Hydro-Elastic Criterion for Practical Design
Hannes Bogaert 1), Mirek Kaminski 2)
1)
MARIN, Hydro-Structural Services, Wageningen, Netherlands &
Delft University of Technology, Ship Structures Laboratory, Delft, Netherlands
2)
MARIN, Hydro-Structural Services, Wageningen, Netherlands
Abstract
A new hydro-elastic criterion which can be used in
practical design of marine structures is proposed in this
paper. By application of this criterion a designer will be
able to decide whether a hydro-elastic problem can still
be addressed by the standard two-step practice whereby
in the first step the hydrodynamic load is determined on
a rigid model of the structure and in the second step the
accompanying structural response is calculated.
The criterion has been developed based on nondimensional numerical calculations using a simple
lumped-mass model of a flexible cone impacting the
free water surface. The penetration velocity is hereby –
in contrast to the assumptions so far in previous investigations – not prescribed, but results from the numerical
model.
In addition the paper will demonstrate that the standard
two step practice does not essentially lead to conservative results toward the response of the structure.
Based on the findings of this paper the research is continued at MARIN and experimental validation is under
way.
Keywords
Hydro-elastic analysis; quasi-static analysis; fluidstructure interaction; design philosophy; slamming;
sloshing; response of marine structure
Introduction
The prediction of the response of ships and other marine
structures caused by wave loads is an important issue in
modern design practice. The prediction methodologies
and capabilities are hereby continuously challenged by
the growing marine transport market. After all these
markets evolve to larger scales and to sever environments whereby the structural integrity and the passengers comfort need to be guaranteed.
The common or standard design practice for the prediction of this response is denoted as the quasi-static analysis. In the quasi-static analysis the hydrodynamic force
is coupled with the rigid body motions and the resulting
hydrodynamic force and the resulting inertia force are
statically imposed on the structure. This implies however that the quasi-static analysis first of all neglects the
coupling between the structural vibrations and the hydrodynamic force. In addition this analysis neglects the
influence of the dynamic properties (natural frequencies, modal shapes and modal damping) in the determination of the structural response and according to the
first omission also in the coupling with the hydrodynamic force.
The hydro-elastic design practice on the other hand
intends to incorporate this complex chain of cause and
effect of the hydrodynamic force, the motions of the
structure (rigid body motions as well as structural vibrations) and the dynamic properties of the structure in the
prediction of the response.
Considering the complexity but at the same time the
correctness of this hydro-elastic analysis the question is
however in the viewpoint of the design process, in the
viewpoint of a business characterized by a short engineering time when it is required to carry out a hydroelastic analysis and what the consequences are of still
applying the quasi-static analysis. This question is addressed at MARIN from a systematic, step by step research philosophy. This approach demands for simplifications and abstractions in the first steps of the research
but allows for the identification of the involved processes, their influences and their mutual relations. The
focus is hereby on only impulsive wave loads such as
slamming and sloshing loads.
In this paper a new hydro-elastic criterion which can be
used in practical design of marine structures, is proposed. By application of this criterion a designer will be
able to decide whether a hydro-structural problem can
still be addressed by the standard quasi-static analysis.
The formulation of this criterion is a progress in answering the above mentioned question and does not imply
that the question can generally be answered as a theorem. After all the criterion contains some simplifications
which are justified and essential for practical design
applications.
First of all the basic principle of the hydro-elastic criterion is introduced. The basic principle translates a practical design application in several impact situations. The
necessity and the effect of hydro-elasticity for these
impact situations is at this stage of the research evaluated by the necessity and the effect of hydro-elasticity
for the lumped-mass model of a flexible cone. The
derivation of the necessity and the effect of hydroelasticity for the lumped-mass model of this flexible
cone is addressed in this paper and is associated with
previous investigations of e.g. Faltinsen (1999) and
Bereznitski (2003).
In order to express the necessity and the effect of hydroelasticity for the impact situations specified by the basic
principle in function of the necessity and the effect of
hydro-elasticity for the lumped-mass model of the flexible cone, a relation between these impact situations and
the flexible cone is established and discussed in the
paper. This relation could only be accomplished by
simplification of these impact situations. Due to these
simplifications the applicability of the hydro-elastic
criterion will accordingly be limited at this stage of the
research. Nevertheless the basis of the criterion is
founded and further research needs to reveal the required accuracy in relation to the usability of the hydroelastic criterion.
