Wave induced extreme hull grider loads on containerships

Wave induced extreme hull girder loads on containerships

J. Juncher Jensen (V, Corresponding Author), Preben

_{Terndrup Pedersen (FL), Bill Shi (V), Sue Wang}

_(M),

Martin Petricic (V), Alaa E. Mansour (FL)

This paper provides simple but rational procedures for_{prediction of extreme wave- induced sectional hull girder} forces with reasonable engineering accuracy. The procedures take intoaccount main ship hull characteristics such as: length, breadth, draught, block coefficient, bow flare coefficient, forward speed and hull flexibility. The vertical hull girder loads are evaluated for specific operational profiles Firstly a quadratic strip theoiy is presented which can give separate predictions for the hogging and sagging bending moments and shear forces and for hull girder loads. Then this procedure is used as a base to derive semi-analytical formulas such that approximate wave load calculations can be performed by a simple spreadsheet program. Due to the few input parameters this procedure can be used to estimate the wave-induced bending moments ai the conceptual design phase. Since the procedure is based on rational methods it can be applied for novel single hull ship types not presently covered by the rules of the classdìcation societies or to account for specific operational profiles.

KEY WORDS Slamming, Whipping,

Wave-induced hull girder response, Quadratic strip Theory, Timoshenko beam,

Stochastic seaway, Extreme loads

INTRODUCTION

The objective of this paper is to present a simplified procedure for determining wave hull girder loads acting on containerships including transient loads such as slamming and green water effects. The high frequency transient loads are combined with lower frequency wave induced loads, and the entire simplified solution is presented in closed-form equations. Whipping accelerations due to forward or stern slamming are also included in the calculation procedure.

The theoretical basis for the present analysis are the procedures described in [1], -[5], supported by information regarding the

effect of weather routing,

_{[6],[7], springing, [8], [9] and,} stochastic correlations between wave and whipping responses, [IO].

High frequency loads such as slamming and green water play an important role in the safety and operability of containerships. There is no well-established procedure for determining the total

response of ships to these loads combined with the low

frequency loads such as vertical bending moment. Because

of

the random nature of these loads, the sOlutions can become complex and not suitable for routine engineering evaluatiOns. At present the main loads on merchant ship hulls are based on rules developed by the Classification Societies. For normal ship types the Classification Societies have developed empirical

design rules for sectional forces based on theoretical and

experimental analyses and feed back from service experience. The Classification rules for initial scantling selection do not explicitly take into account such factors as draft, hull flexibility, operational procedures etc. Only as a possible second step will non-lineas seakeeping analyses be performed [Il].

SMTC-072-2008 Jensen

Assessment of non-linear sectional hull girder loads due to

impulsive wave induced loads due to slamming on flared bows, forward bottom parts and possibly flat stern areas of ships is an

important task during the design phase ofnovel ships.

Thus, for advanced vessels more specific wave-induced hull girder forces have to be determined. This is normally done by specialists using non-linear strip theories and in recent years also numerical calculations based on comprehensive 3D time simulations coupled with finite element models.

Therefore, for novel ship designs there is a need to develop new design oriented prediction tools which can be used in the iñitial

design phase and serve as a gauge for detailed

numerical calculations using numerical methods based on non-linear strip theory or comprehensive 3D time simulations [12].

With this purpose in mind the overall aim of this paper is firstly to present a quadratic strip theory which is suited for numerical prediction of the effect of non-linearities due to bow flare and

for the effect of hull

flexibility and non-linear loads on

springing. Secondly, to show how to use this analysis tool to

derive closed form estimates for the wave frequency hull girder sectional bending moments for ships with fine forms such as

containerships and at the same time to determine the hull girder whipping bending moments due to slamming. The moments caused by whipping are combined with the spatially distributed wave frequency wave-induced bending moment M,,,(x,i) taking into account the phase angles and the effect of damping. The resulting long term Weibull and Gumbel distributions for the peak values taking into account the operational profile of the ship is derived.

The result is a procedure which is more ship specific and

therefore more accurate than the present class rules but just as

easy to use.

The only ship parameters entering the closed form expression

are

Deift University of Technôlogy

Ship Hydromechanics Laboratory

Library

Mekelweg'2, 2628 CD Deift

The Netherlands

Length L of the vessel (either the rule length or the length overall)

Moulded (max) breadth B

Actual draught d (any trim is neglected) Block coefficient C5 (minimum 0.6) Bow flare coefficient C1 (ABS definition)

Freeboard F (used

only in green water load

calculations)

Local dead rise angle a, breadth B,and draught d1 at the location of slamming

Two- and three-node natural periods T,, T3 (used in the decay rate of whipping)

Relative damping _,,

_{, in the two- and three-node}

modes (used in the decay rate of whipping)

In addition the operational profile must be specified in_{terms of} speed and heading distributions, operational areas, total time_at sea and, possibly, effects of weather routing.

THEORETICAL BACKGROUND

The basis for the theory used herein for the estimation of the

wave frequency bending moments and shear forces

is a quadratic strip theory formulated in the frequency domain which originally has been presented in [1]. Therefore, we shall only summarize the governing equations and the assumptions

involved.

Measurements have shown that the springing and whipping deflections are primarily in the form of the lower hull girder vibration modes. That is, the heave and pitch rigid body modes and the two and three node vertical vibration modes. Since it is generally accepted that the lower modes of ship hull vibrations can be determined quite accurately by modeling the hull as a single non-uniform beam, we shall base the equations of motion

in the vertical plane on the following Timoshenko beam

formulatiOn:

aa

EI i +

'iI +

ax[

(

at}]

a

aw /-IGA(i +

-

c)J

=

m,r2.-5t2

--ÍuGAl+17fJ!_

m F(x,t).-q(x,t)

at2 (2.1) =

where EI(x) and pGA(x) are the vertical bending and shear

rigidities, respectively; q(t, x) is the slope due to bending;

w(x,t)

is

the total

deflection; i is an internal damping coefficient; and X is a longitudinal coordinate in an x,y,z -coordinate system fixed with regard to the undisturbed ship so that the z-axis is in the vertical direction. Finally, m(x) is the

hull mass per unit length, mr2(x) is the mass moment of

inertia aboût the horizontaly-axis, F(x,t) is the low frequency

external force per unit length which is nonlinear in w and

q(x,t) denotes the transient slamming load

per unit length along the hull girder.

