is the cross-section warping function reduced to the ¯ wetted surface, and zs is the coordinate of shear centre (Senjanovic´ and Fan, 1992, 1993). Warping of cross-section is smaller than other displacements and it is neglected in further analysis due to the reason of simplicity. In the considered case nx ¼ 0 for the cylinder shell, so one finds the following coefficients for the restoring stiffness matrix: C Hp ¼ Ap1 Iðci cj Þ þ Ap2 Iðwi cj Þ, ij C Hh ij

An1 ¼ Ah1 ; An4 ¼ Ah3 ;

C ij ¼ ðAp1 þ Am1 ÞIðci cj Þ þ ðAp2  Ah2 ÞIðwi cj Þ 0

þ ðAh2 þ Am2 ÞIðci wj Þ þ ðAh3 þ Am3 ÞIðci w0j Þ þ Ah3 ½Iðci w00j Þ 

þ

0 0 Ah4 Iðci cj Þ,

00

0

0

þ An4 Iðci cj Þ þ An5 Iðci cj Þ þ An6 Iðci cj Þ, Cm ij ¼ Am1 Iðci cj Þ þ Am2 Iðci cj Þ þ Am3 Iðci cj Þ, 0

(44)

þ Bn2 ci jj L0 ,

Z

(50)

0

Bh1 Bh2 (46)

Bn1

Y½ðZ  zs Þny  Ynz  ds s

Yny ds

(47a)

s

are pressure coefficients, Z Z½ðZ  zs Þnz þ Yny  ds Ah1 ¼ rg Zs Znz ds Ah2 ¼ rg Zs YZ½ðZ  zs Þny  Ynz  ds Ah3 ¼ rg Zs YZny ds Ah4 ¼ rg

Bn2

ZZ

Y 2 dy dz, ZZ ¼ rg Y 2 Z dy dz, ZZ ¼ rg Z dy dz, ZZ ¼ rg ZðZ  zs Þ dy dz, ZZ ¼ rg Z dy dz, ZZ ¼ rg ZðZ  zs Þ dy dz.

Bp ¼ rg

Bh3

Ap2 ¼ rg

(49)

CH ij ¼ ½ðBp þ Bh3 Þji cj þ Bh1 ki jj þ Bh2 ji wj þ Bn1 wi jj

(45)

where Z

þ

00 Ah4 Iðwi cj Þ.

For the cylinder front head nx ¼ 1; ny ¼ 0 and nz ¼ 0, so that, according to Eqs. (20)–(22), head stiffness contribution reads

00

Ap1 ¼ rg

0 Iðw0i cj Þ

where 0 0 Ah3 Iðci cj Þ

C Hn ij ¼ An1 Iðci cj Þ þ An2 Iðci cj Þ þ An3 Iðci cj Þ 0

(48)

Thus, the complete restoring stiffness of the cylinder shell is expressed with seven different integrals since some of the terms are summed up and some of them are cancelled:

(43)

¼ Ah1 Iðci cj Þ þ Ah2 Iðci cj Þ þ

0

An2 ¼ Ah2 ; An3 ¼ Ah3 , An5 ¼ Ah4 ; An6 ¼ Ah4 .

(51)

There is no gravity contribution to the head stiffness. The following relationship exists between the above coefficients: Bn1 ¼ Bh2 ;

Bn2 ¼ Bh3 .

(52)

Thus, Eq. (50) for the cylinder heads can also be written in the form: CH ij ¼ ½Bp ji cj þ Bh1 ki jj þ Bh2 ðji wj  wi jj Þ (47b)

þ Bh3 ðji cj  ci jj ÞL0 .

(53)

s

are modal coefficients, Z Z½Yny þ ðZ  zs Þnz  ds An1 ¼ rg Zs Znz ds An2 ¼ rg Zs YZ½Ynz  ðZ  zs Þny  ds An3 ¼ rg Zs YZ½ðZ  zs Þny  Ynz  ds An4 ¼ rg s Z YZny ds An5 ¼ rg Z s YZny ds An6 ¼ rg

7. Finite element formulation of restoring stiffness for vertical vibrations Let us consider the ordinary two nodded beam finite element for flexural vibrations (Senjanovic´ and Grubisˇic´, 1991). The shape (interpolation) functions are given by the Hermitian polynomials of the third order: k

wi ¼ haik ifx g; (47c)

s

are normal vector coefficients, and Am1 ¼ gðzm  zs Þm Am2 ¼ gm Am3 ¼ gJ z

(47d)

are gravity coefficients. Quantity m is mass per unit length, Jz is mass moment of inertia around z axis and zm is coordinate of gravity centre on the considered cross-section.

i ¼ 1; 2; 3; 4;

k ¼ 0; 1; 2; 3,

(54)

where aik are the coefficients, x ¼ x=L is the non-dimensional coordinate and L is the element length. Symbols /?S and {?} denote row and column vectors, respectively. The coefficient matrix reads 2 3 1 0 3 2 6 0 L 2L L 7 6 7 aik ¼ 6 (55) 7. 40 0 3 2 5 0

0

L

L

Shear influence on bending is omitted due to the reason of simplicity. In the finite element formulation, the shape functions are treated as natural modes. Thus, the restoring stiffness matrix

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 1322–1338

coefficients are defined according to (34): Z 1 k k fx ghx i dxfajk g, Iðwi wj Þ ¼ Lhaik i

(56)

0

Iðwi w00j Þ ¼

1 ha i L ik

Iðw0i w0j Þ ¼

1 ha i L ik

Z

*

1

k

fx g 0

Z 1

d

xk dxfajk g, 2 dx

d k x dx

0

+

2



d k x dx



(57)

(58)

dxfajk g.

Finally, by employing (30) one obtains 2 3 156 22L 54 13L 6 7 4L2 13L 3L2 7 Ap L 6 6 7, CHp ¼ 156 22L 7 420 6 4 5 Sym: 4L2 2 CHhn ¼ 

36

ð33LÞ

36

4L2

3L

Ahn 6 6 3L 6 30L 4 36 3L 2

6 Am 6 6 Cm ¼ 30L 6 4

36

3L

36

L2

3L

3L

36

4L2

3L 36

Sym:

3L

3

3L

7 7 7, ð33LÞ 5 L2

(60)



4L2

3

7 L2 7 7. 3L 7 5 4L2

0

Iðwi cj Þ; Iðw0i cj Þ,

torsion2bending :

Iðci wj Þ; Iðci w0j Þ; Iðci w00j Þ.

