Hydroelastic modelling of wetdeck slamming on multi-hull vessles

TECHNISCHE UNIVERSITEIT Seheepshydramechaaica Archief Mekelweg 2, 2628 CD Delft Te1:015-786873/Fax:781836

HYDROELASTIC MODELLING OF

WETDECK SLAMMING ON

MULTIHULL VESSELS

DOKTOR INGENIORAVHANDLING 1994:48 INSTITUTT FOR MARIN HYDRODYNAMIKK TRONDHEIM

I' EWE

TECHNISCHE UNIVERSITET Laboretorium voor $cheepshydromechardos Arabia MekeOng 2, MB CD Delft Mt 015- 711111711 015. 751336

Hydroelastic Modelling of

Wetdeck Slamming on

Multihull Vessels

Dr.ing. Thesis

by

Jan Kvalsyold

Department of Marine Hydrodynamics

The Norwegian Institute of Technology

Abstract

Slamming against the wetdeck of a multihull vessel in head sea waves isstudied

an-alytically and numerically. The theoretical slamming model is a two-dimensional,

asymptotic method valid for small local angles between the undisturbed water sur-face and the wetdeck in the impact region. The disturbance of the water surface as

well as the local hydroelastic effects in the slamming area are accounted for. The elastic deflections of the wetdeck are expressed in terms of "dry" normal modes. The structural formulation accounts for the shear deformations and the rotatory inertia effects in the wetdeck. The interaction effects between the local loading and the global rigid ship motions are partly investigated. The findings show that the slamming loads on the wetdeck and the resulting elastic stresses in the wetdeck are strongly influenced by the elasticity of the wetdeck structure.

Acknowledgement

This study has been carried out by supervision from Professor Odd M. Faltinsen whose guidance and encouragement during this work are deeply appreciated.

I am also grateful for the many valuable discussions with my colleagues and friends

at the Department of Marine Hydrodynamics, the Department of Marine

Struc-tures and at MARINTEK.

I thank the staff at SINTEF Industrial Mathematics for their professional guidance concerning computer visualization and optimalization.

This work has received financial support by The Research Council of Norway

(NFR). The computer time on CRAY Y-MP is supported by the Norwegian Su-percomputing Committee (TRU).

Contents

Abstract

1

Acknowledgement

Contents

3

Nomenclature

6 1

Introduction

11

2 The structural formulation of the

wetdeck

17

2.1 The Euler beam model of the wetdeck 20

2.1.1 The eigenvalue problem 22

2.1.2 The governing modal beam equation of motions 93

2.2 The single Timoshenko beam model of the wetdeck 24

2.2.1 The beam eigenvalue problem 26

2.2.2 The governing modal beam equation of motions 28

2.3 The three Timoshenko beams formulation of the wetdeck 29

2.3.1 The beam eigenvalue problem 30

2.3.2 The governing modal beam equation of motions 31

3 The hydrodynamic boundary value problem

33 3.1 The basic formulation of the hydrodynamic boundary value

problem 34

3.2 The symmetric hydrodynamic boundary value problem 36 3.3 The unsyrametric hydrodynamic boundary value problem 37 3.4 The inner solution near the edges of the flow 38

3.5 The acoustic approximation 41

4 The solution procedures of the HBVP

43

4.1 The constant space velocity approximation 44

4.2 The direct numerical solution of the HBVP 45

.

.. .... .

4.2.1 Discussion of the direct numerical method 48

4.3 The Fourier approximation 50

4.4 The unsymmetric solution procedure 53

5 The wetted length of the beam(s)

57

5.1 The symmetric fluid flow case 58

5.1.1 The constant space velocity approximation 58

5.1.2 The Fourier approximation 59

5.2 The unsymmetric local fluid flow case 60

6

Scaling of the hydroelastic response

62

7 Verification of the local hydroelastic analysis

66 8

Interaction between the local loads and the global ship motions

68

8.1 A brief introduction to the computer code FASTSEA 69

8.2 The global coordinate system 70

8.3 The coupled governing equations of motions 70 8.4 The location along the wetdeck of the initial water impact 72

8.5 Vessel particulars 73

8.6 Verification of the steady state ship motions 73

9

Results

75

9.1 The numerical time simulation procedure 73

9.9 The symmetric local fluid flow problem 77

9.3 The coupled hydroelastic slamming analysis 91

9.4 The unsymm.etric local fluid flow problem 94

10 Discussions and recommendations of further work

111

11 Conclusions

121

12 References

123

A The orthogonality conditions of the beam eigenfunctions

127

A.1 The Euler beam model 127

. . . . . . . . . .

...

. ...

....

. . . . . . . . ..,. . ..

A:2 The Timoshenko beam model 130

B The constant space velocity approximation. Modal

coefficients! 135

C The principal coordinate

137

D The

.Fourier

approximation. Modal coefficients

141

E Improvement of the ship

hydrodynamic coefficients

145

'F The coupled governing equations of

motions

150

G The tins

etric local fluid flow.. Modal coefficients

155

H The modal velocity components

159

Calculation of the vertical velocity outside the wetted length

168!

J Analytical evaluation of the singular

integral

171 I

Nomenclature

pa(t) afl(t) nil It AH Amn A k (t) (0, (t), A,n2 (t); An Aki) As ib(t)

A.(0

By, Ewes c(t) ce Cj Cnirn Cr, C?) Con

Cu)

Con

C

dw c(l)

d2G

DW) Ey, E,Y) F(x zyt)1 Fadd(t)

Fm(t)

The position of the right jet The principal coordinate

See equation (2.26)

Hydrodynamic added mass coefficient, of the ship side bills

Fourier cosine coefficient Modal added mass coefficient See section 4.2

Coefficient in the beam eigenfunction Coefficient in the beam eigenfunction The shear area of the beam cross section The position of the left jet

Modal hydrodynamic damping coefficient

Coefficient in the beam eigenfunction Coefficient in the beam eigenfunction, Breadth of the wetdeck

Location of the edge of the symmetric flow (the Jet) The speed of sound in water

Half the wetted length at time step j

Modal restoring coefficient

Coefficient in the beam eigenfunction,

Coefficient in the beam eigenfunction

Coefficient in the eigenvalue analysis Coefficient in the eigenvalue analysis Coefficient in the eigenvalue analysis Coefficient in the eigenvalue analysis

Vertical distance from the z-axis to the wetdeck

Distance from the free water surface to the wetdeck at COG Coefficient in the beam eigenfunction

Coefficient in the beam eigenfunction

Youngs modulus

'Coefficient in the beam eigenfunction Coefficient in the beam eigenfunction A function describing the instantaneous position of the wetdeck

Local hydrodynamic added mass force. Local hydrodynamic damping force

Fe,c(t)

F,,,(t)

FT, F4,3 ) Ftot(t) (t) 9A(x, c(t)) .03(x, c(t)) G(x, z; () GT, Gi3) HT, 11,(!) 3 KT K ke K1, K2, K3 Kti, K57 K6 LB in MB M77177/ Mr N f cam Neig NLms Nsel Nstep

p(x,w,t)

Local excitation force Modal excitation force

Coefficient in the beam eigenfunction Coefficient in the beam eigenfunction

Total local two-dimensional hydrodynamic force Total local hydrodynamic force

Total local hydrodynamic moment (= xsia,F(t))

Acceleration of gravity

Part in the vertical velocity outside the wetted length Part in the vertical velocity outside the wetted length The shear modulus

The Green's function

Coefficient in the beam eigenfunction Coefficient in the beam eigenfunction Coefficient in the beam eigenfunction Coefficient in the beam eigenfunction

The imaginary unit (=

Counter variable

Area moment of inertia in the beam cross section See equation (2.25)

Value of the non-dimensional absolute maximum stress (see equation 10.2) Spring stiffness at the beam ends

Non-dimensional coefficients (see equation (6.4)) Non-dimensional coefficients (see equation (6.4))

Length of the vessel Length of the beam

The beam vibration mode number Mack number

Mass per unit area of the beam

Modal mass coefficient

Rotatory moment at the beam ends The beam vibration mode number Number of Fourier components

