Probabilistic Engineering Mechanics 33 (2013) 26–37

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Probabilistic Engineering Mechanics

0266-89

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n Corr

nische U

E-m

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Stochastic hydroelastic analysis of pontoon-type very large floatingstructures considering directional wave spectrum

Iason Papaioannou a,b,n, Ruiping Gao c, Ernst Rank a, Chien Ming Wang c

a Lehrstuhl fur Computation in Engineering, Technische Universitat Munchen, 80290 Munich, Germanyb Engineering Risk Analysis Group, Technische Universitat Munchen, 80290 Munich, Germanyc Department of Civil and Environmental Engineering, National University of Singapore, Kent Ridge 119260, Singapore

a r t i c l e i n f o

Article history:

Received 1 July 2012

Received in revised form

22 January 2013

Accepted 30 January 2013Available online 13 February 2013

Keywords:

Hydroelasticity

VLFS

Directional spectrum

Random vibration

Response statistics

Distribution of extremes

20/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.probengmech.2013.01.006

esponding author. Current address: Engineerin

niversitat Munchen, 80290 Munich, Germany.

ail address: iason.papaioannou@tum.de (I. Pa

Downloaded from http://www.elea

a b s t r a c t

The hydroelastic response of pontoon-type very large floating structures (VLFS) is obtained by resolving

the interaction between the surface waves and the floating elastic body. We carry out the analysis in

the frequency domain, assuming that the surface waves can be described by a directional wave

spectrum. The response spectra can then be computed by application of stationary random vibration

analysis. Applying the modal expansion method, we obtain a discrete representation of the required

transfer matrices for a finite number of frequencies, while the influence of the wave direction is

obtained by numerical integration of the directional components of the spectrum. Moreover, assuming

a Gaussian input, we can apply well known approximations to obtain the distribution of extremes.

The method is applied to an example VLFS and the effect of different mean wave angles on the

stochastic response is investigated.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Very large floating structure (VLFS) technology is an attractiveapproach to reclaim land from sea. These structures can and arealready being used for floating bridges, floating piers, floatingperformance stages, floating fuel storage facilities, floating air-ports, even habitation, and other purposes [1]. The floatingstructure can be easily constructed, exploited, and relocated,expanded, or removed. These structures are cost-effective andenvironmentally friendly floating artificial islands.

Pontoon-type VLFSs are relatively flexible floating structuresthat behave like giants plates resting on the sea surface. Owing totheir flexibility and large dimensions, their response is governedby elastic deformations instead of rigid body motions. The hydro-elastic response of VLFS is obtained by resolving the interactionbetween the surface waves and the floating elastic body. Variousmethods have been proposed for the hydroelastic analysis ofVLFS [2]. One of the most widely used approaches for performingthe hydroelastic analysis in the frequency domain is the modalexpansion method [3] that utilizes the dry modes of the floatingplate. In the open literature, however, the response is usuallyobtained for distinct wave frequencies and wave angles. In orderto obtain a robust VLFS design against wave-induced deformationsand stresses, it is necessary to account for the stochastic nature of

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wind waves. However, there is limited published work on pre-dicting the stochastic response of VLFS. Hamamoto [4] derivedanalytical expressions for the response of large circular floatingstructures subject to a spectrum of wave frequencies. Chen et al.[5–7] studied the influence of second-order effects of the struc-tural geometry and wave forces on the response of VLFS under twoirregular wave systems coming from different directions.

In this paper, we develop a method for hydroelastic analysis ofVLFS subject to a directional wave spectrum. The analysis is carriedout in the frequency domain by application of the modal expansionmethod. The fluid domain is discretized by the boundary elementmethod, while for the structure we use the finite element methodderived from the Mindlin plate theory, that allows for the effects oftransverse shear deformation and rotary inertia. The derived linearsystem allows for the application of linear random vibration theoryfor the evaluation of response spectra. Assuming that the wind wavecan be described by a Gaussian process, we can estimate thedistribution of extremes and hence obtain mean extreme values ofresponse quantities that are relevant for design.

2. Hydroelastic analysis of VLFS

2.1. Plate–water model

Fig. 1 shows the schematic diagram of the coupled plate–waterproblem. The VLFS has a length L, width B, height h and isassumed to be perfectly flat with free edges. A zero draft is

z

x

L

x

y

LIncident wave

SBz = −H

B

hz = 0 F

HB

�

�HB

�

�

�

�

2A

Fig. 1. Schematic diagram of coupled plate–water problem: (a) plan view and (b) side view.