The boundary conditions to the equations (2.1). express that the bending moments and shear forces are zero at the ends of the

ship.

FORCES BASED ON A QUADRATIC STRIP

THEORY

The quadratic strip theory is formulated in the frequency domain like the usual linear strip theory. This has the advantage that the correct frequency variation of the hydrodynamic added mass and damping terms can be applied and that the extreme value predictions can directly be performed using the well known methods for slightly non-Gaussian responses, e.g. monotonic transformations [13]. Thereby the computational effort becomes several magnitudes smaller thät than for nonlinear time domaiñ analyses. The drawback is that this perturbation procedure

canñot model very

severe non-linearities such as bottom slamming loads and green water loads. Such loads are here treated separately and included in the load term q(x,i).

Calculation of the steady state hydrodynamic forces F(x,t) Us based on the time derivation of the momentum of the added mass of water surrounding the hull. In addition to forces due to a change in momentum of the added mass of water, we include a damping term and a restoring terni, both dependent

on the

relative motion. Thus the fOrce per unit length of the hull acting at position x is taken in the form

F(x

DtL

DtJ

Dt

-f

,ï

dz

where the difference between the absolute displacement of the ship in the vertical direction, w(x,t), and the modified surface

elevation of the ocean, h(x,t), is denoted by ;(X,t)

, the

added mass per unit length by m, and the damping by N. The (3.1)

modification of the real wave elevation h

_(x,

_{t) takes into}

account the exponential decay of the

_{wave particle motion} through the Smith correction factor K.

_{The operator DI Dt is}

the total derivative with respect to time t; that is

Di

ai

ax

where V is the forward speed of the ship. The breadth _{of the}

ship is denoted by B(z,x) and the draft by T(x)

. The

Froude-Krylov fluid pressure is denoted by

p.

If we neglect the z dependence in

_{m, N and B, the force}

expression corresponds to a linear strip theory. However, here

we shall,

as described in [I], evaluate

F(x,t) by

a perturbational method, taking into account linear and quadratic

terms in the relative displacement z and thereby

in the displacement of the hull w and the wave surface elevation h. In order to do this we shall start by evaluating the _waterline breadth B, the added mass m, and the damping coefficient N around =O:

az

B0(x) + ;(x,t)B1(x)

-am

m (,x) m (O,x) + z

h(x,t)=acosy.i

+

a1a[(&

k1)cos(y,1

_+,)-Jk

k

m0(x) +

(x,t)m1(x)

(3.2)

ql(x,t)

=

(t) a. (x)

w(x,t)=

u. (t) v,(x)

The wave surface elevation in the stochastic_{seaway h and the} pressure

p

_{are also expressed as sums of linear terms and}

quadratic terms so that we have, for instance

h(x,i)

=

hw

_{+ h2}

or

(3.5) where a1 are wave amplitudes and

y'1 = -

k, (x + Vt)

- a t + 6',

where k. denotes the wave number in the x-direction, _{and the}

frequency a, is given by w

₌

_{for deep water waves,} where g is the acceleration of gravity.

Similarly, the deflections of the ship hull are expressed as a sum of a linear part and a quadratic part

w =wW

_{+ w}

These assumption lead to

F(x,,t)

=

F(»

_{+ F2}

where F' are linear terms in the displacerient

w _{and the}

(I) (2)

wave surface elevation ., and _{are terms which are}

quadratic in these quantities as well

is

in the linear terms

squared.

DYNAMIC RESPONSE

The solution to Eq. (2.1) and the associated boundary conditions with the excitation given by Eq. (3.1) is approximated as a series

in the form

(4.1)

where u. (t) are coefficients to be determined

_{and where}

{v. (x),a1 (x)}

are eigenfiznctions to the homogeneous, self-adjoining part of Eq. (2.1). These eigenfunctions _{are assumed} orthogonalized so that

nl

mr2 a a

_+mv,v]dx=ÖM

_(4.2)

wherem m, + m11 _{and S, is Kronecker's delta. Furthermore,}

the eigen-functions are assumed to be ordered so that {v0,a0} is the heave mode, {v1 , a1 } is the pitch mode, {v2 , a2 } is the two-node vibration mode associated with the natural frequency

2 and so on.

If we substitute (4.1) in the differential equation_{(3.1), and use} the orthogonality relations (4.2),_{we obtain the following set of} equations of motion:

SMTC-072-2008 _Jensen

(3.3)

N(;,x)

N(O,x) +

_{N(x) + z(x,t)N1(x)}

(3.4)

cos _y'i

_-vii)]

(3.6)

ii1

+cu

+ûu =

(4.3) where

j

= 0, 1, 2,

From the

linear and quadratic transfer

functions of the

responses, the spectral densities and extreme values in statiònary stochastic seaways are easily calculated using for instance _a Gram-Charlier or a Hermite series expansion

_{to model the}

slightly non-Gaussian response. Especially the [-lermite series expansion, suggested by Winterstein [13], seems to provide an effective procedure.

In the following we shall present a simplified solution to this problem by breaking the response down into two parts.

The first part will be a simplified closed form

_{rigid body} solution (j= O and 1) to the quadratic problem formulated above, neglecting transient loads, i.e. q(x,1) = O. This results in an analytical expression for the stochastic distribution of

wave-frequency sectional hull girder fôrces.

The second part will be a closed form approximation to the transient slamming induced hull girder sectional forces, i.e. including the transient slamming loads and green water loads through the load term q(x,t).