Number of eigenfunctions in the normal mode approach Number of surface elements in the boundary element method Number of elements in the least mean square procedure Number of subelements at each boundary element Actual time step in the time simulation procedure Hydrodynamic impact pressure

PaCtt) PC0771 P eqv(4 P P71 Ptot

q.

fi 7' 7' Ft (t) S5(t) Sp i(t), SF 2(t) sv 00 (La (t) ,Un2() V (t) Ve(x,t)

V(t)

V9 (t) V R N (X,1) 14(t) V2(t) 2(X t) W (X t) Wn X

The one-dimensional acoustic pressure Common pressure part of the outer and the inner flow solutions

The equivalent pressure Material constant

Real wave number of the beam Composite hydrodynamic pressure Real wave number of the beam Material constant

Radius of gyration in the beam cross section area

=

\Az

e)2

+

(z ()2 (in the pure

numerical solution procedure)

=

V(x e)2

+

(z + ()2 (in the pure

numerical solution procedure)

Radius of curvature of the waves in the impact region Surface area

Surface area on the wetted length Free water surfaces

See equation (4.27)

Control surface located at infinity Time variable

Time at time step/

Time scale Wave period

U = z Get

Forward speed of the multihull vessel The velocity vector of a boundary element

Modal velocity components (see equations (4.30) and (4.31))

Vertical fall velocity

Effective vertical velocities along the wetted length Mean velocity in space over the wetted length of V;(z,t) Vertical velocity due to the global rigid ship motions Relative normal velocities along the vessel

Mean velocity in space on the wetted length

Mean slope of the velocity curve on the wetted length Relative normal velocity on the wetted length

Elastic beam deflections

Complex wave number of the beam x-coordinate

Q

S

t

-X (-X, t) Xbow COG sic Islam strass(t) ZG Zna

Greek symbols:

awet(I) The slope along the wetdeck

/3(s, t) The angle of the side of an infinitesimal Timoshenko beamelement

fid Deadrise angle of a wedge

-y(x,t) Vortex density

Shear angle

Jet thickness in the inner flow solution

Strain rate

Local coordinate in the boundary element method

c(x t) Wave elevation

Wave amplitude

(slam Minimum wave amplitude to provoke slamming

Th(t) Surge of the vessel

772 (t) Sway of the vessel 773 (t) Heave of the vessel

714(t) Roll of the vessel 775(0 Pitch of the vessel 176(t) Yaw of the vessel

nag (x) Airgap at the initial impact between the undisturbed

free water surface and the wetdeck

nR(x, t) Relative position between the free water surface and the wetdeck

Transformed coordinate on the wetted length be Rotation angle of the beam end

Wave length

ii Water wave number Transformed x-coordinate

x-coordinate (global reference frame) of the vessel bow end x-coordinate (local reference frame) of vessel center of gravity x-coordinate (local reference frame) of the initial contact point x-coordinate (global reference frame) of the initial

contact point

x-coordinate (local reference frame) of the instantaneous maximum stress

z-coordinate

The complex coordinate = x + iz)

z-coordinate (global reference frame) of the COG of the vessel Distance from the neutral axis to the point where the stresses are calculated

'(2

Ca

Crmax CPR Cfvm V RN Cryied Cyd Cye 'Ts 1171 0' eke Oin Co 001St On(X) 44Y)(X) X 1141(4 OW)(X) We WY' W 0 AbbreViations:,

HBVP Hydrodynamic boundary value problem

COG Center of gravity

Local coordinate in the boundary element method. Density of water

Bending stress

Absolute maximum stress

Standard deviation of the relative position

Von Mises stress

Standard deviation of the relative normal velocity Yielding stress

Dynamic yielding stress Static yielding stress

Integration variable for the time Shear stress

Local parameter in the inner jet solution Velocity potential

Even part of the velocity potential

Velocity potential due to the incident waves Velocity potential in the inner jet solution Odd part of the velocity potential

Velocity potential in the outer solution Eigenfunction of the beam vibration mode it Eigenfunction for beam j of vibration mode it Total velocity potential

'Transformed coordinate outside the wetted length Eigenfunction of the beam vibration mode n Eigenfunction for beam j of vibration mode it

Test function for the velocity potential Frequency of encounter

Eigenfrequency of vibration mode it

Angular frequency of the waves

1

Introduction

Impact between water and a ship, i.e. slamming, can. cause important local and

global loads on the vessel. Slamming is often categorized asbottom slamming, bow

flare slamming and wetdeck slamming. By wetdeck is meant the structural part connecting the two side hulls of a catamaran (see Figure 1.1). The local angles be-tween the undisturbed water surface and structure are small in bottom slamming and wetdeck slamming. These types of slamming are characterized by large

pres-sures that are highly concentrated in space and time. In bow flare slamming the water is thrown up along the hull sides. This slamming type lasts longer in time

than the two first types. However, for an extreme flare with small local instanta-neous angles between the hull and the free water surface, the pressure distribution in time and space may be similar as for bottom and wetdeck slamming.

The slamming load level and the pressure distribution in time and space on rigid

blunt bodies are sensitive to the relative normal velocity between the free water surface and the structure. Furthermore, the local angle between the undisturbed

water surface and the structure in the impact area, the entrapment of an air

cush-ion or small air bubbles in the interphase between the water and the structure,

compressibility effects of the fluid and forward speed effects areimportant for the local load distribution. Additionally, the interaction effects between the local im-pact loading and the global ship motions as well as flow separation will influence the slamming pressure. Viscous effects have to be included in the numerical model to predict flow separation from smooth bodies. If the elasticity of the structure is accounted for, the slamming loads will cause local vibrations of the structure.

These vibrations contribute to the local normal velocities as well, and thereby

influence the hydrodynamic loading. Such effects are often referred to as

hydroe-lastic effects. By hydroehydroe-lastic effects are meant that the hydrodynamic pressure is a function of the structural deflections. This means that hydroelasticity is an effect that has to be added to the above list of parameters influencing the slamming load

Figure 1.1: A fish view of the wetdeck of a catamaran. Reproduced with permission from Jens B. Helmers.

study slamming against wetdecks and investigate how fluid-structure interaction influences the hydrodynamic loads and the structural stresses.

Hydroelastic effects may not be of importance for all the slamming categories listed in the above text. Hayman et al. (1991) carried out an experimental study

of slamming against rigid wedges with deadrise angle approximately equal to 30

degrees. Hayman et al. (1991) investigated the effect of using aluminium and a

GRP sandwich material in the wedge. They pointed out that the peak pressure

was not significantly influenced by the elasticity of the structure. However, it is expected that hydroelasticity will become more important when the initial impact

force is large.

A multihull vessel advancing at forward speed U in a head sea wave system may undergo extreme hydrodynamic loading due to wetdeck slamming. Wetdeck slam-ming of multihull vessels occurs most frequently in sea states where large relative vertical motions are present along the vessel due to resonant heave and pitch mo-tions. The large hydrodynamic forces on the structure initiated by slamming will introduce local as well as global hydroelastic effects of the multihull vessel. Only

the local hydroelastic effects are considered in this study. The global elastic effects due to slamming are often referred to as whipping. Whipping has been studied by for instance Kaplan and Dalzell (1993) and Friis-Hansen et al. (1994). Kaplan and Dalzell (1993) modelled the vessel as a beam and used a modal superposition tech-nique for the global elastic deflections. They did not account for any interaction effects between the local loading and the global elastic motions. Friis-Hansen et al. (1994) modelled the vessel as a Timoshenko beam and used slamming pressures based on the work by Ochi and Motter (1973). They developed a new probabilistic

method that accounts better for the non-linearities and demonstrated important

effects of the ship length and the hull flexibility on the response.

Hydroelasticity of ships has been studied by Bishop and Price (1979). They con-sider both the steady state global elastic vibrations, often referred to as springing, and the transient global elastic effects, i.e. whipping.