0

(65)

Separation has to be done for the head stiffness, too. Taking (50) into consideration and following analogy with bending, one finds: 

D

E

F

0

 ,

(66)

where D, E, and F are block matrices of type [4  4]:

1

w2 > > > > > > > ; :j > 2

0

6 6 Bh2 6 D¼6 6 0 4 0

 L62 Bh1

0

4 L Bh1  L62 Bh1 2 L Bh1

0 0 Bh2

6 B L2 h1 2 B L h1  L62 Bh1 4 L Bh1

3 7 7 7 7. 7 5

(67)

The non-zero elements of the other two block matrices E and F are the following, respectively: e21 ¼ Bp  Bh3 ;

f 12 ¼ Bn2 ;

" C¼

The shape functions of beam bending are specified with Eq. (54). For the beam torsion with warping of cross-section, the same shape functions are used (Senjanovic´ and Grubisˇic´, 1991). Thus, all integrals in stiffness matrices CHp, CHh, CHn and Cm represented by Eqs. (43)–(46), respectively, take one of the forms (59), (60) and (61) derived for bending. The finite element equation for the coupled beam vibrations reads (Senjanovic´ and Grubisˇic´, 1991): #  " #( ) ( )   " € q mb mbt kb P U U ¼ þ , (63)  € l mtb mt kt R V V 9 8 w1 > > > > > > > =

2

e43 ¼ Bp þ Bh3 ,

f 34 ¼ Bn2 .

(68)

(69)

Finally, the complete restoring stiffness matrix takes the following form:

8. Finite element formulation of restoring stiffness for coupled horizontal and torsional vibrations

U¼

Iðci cj Þ,

bending2torsion :

(61)

It is obvious that the matrix is asymmetric.

9 8 T 1 > > > > > = < B1 > R¼ ; T > > > > > 2 > ; : B2

Iðw0i w0j Þ; Iðwi w00j Þ,

torsion :

CH Head ¼

The constitution of matrices CHp and Cm is the same as those for the beam mass matrix and the mass rotation matrix, respectively (Senjanovic´ and Grubisˇic´, 1991). This is obvious since both represent load, the former two static and the latter two dynamic load. Matrix CHhn is similar to Cm. Its asymmetry, caused by the elements (733L)* in (60), in the amount of (730L)*, is analyzed in Section 9. The hydrostatic stiffness matrix for the cylinder heads, Eq. (38), takes the following form: 2 3 6 0 Bn  L62 Bh 0 B L2 h 6 7 4 2 6 Bn  Bp 7 0 L Bh L Bh 6 7 H CHead ¼ 6 (62) 7. 6 6 6 0  L2 Bh 0 Bn  L2 Bh 7 4 5 2 4 0 Bp  Bn L Bh L Bh

9 8 Q 1 > > > > > = < M1 > ; P¼ Q2 > > > > > > ; : M 2

where Q is the shear force, w the deflection, M the bending moment, j the cross-section rotation angle, T the torque, c the twist angle, B the warping bimoment, and W the variation of twist angle. The stiffness matrix consists of the bending stiffness and torsional stiffness, kb and kt, respectively. Mass matrix also includes the coupling submatrices mbt and mtb ¼ mTbt . By following constitution of the structural stiffness and mass matrices, it is necessary to split the restoring matrix into bending, torsional and coupling bending–torsion and torsion–bending parts. These four groups are separated with respect to the integrands: bending :

(59)

1327

;

V¼

9 8 c > > > > > 1> > = < W1 > > c2 > > > > > > ; :W >

,

2

(64)

Cb

Cbt

Ctb

Ct

# .

(70)

It is obvious that the coupling of horizontal and torsional vibrations is realized through the restoring stiffness, too. 9. Asymmetry analysis of restoring stiffness By observing formulae (30) for restoring coefficients due to bending, it is obvious that the restoring matrix depends not only on the modal displacement w and cross-section rotation j ¼ dw=dx, but also on the modal curvature k ¼ d2 w=dx2 . However, this is only the first impression. The factors in coefficients I(wiwj) and Iðw0i w0j Þ are commutative and therefore the corresponding matrices are symmetrical, Eqs. (59) and (61). However, the factors in Iðwi w00j Þ are not commutative and the matrix is asymmetric, Eq. (60). The asymmetry is especially pronounced in the case of combined rigid body and elastic modes.

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By applying the integration by parts for a typical coefficient of the asymmetric integral, one can write: Z L dwi dwj dx. (71) Iðwi w00j Þ ¼ ðwi jj ÞL0  0 dx dx In this way, the asymmetric coefficient Iðwi w00j Þ is split into the symmetric integral, the same as in Iðw0i w0j Þ, and the symmetry disturbing part ðwi jj ÞL0 , which is pushed to the beam boundaries. At the same time, we see that the restoring stiffness is not dependent on the curvature any more. The symmetry disturbing term ðwi jj ÞL0 is zero for a clamped beam, but that is not the case for the freely floating ship-like structures. In order to thoroughly investigate the asymmetry of that term, let us consider two rigid body modes and two elastic modes of vertical vibrations (Fig. 3). The mode sample is sufficient since it includes symmetric and anti-symmetric modes. Mode number n is related to the vibration node number. The boundary deflections, wn ð0Þ, wn ðLÞ and angles of rotation, jn ð0Þ, jn ðLÞ, are also specified in Fig. 3. The matrix of the boundary values ðwi jj ÞL0 takes the following form: 2 3 2a0 0 2a2 0 : 6 0 0 2a3 : 7 2a1 6 7 6 7 L 6 0 2a2 0 :7 (72) ½wi jj 0 ¼ 6 2a0 7, 6 7 0 2a3 :5 2a1 4 0 : : : : : where 7an are the boundary values of the rotation angle as specified in Fig. 3. It is obvious that the matrix is not symmetric. As a measure of the matrix asymmetry, the ratio between the upper and lower elements with respect to the main diagonal can be introduced. By applying the formulae for symmetric and antisymmetric modes of prismatic beam (Section 11.2) yields, respectively:

n ¼

anþ2 bnþ2 ½thðbnþ2 lÞ  tgðbnþ2 lÞ ; ¼ an bn ½thðbn lÞ  tgðbn lÞ

n ¼

anþ2 bnþ2 ½cthðbnþ2 lÞ þ ctgðbnþ2 lÞ ; ¼ an bn ½cthðbn lÞ þ ctgðbn lÞ

n ¼ 0; 2; 4; . . .

n ¼ 1; 3; 5; . . .

(73)

(74)

Fig. 4. Assembly in global stiffness matrix.

If en ¼ 1, the matrix is symmetric; while if en ¼ N, the asymmetry is at its extreme. It is obvious that

0 ¼

a2 ¼ 1; a0

lim n ¼ 1,

n!1

(75)

so that the asymmetry of matrix (72) is reduced starting from N to 1 for higher modes. However, in the hydroelastic analysis only the first few natural modes, which considerably contribute to the asymmetry, are of interest. In the case of the finite element application, the global restoring matrix consists of the element restoring matrices according to the scheme shown in Fig. 4. If the structure is prismatic it is modelled by the same type of finite elements and it is obvious that the asymmetric matrix elements are cancelled out and that only the matrix elements in the boundary remain (Fig. 4). They arise from a combination of the deflection and rotation angle shape functions. So, the same effect as in the analytical consideration, ðwi jj ÞL0 in Eq. (71) is manifested in the finite element formulation, which is represented by the term (733L)* in Eq. (60). The symmetry of the restoring matrix is also disarranged by the head contribution, Eq. (62). As result of the above analysis, the complete cylinder restoring matrix can be presented in the following form, summing up Eqs. (30) and (38), and utilizing (71): C ¼ Ap ½Iðwi wj Þ þ ðAm  Ahn Þ½Iðw0i w0j Þ þ Ahn ½wi jj L0 þ ½CH Head ,

(76)

where the last two matrices are asymmetric.