SIMPLIFIED

_{LINEAR AND NON-LINEAR}

WAVE-FREQUENCy RESPONSES

Neglecting hull flexibility, i.e. taking into_{account only heave} and pitch modes (j =O and I) in Eq. (4.3) and neglecting the

bottom slamming pressure q(x,i) and including linear

and quadratic terms in the equations (2.1) and (3.1) - (3.7) given above the wave induced bending moments_{Mgdbodyqrjc and} shear forces Qrigidbody,quadratic have been calculated for a number

of different ship types. Based on these comprehensive numerical calculations an estimation procedure of the long term wave-induced bending moments based on closed form expressions have been suggested in [2]. These expressions are based on analytically determined response operator functions for

simplified ship hull geometries. That is, the same type of

expressions as sought in the present_paper.

In [2] the influence of the ship principal characteristics on the linear part of the vertical wave-induced bending _{moment has} been derived through an analytical analysis of a homogeneously loaded box shaped vessel. Closed form expressions for the

standard deviation of the linear part of the wave-induced

bending moment amidships for ships sailing in long crested

irregular waves modeled by the Pierson - Moskowitz (ISSC) wave spectrum were derived. The result is expressed as

SM =

¡IT

/

g

I4ff2I(JßhI/3

It, _2L

Tg)

L)

where the significant wave height H and the _{zero-up-crossing}

period represent the wave environment and

F(Fn)= I +3Fn2

is a speed dependence factor, valid for F,, <0.3,

F,(Cb)

=[(i

9) +0.6a(2_t9)];

9=2.5(1Cb); Cb=max(0.6,Cb)

is a block coefficient dependent factor and,

I1(i)=3OOO.exp(I4.1.i°

\ j; 6.9+O.I3f

_Jz

I2(Y)=e(l+0.15y)

y=42..L;

V;

13(r)= (o.95_5.4r+1.14T2)

Finally,

fi

_{is the heading with fi = 1 80° for headsea.}

This analytical prediction of the linear

response has been compared with numerous results of direct strip theory calculations in [2] and good agreement is found. A 10_percent reduction is usually assumed to account for short-crested waves,

[19].

Similar closed form expressions are derived for the relative motion in [4],[5].

The frequency response functions cDv,, D9 for heave (w) and

pitch (, for

the vertical wave-induced motions

_{of a}

homogeneously loaded box-shaped vessel (LxBx

_{T) are}

derived analytically by the linear strip theory. By neglecting the

coupling terms between heave and pitch and assuming

a constant sectional added mass equal to the displaced water, the frequencyresponse functions for heave and pitch become,[4]:

cIç =IFI

cIo _IGI (5.2)

with

2

(A2

2'

=[(1_2a2)

+kBa2J

J 2 .

kL

F = KfsIfl(---)

24

[. kL

kL

kL

G = Kf

i sin(---) ---5---cos(--5----)

(k0L)2L[

2 2 2 where ke

=IosßI

is the effective wave number and

f

=J(1_kd)2 (A)2

The Smith correction factorK 5 approximated by

K = exp(k0d)

(5.3)

(5.6)

The sectional hydrodynamic damping is

modeled by the

dimensionless ratio A between the incomiñg and diffracted waves through the approximation, [141

A = 2sin(--kBã2)exp(_kdä2)

where a is related to the encounter frequency

_{W, the wave}

number k, the wave frequency »= _{the ship speed V} and the heading angle

_ß

through

w = cokV cosßmlãjco

(5.8)

For block coefficient C8 less than one, the breadth B is replaced by BC8.

The frequency response functions for heave and pitch

accelerations are simply obtained by multiplication with

_w2.

These frequency response functions for motions and

accelerations have been validated with model _{test results for a} tanker, a container ship and two high-speed mono-hull _vessels and reasonable good agreement is generally found, [4].

The phase angles between the motion components are found

from the equations of motion for heave and pitch.

The

coefficients in the equations are the same, but the amplitudes of the forcing functions are different, and the two motions are9Ø0

degree out of phase. Due to the identical coefficients the phase

angles are the same as well. The phase angle of

the forcing function, i.e. the excitation force in heave or the excitation moment in pitch, relative to the incoming wave is

1kd

f

A2

Sifl(s1,)

kBä3f

COS(jor.e)

where the coefficients follows from the f-function in Equation

(5.5).

The phase angle for the heave and the pitch motions relative to the forcing function is found from the inertia, the damping, and the restoring coefficient in the equation ofmotion:

(54)

cos(e,) = (1 2kdã2)i7

A2

sln(ew)=_k_2

(5.5) _{where the coefficients follows from the ij function in Equation} (5.3). It is seen that this phase angle is between_{0° and 180°.} This ends the discussion of closed form expressions for the linear motions and loads.

The nonlinear part represented by the quadratic terms in Eq. (3.1) give separate predictions for hogging and sagging bending moments. That is, the contribution from momentum slamming caused by immersion and re-emergence of the linearly flared forward and aft part of the vessel is included in the quadratic

(5.7) strip theory described above and therefore also inthe resulting nonlinear sectional forces.

Non-linear numerical calculations in [I] show that the non-linear hogging bending moment peaks become slightly lower

than the linear prediction. On the other hand,

the sagging bending moment peaks become much largerand increase with the bow flare.

_{In [21 an empirical procedure is developed}

whereby first the skewness of the response is approximated by an analytical expression in the wave spectrum parameters H0,T, the flare coefficient C1 and the Froude number F;,:

K3 =

O.26HC1Í1 expí_101

min(

-.5

jcosßl)

(5.11) The flare coefficient C1

_{is here defined as the difference}

between the upper deck area and the water plane area in the forward 20% of the ship hull normalized the freeboard in this

area and with the ship length. Together with the analytical

approximation for the linear standard deviation of the bending moments sM this skewness is used to derive the probability distribution for the peaks of the wave induced sagging bending moments by application of Hermite series, [13]. Hence, the probability P that an individual peak M exceeds a given level m

becomes

SMTC-072-2008 _Jensen

5

(5.9)

The main assumptions behind the present whipping analysis are: The rigid body ship motions are not influenced by the slamming impacts.