Slamming has been widely studied in the literature through the last decades. Re-cent progress in numerical and experimental studies of slamming can be found in

the proceedings of ITTC (1993) and ISSC (1994). Studies of slamming against rigid

two-dimensional bodies can be traced back to the work by von Karman (1929) and

Wagner (1932). Slamming has later been studied by for instance Cointe and Ar-mand (1987) and Zhao and Faltinsen (1992,1993). Cointe and ArAr-mand(1987) stud-ied impact on a circular cylinder that was forced with constant velocity through an initial calm water surface. Zhao and Faltinsen (1993) considered an infinite wedge that was forced with constant velocity through a calm water surface. They solved

the problem numerically and used a boundary element technique to express the

lo-cal fluid flow. They managed to satisfy the exact free surface boundary condition.

Arai and Tasaki (1987) and Arai and Matsunaga (1989) presented a numerical

method for prediction of slamming loads on two-dimensional sections. The gravity effects and the flow separation at a knuckle were accounted for. Faltinsen (1993) compared the theory by Zhao and Faltinsen (1993) with the experimental inves-tigations by Yamamoto et al. (1985). Faltinsen (1993) reported good agreement between the numerical simulations and the experimental results for two of the three pressure gauges. Faltinsen (1993) compared the numerical simulations with the re-sults from the numerical and experimental study by Arai and Matsunaga (1989)

and concluded that the agreement was good. Faltinsen (1993) pointed out that

the retardation of a body during experimental investigations of slamming causes an "added mass effect" that may lead to non-conservative estimates of the impact pressure when scaling to full scale. The effects of three dimensionalities of the ship hull on the impact pressure have been investigated by Watanabe (1987).

Slamming against rigid wetdecks has been studied by Kaplan and Malakhoff (1978)

and Kaplan (1987,1991). Kaplan (1992) reported that wetdeck slamming could cause a hydrodynamic loading in the order of the weight of the vessel or even

larger. This may lead to severe local as well as global damages of the hull structure. Kaplan (1992) points out that the slamming loads have a significant influence on the ship accelerations. The effect of the slamming loads on the ship velocities and the displacements are less pronounced. Kaplan (1992) therefore suggests that the interaction effects between the impact loads and the global rigid ship motions do

not have to be accounted for. However, it is important for theslamming load level

how and where the free water surface hits the wetdeck. This has been pointed out by Zhao and Faltinsen (1992). Zhao and Faltinsen (1992) reported that the global

heave and pitch motions of a catamaran were influenced by wetdeck slamming.

They did not account for any local or global elastic effects of the catamaran. The air cushion effect on the impact has been studied by Koehler and Kettlebor-ough (1977). They studied the airflow in the air layer between a falling body and a viscous free water surface. They found that there is a large pressure gradient in the airflow near the knuckle of the body. This pressure gradient causes an increase of the surface elevation. For small deadrise angles, this will cause an air cushion to be trapped. Compressibility effects in the slamming problem have been investi-gated by Korobkin (1994) and Korobldn and Pukhnachov (1988). They found that

in the initial stage of the impact the flow will be supersonic. The importance of

the forward speed effect on the slamming load level has been pointed out by

Mey-erhoff (1968). Beukelman and Rader (1991) studied the effect of forward speed on the impact pressure on a wedge. They compared numerical simulations with experimental results and concluded that the effect of forward speed increased with the trim angle of the vessel. They obtain good agreement between the numerical simulations and the experiments.

A slamming model where the local hydroelastic effects are accounted for, are not so frequently appearing in the literature. However, the author is aware of the work

by Meyerhoff (1965) who studied slamming against elastic two-dimensionalwedges

penetrating an initially calm free water surface. He modelled half the wedge as a simply supported beam with no shear and rotatory inertia effects. Meyerhoff (1965)

accounted for the retardation of the wedge and concluded that the hydroelastic

effects influenced the response of the wedge.

In this work an idealistic two-dimensional study is carried out. The developed

theory can be applied for moderate forward speeds and when the wetdeck is nearly horizontal in the impact region. The hydrodynamic formulation is based on an

ex-tension of -Wagner's (1932) two-dimensional asymptotic theory. The wetted length

is approximated by two different approaches. The first approach is a generalizer, tion of the von Karman (1929) method while the second is a generalization of the Wagner (1932) method. The main difference between those two methods is that

the latter method accounts for the deformations of the surface due to the slam. Three analytical solution procedures and one pure numerical method have been

applied to evaluate the local fluid loading. In the first solution procedure, the local

fluid flow is assumed to have symmetry properties and the vertical velocities on the

wetted length are assumed constant in space. The second solution procedure ac-counts for an arbitrary vertical velocity variation along the wetted length but still the fluid flow is assumed to be symmetric. The third solution procedure accounts for an unsymmetric local fluid flow. The vertical velocities on the wetted length

are then approximated by a linear velocity variation in space. The pure numerical solution procedure is based upon Green's second identity. For the analytical solu-tion approaches, the local hydrodynamic loading is expressed in terms of analytical

functions. This ensures the stability and the accuracy of the time integration of

the hydroelastic response.

Three different structural formulations of the wetdeck have been investigated. These are the Euler and the Timoshenko single beam models and a set of three Timoshenko beams The Timoshenko beam model accounts for the shear

deforma-tions and the rotatory inertia effects in the wetdeck. Such effects are not covered

different beam ends boundary conditions is investigated as well.

The interaction effects between the local hydrodynamic loading and the global rigid ship motions have been studied in the initial stage of the water impact.

The hydroelastic response of the wetdeck is scaled by separating the water impact into three time phases. Each phase has its own time scale. The three time phases can be called the compressibility phase, the structural inertia phase and the added mass-restoring phase. The compressibility phase comes first. The speed of sound

in water is then important. The last time phase is the added mass-restoringphase.

The inertia due to the added mass of one beam and the restoring loads due to the

elastic stresses in the wetdeck determine the time scale. The absolute maximum stress occurs in this phase. The hydroelastic response is scaled by using the time scale associated with the third impact phase. When the impact velocity is not too

small, it is found that the absolute maximum von Mises stress cynic= is not sensitive

to the local details of the wave shape and that cr,,x is linearly proportional to the impact velocity. This should be further investigated for more complex type of elastic modelling of the wetdeck. However, the findings indicate a simple way

to describe the statistics of the stresses and the voluntary speed reduction of a

multihull vessel, due to slamming loads on the wetdeck.

Parts of this work have been published in proceedings film international

confer-ences and is submitted for publication in a journal. The two symmetric local

fluid flow approaches together with a simply supported Euler beam model for the wetdeck are presented in Kvalsvold and Faltinsen (1993a). Arbitrary beam ends

boundary conditions as well as a limited study of the interaction effects between

the local loading and the global rigid ship motions. in the initial impact phase.

are reported by Kvalsvold and Faltinsen (1993b). The effects of the shear defor-mations and the rotatory inertia on the hydroelastic response are briefly reported in KvaLsvold and Faltinsen (1994a) and more in detail by Kvalsvold and Faltinsen (19941)). The set of three Timoshenko beams together with the unsymmetric local fluid flow is submitted for publication in a journal (Kvalsvold and Faltinsen 1995).

The scaling of the hydroelastic response is considered by Kvalsvold and Faltinsen

2

The structural formulation of the

wetdeck

A detail of the wetdeck structure of a multihull vessel is shown in Figure 2.1.

The righthanded xyz-coordinate system is a local coordinate system that moves with the forward speed U of the vessel. The z-axis is parallel' to the longitudinal stiffeners and is pointing towards the stern of the vessel. The y-axis is parallel to

the transverse stiffeners and is pointing towards the starboard side. The z-axis is pointing upwards. The origin of the local coordinate system is located at the

midpoint between two of the transverse stiffeners in the wetdeck.

The part of the wetdeck between two of the transverse stiffeners is modelled as a beam with length LB corresponding to the distance between the transverse stiffen-ers. This means that the beam deflections are dominated by those of the longitudi-nal stiffeners. Local deformations of the plate field between two of the longitudilongitudi-nal stiffeners is a three-dimensional effect that is notcovered by this two-dimensional

analysis. Such effects are discussed in chapter 10. The transverse stiffeners are

assumed to be much stiffer than the longitudinal stiffeners, so that the vertical

deflections at the transverse stiffeners can be disregarded. The properties of the beam are assumed constant along its length.