10. Incorporating restoring stiffness into ship hydroelastic model

Fig. 3. Rigid body and elastic natural flexural modes.

The restoring stiffness can be incorporated in the static and dynamic ship analysis in different ways, as described in Appendix B. It can be included in hydroelastic model at the very beginning, i.e., at the level of finite elements or later on in the modal synthesis. The latter is preferable, since the hydrostatic pressure is treated in the same way as hydrodynamic pressure and damping. Dry natural modes can be determined by 1D or 3D FEM analysis. From the global point of view, it does not matter which structural model is used. An advantage of the 3D structural model, when compared to the 1D model, is that not only the global response of ship hull is obtained, but also the substructure response and the local effects, depending on the finite element mesh density. In both approaches, natural modes of the wetted surface are defined in order to determine the restoring, inertia and damping

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forces (Figs. 5 and 6). For the latter two, the 2D strip theory or 3D radiation–diffraction theory can be applied. If restoring stiffness is incorporated in 1D model at the very beginning, i.e., at the finite element level, then some correlation to the hydrodynamic strip model can be drawn. The strip model consists of a number of pseudo-cylinders, each with a different but uniform cross-section (Fig. 7). Hydrodynamic pressure is calculated for each cylinder assuming its infinite length, as a 2D problem. However, in hydrostatic analysis, pressure forces are determined not only on the cylinder bottom and side shell but also on its heads (Fig. 8). Thus, the finite elements are axially loaded proportionally to the cylinder cross-section area. After the finite elements are assembled, the axial forces on the union of the two joined cross-sections are cancelled, and only the axial force due to the pressure acting on the wetted part of the cross-section remains (Fig. 8). Fig. 8. Assembly of pseudo-cylinders.

11. Illustrative example 11.1. Barge particulars The application of the developed method for determining the modal restoring stiffness for vertical and coupled horizontal and torsional vibrations (Sections 5 and 6) is illustrated in case of a flexible barge consisting of 12 equal pontoons (Remy et al., 2006). The pontoons are connected by means of a steel rod somewhat above the deck level, as shown in Fig. 9. In the hydrostatic analysis, the barge is treated as a monohull with the following main characteristics: Fig. 5. The first torsional mode of the wetted surface, 1D FEM model of 11400 TEU container ship.

Young’s modulus of rod: Shear modulus of rod: Moment of inertia of rod cross-section:

E ¼ 2.1 1011 N/m2 G ¼ 0.808  1011 N/m2

Polar moment of inertia of rod crosssection: Bending stiffness of rod: Torsional stiffness of rod: Pontoon length Barge length (pontoons+clearances): Total mass (pontoons+equipment): Distributed mass: Radius of gyration in roll: Polar moment of inertia of distributed mass: Distance of gravity centre from torsional centre: Polar mass moment of inertia about torsional centre: Radius of inertia:

It ¼ ða4 =6Þ ¼ 16:67  1010 m4

Iy ¼ Iz ¼ ða4 =12Þ ¼ 8:33  1010 m4

EI ¼ 175 Nm2 GIt ¼ 135 Nm2 l ¼ 0.19 m L ¼ 2.445 m M ¼ 171.77 kg m ¼ M/L ¼ 70.253 kg/m ix ¼ 0.225 m J 0t ¼ mi2x ¼ 3:556 kg m c ¼ 0.144m J t ¼ J 0t þ mc2 ¼ 5:013 kg m r ¼ ðJ t =mÞ1=2 ¼ 0:267 m

Fig. 6. The first torsional mode of the 3D FEM model of 11400 TEU container ship.

The barge longitudinal section, with the coordinates of the actual points for restoring stiffness determination, is shown in Fig. 10. 11.2. Natural flexural vibrations

Fig. 7. Strip model of hydrodynamic analysis (A new generation STENA tanker with two engine rooms).

Natural vibrations for a prismatic pontoon can be determined analytically, as elaborated in Senjanovic´ et al. (2007a). The origin of the coordinate system is located in the middle of the length. Thus, lpxpl, where l ¼ L/2. Symmetric modes:   1 chðbn xÞ cosðbn xÞ wn ¼ þ ; n ¼ 0; 2; 4; . . . 2 chðbn lÞ cosðbn lÞ

b0 l ¼ 0;

b2 l ¼ 2:365;

b4 l ¼ 5:497

(77)

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Heads: C Hp 33

¼ C Hhn 33 ¼ 0.

(82)

It is obvious that the total restoring coefficient C 33 ¼ rgLB is equal to that determined by the ship hydrostatics, since LB is waterplane area AWL. The pitch mode yields: w5 ¼ x; j5 ¼ 1, where lpxpl. The corresponding coefficients are the following: Bottom: C Hp 55 ¼ rg

BL3 12

C Hhn 55 ¼ 0

Fig. 9. Barge cross-section.

Cm 55 ¼ rgLBTðzN  zG Þ

(83)

Heads:

T ¼  r gLBT z þ C Hp N 55 2 LBT 2 2 LBT 2 . ¼ rg 2

C Hh 55 ¼ rg C Hn 55

(84)

Hn It is obvious that C Hh 55 and C 55 are cancelled. 3 Since ðBL =12Þ ¼ IWLY is the longitudinal moment of inertia of waterplane area and V ¼ LBT is the displaced volume, the restoring coefficient yields 

 T þ zG . (85) C 55 ¼ rg IWLY  V 2

It is identical to the hydrostatic expression, since the coordinate of the centre of buoyancy reads zB ¼ T=2. The coordinate of neutral line, zN, disappears in C55 and that is physically correct for the rigid body modes. Concerning coefficients of mixed modes, w3 and w5, their values are zero since the modes are orthogonal. As a result, the corresponding restoring coefficients yield C 35 ¼ C 53 ¼ 0.

Fig. 10. Barge longitudinal section.