The dynamic transients are such that they are without significant effects from previous waves

The mathematical model for the flexible

_hull

_{is based on}

Es.(2; 1) and we shall superimpose the effect of the bottom slammiñg load q(x,t) on the previously described _nonlinear wave-induced loads calculated for the rigid vessel.

Assuming the flexible displacements and velocities to be zero at the time of impact, t = O,_{the transient solütion to Eq. (4.3) for}

j=2,3,... isfoundas

u. (t)

=_L je__T)

siny (tr)

_{q(x,r) v (x) dxdr}

(6.1)

where

/3

ç2

=Jc? -fi12

(6.2) The vertical bending moment distribution afterthe occurrence of the slamming event is then calculated as

M(x,t)

MÇjgjd_{body quadratic}

(x,t)

+

± u,

[m,r2 (Ç)a

(Ç)

-

(xÇ)mv (c)] dÇ

(6.3) provided the slamming pressure is

acting forward of the

considered section at x. In Eqs. (6.1) and (6.3)

_{a (x), y. (x)}

are the same eigenfunctions as used in Eq.(4.-3).

In the following we shall seek a simplified analytical solution to the transient bending moment expressed by Eq. (6.3).

Two types of slamming loads are considered. The first is_bottom slamming which occurs when the relative displacement_{at any} section exceeds the draught so that the local section emerges and then reenters the water. The magnitude of the bottom _pressure impulse is here estimated from the slamming load per unit length. Its value can be derived from calculations perfòrmed in [151 for the instantaneous force q on the wedge:

q(t)

3

'

p±3t

_(6.4)

4tan a

where the wedge angle is denoted a and with

_{p, i, t being}

the density of water, the vertical relative velocity at the

slamming location, and the time measured from the wedge apex hitting the water, respectively. For bottom slamming the time at maximum immersion including account for the water up rise on

J=2

P[Mm]=expí!u2(m))

(5.12) where u is given by

1+ .Ji + 4(z

+

2 1(3

-

_{m i - -i} with

)=--, m

=-

and

_{r=(V1+/c3 /18)}

6

_sMr

In the_{subsequent analysis the hogging bending moment is taken}

as the

linear result. Whipping can make this assumption

somewhat non-conservative, but further evaluations of full-scale data are needed to clarify this topic.

WHIPPING ANALYSIS

Assessment of non-linear sectional hull girder loads due to impulsive wave induced loads caused by slamming on flared bows, forward bottom parts and possibly flat stern areas of ships is an important task during the design phase of novel ships. In the following we shall analyze the combined effects of such slamming induced forces and the wave-induced rigid body_loads derived above. Traditionally two types of slamming forces have been considered. That is a) bow flare impact which is modeled by the momentum slamming effect due to rapid immersion of the flared bow and b) the bottom slamming effect _{caused by} sudden impact of the relatively flat stem or stern bottom. Part of the non-linear effect on the hull girder sectional forces due to immersion of the flared bow is included in the _{quadratic strip} theory and is the main cause for the skewness given by Eq. (5.11). Since the rise time for this momentum slamming is of the order of the wave period i.e. typically with _{a rise time of I s}

or more the dynamic effects of this momentum slamming

caused by the linearly flared bow section

is very limited. Therefore, in the following transient response analysis we shall only consider bottom slamming and slamming caused by large changes of curvature of the bow above the still water plane with rise times which are relatively small compared to the lowest hull girder natural period. That is, we shall consider situations where a slamming impact force is delivered on to the stern _{or stem of} the ship and produces a shutter throughout the entire hull. _This results in whipping stresses and a sudden deceleration, which are superimposed to the steady state wave-induced sectional hull stresses and accelerations. Whipping caused by emergence of the bow or stern which acts like a negative bow flare is not considered in here. But this may have an effect on the whipping response of full form ships.

The calculation of slamming effects requires consideration

_of

hull flexibility. The maximum slammiñg loads do not typically occur when the wave induced loads are the largest, and such

phasing needs to be considered in the combination of the

calculation of the combined load effects.

a section with dead rise angle a and local breadth B, is

approximately

2B tana

(6.5)

3

2z

taking into account that the water up-rise is about half the

immersion of the section according to the results shown in [15].

The second type of slamming load is caused by a rapidly

decreasing angle of the bow sections with the horizon, both in the transverse and in the longitUdinal direction. That is the angle the wave hits the shell plating under. It is noted that the quasi static calculation procedure presented above includes the 2D

effect of a linear increase of the waterline breadth with

immersion and that the momentum bow flare slamming considered here is confined to the effect of the large waterline increase most container vessels have in the fore ship above the

still water level in order to increase deck _{area and to protect} cargo on the deck from water impact. A mathematical

representation of this momentum slamming loadcan be derived

considering the bow forward of FP as a simple half cone (which often will model the bow fairly well since it is surface which can

de developed from flat plates). Then, the momentum impact of that body can be determined by a von Karman approach where

the added mass of a cone is taken to be equal to added mass of a

disc having the same radius as the cone's penetration of the still water plane, i.e.

r

=

z cotß

_{where z is the relative immersion}

and ß is angle between a horizontal plane and the_{cone surface.} The added mass of a disc in an infinite fluid is 8/3 ? p Thus for a half cone neglecting the Wagner correction, we find

M=pr

According to [16] the slamming load terms proportional to the forward velocity Vplay a less significant role for the whipping loads. Therefore, neglectingthe effect of forward speed Von the slamming load for a given vertical velocity ± we find the impact of the bow flare slamming load as

'impac: ₌

f F(t)dt

₌

f D(Mi)

=

p(z z)cot3ßi

=p(,3

ij3)±

The problem now remains to choose the vertical position of the section which generates the bow flare slamming impact. This position will depend on impact velocities, natural frequencies of

the hull girder, the flare angle ß andso on. Numerical studies

are needed to determine appropriate values.