Basic Principle of Hydro-Elastic Criterion
The practical design applications under consideration
are the prediction of the response of ships and other
marine structures caused by external impulsive loads
(slamming and green-water on deck) as well as by internal impulsive loads (sloshing). Several types of
slamming are herby considered, namely forward bottom
slamming, aft body slamming, flare slamming, wave
slap and wet-deck slamming. In the proposed hydroelastic criterion the marine structure is divided into
successive structural levels, namely from a local structural level to a global structural level. This division
depends on the type of impulsive load and the location
of impact. An overview for the response of a ship
caused by the above mentioned types of slamming, by
green water on deck or by sloshing is given in Table 1.
The basic principle of the hydro-elastic criterion can
now be formulated as followed: The evaluation of the
response of one of the above mentioned structural levels
takes the influence of the remaining part of the accompanying successive level into account. In contrast to the
considered structural level the remaining part of the
accompanying successive level is hereby considered to
be rigid (non-deformable).
If for example the response of a panel caused by forward bottom slamming is investigated than the influence of the remaining of the double bottom which includes this panel, is taken into account. The panel is
hereby considered to be flexible (deformable) while the
remaining part of the double bottom is considered to be
rigid.
Since for example a plate is supported by stiffeners
forming together a panel and this panel is supported by
a double bottom, the plate is in principle not only supported by the reaming of the panel which includes this
plate but as well by the adjoining panels and other parts
of the double bottom. Following this line of reasoning
this implies in principle that the plate is supported by all
other parts of the ship. However in order to simplify the
problem and the utilization of the hydro-elastic criterion, the boundary of influence is set by the basic principle to the adjacent successive construction level which
includes the previous. In Fig. 1 the division into structural levels in case of forward bottom slamming is illustrated. The basic principle of the hydro-elastic criterion
is illustrated for the evaluation of the response of a
plate, a panel, a double bottom and a section in case of
forward bottom slamming in Fig. 2. This results in four
different impact situations wherefore the necessity and
Table 1: Division of a ship in successive structural levels
local
↓
global
local
↓
global
local
↓
global
forward bottom
slamming
plate
panel
double bottom
section
ship
aft body slamming
flare slamming
plate
panel
double bottom
aft peak
ship
plate
panel
bow section
ship
wave slap
e.g. at the bow
plate
panel
bow section
ship
wet-deck slamming
green water
plate
panel
double deck
section
ship
plate
panel
section
ship
sloshing
containment system:
- one CS1 box
- one NO96 box
- one MARIK III box
panel covered with containment system
double side shell covered with containment system
ship
Fig. 1: Forward bottom slamming - division into
structural levels
the effect will be evaluated by means of the lumpedmass model of the flexible cone.
Lumped-Mass Model of Flexible Cone
In the last decade the effect of hydro-elasticity on the
local response due to impulsive loads were studied
theoretically and experimentally by Haugen and
Faltinsen (1999), Faltinsen (1999) and Bereznitski
(2003). All conclude that hydro-elasticity is important
for small values of the ratio between the duration of
the loading and the longest natural period of vibration. This ratio is directly related to the impact velocity, the deadrise angle, the structural stiffness and air
entrapment. However the precise values of this ratio
for which a hydro-elastic analysis for the beam model
(Haugen and Faltinsen (1999) and Bereznitski (2003))
or the orthotropic plate model (Faltinsen (1999)) is
important and the error made by the quasi-static
analysis could only be established for imposed global
velocities which are constant during impact. The
physical implication of a constant global velocity
during impact is a fluid impact of a beam or an
orthotropic plate which is supported by an infinite
large mass. This physical incorrectness and the close
link between the retardation of the velocity of the
object of impact, the mass and the deadrise angle of
the object of impact and the hydrodynamic force on
the object of impact underline the importance of a
hydro-elastic model whereby the velocity of the structure during impact is not imposed to the model but
results from the numerical model. In other words in
order to correctly specify the necessity and the effect
of hydro-elasticity a model is required with full coupling between the motions (rigid body motions and
structural vibrations) and the hydrodynamic force.