Anti-symmetric modes:   1 shðbn xÞ sinðbn xÞ þ ; wn ¼ 2 shðbn lÞ sinðbn lÞ

b1 l ¼ 0;

b3 l ¼ 3:925;

lim bn l ¼

2n þ 1 p, 4

n!1

n ¼ 1; 3; 5; . . .

b5 l ¼ 7:068

(78) (79)

on ¼

l

2

rffiffiffiffiffi EI m

(80)

11.3. Rigid body stiffness coefficients The rigid body modes are denoted with index i ¼ 1; 2; . . . ; 6 and the elastic modes with i ¼ 7; 8; . . . ; in accordance with the usual hydrodynamics notation. Only static pressure acting on the barge bottom and the barge front and aft heads is relevant for the barge hydrostatic stiffness in the case of vertical vibrations. The pressure forces on the barge sides and adjacent pontoon heads are in equilibrium and therefore cancelled. For the bottom panels, the normal vector is n ¼ k, while for the aft and fore head panels n ¼ 7i, respectively. The origin of the coordinate system is located at the waterplane below the centre of gravity (Fig. 10). For heave, where natural mode is w3 ¼ 1; j3 ¼ 0, one finds: Bottom: m C Hhn 33 ¼ C 33 ¼ 0

i ¼ 7; 8; . . .

(86)

while at the level of the centre of gravity one finds

is the natural frequency of the nth mode.

C Hp 33 ¼ rgLB;

According to (29), the bottom elastic mode yields hiK ¼ ji ðzN þ TÞi þ wi k;

where ðbn lÞ2

11.4. Stiffness coefficients of the elastic modes

(81)

hiG ¼ ji ðzN  zG Þi þ wi k;

i ¼ 7; 8; . . .

(87)

where wi is the barge deflection and ji is the rotation angle of cross-section. By employing expressions for hydrostatic stiffness and for gravity contribution from Section 5, the following formulae for the elements of the restoring matrix are derived, where i; j ¼ 7; 8; . . . Pressure: C Hp ¼ rgB ij

Z

l

wi wj dx.

(88)

l

Modes and normal vector: ¼ rgBTðzN þ TÞ C Hhn ij

Z

l

wi kj dx.

(89)

l

Gravity: Cm ij ¼ rgBTðzN  zG Þ where l ¼ L/2.

Z

l

l

ji jj dx,

(90)

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 1322–1338

For the barge heads, according to (38), one finds: C Hp ij

¼ r p ½ji ðlÞwj ðlÞ  ji ðlÞwj ðlÞ,

(91)

C Hh ij ¼ r h1 fr h2 ½ki ðlÞjj ðlÞ  ki ðlÞjj ðlÞ  ½ji ðlÞwj ðlÞ  ji ðlÞwj ðlÞg,

(92)

C Hn ij ¼ r n ½wi ðlÞjj ðlÞ  wi ðlÞjj ðlÞ,

1331

that renders analytical approach very useful. Further step is to substitute expressions (77) and (78) for vibration modes into formulae (88) up to (93) and then to solve the modal integrals. Again, it is possible to do that analytically as shown in Senjanovic´ et al. (2007a). However, nothing special is obtained by that step. Therefore, we take another view of this problem from the numerical side (Hydrostar, 2006; Tomasˇevic´, 2007).

(93) 11.5. Numerical analysis of restoring stiffness

where

T , r p ¼ rgBT zN þ 2 BT 2 ; 2 2 BT . r n ¼ rg 2

r h1 ¼ rg

r h2 ¼

T2 4 þ zN T þ z2N , 3 2 (94)

The above formulae, derived for elastic modes, are general and therefore applicable for rigid body modes, as well as for their combined products. The above formulae are the range of the analytical procedure for determining stiffness coefficients. Their constitution makes it possible to see the influence of each barge particular, something

Table 1 Modal parameters Frequency, o (rad/s)

Mode no. Vertical/horizontal 1 2 3 4 5

Mass, m

vibrations 5.91 16.28 31.91 52.75 78.79

Torsional vibrations 1 6.67 2 13.34 3 20.03 4 26.74 5 33.48 Coupled H and T vibrations 1 5.73 2 7.89 3 14.40 4 16.13 5 23.63

Stiffness, k

42.92 42.94 42.95 42.95 42.96

1498 11,383 43,745 119,521 266,663

6.12 6.11 6.09 6.06 6.03

272 1088 2445 4337 6756

44.68 2635.30 58.24 384.36 115.40

1466 163,864 12,075 100,305 64,446

As a result of the numerical calculation, Table 1 shows natural frequencies o, modal mass m and modal stiffness k for the first five natural modes of vertical, horizontal, torsional and coupled horizontal and torsional dry vibrations. Vertical and horizontal vibrations are identical due to the same rod flexural bending stiffness in both directions. Modal parameters m and k are determined for the normalized modes so that maximum deflection and twist angle in the uncoupled vibrations take unit value (Senjanovic´ et al., 2007a). Since the mass distribution is uniform, modal masses are constant, while the modal stiffness is rapidly increased. The natural modes of coupled horizontal and torsional vibrations are normalized in such a way that maximum barge deflection at the level of gravity centre is unit (Senjanovic´ et al., 2007a). However, this normalization does not help to ensure uniform variation of modal parameters m and k. The computed modal restoring stiffness matrix is shown in Table 2. The first submatrix of the type [6  6] belongs to the rigid body modes. The next two submatrices of the type [5  5], on the main diagonal, are the result of the vertical, and coupled horizontal and torsional vibration modes, respectively. The remaining out of diagonal submatrices are generated by products of rigid body and elastic modes. The null sub matrices show that there is no interference between vertical and coupled horizontal and torsional vibrations. It is obvious that the restoring stiffness matrix is positively definite. Because the values of the modal masses of vertical vibrations are constant, it is possible to analyze the constitution of the stiffness matrix. Its diagonal elements grow up and the asymmetry is noticeable. Since constant modal mass is not achieved by normalization of coupled modes, the diagonal elements in the restoring matrix for coupled horizontal and torsional vibrations take jumping values. Therefore, it is not possible to say anything about the level of matrix asymmetry. In the coupled vibrations, there is no uniformity.

Table 2 Modal restoring stiffness 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