Three different structural formulations of the wetdeck have been carried out in this study. For the first and the second structural formulation, the partof the wetdeck

between two of the transverse stiffeners is modelled as a single Euler beam and

a single Timoshenko beam, respectively. Neither the shear deformations nor the

rotatory inertia effects in the wetdeck structure are accounted for by the Euler

beam model. Such effects are covered by the Timoshenko beam model. The Euler beam is often referred to as the elementary beam. The wetdeck is modelled as a set of three Timoshenko beams, each of length LB, in the third structural formulation.

This implies that a broader part of the wetdeck structure and the interactioneffects

between the different parts of the wetdeck are more properly accounted for.

Longitudinal stiffener

Wetdeck

Wave profile

Figure 2.1: A detail of the wetdeck structure of a multihull vessel. The xyz coordinate system is a local coordinate system defined more clearly in Figure 2.2 for the single beam models and in

Figure 2.3 for the multiple beam formulation.

ends to account for the restoring moment of the part of the wetdeck structure outside the modelled beam(s).

ko is the spring stiffness that is related to the

restoring moment Mr by Mr --= 1c9Obe. Obe is the rotation angle at the beam end.

No axial force effects of the modelled beam(s) are considered. The beam deflections

are assumed to be small compared to the single beam length LB so that a linear theory can be applied for the structural model. The curvature of the wetdeck in the water impact region is not accounted for in the structural formulations. Such effects

may become important for initially buckled wetdeck panels due to fabrication or caused by earlier slams. The material and the geometrical properties of the three modelled beams in the third structural formulation are assumed to be equal.

This study accounts for the inertia effects of the wetdeck structure limited to the modelled beam(s). No inertia effects of the wetdeck structure outside the modelled beam(s) are considered. Such effects are believed to become important only when the edge of the flow approaches one of the rotatory springs.

It is assumed, for the single beam models, that the crest of a regular wave system

undisturbed free

water surface

Figure 2.2: The definitions of the single beam model and the local coordinate system. ?co is the spring stiffness of the rotatory springs and riag(x) is the a.irgap between the undisturbed free

surface of the waves and the wetdeck at the moment of initial waterimpact.

hits the wetdeck at the mid point between two of the transverse stiffeners in the

wetdeck. Furthermore, it is assumed that the wetdeck is horizontal in the impact

region, so that both the beam deflections as well as the local fluid flow are

sym-metric about a vertical axis through the plane of the initial water impact. The

local coordinate system and the definitions of the single beam model are shown in Figure 2.2. 77a9(x) is the airgap between the undisturbed free surface of the waves and the wetdeck at the moment of initial water impact.

For the case when the wetdeck is modelled as a set of three Timoshenko beams, unsyrametry of the local fluid flow is accounted for as well. It is then assumed

that the free surface of the waves hits the wetdeck at the x-position xi. The local coordinate system together with the definitions of the three beams are reproduced in Figure 2.3. a9(x) accounts for the relative orientation between the undisturbed free surface of the waves and the wetdeck at the moment of initial water impact.

From left to right, the three beams are referred to as beam one. two and three.

respectively. It is assumed for simplicity that xi, is located on beam two.

The next three sections are addressed to further details about each of the three

structural formulations.

LB Lon

free surface of the waves

Lon,

LB

Figure 2.3: The definition of the set of three Timoshenko beams and the local coordinate system. k9 is the spring stiffness of the rotatory springs and 77,39(x) is the airgap between the undisturbed free surface of the waves and the wetdeck at the moment of initial water impact. The three beams

are numbered from left to right as beam one, beam two and beam three. respectively.

2.1

The Euler beam model of the wetdeck

Define w(x, t) as the elastic beam deflection, so that w(x,t) is to be interpreted as the difference between the actual vertical position of the wetdeck and the vertical

position due to the global rigid heave and pitch motions of the vessel. w(x.t) is assumed to be sufficient small compared to the beam length L8, so that linear

theory can be applied for the beam equation of motions. Furthermore, let MB

be the mass per unit length of the longitudinal stiffener together with the flange and divided by the width of the flange. The width of the flange is equal to the distance between the longitudinal stiffeners in the wetdeck (see Figure 2.1).

El is

the bending stiffness, so that E is the Youngs modulus and I is the area moment of inertia of the beam cross section area and divided by the width of the flange. Seventy percent of the width of the flange is used as the effective flange when calculating I. The governing beam equation of motions, satisfying the vertical

force equilibrium as well as moment equilibrium of an infinitesimal beamelement,

is then revealed as (see for instance Timoshenko 1974 or Clough and Penzien1975):

w(x,t) =

for

x=

LB 2 Ice iaw(x,t) ±02w(x,t) ax2 for x

El

ax (2.3) (2.4)

g(t) is the local vertical accelerations due to the rigid heave and pitch 'motions

of the vessel. It is assumed that the beam length is small compared to the ship length. Then the variation along the beam of the local vertical accelerations due

to the global rigid ship motions can be disregarded. p(x, w,t) is the hydrodynamic

impact pressure that is a function of time, space and the beam deflections. The

dependency on the beam deflections is due to the hydroelastic effects as discussed

earlier in the text. t is the time variable and dot stands for the timederivative.

The beam deflections are expressed in terms of the beam's "dry"normal modes

in this work. By "dry" modes is meant that the effect of the surrounding water

on the eigenfrequencies and the eigenfunctions of the beam is disregarded. This modal superposition technique to express the response of a dynamic system has

been widely used in the literature (see for instance Timoshenko 1974, Clough and Penzien or Bishop and Price 1979). Thus:

CC

fig

Wfr,

E a,2(t)0(x) E an(t)4,(x)

(2.2)

n=1 n=1

a(t) is the principal coordinate and ik(x) is the "dry" eigenfunctionof the beam

vibration mode n. N9 is the number of eigenfunctions used in the normal mode

approach. To achieve' an expression of the eigenfunction #0,.(x) and the

eigenfre-quency Lon one needs to solve. an eigenvalueproblem.

It would have. been better to use "Wet" modes instead of "dry" modes By

'wet'

modes is meant that the effect of the surrounding water is accounted for when

evaluating the eigenfunctions and the eigenfrequencies. That would probably cause

a faster convergence of the series described by equation (2.2). However, "dry"

modes are used in this work since the wetted length and by that the effect of the

surrounding water changes rapidly as a function of time.

The beam ends boundary conditions are

t) =

Physically, the boundary conditions are interpreted as no vertical deflections at the transverse stiffeners and continuity of the bending moment at the rotatory springs

at the beam ends

2,1.1

The eigenvalue problem

tbn(x) and con are determined by an eigenvalue analysis. The eigenvalue problem is characterized by the governing equation of motions (2.1),, with frs,(t) and p(x,w, t) equal to zero, together with the beam ends boundary conditions in equations(2.3)

and (2.4). The eigenfunctions are then determined by solving a homogenous

dif-ferential equation (see for instance Timoshenko 1974, Clough and Penzien 1975

or Hildebrand 1976). The beam deflection w(x, t) is assumed to have the general

.form:

w(x,t) = Co* e'elwn't (2.5)

Con is a constant and W,,, is the wave number that is not yet known.

is the

imaginary unit. The assumed solution of w(x,t) is substituted into the governing equation of motions (2.1) with 1(t) and p(x, w, it) equal to zero. This leads to the

following equation (Cg, 0,0)1

wf

Pein =O (2.6)

)9,1 is expressed as:.

m Ewn2

Pn =

_El

Equation (2.6)'is the relation between the wave 'numbers and the eigenfrequencies

for the Euler beam and is therefore interpreted as the dispersion relation. The four possible solutions

we,

w(3)1, we ofW', are:

w$1),(2)

_{= ±ipt,}

_(2.'8)

wp),(4)

±Pn. (2.9).