The creation of this model inherently calls for simplification of the fluid impact problem. Simplification is
possible at three levels: the object of impact, the dynamic structural properties of the object and the conditions of impact.
Fig. 2: Forward bottom slamming - evaluation of the
response of plate, panel, double bottom and section
according to the basic principle of the hydro-elastic
criterion
The above mentioned previous investigations considered a beam or an orthotropic plate as object of impact. Although these objects of impact seem to be
simple in nature compared to a marine structure, the
investigations could not lead to conclusive answers
with respect to the main question of this paper. As a
consequence in this investigation the object of impact
is further simplified toward a three-dimensional cone
which has the mathematical benefit of being axisymmetric.
The dynamic structural properties of the cone are in
addition simplified toward a lumped-mass model with
two degrees of freedom, whereby the flexibility of the
cone is represented by a linear spring. The degrees of
freedom are in the direction perpendicular to the free
surface. The lower part of this system is an undeformable cone which directly impacts the free surface. The upper part which is suspended over the
lower mass by way of the spring, is as well unde-
formable. A schematic presentation of the lumpedmass model of the flexible cone is given in Fig. 3.
The conditions of impact form the final level of simplification. The hydro-elastic model considers an
impact with no air entrapment, no compressibility of
the fluid and an axisymmetric impact. Furthermore
only water entry stage and not the possible water exit
stage is investigated.
Fig. 3: Schematic presentation of the lumped-mass
model of the flexible cone
The dynamic equations of this hydro-elastic model
are formulated as followed:
⎧m1z1 = m1g − k ( z1 − z2 ) − Fimpact ( z1, z1, z1 )
⎪
⎪m1z2 = m2 g + k ( z1 − z2 )
⎪
⎨ With application of following initial conditions
⎪ z ( 0) = z ( 0) = 0
2
⎪ 1
⎪ z1 ( 0) = z2 ( 0 ) = Vimpact
⎩
(1)
Whereby :
z1 : displacement of lower part of cone
z2 : displacement of upper part of cone
m1 : mass of lower part of cone
m2 : mass of upper part of cone
(
k : spring stiffness
g : acceleration of gravity
)
Fimpact z1 , z1 , z1 : impact force on lower part of cone
In order to specify the necessity and the effect of
hydro-elasticity not only a hydro-elastic analysis is
required but as well a quasi-static analysis of the cone
impacting the free surface. In a quasi-static analysis
the impact force is evaluated on a rigid model of the
cone and is accordingly only coupled with the rigid
body motions. The rigid counterpart of the lumpedmass model of the flexible cone is the impact of an
undeformable cone with mass (m1+m2) and only one
degree of freedom in the direction perpendicular to
the free surface. The equation of motion of the rigid
model is written as followed:
⎧( m + m ) z = ( m + m ) g − F
1
2
impact (z, z, z)
⎪ 1 2
⎪With application of following initial conditions
⎨
⎪ z (0) = 0
⎪ z ( 0 ) = V
impact
⎩
(2)
Once the impact force and the acceleration of the
rigid body are specified, they are statically imposed
on the lumped-mass model of the flexible cone. Consequently it is assumed that at each time instant the
impact force and the inertial force are in equilibrium
with the elastic force. The structural deformation (z1z2) is accordingly specified as followed in the quasistatic analysis:
⎧
m2 ( z−g)
⎪ z1− z2 =
k
⎪
⎪or
⎨
z − g )+ Fimpact
⎪ z − z =− m1 ( ⎪1 2
k
⎪
both
resulting
in
the
same deformation
⎩
(3)
Kim et al. (1996) as well as Lafrati et al. (2000) studied also the impact of a lumped-mass model with two
degrees of freedom onto the free surface. However
the lower part of the model is a two-dimensional,
undeformable wedge in these studies. Moreover both
investigations where not aimed at answering the main
question of this paper.