0.00 0.00 0.00 0.00 0.04 0.00 0.06 0.02 0.18 0.75 1.39 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.06 1.17 0.15 0.47 0.34 0.00 0.00 14391.27 0.00 0.05 0.00 664.44 2.72 1564.62 30.56 2412.44 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 258.28 0.00 48.57 0.00 0.00 0.00 0.00 0.00 115.49 106.19 43.42 179.08 155.32 0.00 0.00 0.03 0.00 6993.42 0.00 24.84 1139.09 66.84 2178.19 155.79 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.83 0.31 0.24 1.20 80.50 0.00 825.42 0.00 13.68 0.00 3746.22 75.97 1147.41 156.05 1910.16 0.00 0.00 0.00 0.00 0.00 107.09 0.00 4.02 0.00 1399.26 0.00 79.62 4726.75 191.17 1312.38 323.24 0.00 0.00 0.00 0.00 0.00 211.12 0.00 1940.45 0.00 48.83 0.00 1351.09 206.92 6387.51 365.18 1461.09 0.00 0.00 0.00 0.00 0.00 267.67 0.00 43.51 0.00 2698.18 0.00 196.64 1475.89 406.58 8764.37 612.07 0.00 0.00 0.00 0.00 0.00 411.13 0.00 2983.64 0.00 159.83 0.00 2259.07 435.79 1551.44 697.78 11864.34 0.00 0.00 0.00 0.00 0.00 0.00 140.60 0.00 59.57 0.00 15.08 0.00 0.00 0.00 0.00 0.00 86.63 29.63 12.66 32.58 51.46 0.00 1013.88 0.00 138.17 0.00 1731.91 0.00 0.00 0.00 0.00 0.00 927.90 73570.27 4878.78 472.83 1289.61 0.00 68.59 0.00 8.94 0.00 70.81 0.00 0.00 0.00 0.00 0.00 61.22 3055.25 562.78 31.38 79.02 0.00 755.77 0.00 100.07 0.00 411.61 0.00 0.00 0.00 0.00 0.00 1135.09 917.02 361.67 9497.88 2336.59 0.00 39.47 0.00 98.11 0.00 185.11 0.00 0.00 0.00 0.00 0.00 333.62 410.61 153.59 1206.95 1920.31

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Table 3 Restoring stiffness contribution to barge stiffness Mode no.

Vertical vibrations

Coupled H and T vibrations

Cii/ki

Cii/ki

Heave Pitch Roll

N N

1 2 3 4 5

2.501 0.415 0.146 0.073 0.044

Dominant mode

N 0.059 0.449 0.047 0.095 0.030

H1+(T0) T1+(H0) H2+(T3) T2+(H1) H3+(T4)

In further consideration, it is interesting to investigate the contribution of the restoring stiffness to the total stiffness. For this purpose, the ratio between diagonal restoring matrix elements Cii, which are dominant, and the structure modal stiffness ki can be used as an indicator. The obtained results are presented in Table 3. The restoring stiffness is rather high for rigid body motion, while the structural stiffness is zero. For the first vertical elastic mode, the restoring stiffness is quite high with respect to the structural stiffness, and this ratio gets rapidly reduced for the higher elastic modes. Concerning the coupled vibrations, it is necessary to separately consider primary horizontal and torsional modes. In this context, both corresponding sets decrease uniformly in the higher modes. Values of the dominant horizontal modes are lower than those of torsional modes, since the uncoupled horizontal modes do not cause hydrostatic reaction.

12. Observation and discussion The restoring stiffness is a very important parameter in the ship hydroelasticity analysis. It combines the hydrostatic part with the gravity contribution. Therefore, the expression restoring stiffness is used throughout the paper instead of hydrostatic stiffness which is more common in the literature. Different formulation of the restoring stiffness motivated a detailed investigation of this interesting problem. We started with the definition of the actual stiffness and relative stiffness, as well as the actual force and the modal force. These notions have to be distinguished. Since a modal force represents work of the actual force on modal displacement (a scalar value), it is equal to the product of the relative modal stiffness and relative mode amplitude. The mode amplitude is actually an unknown weighting coefficient in the modal superposition method, which determines participation of each dry natural mode in the wet natural vibrations and forced response of the elastic floating structure. If the assumed modes have higher amplitudes the modal stiffness is higher while the weighting coefficients will take proportionally lower values and vice versa. This is the explanation why the modal restoring stiffness is relative. In a straight-forward formulation of the modal restoring stiffness, there is doubt whether it is necessary to take the mode variation into account, together with the pressure, normal vector and gravity parts, or not. The reason for this doubt lies in the fact that the modal force with respect to the still water (reference configuration) already seems to be a variation. The necessity for the mode variation is made clear within the detailed step-by-step formulation of the modal restoring stiffness in Section 4. Without the mode variation, the correlation between numerical results and model tests of a flexible barge would not be so good (Malenica

et al., 2003; Remy et al., 2006; Tomasˇevic´, 2007; Senjanovic´ et al., 2007a). In one of the future papers, it will be shown that asymmetric rigid body terms of matrices C Hp and C m ij are ij Hh cancelled by those of C Hn and C , respectively, resulting in ij ij such a way in the expected diagonal rigid body restoring stiffness matrix. The restoring stiffness is defined in accordance with Malenica (2003). It is further applied to a 3D model of ship-wetted surface, determined by the dry beam natural modes of ship hull. In the next steps, the modal restoring stiffness is incorporated in the modal superposition formulation for solving ship response. The beam finite elements of restoring stiffness for vertical as well as for coupled horizontal and torsional vibrations are developed to analyze the influence of the involved ship parameters on that stiffness. The beam model also makes it possible to analyze the asymmetry of the restoring stiffness. Huang and Riggs (2000) tries to offer a consistent definition of the restoring stiffness based on a sophisticated structural approach. As noticed in Section 2, the pressure variation in Eqs. (4a) and (5) are identical. The mode variation is not taken into account in (4a). The normal vector variation in (4a) is of the following form: ZZ rg Z½hjl;l hik  hjk;l hil nk dS S

¼ rg "

ZZ S

(" Z

#

!

qhjy qhjz i qhjx i qhjx i þ h  h n h  qy qz x qy y qz z x #

!

j qhjy i qhjx qhjz i qhy i hx þ þ hy  h n þ  qx qx qz qz z y

" þ 

j

! # )

qhjz i qhjz i qhjx qhy i h  h þ þ h n dS. qx x qy y qx qy z z

(95)

It is obvious that Eqs. (95) and (22) are identical, but presented in different forms. Thus, the integrand functions, due to the normal vector variation in formulations Eqs. (4a) and (5), are commutative with respect to the subscripts. This fact leads to the following identity: ZZ ZZ rg Z½hjl;l nk  hjl;k nil hik dS ¼ rg Z½hjl;l hik  hjk;l hil nk dS. (96) S

S

The contribution of the normal vector variation to the restoring stiffness specified in form (95) is more preferable than (22), because it gives the modal products in the explicit form similar to the mode variation, Eq. (21). The structural term in Huang and Riggs (2000) takes internal forces into account. They consist of the gravity forces and membrane forces (stresses) in the ship structural elements due to hull deformation in still water. Contributions of the gravity forces to the restoring stiffness in (4a) and (5) are the same. The membrane forces are used for formulating the geometric stiffness matrix of ship structure, which has to be considered in the context of complete ship structure stiffness. The geometric stiffness has rather small influence on the natural ship structure vibrations, so that it can be neglected. It is an influencing parameter in the slender offshore structure vibrations, as demonstrated in Appendix C. In the ship strength analysis, we distinguish static and dynamic loads and the restoring stiffness can be taken into account in different ways, Appendix B. The static and dynamic calculations are performed separately and the results are superimposed. In dynamic analysis, the ship static equilibrium condition in still water is the referent state. The static deformation of the ship-wetted surface has repercussions on the restoring stiffness. This effect can be included in the analysis as an initial