In the following it is assumed that in order to develop a

slamming impact the vertical velocity shall also be larger than the classical threshold value by Ochi. In addition the duration of

SMTC-072-2008 Jensen

or

the slam t, must be small compared to the relevant whipping

period in order to cause dynamic amplification. Hence, for

bottom slamming we have

=

max

(O.o93JZ

--

tana)

(6.7)

Denoting the effective longitudinal extent of the slamming _area

ßL and the longitudinal centre of pressuré_{x0 then the weighted}

impulses of the bottom slam, see Eq.(6.4) is:

I,

=

v,(x0)ßL

q(t)dt

=

--4-ßL. p. v(x0)B,2±

(6.8)

Now, using eigenfunctions (a,,v,) normalized such _{that the}

vertical deflections V,

(L)

equals one, then the uncoupled one

degree of freedom equations (4.3) can be expressedas

ü,

+1,2u,

=

P(t)/M,

(6.9)

where

=17/

Ç,;

M,

=

r[m,r2a,2 +m,v,]

dx

and

¡(t)

denotes the generalized slamming load, i.e

J(t)

=

v,(x0)ßLq(t)

Let us introduce the damping corrected frequency

û

=

cJi _2

, then the free vibration solution _{can be}

written

u, (t)

=

u, (0)e" cos(,t) +

z,(0)+ ,û,u,(0)

e" sin(O.t)

(6.10)

û,

Let us also introduce a characteristic time 7. as the smallest

positive value of t satisfying the condition z, (t) = O for

(6.6) u,(0) = O and ú,(0) O. Then by integration Eq. (6.10) we

get the following expression for the time _T to maximum values of the coordinate fUnctionsu,.

û,(7.)

=

O

= 4,Q,

z(0)

sin(,7.)

tan(o,.T )

' =

Jj

Let us now assume that a slamming load P(t)> O is acting

during a time t.. where O

t,

Let us also assume that no whipping motion is present at t

=

O, that is

ui(0)

=

ù1(0) =O

Then we can integrate Eq. (6.10) withrespect to time t and get

ú1(t5)=

.!!_

_{2û1u1 (ti) Ç2}

_{' u1(t)dt}

with

Ii

=

_{j(t)dt=.ßLpv,(xo)B,2±}

If t,

«7

then we find the following approximation to the velocity at the end of the application of the slamming pressure

I. u. (t5) =

After the time of impact the motiòn will continue

as a free

motion described by Eq. (6.10) . Since 15« T we can use the approximation that u(0)

=

O and that ii, (0)

I, I

M1. The maximum modal displacement is then found for t= as

UI =

e°'

.sin(O 1T )

' _Ç21A4,

where Eq.

(6.11)

has been used

together with standard

trigonometric relations.

Note, that T, denotes the rire time from the time of impact to the maximum value of the response. For superposition with the wave frequency induced loads also the phase angle for the time of maximum impact pressure needs to be considered. Here these

phase lags are taken relâtive to peaks in the wave-induced

sagging bending moments. Positive means that the whipping bending moments occur before these peak values. With this definition the phase lag for maximum bottom slamming pressure is taken as 2 7r ¡3 and the phase lag for the momentum bow flare

slamming is taken as it /6. These phase lags are based

on numerical calculations using a non-linear strip theo!)'program.

We shall now use this result to derive an approximation to the spatial distribution of bending moments along the hull girder for the case where the numerical hull girder bending frequencies fl and mode shapes are not known. For the lowest natural vibration mode shapes the shear deformations can be neglected. Then the

two-node vibration mode for a free-free beam with constant (6.11) mass and stiffness properties is

(6.12)

y (x)

2

(sinh k + sin /cx)(cosh h/cL - cos kL)

2(cosh kLsinkLsjnh kLcoskL)

(cosh /cx + coskx)(sinhkL sin kL)

2(coshkLsinkLsinhkLcoskL)

where ¡cL =4.73004074 and the mode shapes are normalized

such that

v2(0) = v2(L) O. This solution _{satisfies the}

orthogonality conditions (10) with heave and pitch motions for the two first modes.

Here a more realistic non-homogeneous mass distribution _m(x) is needed for calculation of the spatial distribution of the hull

girder sectional forces. For a symmetric

mass distribution,

increasing linearly from zerb to 0.3L forward and aft

and

constant on the remaining 40 per cent middle section, the

resulting mode deflection becomes as shown in Figure 1.

05

-OE5

Figure 1. Mode shape for the two-node vibration mode for_a uniform (thin line) and a non-uniform beam (thick line)

This two node vibration mode for the non-uniform beam can be approximated by y2 (x) L )3

1.23

_-

4.58(..J

I

v2(x+)= v2(--x)

O-O.5

L

using a least-square fit. Similar results for the 3-node modeare

shown in Figure 2.

(6.13)

Figure 2. Mode shape for the three-node vibration mode for a uniform (thin uñe) and a non-uniform beam (thick line)

The maximum values for the accelerations in the _individual vibration modes at the forward perpendicular given _by

.(x,T) =

"

" (\ D

C

24

(6.15) Assuming that the wave frequency _{wave induced acceleration} and the whipping induced accelerations are normal distributed with Rayleigh distributed peaks

_{we can express the most}

probable largest value, MPL, of each of the three _acceleration components in a short term sea state with duration T _as

MPL

= s0 (x)

2 in

-MPL..

_{whip,inode ace}

=

ßL

24 ¡

iM

¡=2,3

M1

T

°5s±vI(xo)I(x)I21n(77h.