The axisymmetric hydrodynamic problem of Eq. 1
and Eq. 2 is evaluated in the framework of the Wagner theory and is based on the work presented in
Scolan and Korobkin (2003) which is on its turn
founded on Wagner (1932), Armand and Cointe
(1986) and Zhao and Faltinsen (1993). In this frame
work the hydrodynamic force is defined as followed:
c( t )
Fimpact ( z , z , z ) = 2π ∫ pouter ,root ( r , t ) rdr
0
∞
+2π ∫ p jet ( r , t ) rdr
c ( t )+
(4)
Whereby:
r : radial position on cone at which pressure
is evaluated
c ( t ) : radial position of contact line of fluid
domain on cone defined as
4
z
π tan β
(
pouter ,root : pressure at contact region 0 ≤ r ≤ c ( t )
(
p jet : pressure in jet region r > c ( t )
)
)
The pressure at the contact region and the pressure in
the jet region are hereby mathematically expressed as:
pouter , root ( r , t ) =
2
π
the dynamic equations of the hydro-elastic model can
be rewritten in non-dimensional form as:
2
ρ c ( t ) − r z
2
(
+
ρ
π
z
dc
dt
τ2
1
2
(1 + τ 2 )
2
(
1−
r
c (t )
2
−
2
2
1−
)
r
c (t )
(5)
1
dc 2
p jet ( r , t ) = 2 ρ ( )
dt
)
⎧C z ' = C G − C z ' − z ' − F '
impact
⎪ m1 1 m1
2
k 1
⎨
⎪α Cm1z '2 = α Cm1G + Ck z '1 − z '2
⎩
1
dc 2
+2 ρ ( )
dt
τ2
1
2
(1 + τ 2 )
(6)
Whereby:
τ : specified by the following equations
⎧
δ
⎪r − c ( t ) = π ( − log τ − 4 τ − τ + 5)
⎪
2
⎨
1 z c ( t )
δ
=
⎪
2π dc 2
⎪
( )
⎩
dt
(7)
Eq. 4~7 are valid for the impact force on the rigid
model (Eq. 2) as well as for the impact force on the
lumped-mass model (Eq. 1). They differ only by the
independent variables, namely the penetration depth,
the velocity and the acceleration of the rigid cone as a
whole respectively of the lower rigid part of the flexible cone.
By introducing the following four non-dimensional
parameters:
1. The ratio of the gravity force relative to the
buoyancy force
m
(8)
Cm1 = 1
3
ρR
2. The ratio of the deformation energy relative to
the kinetic energy of the fluid
k
Ck =
(9)
2
ρVimpact R
3. The ratio of the gravity force relative to the inertial force. This parameter is the reciprocal of the
square of the Froude number.
gR
(10)
G=
2
Vimpact
4. The ratio of the upper mass of the cone relative
to the mass of the lower part of the cone.
m
α= 2
(11)
m1
(
)
(12)
Likewise the quasi-static model given by Eq. 2 and
Eq. 3 can be expressed as function of these four nondimensional parameters. Since moreover the impact
force (Fimpact) is a function of the deadrise angle of the
cone (β), it can be concluded that the hydro-elastic
and the quasi-static model are characterized by the
following parameters: Cm1, Ck, G, α and β. In the
derivation of the hydrodynamic force (Eq. 4~7) the
influence of gravity is however neglected in the dynamic boundary condition at the free surface and
consequently as well in the linearised Navier-Stokes
equations. This implies that Eq. 4~7 can only be
combined correctly with either Eq. 1 or Eq. 2~3 if the
influence of gravity is as well neglected in Eq. 1 and
Eq. 2~3. In other words the derived hydro-elastic and
quasi-static models are applicable for impact situations whereby the non-dimensional parameter G is set
to zero.
Necessity and Effect of Hydro-Elasticity for
Lumped-Mass Model
The investigation of the main question demands for
the comparison of a hydro-elastic analysis with a
quasi-static analysis in order to evaluate the necessity
and the effect of the application of a hydro-elastic
analysis. The indicator – also denoted as the operational definition - used in this research to observe the
necessity and the effect of hydro-elasticity is the reduction in the maximum of the absolute value of the
deformation of the cone. The deformation of the cone
is indicated by the deformation of the spring, namely
z1-z2. Moreover the reduction is indicated by considering the difference between the deformation obtained
by applying a quasi-static analysis and by a hydroelastic analysis relative to the quasi-static analysis.