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 1322–1338

deformation (imperfection) of the wetted surface. In that case, the deformed wetted surface has to be taken into account in the governing formulae for restoring stiffness instead of the designed form. However, there is no sense in doing it since, on the other hand, linear hydrodynamic theory does not take the wave profile into account, which is much higher than the ship static deformation. Furthermore, the hydrostatic pressure and the hydrodynamic pressure, are both types of load and should be treated in the same way. The second step in hydroelastic analysis is to determine the structure strength. It depends on the chosen mesh density of FEM model and the number of involved natural modes, whether the substructure and structural element strength will be covered or not. For example, if 1D FEM model is to be applied, only sectional forces will be obtained. Since the calculation of stress concentrations used in fatigue analysis requires very fine FEM models, the application of substructure technique is necessary. Needed boundary conditions for submodels can be determined both by 1D FEM model and 3D global FEM model. In Huang and Riggs (2000), it is proved, by employing Stokes’ theorem, that the restoring stiffness matrix is symmetrical. However, asymmetry analysis of the beam restoring stiffness matrix performed in Section 9 leads to the opposite conclusion. This is confirmed by illustrative numerical example, in which asymmetry is rather pronounced. The asymmetry of the restoring stiffness matrix can cause some non-linear effect, which is expected to be negligible in domain of small amplitudes. The dynamic system is conservative, i.e., there is no loss of potential energy due to asymmetry.

13. Complete restoring stiffness formulation for floating structures After the present restoring stiffness formulations have been analyzed, the following improved and complete expression can be specified for ships and monohull offshore structures: Hh Hn m 0 C ij ¼ C Hp ij þ C ij þ C ij þ C ij þ C ij ,

(97)

where C Hp ¼ rg ij C Hh ij C Hn ij

¼ rg ¼ rg

ZZ S ZZ S ZZ

j

i

h3 hk nk dS j i

Zhl hk;l nk dS j

i

j

i

Z½hl;l hk  hk;l hl nk dS

S

Cm ij

ZZZ ¼g rs hi3;k hjk dV

C 0ij ¼

ZZZV

s0kk ½hik;l hjk;l  hik;k hjk;k  dV

(98)

V

and further

s011 ¼ s022 ¼ rgZ s033 ¼

P  rgðZ  Z b Þ þ g Az

Z

Z

Zb

rs dz.

(99)

Formulae (98) represent the restoring stiffness contribution from the hydrostatic pressure, C Hp , natural mode, C Hh ij , normal vector, ij Hn m C ij , gravity, C ij , and the hydrostatic (global) geometric stiffness, C 0ij . The last one is caused by the structure tension or compression and takes a respective value only for slender structures, while it is negligible for the ship hull.

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The restoring stiffness can be completely calculated as a statically determined task. The above formulation is applicable to both 1D and 3D FEM structural models. If the geometry of an offshore structure is complex, as for instance semi-submersible platforms, it is not possible to extract the hydrostatic geometric stiffness, C 0ij , and the general geometric stiffness should be determined for a structure in still water as a structural problem and added to the ordinary stiffness for calculation of dry natural modes, Appendix C.

14. Conclusion Ship vibrations caused by the main engine and propeller are of higher frequency and amplitude usually within 1 mm. Therefore, restoring stiffness is neglected in the analysis of ship vibrations caused by internal excitations. However, large container ship response to wave loading is in the lower frequency domain, with rigid body displacement of several meters and elastic mode amplitude of the order of 1 m. In this case, the restoring stiffness plays a very important role and therefore has to be determined as accurately as possible. By condensing the modal restoring stiffness to beam model, many physical aspects become clearer, as for instance the physical meaning of stiffness formulation, existence of asymmetry (which is transferred to the model ends and which depends on the boundary conditions), etc. The development of the beam finite elements with restoring properties also helps to see the influence of different ship parameters. However, the most convenient way is to incorporate the restoring stiffness directly in the modal dynamic model, as it is usually done with wave loads and damping. Huang and Riggs (2000) seems to offer the best formulation of the restoring stiffness up to now. However, the analysis of the formulation given in Malenica (2003) shows that the mode variation is missing in Huang and Riggs (2000) and that the restoring stiffness matrix is not symmetric. The asymmetry is especially pronounced in coupling of the rigid body and elastic modes. Also, it is shown that geometric stiffness, as result of a prestress field due to ship deformation in still water, is negligible for ship structures. It is an influential parameter only for slender offshore structures. The modal restoring stiffness presented in Malenica (2003) for ship structures and analyzed in this paper is quite reliable. This is confirmed by the correlation analysis of the calculated and the model test results for a very flexible barge, for which rigid body and elastic response are of the same order of magnitude and deflection follows more or less the wave profile (Malenica, 2003; Remy et al., 2006; Tomasˇevic´, 2007; Senjanovic´ et al., 2008). After such validation, the formulated restoring stiffness, Eq. (99), can be successfully applied in the hydroelastic analysis of very large container ships. Container ships are slender vessels with large deck openings and the effect of hydroelasticity is very important. Along these lines the development of software tools and design assessment procedures by classification societies for the evaluation of the effects of hull flexibility on dynamic loads is recommended. Such design practice would enhance the safety margin in the design of this type of ships. In future, the authors intend to do more work in an attempt to clarify this point in further. They will concentrate on investigating the usefulness of 1D (beam-like) approaches coupling beam dynamics with hydrodynamic theory for hull girder strength and fatigue analyses. In order to verify the results of the theoretical investigation, it is necessary to correlate them with model tests and full-scale measurements.

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Appendix A. Variation of body surface normal vector Body surface can be represented in one of the following vectorial forms (Kreyszig, 1993): r ¼ Xðy; zÞi þ yj þ zk, r ¼ xi þ Yðx; zÞj þ zk, r ¼ xi þ yj þ Zðx; yÞk,

(A.1)

where the second form is conventional in naval architecture. Arcs of differential surface in the first case yield



qr qX dy ¼ i þ j dy, qy qy

qr qX i þ k dz. ds2 ¼ dz ¼ qz qz

ds1 ¼

(A.2)

Differential surface reads dS ¼ ds1  ds2 ¼



qX qX j k dy dz ¼ Nx dSx ¼ n dS, i qy qz

(A.3)

where dSx ¼ dy dz, i.e., orthogonal projection of dS onto y, z plane, Nx is the corresponding normal vector while n is the unit normal vector. In the second and third surface representation (A.1) in a similar way one finds dS ¼







qY qY ijþ k dx dz ¼ Ny dSy ¼ n dS, qx qz





qZ qZ i j þ k dx dz ¼ Nz dSz ¼ n dS. dS ¼  qx qy

(A.4)

(A.5)

(A.6)

where h ¼ hX ðx; y; zÞi þ hy ðx; y; zÞj þ hz ðx; y; zÞk.