The four solutions of Wn together with the assumed solution of w(x,.t) lead' to the following expression for the eigenfunction 42(x):

i

-114,(x)= A siti(Pmr) + Ba cos (mix" + CR sinh (pRx), + DR cosh (px)' (2.10)

Bm, C,, D coefficients that are not yet known The eigenfrequency

and three of the four coefficients k, BR, CR, are solved in order to satisfy the. four beam ends boundary conditions in equations (2.3) and (2.4),. In order for the

solutions of k,

1) to be non-trivial, the coefficient determinant from the

four by four equation system, describing the four beam ends boundary conditions,

has to be zero. The coefficient determinant is a function of the eigenfrequency tan. There is an infinite number of w, that causes the coefficient determinant to

be zero. This determines the eigenfrequencies w for n = 1, Do. The fourth of the

coefficients An, 13,,, can be chosen arbitrarily. A normalizing procedure by

requiring that the sum of the coefficients An, B, C, Dr, for each n has to be equal to 1, is used in this work.

2.'1:2

The governing modal beam equation of motions

The modal solution of w(x,t), expressed by equation (2.2), Is substituted into

equation (2.1) that is multiplied by TPm(X) and integrated of the length of the beam

0,(x) is the eigenfunction of an arbitrary beam vibration mode in. By usingthe

orthogonality conditions of the beam eigenfunctions, the governing modal beam equation of motions of vibration mode m becomes:

L,

2 c(t)

.111,,,,nicin (0+ C,,,,a,,,(t): + Matig(t)

I

IP,(x)dx =

j p(x,

w ,t)11),,(x)dx (2.11)

La _c(t)

- 2

M is the modal mass coefficient and Cm, is the modal restoring coefficient 2c(t) is an approximation of the wetted length of the beam. How to determine

c(t) will be focused on in chapter 5. Generally, and can be written as,

respectively:

Mm,, =

It'

ERIC(x)Ibrt(x)dx (2.12)

An, are

X

Figure 2.4: An infinitesimal timoshenko beam element,

2

f

al'On (X)

Cmn =

E I 21,,,,(x)ctx

ax4 (2.13),

By utilizing the orthogonality conditions of the eigenfunctions

rP(x), it can be

shown that Mum and Cum are equal to zero for in 7l. Details about the

orthogo-nality conditions of the eigenfunct ions for the Euler beam arerevealed in Appendix

A. It may be shown as well from the orthogonality conditions that Cm, ,= curl M,,,.

2.2

The single Timoshenko beam model of the

wetdeck

The force and the moment equilibrium of the infinitesimal Timoshenko beam

el-ement shown in Figure 2.4 are satisfied through the following governing coupled

beam equations of motions (see for instance Timoshenko 1974 or Clough and

Pen-zien 1975): MB (111(X,00 79(0) + GA,

(8

13(x, t) 82w(x, t) Ox ax2

=

w

0

11

4Q+aQ/ax dx

M+3M/ax dx

dx

2 p(x, (2.14)

M Br' ,j(x, t) + GA,

((x

t) aw(x, E1.82

(x't)

= 0 (2.15)

ax a x2

G is the shear modulus. A, is the shear area divided by the width of the flange and r is the mass radius of gyration of the Timoshenko beam element. The remaining symbols are defined in connection with the Euler beam equation of motions (see equation 2.1). /.3(x ,t) is the slope of the deflection curve due to bending. The total

rotation angle at') of the beam element is related to the shear angle s(x, t) as well as to 0(x, t) through the relation (see Figure 2.4):

aw(x ,t)

= rys(x ,t) + 13(x ,t) (2.16)

ax

The solutions of w(x ,t) and /3(x, t) are expressed in terms of the beam's "dry" normal modes:

w(x,t) E an(t)7pn(x)

E an(t)-0(x)

n=1 n.1

3(x, t) = E an(t)(x) E an(t)¢n(x)

(2.18)

n=1 n=1

11)(x) and On (2) are the eigenfunctions of vibration mode n. On(x) is related to 'On (x) through the coupled equations of motions (2.14) and (2.15) with 14(t) and

-p(x, w. t) set equal to zero. One should note that the same principal coordinate

an(t) applies for both w(x, t) and 0(x, t) in equations (2.17) and (2.18). A detailed proof of this is revealed in Appendix C. The four beam end boundary conditions needed to determine the eigenfunctions On(x) and On(x) as well as the

eigenfre-quency- wn of vibration mode ri are:

LB

W(X,t) = 0 for

x = ±-2

Ice3(x t) ± a 43(x =

0LB

for

X =

±-EI

x 9

The beam ends boundary conditions are physically interpreted as no vertical de-flections and continuity of the bending moment at the beam ends.

(2.17)

(2.19)

2.2.i

The beam eigenvalue problem

To derive expressions of ibn(x) and q5(1-) one needs to solve the eigenvalue problem

characterized by equations (2.14) and (2.15) together with the beam ends boundary conditions in equations (2.19) and (2.20). In the eigenvalue analysis,p(x,w, t) and 1.79(0 are set equal to zero. The eigenfunctions are obtained as the solutions of a 'coupled homogenous system of differential equations (see for instanceTimoshenko

1974, Clough and Penzien 1975 or Hildebrand 1976). The solutionsof w(z-, t) and 13(x, t) are written in general form as:.

w(x,t) =

(2.21))

= Conewa e't

(2.22)

Co_ and Co, are constants Wn is interpreted as the wave number that is not yet known and i is the imaginary unit. By substituting equations (2.21) and (2.22)

into equations (2.14) and (2.15) two equations are obtained,, from which Co and

Co_ are solved.

(MBu

GA,W2)

[ GA,Wn

Con] _11 0 I

_(2.23)

GAsWn

HA/87.24 ± GA, A EIW;D

Co, i

[ 0

In order for the solutions of Co, and Con to be non-trivial the coefficient

determi-nant in equation 0.23) has to be zero. Thus:

W79. wn2,(Kn2 r2(4) Kn2r24

4.=i0

(2.24)

Here

KT! = MBwn2

GA,fl

(2.25)

an4- = AI/34 (2:26)t

El

Equation (2.24) is the relation between the wave number Wn and the

eigenfre-quency can and is therefore interpreted asthe beam dispersion relation. The four

V.A4G7,34,11:

Li)n < is implicitly assumed in the!expression of qn. For ca;,

,

is,

Aiinr2 MBrn expressed as

\

+ r2a;4 (K,?

r242

2 4 2 -r ;an 2.31)

Each of the wave numbers described, in equations (2.27) and (2.28) will, together with the assumptions made in equations (2.21) and (2.22), express possible solu-tions of the eigenfwacsolu-tions. The eigenfancsolu-tions are then written as alinear

combi-nation of each of the possible solutions. This leads to the following expressions, of

(x) and q5(x) for wn <

An sin(pnx) + Bn cos(px) + sinh(qnx) + D. cosh(qx) (2.32)1

95,-,(x) :th

En sin(px)+ F cos(px)+ Gn sinh(qnx) + Hcosh(qx) (2.33)

'For wn >

wal),(2) ±iph (2.27)

±qn

for

< v mi,r2/ GA

wp),(44p

The real wave numbers pn and

±igh qn are: (2.28) for wh >

Vera

Pn =

and

+ r2a4n r2a4 2 n -4 _(2.29) 2 (km qn K,2, + r1/4:1, -r r2a,i) 2

+

(2.30) 2 (K,2, 2 >

=

2 =

Amsin(pz) + B,, cos(p,x) +

sin(q,x)+ Dncos(q;)

(2'.34)

0.(x)

.= En sin(p,,x) + Fn cos(p,x) Gnsin(q,,x) Thecos(q,x) (2.35)

The coefficients En, En, Cn, Hn are related to. the coefficients through

the coupled beam equations of motions (2.14) and (2.15) with p(x, w, t) and Vg(t)

set equal to zero. The coefficients An, En, Cm, Dn together with the eigenfrequency