The operational definition is accordingly formulated
as:
reduction =
(
)
(
max z1 − z2
− max z1 − z2
q−s
h −e
(
max z1 − z2
q−s
)
)
(13)
Whereby:
q − s : basedon quasi-static analysis
h − e : based on hydro-elastic analysis
This operational definition is close to the one applied
in Bereznitski (2003). However in Bereznitski (2003)
the deformation was indicated by the deformation at
the center of the beam. The operational definitions in
Haugen and Faltinsen (1999) and Faltinsen (1999)
were indicated by a non-dimensional maximum stress
and did not directly include the difference in method
of analysis, namely a hydro-elastic or quasi-static
analysis.
Since the Wagner framework considers the initial
stage of the impact of a blunt, convex body onto an
inviscid, irrotational and incompressible flow with no
air entrapment, the defined operational definition can
only correctly be evaluated by the developed models
(Eq. 1~7) if the following conditions are fulfilled:
1.
2.
5 ≤ β ≤ 20
2
∂ z1 − z2
= 0 take place before one of the
2
∂t
following events occur:
a. the flow becomes compressible
b. the velocity of the cone changes direction
which would result in a water exit instead of
a water entry event
c. the jet reaches the chines of the cone. As a
consequence the influence of flow separation at the chine of the cone on the response
can not been evaluated
These conditions define the numerical domain of the
developed hydro-elastic and quasi-static model and
accordingly limit the possible combinations of the
non-dimensional parameters (Cm1,Ck,α,β) that can be
investigated.
The values of the operational definition evaluated for
combinations of these non-dimensional parameters
which are included in the numerical domain are first
of all expressed as function of the following nondimensional hydro-elastic ratio (HER):
HER =
rise time of impact force on rigid cone
(14)
longest natural period of dry cone
The rise time of the impact force (Trise time) is defined
as the time between the moment of impact and the
time at which the impact force attains its maximum
value. Furthermore the longest natural period of a dry
cone (Tnatural period) is specified as:
1 m1m2
Tnatural period = 2π
k m1 + m2
(15)
The resulting relation between the values of the operational definition and the hydro-elastic ratio (Eq.
14) is visualized in Fig. 4 for three different values of
the parameter α. Although a different relation between the operational definition and the hydro-elastic
ratio is applicable for each value of the parameter α,
each relation can be subdivided into the following
three regions:
1. Region of positive reduction: In this region the
maximum of the absolute value of the deformation of the spring of the cone is larger when it is
evaluated by a quasi-static analysis instead of a
hydro-elastic analysis. In other words in this re-
gion the quasi-static analysis overestimates the
deformation.
2. Region of negative reduction: In this region the
maximum of the absolute value of the deformation of the spring of the cone is smaller when it is
evaluated by a quasi-static analysis instead of a
hydro-elastic analysis. In other words the quasistatic analysis underestimates the deformation
3. Region of zero reduction: In this region the
maximum of the absolute value of the deformation of the spring of the cone remains the same
when it is evaluated by a quasi-static analysis instead of a hydro-elastic analysis. In other words
in this region the quasi-static analysis correctly
defines the deformation.
Fig. 4: The reduction of the maximum deformation
as function of the ratio between the rise time of the
impact force on a rigid cone relative to the longest
natural period of a dry cone
The values of the hydro-elastic ratio which separate
the region of positive and negative reduction and the
region of negative and zero reduction are as well
given in Fig. 4 as respectively point 1 and point 2.
Collecting these values of the hydro-elastic ratio for
all other investigated values of the parameter α results in Fig. 5. This figure presents the three regions of
reduction as function of the hydro-elastic ratio and the
parameter α whereby the corresponding values of the
reduction are indicated by contour lines.
Finally the hydro-elastic ratio is formulated as function of the non-dimensional parameters (Cm1,Ck,α,β).
To this end this ratio is now expressed in terms of the
non-dimensional forms of Trise time and Tnatural period.
Given Eq. 15 the non-dimensional longest natural
period is hereby related to the parameters Cm1 and Ck
in the following way:
Vimpact
'
T natural period =
Tnatural period
R
= 2π
α
(1 + α )
Cm1
(16)
Ck
The non-dimensional rise time of the impact force on
a rigid cone is covered by Eq. 2, Eq. 4~7 and can be
defined within the numerical domain of the developed
quasi-static model by the following relation:
(
)
b
'
T rise time = a1 (1 + α ) Cm1 1
⎧
tan β
⎪a1 ≈
2
with ⎨
0.195π
⎪b ≈ 1 3
⎩1
(17)
value of the deformation but as well to the oscillating
behavior of the deformation. This investigation of the
extent of these consequences and when they can be
neglected, has not been undertaken in these first steps,
but will be essential for further development of the
hydro-elastic criterion. As a consequence at this moment the stringent definition of the word required is
applied for formulation of the hydro-elastic criterion.