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–37 27

assumed for simplicity. The water is treated as an ideal fluid(inviscid and incompressible) and its flow is irrotational. Thewater domain is denoted by D. The symbols OHB, OF and OSB

represent the plate domain, the free water surface boundary andthe seabed boundary, respectively. The free and undisturbedwater surface is at z¼0 while the seabed is found at z¼�H.Assuming an incident wave fI with a circular frequency o, height2A and wave angle y enters the computational domain, the watermotion and plate deflection will oscillate in a steady stateharmonic motion in the same frequency o. The deflection w ofthe plate is measured from the free and undisturbed watersurface.

2.2. Equations of motion for floating plate

The VLFS is modeled as an isotropic and elastic plate based onthe Mindlin plate theory [8]. The motion of the Mindlin plate isrepresented by the vertical displacement w(x,y), the rotationcx(x,y) about the y-axis and the rotation cy(x,y) about the x-axis.The governing equations of motion for the Mindlin plate (afteromitting the time factor e� iot) are given by

k2Gh@2w

@x2þ@2w

@y2

!þ

@cx

@xþ@cy

@y

� �" #þrpho2w¼ pðx,yÞ ð1Þ

Dð1�nÞ

2

@2cx

@x2þ@2cx

@y2

!þð1þnÞ

2

@2cx

@x2þ@2cy

@x@y

!" #

�k2Gh@w

@xþcx

� �þrp

h3

12o2cx ¼ 0 ð2Þ

Dð1�nÞ

2

@2cy

@y2þ@2cy

@x2

!þð1þnÞ

2

@2cy

@y2þ@2cx

@x@y

!" #

�k2Gh@w

@yþcy

� �þrp

h3

12o2cy ¼ 0 ð3Þ

where G¼E/[2(1þn)] is the shear modulus, k2 is the shearcorrection factor taken as 5/6, rp the mass density of the plate,h is the thickness of the plate, D¼Eh3/[12(1�n2)] is the flexuralrigidity, E is Young’s modulus and n is Poisson ratio. The pressurep(x,y) in Eq. (1) comprises the hydrostatic and hydrodynamicpressure, i.e.

pðx,yÞ ¼ �rgwþ iorfðx,y,0Þ on OHB ð4Þ

where r is the mass density of water, i the imaginary number(i¼

ffiffiffiffiffiffiffi�1p

), g the gravitational acceleration and f(x,y,0) the velo-city potential of water on undisturbed water surface. At the freeedges of the floating plate, the bending moment, twistingmoment and transverse shear force must vanish, i.e.

Bending moment: Mnn ¼D@cn

@nþn @cs

@s

� �¼ 0, ð5aÞ

Twisting moment: Mns ¼D1�n

2

� �@cn

@sþ@cs

@n

� �¼ 0, ð5bÞ

Shear force: Qn ¼ k2Gh@w

@nþcn

� �¼ 0 ð5cÞ

where s and n denote the tangential and normal directions to thecross-section of the plate, respectively.

2.3. Equations of motion for water

The water is assumed to be an ideal fluid and has an irrota-tional flow so that a velocity potential exists. Thus the singlefrequency velocity potential of water must satisfy Laplace’sequation [9]:

r2fðx,y,zÞ ¼ 0 in D ð6Þ

and the boundary conditions

@f@zðx,y,0Þ ¼�iowðx,yÞ on OHB ð7Þ

@f@zðx,y,0Þ ¼

o2

gfðx,y,0Þ on OF ð8Þ

@f@zðx,y,�HÞ ¼ 0 on OSB ð9Þ

Eq. (7) implies that no gap exists between the plate and thewater-free surface whereas Eq. (8) is derived from the linearizedBernoulli equation where the pressure is taken as zero at thewater surface. Eq. (9) is the boundary condition at the seabedwhich expresses impermeability, i.e. no fluid enters or leaves theseabed and hence the velocity component normal to the seabedis zero.

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–3728

The wave velocity potential must also satisfy the Sommerfeldradiation condition as 9x9-N [10]

limxj j-1

ffiffiffiffiffiffixj j

p @

@ xj j�ik

� �ðf�fIÞ ¼ 0 on O1 ð10Þ

where x¼(x,y,z) and O1 represents the artificial fluid boundary atinfinity. The wave number k satisfies the dispersion relationship

k tanhðkHÞ ¼o2

gð11Þ

and fI is the incident velocity potential given by

fI ¼ Ag

ocoshðkðzþHÞÞ

coshkHeikðx cosyþy sinyÞ ð12Þ

where A is the wave amplitude and y is the incident wave angle asshown in Fig. 1.