44, are determined in order to satisfy the four beam end boundary conditions in equations (2.19) and (2 20). Non-trivial solutions of the four coefficients are

achieved when the coefficient determinant of the four by four equation system, expressing the beam ends boundary conditions, is set equal to zero. There is an infinite number of w,, that causes the coefficient determinant to be zero. This,

determines the eigenfrequencies wn for Ti = 1, oo. To each con, three of the four

coefficients An, Bn, C, D are uniquely determined, while the fourth may be

chosen arbitrarily. Here, the fourth coefficient is selected in order to normalize the eigenfunction Ihr,(x),so that the sum of the coefficients An, Bn, C, D,, is equal to 1 for each it

2.2.2

The governing modal beam equation of motions

The normal mode formulations of w(x, t) and 0(x, t) described by equations (2.17)

and (2.18) are substituted into equations (2.14) and (2.15) that are multiplied by tpm(x) and 4),(x), respectively. lb, (x) and Ø(x) are the eigenfunctionsof an

arbi-trary beam vibration mode in. The arising equations are added and integrated over

the length of beam. By using the orthogonality conditions of the eigenfunctions

the governing modal beam 'equation of motions for vibration mode in becomes:,

c(t)

Mnuna,(t)

'C,,ani(t) + Mstc,(t)

f '0,(x)dx

=

f p(x,w,t)71),,,(i)dx

La -c(t)

andCra' denote the generalized mass and restoring coefficients, respectively,.

Mtn. and Cm are expressed as:

=

+

+

2

Mmn = MB

[Oni(x)7,14t(x)+ r20,(x)¢,i(x)]dx (2.37) La 2 a 6n( x)

f

El

z cbm(x) ax2 2 7Prn x ) '4L (I) +G

As(a

ax

q5m(x))a

(

ax 0,,(x))1dx (2.38)

Due to the orthogonality conditions of the beam

eigenfunctions. M and C,

are zero for m7L- m. It also follows from the orthogonality conditions that

=

wm2m. Details about the orthogonality conditions of the beam eigenfunctions

are shown in Appendix A.

2.3

The three Timoshenko beams formulation of the

wet-deck

The set of three Timoshenko beams formulation of the wetdeck has similarities to the single Timoshenko beam model derived in section 2.2. An abbreviated theoretical formulation of the set of three Tirnoshenko beams will therefore be concentrated on in this section. In particular, the differences between the two

formulations are emphasized.

The governing equations of motions of a Timoshenko beam element are shown in

equations (2.14) and (2.15). The assumed normal mode formulations of w(x,t)

and 0 are expressed by equations (2.17) and (2.18). To determine the

eigenfunc-tions 11)7,(x) and 07,(x) as well as the eigenfrequency Lan, four beam end boundary

conditions are required for each of the three beams. This implies that a totalof

twelve beam boundary conditions are needed to determine the eigenf-unctions of

the three Timoshenko beams. Let xl, x2, x3 and x4 be defined as xi

=-x2 = If-, x3 =

.1?- and x4 = - , respectively. The beams' boundary conditions then become:

The superscript j means that w(x, t) and 0(x, 0 are referred to the beam j (see

Figure 2.3). The beams' boundary conditions can physically be interpreted as continuity of the bending moment at the beam ends and no beam deflections at the transverse stiffeners located at x .±L22- and x = ±3+-. Furthermore, continuity of the bending moment as well as the rotational angle /3(x, t) are required at the two transverse stiffeners.

2.3.1 The beam eigenvalue problem

Expressions of the eigenfunctions 71.1/4(x) and 41/2.(x) are derived in a similar way as

for the single Timoshenko beam model. This means that the expressions of the

deflection, the dispersion relation and the wave numbers are expressed by equations

(2.21) to (2.31). One should note that the wave number may be different for the

three beams, since the beams' properties may vary_ However, the eigenfrequency

is the same for the three beams One should also note that the expressions' of

the eigenfunctions are different for the three beams but dependent on each other

through the beam boundary conditions in equations (2.39) to (2.43). Therefore, the

eigenfunctions are written as 1141)(x) and 0,Y)(x) where the superscript j = 1, 2, 3)i

is introduced in order to distinguish between the three beams (see Figure 2.3 on

.page 20). For wn < Atl, 0,(232(x) and øW(') are written as:

7PS23)(x) iginsin(px) + 143) cos(px) + CP.sinh(qx) + D,(e)cosh(qx)

(2.44)

(bW).(x), = gj)sin(p,,i)+

cos(px) + di) sinh(q.x)+ 111j) cosh(qmi).

(2.45) =- 0 = 00+1)(x, 0

for x-= x,x1 (j = 1,2,3)

for 'x = xi (j = 1,2)

(2'.39) (2.40) Bow (x,0 3,30±1) (X ,t) for X = Xj+1 (j = 1, 2)i (2.41) Ox ax .916(1)(x,0

+

0

for _3L-2B (2:42)

El

ax

x=

ko , 8,80)(x,

0 for

z=3LB

(2.43) I00)(x,,t) ax 2 w(j)(x, t) ,t) 2 t)

+

= (j

=

For wry > V Mar'

14j)(x) AW) sin(px) + cos(pnr) + C,c.3) sin (q,x) + D$,-1.) cos (qx)

(2.46)

0(2)(x) = 43) sin(pmx) + FnY) cos(p,x) + sin(qx) + 11;23) cos(qi,x) (2.47)

The coefficients EP) FP) , , HP) are related to the coefficients Al), ,

op

, D,Y) through the coupled beam equations of motions (2.14) and (2.15) with

p(x,w,t) and V9(t) equal to zero. The coefficients AW), BO, CP), DP) and the

eigenfrequency wn are determined in order to satisfy the twelve beam boundary conditions in equations (2.39) to (2.43). The coefficient determinant is required

equal to zero to obtain non-trivial solutions of the twelve coefficients. There is an infinite number of wn that causes the coefficient determinant to be zero. This determines the eigenfrequencies wn for it = 1, cc. To each wn, eleven of the twelve

coefficients A4), B,V), C,Y) , DW) (j = 1,2,3) are uniquely determined, while the twelfth may be chosen arbitrarily. Here, the twelfth coefficient is selected in order to normalize the eigenfunctionV4ii)(x) so that the sum of the coefficients AW), ,

CV), DO (j = 1,2, 3) is equal to 3 for each vibration mode n.

2.3.2

The governing modal beam equation of motions

The normal mode formulations of w(x,t) and (x. t) described by equations (2.17) and (2.18) are substituted into equations (2.14) and (2.15). The arising equations are multiplied by Ibm(x) and Om(s), respectively, added and integrated over the total length of the three beams. If the orthogonality conditions of the

eigenfunc-tions are utilized, the governing modal equation of moeigenfunc-tions of the vibration mode

ya becomes:

3La

a(t)

mn,,a,(t) + c,,,,arn(t) + mi31.79(t) f 7Pm(x)dx = P(x.w ,011),,,(x)dx (2.48)

3LR b(t)

- 2

and Cm,, are called the generalized mass and restoring coefficients, respec-tively. a(t) b(t) is an approximation of the wetted length of the beams. How to

Due to the orthogonality conditions of the beam eigenfunctions, Ann and Cm are zero for in 0ii. It also follows from the orthogonality conditions that Cm,

The orthogonality conditions for the beam eigenfunctions are reproduced in detail in Appendix A for the single Timoshenko beam model. The orthogonality

conditions for the eigenfunctions of the set of three Timoshenko beams can be

derived similarly.

determine a(t) and ib(t) will be discussed in chapter 5. Mm,. and Cmn are expressed

as:

2

MB f

tiPm(Z)0n(X),

x2,0,(x)0(x)idx

(2.49)

3L p

Ginn =

El

at2(z)

±GAsi(atfr on?) _{om(x))(atibn(r)ax}

ø())]d

x (2.50)

=

3 L

3

The hydrodynamic boundary value problem

The hydrodynamic pressure p in the governing modal beam equations of motions (2.11), (2.36) and (2.48) is determined from a hydrodynamic boundary value prob-lem (HBVP). The HBVP used in this work is a generalization of the HBVP Wagner (1932) used. The theory by Wagner (1932) is an asymptotic theory valid for small local angles between the undisturbed water surface and the body. One may ques-tion how well Wagner's theory with a jet flow may express the local fluid flow in the impact area for a large forward speed of the vessel. This has been pointed out by Meyerhoff (1968) and is further discussed in chapter 10.