(6.16) Here we have taken into account the relative occurrence of whipping, and s. is the standard deviation of the linear wave induced acceleration and _{; the associated standard deviation of} the relative velocity at the centric of the slamming pressure. Still neglecting inertia effects associated shear deformation the approximate whipping bending moment

_{can from (6.3) be}

determined as

Mwhipping (x,t)=

-X[

(x

-

Ç) m,v (c)]

dÇ

and the maximum values of whipping bending moments can be expressed as

SMTC-072-2008 _Jensen

0 0.1 0.2 0.3 0.4 0.5 0.5 0.1 0.8 0.9

Figure 3. Normalized bending moment variation_{S2(x)/LM2 for a} uniform (thin line) and a non-uniform beam (thick line) in

two-node mode 0.5 0.8 H 'rIII:l¼ 07 08 0.8 Mwhjpping (x, T) =

ßLpv(x0)

(6.17) where S,.(x) =

fm(,(u)(u_x)du

The corresponding result for the shear force is

N 2

Q(x,T)=

_{fJ3Lpvj(xo)Bj±(xo)Le21TG(x)}

G1(x) = f m(u)v,(u)du

(6.18) Based on the approximation to the two node vibration mode depicted in Figure 1, the variation of S2(x)/LM2 and G2(x)/M2

are shown in Figure 3 and Figure 4, respectively.

It is noted that the variation S2 (x)/L M0 for the_{non-uniform beam}

approximation of the hull rather well resembles the _numerjòaj results for the spatial distribution of the _{wave induced bending} moments obtained from the second order strip theory.

Figure 4. Normalized shear fòrce variation_{G2(x)/M2 for a} uniform (thin line) and a non-uniform beam (thick line) in two-node mode

As the whipping bending moment depends linearly on the relative vertical velocity the standard deviation

Sm_whipping

(x,7)

= N .2

S.(x)

, (6.19)

ßL2Bips2(xo)v(xo)ç2

e'''

1=2

24

LM,

can be added directly to the standard deviation of the wave-induced bending moment. It is noted that the correlation is taken

care of by the exponential decay factor in the

_whipping

response. The standard deviation s (x0) of the relativemotion

at the location x0 for a, ship sailing in ocean waves modeled by JONSWAP or Pierson-Moskowjtz spectra is determined as described in Section 5 of the paper by Eqs. (5.2)-(5.10).

The relative occurrence of whipping due to bottom slamming is taken as

I

i(v

2

Tlwh,i

=exp{--I

2 s2

where the last term follows from the requirement that the bottom must be out of the water prior to a slam. Fiñally, the probability P(MW+Wh > m) that

the combined wave and whipping

bending moment exceeds a value m in a stationary stochastic condition is written

(6.20)

P(MW+Wh> m)

=(1

7Wh,2)P(M > m)

(6.21)

+(wh2 l7Wh,3)P(MWh2 > m)+7Içh3P(MWh3 > m) Here P(MW > m) is calculated using the standard_deviationSM

and skewnessic3_{as calculated by Eqs. (5.1) and (5.11), whereas}

P(MWh > m) applies the same skewness, but

a standard deviation equal to + SbS .The subscript i = 2,3 sign ifs' the relevant 2 and 3 node properties (natural _{frequency, damping} factor, mode shape and bending moment variation).

P(M,VhI > m) is calculated with the 2-node (i = 2) or the 2 and 3-node modes (i =3) included in Eq (6.19).

For container vessels green water loading on the forecastle deck

is of major concern since green water can cause various

damages to deck equipment and damage or even throw

contaiñers over board. Besides, the vertical component of this loading can add to the dynamic vertical hull girder loading. To evaluate this longitudinal load effect simplified expressions fôr the effect of green water loads on the wave-induced hull girder bending moments are determined principally as in [3] and added to the combined response assuming bottom slamming and green

water loads are mutual exclusive. The only modification is that a threshold vertical velocity

(O.O93jZ)

is included together with a dynamic amplification factor of_{0.5 and the phase lag is}

taken to be 5

ir

/3. _{Green water loads are fürthermore only}

considered on sections in the forward twenty per cent of the

vessel.

LONG TERM PREDICTIONS

The long term analysis deals with the probability distributions of the extreme peaks taken over a period of typically 20 to 25 years. The basic assumption in the analysis is that the process

can be modeled as a sequence of stationary

_{processes with} independent peaks. With

_{a given operational profile and a}

relevant wave scatter diagram the long term distribution of the non-linear wave-induced wave frequency loads and

accelerations and the whipping loads can be calculated. The operational profile specifies the relative occurrence of

short-term conditions with constant forward speed, heading,

significant wave height and up-crossing period. Wave scatter

diagrams for the North Atlantic, the North Pacific, or the

Asia/Europe trades are generally used. These_{long term scatter} diagrams can be modified in order to account for the effect of weather routing. Significant effects of weather routing on the long term wave scatter diagram have been found in [6], [7] and [18], reducing the predicted extreme values significantly. Having determined numerically the long-term _{individual peak} distribution of a response R by averaging as discussed above, it

is

_{often useful to approximate the result to an analytical}

distribution. Here the Weibull distribution

P[Rr]=.LxpI_(!

_y(0)

(7.1)

(

a)

with the two free parameters a, b generally provides a good fit. v(r) is the up-crossing rate. From _{this expression the extreme} value distribution follows from the Poisson _{up-crossing model} or, asymptotically, the Gümbel distribution:

P[max(R)

_<rio

_{<t <T]}

=

exp {v(r)T}

=

_{exP{_Nexp[_(L)J}}

exp{exp(_c(r r,))};

rmp,

=

a/i7;

c

=

_{(1n N)1}

Here T and N are the total time and total

_{number of peaks,} respectively. rmp, is

denoted the most probable maximum

response and has a return period of lIT

VALIDATIONS

A comprehensive survey of the literature was conducted on

full-b)

scale measurements of global hull girder loads and stresses on container ships. The objective of the

_{search was to identif'}

results that are suitable for comparison with results predicted by the procedure developed in this paper.