With Eq. 16. and Eq. 17 the hydro-elastic ratio can
accordingly be approximated as:
HER ≈
1
12
53 C
k tan β
13
α
Cm1
(1 + α )
(18)
Fig. 6: Necessity and effect of hydro-elasticity for
lumped-mass model of flexible cone expressed as function of the non-dimensional (Cm1,Ck,α,β)
Relation between impact situations of hydroelastic criterion and impact situations of flexible cone
Fig. 5: Three regions of reduction as function of the
hydro-elastic ratio and the parameter α
The combination of the description of the three regions of reduction as function of the hydro-elastic
ratio and the parameter α (Fig. 5) with the expression
of the hydro-elastic ratio as function of the nondimensional parameters (Eq. 18), results in Fig. 6.
This figure directly denotes as function of the nondimensional parameters (Cm1,Ck,α,β) to which extend the quasi-static analysis either overestimates,
underestimates or correctly defines the impact of the
lumped-mass model with respect to the hydro-elastic
analysis. Fig. 6 expresses accordingly the necessity
and the effect of hydro-elasticity for the lumped-mass
model of the flexible cone within the numerical domain of the developed models. This statement is only
true if the most stringent definition or interpretation
of the word required is applied, namely it is required
to carry out a hydro-elastic analysis if the quasi-static
analysis either over- or underestimates the deformation of the spring. However in the viewpoint of the
hydro-elastic criterion the word required should as
well taken the extent of the consequences into account. After all the effort of a hydro-elastic analysis is
only required if the consequences of not applying a
hydro-elastic analysis can not be neglected. The extent of a consequence is hereby not only related to the
The necessity and the effect of hydro-elasticity for the
impact situations specified by the introduced basic
principle of the hydro-elastic criterion can finally be
evaluated as function of the derived necessity and the
effect of hydro-elasticity for the lumped-mass model
of the flexible cone, by specification of a relation
between these impact situations and the flexible cone.
This relation can come about by application of the
three levels of simplification, namely the object of
impact, the dynamic structural properties of the object
of impact and the conditions of impact, as used in the
derivation of the lumped-mass model of the flexible
cone.
1. Object of impact
Since the lumped-mass model considers a threedimensional axisymmetric cone the identified impact
situations by the basic principle of the hydro-elastic
criterion which have not essentially an axisymmetric
cone shape, need to be considered as a geometrically
axisymmetric cone in their relation to the flexible
cone.
2. Dynamic structural properties of object of impact
The dynamic structural properties of the flexible cone
were simplified toward a lumped-mass model with
two degrees of freedom. The dynamic structural
properties of the impact situations of the hydro-elastic
criterion need accordingly as well be simplified toward a lumped-mass model with two degrees of freedom in their relation to the flexible cone. In contrast
to the flexible cone these impact situations however
consist of a flexible (considered structural level) as
well as of a rigid (remaining of successive structural
level) part. As a consequence the translation toward a
lumped-mass model will be different for these impact
situations than for the flexible cone. The following
procedure is applied:
- The flexible part of the impact situations, namely
the structural level for which the response is evaluated, is simplified to a lumped-mass model with a
lower and a upper mass equal to half of the mass of
the flexible part (mflexilbe part). The flexibility of this
part is hereby represented by a linear spring with a
spring stiffness specified as:
2
k = ωflexible part mflexible part
(19)
- The influence of the rigid part, namely the remaining of the successive level, is translated to the
lumped-mass model in the following way:
- The first natural frequency of the flexible part
(ωflexible part) takes the boundaries into account,
namely the flexible part is considered to be fully
supported in the direction of the impact whereby
all other restrains are neglected.
- The mass of the remaining part of the successive
level is added to the mass of the upper part of the
lumped-mass model and accordingly influence
the non-dimensional parameter α, whereby α increases as the mass of the remaining part of the
successive level increases relative to the mass of
the considered structural level.