Some assumptions are made before defining the HBVP. First, the accelerations in

the fluid are assumed to be much greater than the acceleration of gravity g. The

disturbances of the free surface of the waves caused by the side hulls of the vessel

are not considered when solving the HBVP. Further, no airpocket is allowed to be trapped between the water surface and the wetdeck. This will be discussed in chapter 10. Cavitation is not considered, but could actually occur for high impact velocities some time after the initial stage of the fluid flow.

Let 0 be the velocity potential in the fluid due to wetdeck slamming. At each time instant and for a known wetted length, a linearized hydrodynamic boundary value problem for th is defined. However, the development in time of the hydrodynamic boundary value problem is non-linear since the wetted length is a priori unknown and part of the solution of the velocity potential.

First, a basic HBVP for 0 is formulated. Next, a symmetric and an unsymmetric

simplified HBVP are defined. The symmetric HBVP has been applied with the

single beam models. while the unsymmetric HBVP has only been applied with the set of three Timoshenko beams

3.1

The basic formulation of the hydrodynamic boundary

value problem

The hydrodynamic boundary value problem for 0 has been formulated by Wagner

(1932). Here, his formulation is generalized to account for an elastic wetdeck

structure as well. The local angles between the undisturbed water surface and the

wetdeck structure have to be small, in order to apply 'Wagner's theory. Wagner

(1932) models a jet flow at the intersection between the free water surface and the structure. Here, a simplified two-dimensional HBVP, often referred to as the outer solution, is set up to express the fluid flow in the impact region. By outersolution,

it is meant that the details of the jet flow are not accounted for.

Let th I be the velocity potential due to the incident waves. The total velocity

potential 4) is then defined as:

= + Ux +

(3.1)

Assuming irrotational flow and incompressible fluid, the velocity potential th satis-fies the Laplace equation, V20 = 0 in the fluid domain. It is assumed that the fluid velocities due to the potentials 0 and 01 are small. Products of fluid velocities are

then neglected. This means that 17/012 has to be smallcompared to 2-). This is not satisfied close to the edge of the flow and is discussed in chapter 10. Assum-ing gravity free flow as well, the free surface boundary condition of the velocity

potential becomes:

ao ,acb

+ v (3.2)

Ot ax

The kinematic boundary condition on the wetted surface will now be discussed.

Define the function F(x, z, t). so that F(x, z, t) = 0 on the wetdeck structure of

the vessel. The coordinates are referred to the local coordinate system (see Figure

2.2). Then (see for instance Watanabe 1987):

F(x, z, t) = z i73(t) + (x xcoG)775(t) (dwet(x) + w(x, t)) (3.3)

XcoG is the x-coordinate of the center of gravity of the vessel. 7/3(t) and n5(t) are the heave and the pitch motions of the vessel, respectively. n3(t) and n5(t) are

referred to the global coordinate system defined in Figure8.1. dwet(x) is the vertical

distance from the x-axis to the rigid wetdeck structure. By requiring that a fluid

particle on the part of the wetdeck in contact with water stays on the wetdeck, the boundary condition on the wetted length becomes:

OF

+ Vt.

=0

on F(x, z, t) = 0

Equation (3.4) is evaluated and rewritten with respect to as:

173 (t) (x rcoG)715(t)

(u +ao

±aq51 715 (t) wet(x)

aw(x't))

au.,(x,t) ath on F(x, z, t) = 0 (3.5)

at

az

awet (X) is the slopealong the rigid wetdeck. 725(0, ceut(x) and are assumed

sufficient small, so that cos (775(t) awet(x)

at't)) Pt 1.

It is assumed that

&wax and ao,/ax are an order of magnitude smaller than U. This is not satisfied

for wax close to the edge of the flow. There an inner solution has to be defined. This is discussed later in the text. The penetration depth of the wetdeck into the

free water surface is assumed to be small compared to the wetted length. The

boundary condition is then transferred to z = 0 in the local coordinate system.

Thus, the boundary condition on the wetted length becomes:

acbav)(x,t)

= ii3(t)

(x

xcoG)i)5(t) u

(715(t) aivet(x)

az ax

aw(x,t)

a6,

on z = 0 (3.6)

at

az

In order for Wagner's (1932) theory with a jet flow to be valid, the parts of

2

proportional to U have to be small compared to the remaining parts. This is

not satisfied for a large forward speed or a large local instantaneous angle of the

wetdeck. Additionally, this is not satisfied when the vertical vibration velocity

°.-2.--e(eV1 cancels the velocities due to the global rigid ship motions and the vertical

velocity of the undisturbed free water surface. The importance of the forward

speed effects on the flow has been pointed out by Meyerhoff (1968).

(3.4)

az

The free surface boundary condition in equation (3.2) is further simplified. An estimate of the ratio between(1 and atat is Fa. T is a characteristic time scale_Ox and LB is used as the length scale. Fa. will be small for realistic values of U,

T

and LB. This means that Ut is small 'compared to

2 and that the free surface

boundary condition can be approximated by r= 0 or 0 = 0. The free surface

boundary condition is transferred to z = 0.

The hydrodynamic pressure on the wetted length Is estimated by the

pc: terra

in the Bernoulli equation. This follows from the above discussion.,

3.2

The symmetric hydrodynamic boundary value

prob-lem

A simplified two-dimensional symmetric HBVP' can be set up to express. the fluid flow in the impact region.. The boundary conditions for 0 will now be discussed. The kinematic boundary condition expressed by equation (3:6) on the wetted length is further simplified. It is assumed that the crest of a regular wave system hits the

wetdeck at the mid point between two of the transverse stiffeners. The vertical velocities due to the waves are neglected in the water impact region. This is,

reasonable as long as the line of initial contact between the. free surface of the

waves and the wetdeck is located along a wave crest. The wetdeck is assumed to be horizontal (n5(t) r4.1 oe(x)) in the impact region at

the initial impact. The

order of magnitude of the ratio between Uawg,t) and 19 Vt'`) in equation (3.6) is

UT _{UT will be small for realistic values of T and LB. Furthermore, the} LB

variation in space in the impact region of the local vertical velocities

due to the

global rigid ship motions is neglected. The boundary conditions for 0 are then V(t) -r- fv(x,,,t) a Ve(x,t) on the wetted length of the beam and 0 ='0 on

the free water surface. Following the assumptions in the above text, the only contribution to V(t) comes from the global heave and the pitch motions of the

vessel. Vc(x, t) is defined as the effective velocity. 2c(t) is an approximation of

the wetted length of the beam. c(t) is unknown and part of the solution of the impact problem. How to evaluate c(t) is focused on in chapter 5. The simplified

two-dimensional symmetric HBVP is reproduced in Figure 3d.

(t) =0

Figure 3.1': The symmetric hydrodynamic slamming Model.

Close to the edges of the flow(Ix' = c(t)) the outer solutionbreaks down clue to a

singular hydrodynamic pressure. A possible explanation to this is the inconsistent

boundary condition used in the HVBP: On the wetted

length at !xi = c(t)-, the

vertical velocity is directed downwards. Outside the wetted length at Ix' = c(t)+, the vertical velocity is directed upwards. This means that there is a jump in the

vertical velocity at x = ±c(t). Close to the edge of the flow, an inner solution has to be defined. The outer and the inner solutions can be matched in the same

way as shown by Cciinte and Armand (1987) for impact of a rigid and blunt body.

Here, only the outer solution is considered. The reason is that the details of the inner flow do not influence the solution of the hydroelastic problem. However,, it is important for the outer solution to have the correct singular behaviour near

the edges of the flow, in order to match an inner local solution. Therefore, an

abbreviated reproduction of the inner solution is presented in a separate section.

3.3

The unsymmetric hydrodynamic boundary value

prob-lem

For the unsymmetric HBVP as well, the' vertical velocities due to the waves are neglected in the impact region. This is a reasonable assumption for long waves and when the wetdeck hits close to a wave crest. in principle the verticalvelocities due

to the waves can be incorporated in this formulation. 4tr.