Differential element is defined as follows: 



qr qr qh qr  þ  dS~ ¼ ds~ y  ds~ z ¼ qy qz qy qz



 qr qh qh qh  þ  dy dz. þ qy qz qy qz

(A.9)

The first product in (A.9) represents the differential element of the undeformed surface (A.3), the last product is a small negligible quantity of higher order, while the second and third products are related to the variation of differential element due to modal deformation. Thus,

qh qr qr qh  þ  dSx . dðdSÞ ¼ dS~  dS ¼ (A.10) qy qz qy qz Similar expressions can be derived for the other two definitions of body surface (A.1):

qh qr qr qh  þ  dSy , dðdSÞ ¼ (A.11) qx qz qx qz

dðdSÞ ¼





qh qr qr qh  þ  dSz . qx qy qx qy

(A.12)

By substituting (A.1) into (A.10), (A.11) and (A.12), respectively, and taking into account relation

dðdSÞ ¼ dn dS

The corresponding quantities for the third case of body surface definition are shown in Fig. A.1. The deformed body surface by mode h is represented by vector r~ ¼ r þ h,

In the first surface definition (A.1), one can write for arcs of differential element of deformed surface:

qr~ qr qh ds~ y ¼ dy ¼ þ dy, qy qy qy

qr~ qr qh dz ¼ þ dz. (A.8) ds~ z ¼ qz qz qz

(A.13)

and (A.7), one finds the following three expressions for the normal vector variation depending on the body surface definition (A.1):



qhy qhz qhx qhz qhz þ nx i þ  nx þ ny  nz j ðdnÞx ¼ qy qz qy qz qy

qhx qhy qhy n  n þ n k, (A.14) þ  qz x qz y qy z

(A.7) ðdnÞy ¼

ðdnÞz ¼









qhz qhy qhz qhx qhz n  n  n iþ þ n j qz x qx y qx z qx qz y

qhy qhx qhx n  n þ n k, þ  qz x qz y qx z



(A.15)



qhy qhy qhz n  n  n i qy x qx y qx z

qhx qhx qhz nx þ ny  nz j þ  qy qx qy

qhx qhy þ n k. þ qx qy z

(A.16)

The derived expressions are different in spite of the fact that they should represent the same quantity. Almost a half of their terms are common. Due to practical reason, a unified expression should be formulated. If we want to keep all essential terms of formulae (A.14)–(A.16) in such a formulation without their repeating, then let us refer to the mathematical logic and apply the set theory (Kreyszig, 1993). In this case, the unified expression for variation of normal vector is represented by the union of sets (A.14)–(A.16): Fig. A.1. Definition of normal vector variation (dn)z due to modal deformation h for the explicitly defined body surface Z ¼ Z(x, y).

dn ¼ ðdnÞx [ ðdnÞy [ ðdnÞz , which is illustrated by Venn diagram in Fig. A.2.

(A.17)

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stiffness is not taken into account in the ship vibration analysis due to the reason of very small amplitudes. The ship hydroelasticity couples the ship motion in waves and the wave induced vibrations (springing and slamming). Therefore, the rigid body modes and elastic modes are involved in ship response. Restoring stiffness, as the basic parameter in ship motion, Eq. (B.3), couples rigid body with elastic modes as a bridge. Its influence on higher elastic modes is reduced. The governing modal differential equation yields ðk þ CÞd þ ðb þ BðoÞÞd_ þ ðm þ AðoÞÞd€ ¼ Fw ,

(B.5)

where k is the hull stiffness, b the structure damping, m the ship mass, C the restoring stiffness, B(o) the hydrodynamic damping, A(o) the added mass, and Fw is the wave load.

Appendix C. Geometric stiffness of floating structures Fig. A.2. A Venn diagram for variation of unit normal vector in explicitly represented body surface.

Formulation (A.17) is used in Eq. (21). It is interesting to point out that the following relation also exists in the considered case

dn ¼ 12½ðdnÞx þ ðdnÞy þ ðdnÞz .

(A.18)

Appendix B. Manipulation with restoring stiffness In longitudinal strength analysis of a ship in still water, the ship hull is considered as a non-prismatic beam loaded by ship weight and buoyancy. The problem is statically determined and sectional forces (bending moment and shear force) and hull deflection are determined by successive load integrations. For the above purpose, the 1D FEM model could be used utilizing the matrix bending equation: Kd ¼ Fsb ,

(B.1)

where K is the hull bending stiffness, d is the displacement vector, and Fsb is the load vector comprising ship weight and buoyancy. A ship in still water can be considered as a beam on elastic support. In that case, buoyancy is not treated as a load anymore, but as the foundation reaction. The corresponding equation reads ðK þ CÞd ¼ Fs ,

(B.2)

where C is the buoyancy stiffness matrix and Fs is the ship weight vector. In the FEM formulation (B.1) and (B.2), the sectional forces are determined by retracking to the finite element equations with the known hull displacement vector d. In ship motion in waves, the ship hull is considered as a rigid body. Static equilibrium is a referent state for dynamic analysis. The governing matrix differential equation for 6 d.o.f. is of the following form: Cd þ d_ þ MðoÞd€ ¼ Fw ,

(B.3)

where C is the restoring stiffness of the rigid body modes, B is the hydrodynamic damping, M(o) is the virtual mass (ship mass and added mass), and Fw is the wave load. The matrix equation for ship hull vibrations reads Kd þ Bsh d_ þ MðoÞd€ ¼ Fep ,

(B.4)

where Bsh is the structural and hydrodynamic damping, and Fep is the vibration excitation (engine, propeller, etc.). The restoring

Ships and floating offshore units are thin-walled structures consisting of beams, plates and shells. The geometric stiffness is a correction of their bending stiffness due to the influence of axial and inplane loads. The determination of the geometric stiffness is a structural problem. Generally, complex structures are analyzed by 3D FEM models. First, strength analysis for the structure in still water is performed. Then, the membrane forces in structural elements are extracted to create the geometric stiffness, K0. Furthermore, it is natural and logical that the geometric stiffness is added to the ordinary structure stiffness, K, forming in such a way the equivalent (effective) bending stiffness, K* ¼ K+K0. The next step is the calculation of natural modes from the eigenvalue problem ðK  o2 MÞh ¼ 0,

(C.1)

as necessary for hydroelastic analysis. The geometric stiffness is related to the elastic modes as well as the ordinary stiffness. It is a structural parameter usually relevant for the structure stability, ultimate strength and collapse. The role of the geometric stiffness in linear hydroelasticity requires special attention and consideration. In some cases, one part of the geometric stiffness, common to all structural elements, can be determined in a simpler way, as a statically determined problem, and in other cases its influence is negligible. A ship vibrating in still water is exposed to compression in axial direction due to the hydrostatic pressure. The axial pressure force on hull cross-section reads ZZ F ¼ rg Z dy dz. (C.2) It increases the hull deflection, i.e., decreases the hull resistance to bending. Equilibrium of the moments acting on a beam infinitesimal element is (Fig. C.1): dM ¼ Q dx þ F dw,

(C.3)

so that taking M ¼ EIw00 into account, yields 3

Q ¼ EI

d w dw . F dx dx3

(C.4)

In order to estimate the influence of force F on bending, let us assume sinusoidal modal deflection due to the reason of simplicity: w ¼ a sin

npx L

;

n ¼ 1; 2; . . .