2.4. Modal expansion of plate motion and water velocity potential

Eqs. (1) and (4) indicate that the response of the plate w(x,y) iscoupled with the fluid motions or velocity potential f(x,y,z).On the other hand, the fluid motion can only be obtained whenthe plate deflection w(x,y) is specified in the boundary conditionat the fluid side of the fluid–structure interface, as given in Eq. (7).In order to decouple this fluid–structure interaction problem intoa hydrodynamic problem in terms of the velocity potential and aplate vibration problem in terms of the generalized displace-ments, we adopt the modal expansion method as proposed byNewman [3]. According to this method, the deflection of the platew(x,y) is expanded as a finite series of products of the modalfunctions cw

l ðx,yÞ and corresponding complex amplitudes Bl:

wðx,yÞ ¼XN

l ¼ 1

Blcwl ðx,yÞ ð13Þ

where N is the number of terms in the series. The single frequencyvelocity potential f of water can be separated into the diffracted partfD and the radiated part fR based on the linear potential theory. Theradiated potential fR can be further decomposed as [11]

fðx,y,zÞ ¼fDðx,y,zÞþfRðx,y,zÞ ¼fDðx,y,zÞþXN

l ¼ 1

Blflðx,y,zÞ ð14Þ

where fl(x,y,z) is the radiation potential corresponding to the unit-amplitude motion of the lth modal function and Bl is the complexamplitude which is assumed to be the same as those given inEq. (13) [3]. The diffracted potential fD is taken as the sum of theincident potential fI and the scattered potential fS which representsthe outgoing wave from the body.

2.5. Hydrodynamic analysis

By substituting the expanded plate deflection and velocitypotential [Eqs. (13) and (14)] into Eq. (6) and the boundaryconditions [Eqs. (7)–(10)], we obtain the decoupled Laplace’sequation and boundary conditions for each of the uni-amplituderadiation potentials (i.e. for l¼1, 2,y, N) and the diffractionpotential (i.e. for l¼D)

r2flðx,y,zÞ ¼ 0 in D ð15Þ

@fl

@zðx,y,0Þ ¼

�iocwl ðx,yÞ for l¼ 1,2,. . .,N

0 for l¼Don OHB

�ð16Þ

@fl

@zðx,y,0Þ ¼

o2

gflðx,y,0Þ on OF ð17Þ

@fl

@zðx,y,�HÞ ¼ 0 on OSB ð18Þ

limxj j-1

ffiffiffiffiffiffixj j

p @fl

@ xj j�ikfl

� �¼ 0

for l ¼ 1,2,. . .,N

for l¼ Son O1 ð19Þ

We then transform Eqs. (15)–(19) into an integral equation byusing Green’s second identity [1]. The resulting boundary integralequation is

flðxÞþZOHB

@Gðx,nÞ@z

flðnÞdn¼ZOHB

Gðx,nÞ@flðnÞ@z

dn, ð20Þ

where x¼(x,y,z) is the source point and n¼(x,Z,z) the field point.G(x,n) is a free-surface Green’s function for water of finite depththat satisfies the seabed boundary condition, water free surfaceboundary condition and boundary at infinity and is given by [12]

Gðx,nÞ ¼�X1

m ¼ 0

K0ðkmRÞ

pHð1þðsin2kmH=2kmHÞÞcoskmðzþHÞcoskmðzþHÞ

ð21Þ

where km is a positive root number satisfying km tanhðkmHÞ ¼

�o2=g, with mZ1 and k0¼ ik. K0 is the modified Bessel functionof the second kind and R represents the horizontal distancebetween x and n.

By further introducing the boundary conditions for @jl=@z intoEq. (20), one obtains

flðxÞþZOHB

@Gðx,nÞ@z

flðnÞdn¼

ROHB�ioGðx,nÞcw

l ðnÞdn f or l¼ 1,2,. . .,N

4pfIðxÞ f or l¼D

(

ð22Þ

The integral equation given by Eq. (22) can be solved bydiscretizing the surface of the boundary OHB using the boundaryelement method. Within this study the constant panel method[13] is applied.

2.6. Solution for radiated and diffracted potentials

By rearranging Eq. (22), the radiated potential fR and dif-fracted potential fD can be written in matrix forms as

ffRgq�1 ¼ ½~fR�q�NfBgN�1 ¼ ½

~f�q�qfwgq�1

¼ ð�io½½I�þð@½G�=@zÞ��1½G�Þq�q½cw�q�NfBgN�1 ð23aÞ

ffDgq�1 ¼ I½ �þ@½G�

@z

� ��1

q�q

4pffIgq�1 ð23bÞ

where [G] is the global matrix for Green’s function, [I] the identitymatrix and ½cw� the matrix containing N eigenvectors (thatcorresponds to the deflection w of the plate) obtained by per-forming a free vibration analysis on the Mindlin plate.The subscripts in Eqs. (23a) and (23b) denote the size of thematrix, where q is the total number of degrees of freedom in theplate domain and N the total number of modes.