In this respect the reaming part of the successive
level does not only influence the dynamic properties of considered structural level but as well the
motions and the coupling to the hydrodynamic
force of the considered structural level.
stage and chines dry stage are described by these
models. The impact conditions of the identified impact of the hydro-elastic criterion are accordingly as
well restricted to the above mentioned impact conditions in their relation to the flexible cone. An axisymmetric impact of a lumped-mass model of the
cone implies that the dead-rise angle (β) of the cone is
not only geometrical axisymmetric but as well axisymmetric with respect to the free surface. In the
relation to the flexible cone this deadrise angle represents the angle between the object of impact and the
fluid surface of the identified impact situations. For
these impact situations this angle is not essentially
axisymmetric but is however in its relation to the
flexible cone assumed to be geometrically axisymmetric as well as axisymmetric with respect to the
fluid surface.
The resulting relation is summarized in Fig. 7. Although the applied simplifications in this relation
limit in principle the applicability of the hydro-elastic
criterion, only further research can reveal the required
accuracy in relation to the usability of the hydroelastic criterion and can accordingly reveal the extend
to which these simplifications need to be reduced. At
this stage the basis of the hydro-elastic criterion is
elaborated and can be summarized in the following
three steps:
1. Specification of the impact situations for the
evaluation of the response of as structure by division in structural levels and application of the basic principle of the hydro-elastic criterion.
- The lower part of the lumped-mass model which
directly impacts the free surface, is undeformable
and has according to the first level of simplification
an axisymmetric cone shape. The radius of the cone
is hereby equal to the half of the smallest of the
breadth or length of the considered structural level.
Since only the considered structural level and not
also the remaining part of the successive level is
geometrically transformed into the lower part of the
lumped-mass model where the hydrodynamic force
acts on, this translation can only describe the impact situations where the considered structural level
is located at the position of impact.
3. The conditions of impact:
The derived models (Eq. 1~7) of the lumped-mass
model of the cone consider an impact with no air
entrapment, no compressibility of the fluid and an
axisymmetric impact. Furthermore only water entry
Fig. 7: Relation between impact situations of hydroelastic criterion and impact situations of lumped-mass
model of flexible cone
2. Specification of the non-dimensional parameters
of the lumped-mass model of the flexible cone by
relating the identified impact situations to flexible cone according the relation summarized in
Fig. 7.
3. Specification of the necessity and the effect of a
hydro-elastic analysis by means of Fig. 6 and the
defined non-dimensional parameters.
elastic criterion, this investigation of consequences
form the next steps in the systematic, step by step
research at MARIN toward the necessity and the
effect of hydro-elasticity for practical design.
At the moment of writing MARIN works at the experimental validation of the proposed hydro-elastic
criterion.
References
Conclusions
The new hydro-elastic criterion proposed in this paper
enables a designer to decide whether a hydro-elastic
problem can still be addressed by the standard quasistatic analysis. The criterion subdivides the marine
structure into successive structural levels based on the
type of impulsive load and the location of impact.
Within the proposed hydro-elastic criterion the
evaluation of the response - caused by impulsive
wave loads - of such structural level takes the influence of the remaining of the accompanying successive level into account.
In this criterion use is made of the derived necessity
and effect of hydro-elasticity for the impact of a
lumped-mass model of a flexible cone for the evaluation of the necessity and the effect of hydro-elasticity
for the above mentioned identified structural levels.
This necessity and effect can be expressed as function
of four non-dimensional parameters which characterize the impact of the lumped-mass model of the flexible cone onto the free surface and can be summarized
into three regions, namely a region where the quasistatic analysis overestimates the response, a region
where the quasi-static analysis underestimates the
response and finally a region where the quasi-static
analysis correctly defines the response.
In this formulation of the necessity of a hydro-elastic
analysis the stringent definition of the word required
is applied, namely it is required to carry out a hydroelastic analysis if the quasi-static analysis either overor underestimates the deformation. However the word
required in the formulation of the necessity of hydroelasticity should as well taken the extent of the consequences into account. The investigation of the extent of these consequences and when they can be
neglected is essential for further development of the
hydro-elastic model. Together with further impact
model developments and research toward the required
accuracy in relation to the usability of the hydro-
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