The search revealed that there exist some full-scale data, but few were reported in such details that they could be used to conduct an accurate comparison with the analysis results _{developed in}

this

_{project. The main reason in most cases}

_{is missing}

information necessary to conduct the comparison, such as some particulars and geometric characteristics of the containership. In

other instances there was

no or little

information on the environmental and operating conditions (speed and heading) under which the full-scale measurements were collected. Two studies were identified where only few assumptions had to

be made in order to complete the input required for the

procedure.

The first study considers the thorough measurements described and analyzed by Okada et aI [17]. Three years of measurements have been recorded on board a container ship on route between Europe and Japan. The recorded data includes strain and wave height measurements. It is

_{shown that the wave height}

distribution _{resembles those found from standard scatter} diagrams for this route However, wave heights larger than 10m have not been measured. In thepresent study, the AsialEurope scatter diagram is therefore applied together with the _weather routing modification detarmined in [6], [7]. Similar or even greater differences between measured wave data and standard scatter diagrams are observed in [18]. The operational profileis taken in accordance with the fairly detailed information given in [171, basically assuming service speed (24.5 kt) when the significant wave height is less than 6 m and a reduced speed of 6 kt for higher sea states. The heading is taken as slightly

non-uniform with larger weights

_{on head and near head sea}

conditions.

The main dimensions of the container ship

are length L Lb=

283.8 m, Breadth B = 42.8 m, draft d= 14 m and depth D = 24.4 m. The block coefficient Cb and the bow flare coefficient (ABS definition) C1 are not known, but taken

_{to be 0.6 and 0.2,}

respectively, as these are typical values for

_{làrge container} vessels. The section modulus W at the position of the strain gauges is W =47.183 m3 (from private communication with Dr. Okada). Hereby, all the required input data is defined for the present procedure as regards the rigid body response. However, reference [17] also contains a shOrt discussion of the importance of hydro-elastic effects (whipping and springing). They find whipping single stress amplitudes as high as 30 MPa and by use of a cut-off frequency below the two-node _{frequency (at 0.68}

Hz), they derived numerically from the

_{measured records} individual peak stress distributions for the deck stresses with and without whipping (and springing) included. The results are presented in their Fig. 33, showing a marked increase in the extreme stresses (at 10 lével) when whipping is accounted for. These results can, to some extent, be used to validate the present procedure and it has been found that using the following_data: For the slamming impact calculations the local bottom breadth

B, _{20 m, local draft d, =10 m, local dead rise}

_{angle a =5}

degrees and relative longitudinal extent fi = 0.07 yield results iñ fair agreement with the measured values, Figure 5. lt is noted that the. comparison deals with stress amplitudes defined as the average of the sagging and hogging stress amplitudes. The hogging peak following a sagging peak is taken as 80 per cent of

the hogging peak at the same probability level in order to

account for the correlation between sagging and hogging peaks. This percentage is

_{chosen from the}

_{linear autocorrelation} function of the bending moment. Furthermore, _{the calculated} stresses in long-crested waves are reduced by IO per cent to account for real ocean waves being short crested, [19]._{With this} in mind the example shows that the _{present procedure yields} predictions in good agreement with available _measurements

both for the rigid body and the whipping bending

_moment response. Especially it is

_{noted that the rapid increase in}

whipping stresses at low probability levels ₍₁₀

_{to 1O) is}

captured by the present procedure. No theoretical _calculations are included in [17].

Long term distnbutn of longmidìzml tending stress with_{c1d-offfrcqcy 03 and 1.0Hz.}

Figure 5. Comparison between measured and predicted stresses in the deck of a container ship.

Full-scale measurements presented by Miyahara et al. in [181 have been collected over a period of two and a half years on board a post-Panamax containership on route between Japan and Europe. The main dimensions of the ship

_{are length L L=}

281.4 m, Breadth B = 40.0 m, draft d _{l2.Sni and depth D =} 24.2 m. The block coefficient Ch and the bow _{flare coefficient} (ABS definition) C1 are not known from [18], but are taken tobe 0.6 and 0.2, respectively, as in the previous _{comparison. The} section modulus W at the position of the strain gauges is not given in [18] and is, therefore, calculated using ABS Rules which give a value of W = 46.63 m3. For the calcülátion of bottom slamming loads,, the longitudinal extent of slamming is taken to be ß= O6L, the local breadth of slamming_{B, = 15 m,} local draught d, = 12.5 m and the deadrise angle at slamming a = 10 deg. The operational profile of the vessel is derived from the

data given

in [18]. _{Frequency-speed histogram has been}

modified to fit the input format of the

_{spreadsheet program.} Also, the time spent in harbor (with zero speed) was Climinated

ulalive Pmbabiisy

i:

os i .uo7

i ro

i .1o5 i E404 i i no2

iioi

LEq.ct

SMTC-072-2008 _Jensen

Il

12

E

from the analysis. Heading-wave height histogram has been normalized with respect to the time spent in each speed interval. An Euro-Asia scatter diagram is applied together with four different weather routing schemes, _{namely: I) weather routing} derived from [18], 2) weather routing based on observations on containership in North Atlantic, 3)

weather routing with

stepwise decrease of the significant

_{wave height and 4) no}

weather routing. Scheme 3) was generated to further

demonstrate the effect of weather routing on the results. Scheme 4) was used as the default weather routing scheme in subsequent sensitivity analysis.

_{Figure 6 shows comparison}

_between

measured and predicted stresses in the deckof a containership for all four weather routing schemes. _{Whipping contributions} are not included as they were filtered out from the measurements presented in [18]. In general, the resultscompase

well with the full scale data, provided a reasonable weather routing scheme, 2) or 3), is applied.

a

n-E

î

Cumulative Probabi ity

Figure 6. Comparison between measured and predicted stresses in the deck due to hogging and sagging BM for various weather routing schemes

SENSITIVITY STUDIES

The sensitivity analysis was performed on the containership from [18] in order to assess the impact of various slamming parameters, scatter diagrams, operational

_{life and weather}

routing schemes on the cumulative probability of the bending moment (in hogging and sagging) and on the most probable value of the bending moment. The values stated in section 8 were used as the default values. These are the results of this analysis:

1) _{Impact of the extent of slamming}

!' 10000

9000 8000

'.