-c(t)

k

c(t)

a 4ilaz=m+3w(x,t)at

X

0=0'

A simplified two-dimensional HBVP (see Figure 3.2) IS set up to express the fluid

2

V

=0

b(t)

a(t)

la 4ilaz=v0H-aw(x,t)at

Figure 3.2: The urtsymmetric hydrodynamic slamming model.

flow in the water impact region. The boundary conditions for 0 are

t = V(x,t)+

ii)(x,t) on the wetted length of the beams and 0 = 0 on the free water surface.

9"(x, t) is expressed as (see equation (3.6)):

aw(x,t)]

(x, t) i73

zcoc)175(0

U1715(t) awe& ax (3.7)

Equation (3.7) can be further simplified. The order of magnitude of the ratio

between U_(_lawa:3 and atax,t)et is small as 'discussed in the above text. The variation

in space in the impact region of the term i73(t) (x xcoG)ns(t) is neglected.

Furthermore, the curvature of the wetdeck in the impact region is small so that

a(x) is approximately constant. Then

(x, t) becomes independent of space. Consequently, the boundary condition on the wetted length of the wetdeck is

2 =

V(t) + t) Ve(x,t). How to evaluate the wetted length ,a(t) b(t) will be

focused on in chapter 5.

Similarly as for the symmetric HBVP, this Outer solution breaks down near the edges of the flow (x = a(t) and x = b(t)). For further discussion, references are

made to the previous section.

3.4

The inner solution near the edges of the flow

The inner solutions near the edges of the flow Will now be focused on. By inner

solutions it is meant the details of the flows near x = ±c(t) for the symmetric

0=0

= (x

HBVP and near x = a(t) or x = b(t) for the unsymmetric HBVP. To exemplify

the inner flow solution, the symmetric HBVP is considered and the details about the flow near x = c(t) are investigated.

The inner solution and the asymptotic matching to an outer solution have been considered by for instance Zhao and Faltinsen (1992). However, the origin of the inner solution can be traced back to Wagner (1932). Here, only a brief introduction to the inner flow solution is presented. The derivations are more or less the same as in Zhao and Faltinsen (1992).

Assume for simplicity that the vertical velocity is constant in space over the wetted

length of the beam and denote it by Ve(t). The outer solution of the velocity potential on z = IV and for 1x1 < c(t) is then expressed as:

thout = e(t)\ c2(t) x2 (3.8)

A first term inner expansion (x + c(t)) of the outer solution is evaluated as:

(14,,TS =V e(t)/2c(t)/c(t) x (3.9)

This means that the velocity potential near the edge of the flow is proportional to the square root of the distance from the edge.

-The inner flow velocity potential Oin on z = 0- is expressed in parametric form as (see Zhao and Faltinsen 1992):

) 6 dc(t)

dc(t

0,Th =

dt 7r + 111171 1 71) + dt c(t))

6 is the jet thickness that is not yet known and In is a parameter that is related

to the local coordinate x by:

x c(t) = ln 171 4 I I

H +

5) (3.11)

An outer expansion of the inner solution jr(1) of the velocity potential is achieved by

letting x c(t)

co

_(HI oo). By substituting equation (3.11) into equation

(3.10) and let co, it follows that:

(3.10)

V

-r

-

-Aezp

₌

_4dc(t)

_VI

= 4

dc(t) _{,/c(t) x}

dt iv dt ir

(3.12)

In equation (3.12), it is used that x c(t)

471 when

cc. This follows from equation (3.11).

The outer expansion of the inner solution in equation (3.12) has to match the inner expansion of the outer solution expressed by equation (3.9). This determines the jet thickness 6 as:

6 = 71-V2 (t)c(t)

8('dcd(tt))2

(3.13)

A solution of the velocity potential that is valid in both regions is achieved as the sum of the outer and inner solutions (/)t and Om, respectively, minus the common

term expressed by equation (3.12) with the jet thickness due to equation (3.13).

Similarly, a composite solution of the hydrodynamic pressure pi& valid in both flow

regions is achieved as:

Ptot = Pin + Pout Pcorn (3.14)

Pin, Pout and Pcurn are the innerflow hydrodynamic pressure, the outer flow

hydro-dynamic pressure and the common hydrohydro-dynamic pressure terms, respectively. pin and pcon, are expressed as, respectively (see Zhao and Faltinsen 1992):

Pin -= 2p

Pconi

p17(t)

V2c(t)(c(t) x)

In the outer fluid flow. Pin cancels peso, so that ptot = pout.

Pout cancels peon, so that Apt = Pin.

(d0)2

%ATI

dt )

(1 )2

c(t)?

(3.15)

(3.16)

In the inner fluid flow,

In this work, details of the inner fluid solution are not considered. The reason is

that the inner flow solution will not influence the hydroelastic response. The only reason that the inner flow solution is focused on in this section is to emphasize that

fri --+

±

the outer flow velocity potential near the edge of the flow has to be proportional

to the square root of the distance from the edge. By using the pTI2r term in the Bernoulli equation, this causes a hydrodynamic pressure that will match the inner

solution.

3.5

The acoustic approximation

The fluid will be compressible at the initial stage of the impact. This means

that

the two-dimensional Laplace equation cannot be applied as the governing equation for the fluid flow. The two-dimensional wave equation should rather be used. This means that the velocity potential 6 has to satisfy:

(920 a20 1 520

(3.17)

ax2 az2 c2 fag

Ce is the speed of sound in water. If the velocity potential is harmonic in time. equation (3.17) is better known as the Helmholtz equation.

Equation (3.17) can be simplified by assuming that the variation in the x direction

of the velocity potential is small compared to the variation in z direction and

the variation in time. This means that the first term inequation (3.17) is small

compared to the other terms. It is assumed that the variation of the vertical velocity on the wetted length is small, so the boundary condition on the wetted length can be approximated by

yen -yew is the mean vertical velocity

in space over the wetted length. Equation (3.17) is then reduced to the

one-dimensional wave equation:

a2th 1 320

(3.18) az2

at2 =

This means that the solution of the velocity potential has to be of the form 0(u) where it = z cet. The positive sign is used since one is only interested in the

acoustic waves that are propagating in the negative z direction. The hydrodynamic

pressure, also known as the one-dimensional acoustic pressure pac(t), on the wetted

length is then evaluated as:

, ad) ao au ad) au

This means that at the initial stage of the impact, acoustic effects can be accounted

for in a simplistic way by setting the hydrodynamic pressure equal to the

one-dimensional acoustic pressure.

4

The solution procedures of the HBVP

The hydrodynamic boundary value problems (HBVP) in Figures 3.1 and 3.2 have

been solved analytically and the hydrodynamic loading is expressed in terms of

analytical functions in this work. Additionally, a pure numerical solutionprocedure

has been applied for the symmetric HBVP in Figure 3.1.

The HBVP in Figure 3.1 has been solved by using four different approaches. The first is a simplified method where the vertical velocities in the water impact region are assumed independent of space and only dependent of time. That particular solution approach has only been combined with the single Euler beam model for the wetdeck. The solution procedure is called the constant space velocity approxi-mation. The second solution method is a pure numerical solution approach based upon Green's second identity. Inthe third and the fourth solution procedures the

vertical velocities on the wetted length of the beam are rewritten in terms of a

Fourier cosine series. In the last two solution methods the Euler beam model and the single Timoshenko beam model have been used for the wetdeck, respectively.

The solution procedures are called the Fourier approximations. The unsymraetric HBVP in Figure 3.2 has only been combined with the wetdeck model consisting

of the set of three Timoshenko beams An overview of the different solution pro-cedures is shown in Table 4.1. Each of the five solution propro-cedures will now be focused on in detail.

Table 4.1: An overview of the solution procedures.

HBVP

Structural model

Single beam (symmetric) Three Timoshenko beams

Euler Timoshenko Symmetric Unsymmetric

Constant velocity 1

Numerical solution

Fourier approximation 3 4