(C.5)

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simplicity: F ¼ 12rgLA ¼ 1570 kN.

(C.10)

The equivalent stiffness, for n ¼ 1, reads EI ¼ ð1 þ 0:1207ÞEI.

Fig. C.1. Differential element of bending beam.

By substituting (C.5) into (C.4) one finds   2  npx np np . a cos Q ¼ EI F L L L

(C.6)

Based on the term in the square brackets, it is possible to specify the equivalent bending stiffness: EI ¼ EI 



L 2 F. np

(C.7)

The second term in (C.7) is the geometric stiffness of ship hull. It is interesting to analyze its influence on the reduction of the hull bending resistance. For this purpose, let us consider a pontoon with ship-like cross-section. A 7800 TEU container ship is at disposal with the following parameters (Senjanovic´ et al., 2007b): Length: Breadth: Draught: Moment of inertia of cross-section: Young’s modulus:

Lpp ¼ 319 m B ¼ 42.8 m T ¼ 14.5 m I ¼ 670 m4 E ¼ 2.1 108 kN/m2

Thus, assuming the rectangular cross-section, one finds for the axial force F ¼ 12rgBT 2 ¼ 45  103 kN.

(C.8)

Matrix C0 is similar to the gravity matrix Cm, Eq. (61). It could be incorporated in 1D FEM model for calculating dry natural vibrations. Determining the geometric stiffness due to axial pressure force is a statically determined task, and can be realized by following the general beam geometric stiffness definition. By substituting (C.2) into (C.12) and taking into consideration that beam is bended in vertical and horizontal planes, yields # ZZ Z L" i qwz qwjz qwiy qwjy C 0ij ¼ rg Z dy dz þ dx. (C.14) qx qx qx qx 0 However, the formulation of the beam geometric stiffness for vertical offshore structures requires an additional consideration. The axial (vertical) compression force, according to Fig. C.2, reads Z Z ZZ Z Z ðqw  qs Þ dz  P ¼ g ðr  rs Þ dz dx dy  P FZ ¼ ¼g

ZZ 

(C.9)

So, the bending stiffness is reduced by 0.33% only. For the higher modes, the reduction approaches zero, (C.7). This numerical example shows negligible influence of the hull geometric stiffness on the ship flexural vibrations. If, for instance, a floating airport with a length of 5 km is considered, then the bending stiffness reduction of 50% can be expected. In case of a monohull offshore structure, the problem is similar. Let us consider a tower buoy simply supported at the sea bottom with the following parameters: Height: Diameter: Thickness: Cross-section area: Moment of inertia of cross-section:

In the finite element formulation, the vibration modes are the shape functions (54), and Eq. (C.12) leads to the geometric stiffness matrix of beam element, well known in stability analysis (Holand and Bell, 1970; Senjanovic´, 1998; Senjanovic´ et al., 2006): 2 3 36 3L 36 3L 6 7 4L2 3L L2 7 F 6 6 7. C0 ¼  (C.13) 6 36 3L 7 30L 4 5 Sym: 4L2

Zb

The equivalent bending stiffness for the first mode, n ¼ 1, reads EI ¼ ð1  0:0033ÞEI.

(C.11)

In this case, the geometric stiffness increases the tower bending stiffness by 12.07% and this is considerable. The modal geometric stiffness for the beam idealization of ship hull can be determined as follows. In the energy approach to structural problems, the stiffness matrix is formulated as a double value of the strain energy (Bathe, 1996). If Eq. (C.4) for the ith mode is multiplied by dwj =dx and integrated along the hull, the first term, using the integration by parts, results in the bending stiffness, and the second one in the geometric stiffness: Z L dwi dwj C 0ij ¼ F dx ¼ FIðw0i w0j Þ dx. (C.12) 0 dx dx

L ¼ 100 m D ¼ 2m t ¼ 0.02 m A ¼ 3.14 m2 I ¼ 0.0628 m4

Here, we are faced with the linearly distributed tension force. Its average value is taken into account due to the reason of

rðZ  Z b Þ 

Z

Zb

Z Zb

 rs dz  P,

(C.15)

where P is a pre-stress (tension) force. Thus, the beam geometric stiffness according to (C.12), yields # Z H " i qwx qwjx qwiy qwjy 0 C ij ¼  þ FZ dz, (C.16) qz qz qz qz 0 where H is the height of the structure. The generalized form of (C.14) and (C.16) for an elastic body with modes hi, adopted for application in 3D FEM model, can be presented as follows: # ZZZ " i qhy qhjy qhiz qhjz qhix qhjx qhiz qhjz þ þ þ C 0ij ¼ rg Z dV qx qx qx qx qy qy qy qy V

ZZZ

P  rgðZ  Z b Þ þ g Az V " #) i j qhix qhjx qhy qhy þ dV qz qz qz qz

Z

Z

þ

Zb

rs dz



(C.17)

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Fig. C.2. Axial force in vertical slender structure.

Quantity Az is the body cross-section area. The structure density can be declared as the ratio

rs ¼

mz , Az

(C.18)

where mz is the vertical mass distribution per unit height. The geometric stiffness matrix (C.17) is symmetrical, since the integrand product functions are commutative. The mode xderivatives in the first volume integral are related to the ship hull as a beam structure. The derivatives of the mode components hz are relevant for the artificial floating islands (airports, power plants, recreation areas, etc.) as plate structures. The second volume integral holds the tower-like offshore structures. C 0ij is formulated as a statically determined hydrostatic problem and therefore can be called the hydrostatic geometric stiffness. Finally, it can be treated as an additional part to the restoring stiffness (5) caused by elastic modes only. A ship in still water is in sagging or hogging state. In the former case, the structural elements above the neutral line are compressed, while those below it are in tension. Thus, the stiffness of the upper elements is reduced, and of the lower elements is increased and vice versa for hogging. Those two effects are cancelled in linear domain and only geometric stiffness contribution due to the axial hydrostatic pressure component remains. The same effect occurs in the other monohull structures. In case of a complex structure, the complete geometric stiffness, K0, has to be determined and included in the effective stiffness, K*. References Bathe, K.J., 1996. Finite Element Procedures. Prentice-Hall. Bishop, R.E.D., Price, W.G., 1979. Hydroelasticity of Ships. Cambridge University Press. Hirdaris, S.E., Price, W.G., Temarel, P., 2003. Two and three-dimensional hydroelastic analysis of a bulker in waves. Marine Structures—Special Issue on Bulk Carriers 16, 627–658.

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