7000 6000 o . 5000 0.03

*-- Extent of amming O.06L (&fault) s--- Extent of amming O.04L

a--Extent ofslamming=O.08L

Cumulative Probability

Figure 7. Cumulative probability of the stress in the deck due to hogging and sagging BM for different values of the extent of slamming

0.04 0.05 0.06 0.07 0.08 0.09 Extent of Slamming [% of L]

Figure 8. Variation of Most Probable Value (MPV) of BM with extent of slamming

2) _{Impact of the local breadth of slamming}

Cumulative Probability

Figure 9. Cumulative probability of the stress in the deck due to

Scale Measurements as in North Atlantic R.

WRasrnref[lô]

a-- Full

....

4--W R.

No W

L:

i j

PO-= ilE Local Breadth = 15m (default)

e Local Breadth = IO m

aLocaJBreadth=2Om

-

hogging and sagging BM for different values of the local

breadth of slamming 12000 10000 8000 6000 4000 2000 o 8 10 12 14 16 18 20 22

Local Breadth of Slamming [m]

Figure 10. Variation of MPV of BM with local

_{breadth of} slamming

3) _{Impact of the deadrise angle at slamming}

Cumulative Probability

Figure 11. Cumulative probability of the_{stress in the deck due} to hogging and sagging BM for different values of the deadrise angle at slamming 8000 7500 7000 6500 6000 0 Cumulative Probability

Figure 14. Cumulative probability of the stress in the deck due to hogging and sagging BM for different scatter diagrams

7000

6000 5000 4000 3000 2000 000

- -

.

---

Sagglng

e Hoing

-*-- Deadrise Angle at Slam. = IO deg (default)

Deadrise Angle at Slam. = 5 dog Deadrise Angle at Slam. = IS deg Deadrise Angle at Slam. = 20 deg

--

---*

o. i. 07

I. ;tLÌ_r!'''oI

I.,

Euro Asia (default)

Atlantic Pacific

*

-North

+--VNorth

SMTC-072-2008 _Jensen 13 5 10 15 20 25

Deadrise Angle at Slamming [deg]

Figure 12. Variation of MPV of BM with deadrise angle at slamming

4) _{Impact of the operational life}

O _{iO 15 20}

25 30 35

Operational Life [years]

Figure 13. Variation of MPV of BM with operationallife

0

Eurn - As North Atlantic North Pacific

Operatknal LÎ [25 years]

Figure 15. Dependence of MPV of BM on scatter diagram

CONCLUSIONS

The paper presents simple but rational

_{procedures for the} prediction of dynamic hull girder responses to wave frequency load and slamming load. The developed procedures have been applied to container carriers and the results are comparable with full scale measurements. The following findings are drawn from the analysis results of the applied vessels:

Closed forni expressions for the wave bending moment as those presented in the paper can predict measured full-scale results with reasonable accuracy. The formulas take into account main ship hull characteristics such as: length, breadth, draught, block coefficient, bow flare coefficient, forward speed and hull

flexibility.

The effect of flare coefficient on slamming loads and other dynamic loads is important. In general, the larger the flare coefficient, the larger the bow slamming loads.

The impact of weather routing on life-time maximum loads can be important, but the effect of_{operational profile in} terms of ship speed/heading combination is less important.

The impact of the choice of scatter diagram used in estimating the wave loads is important. In general, North Atlantic and North Pacific ship routes give approximately the

same magnitude

of

life-time wave loads, whereas the Asia/Europe route produces considerably lower values of wave

loads.

It is important to include the influence _{of whipping} stresses in the determination of the overall response.

The present paper deals with easy prediction of short and long term extreme loads in the vertical plane, i.e. the symmetric modes. Development of a similar procedure for estimation of the fatigue loading must in addition to the procedures described here also include research leading to a rational but similar simple procedure for calculation of hull girder springing caused by linear and non-linear wave excitation.

Sasng

Hogg,n8

Another subject for future research could be development of initial design oriented procedures for the_{coupled horizontal} bending and torsion response. For the case of containerships and open bulk carriers these anti-symmetric modes are important for fatigue evaluation of hatch corners on the deck structure, see for

instance [12].

DISCLAIMER

The views expressed in this paper are of the authors and do not necessary represent those of the American Bureau of Shipping.

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Bending Moments

in Ships- a Quadratic Theoiy" Trans. RINA, Vol 121, 1979, _pp

151-165.

Jensen, J.J., Mansour, A.E. "Estimation of Ship Long-term Wave-induced Bending Moment usingClosed-form Expressions". Trans. RINA, pp. 4 1-55, 2002.

Jensen, J. J., Mansour, A.E. "Estimation of the Effect of Green

Water and Bow Flare Slamming on the Wave-Induced Vertical Bending Moment Using Closed-Form _{Expressions", Proceedings}

3rd

International _{Conference on Hydroelasticity in Marine} Technology, _{pp. 155-161. Ed. R. Eatock Taylor, The Oxford} University, Oxford, Sep. 2003.

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12000 10000 8000 6000 4000 2000

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, Fourth International Conference _on

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2006.

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pp. 1772-1790, 1988.

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Bodies", J. Fluid Mechanics, Vol. 246, pp 593-6 12, 1993.

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13,no 1, Jan, 2000, pp 25-51

Okada, T., Takeda, Y., Maeda, T. "On board measurement of stresses and deflections of a Post-Panamax containership and its feedback to rational design", Marine Structures, Vol. 19, p

141-172, 2006

Miyahara, K., Miyake, R., Abe, N.,_{Kumano, A., Toyoda, M.,} Nakajima, Y. "Full-scale Measurements _{on Hull Response of a} Large Container Ship in Service", Proc. OMAE2006, Paper no.

92233, June 4-9, Hamburg, Germany

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SMTC-072-2008 _Jensen