2.7. Solution for plate–water linear equation

The plate equation is solved by using the finite element (FE)method with 8-node Mindlin plate elements that use the sub-stitute shear strain method to avoid shear locking and spuriouszero energy modes [14]. By assembling the coupled plate–waterEqs. (1)–(3) into the global form, we obtain

ð½Kf �þ½Ks�þ½Kw��o2½M��o2½Mw��io½Cw�Þq�qfwgq�1 ¼ fFgq�1 ð24Þ

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–37 29

where [Kf], [Ks], [Kw], [M], [Mw] and [Cw] are the global flexuralstiffness matrix, global shear stiffness matrix, global restoringforce matrix, global mass matrix, global added mass matrix andglobal added damping matrix, respectively. As discussed inSection 2.4, the displacement vector {w} may be expanded in anappropriate set of modes [c] as

fwgq�1 ¼ ½c�q�NfBgN�1 ð25Þ

Note that [c] can be obtained by performing a free vibrationanalysis on the Mindlin plate where ½c� ¼ ½cw ccx ccy �T . [cw] isthe matrix containing N eigenvectors corresponding to the platedeflection, ½ccx � and ½ccy � are the matrices containing N eigenvec-tors corresponding to the rotations about y- and x-axis, respec-tively. By substituting Eq. (25) into Eq. (24), we obtain

½c�TN�qð½Kf �þ½Ks�þ½Kw��o2½M��o2½Mw��io½Cw�Þq�q½c�q�NfBgN�1

¼ ½c�TN�qfFgq�1 ¼ fFDgN�1 ð26Þ

where {FD}¼[c]T{F} is the generalized exciting force. By using thecomputed velocity potentials [Eqs. (23a) and (23b)], the elementsof the global matrices for the added mass [Mw], the addeddamping [Cw] and exciting force {FD} can be calculated by

½Mw�q�q ¼�ro

Imð½ ~f�q�qÞ ð27aÞ

½Cw�q�q ¼ r Reð½ ~f�q�qÞ ð27bÞ

fFDgN�1 ¼ ior½c�TN�qffDgq�1 ð27cÞ

Upon solving the coupled plate–water Eq. (26), we obtain thecomplex amplitudes {B} and then we back-substitute the ampli-tudes into Eq. (25) to obtain the deflection and rotations of theplate {w} and hence the stress resultants.

3. Stochastic formulation

3.1. Directional wave spectrum

Assuming that the irregular (random) wind waves can bedescribed by a zero mean stationary Gaussian process, they canbe completely specified by the directional wave spectrum S(o, y),which represents the distribution of the wave energy in thefrequency domain o as well as in direction (wave angle) y.The directional spectrum is generally expressed in terms of theone-dimensional frequency spectrum S(o) as

Sðo,yÞ ¼ SðoÞDðy9oÞ ð28Þ

where D(y9o) is the directional spreading function and representsthe directional distribution of wave energy for a given frequencyo. The conditioning of D(y9o) on o implies that the distributionof wave energy in direction (in general) varies with frequency.The function D(y9o) has the following normalization propertyZ p

�pDðy9oÞdy¼ 1 ð29Þ

The sea surface Z(x, y, t) can then be modeled by linearsuperposition of monochromatic waves of all possible frequenciesapproaching a point from all possible directions, i.e.

Zðx,y,tÞ ¼ ReX

j

Xl

Ajl eiðkjx cosylþkjy sinyl�oj tþ ejlÞ

24

35 ð30Þ

where kj, oj denote the wave number and correspondingfrequency of the jth wave component traveling in the directionyl and ejl are independent random variables uniformly distributedin [0,2p]. The amplitudes Ajl are obtained from the directional

spectrum as

Ajl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Sðo,yÞDo Dy

pð31Þ

where Do and Dy represent the change in wave frequency and indirection between two consecutive waves. In this study, we usethe (one-sided) one-dimensional frequency spectrum proposedby Bretschneider and further developed by Mitsuyasu for thedescription of fully developed wind waves [15], i.e.

SBMðoÞ ¼ 0:257H21=3T�4

1=3

o2p

� �5

exp �1:03 T1=3o2p

� �4� �

ð32Þ

where H1/3 is the significant wave height and T1/3 is the significantwave period. Also, we assume independence of the directionaldistribution on the wave frequency and adopt the followingdirectional spreading function given by Pierson et al. [16]

Dðy9oÞ ¼DðyÞ ¼2p ðcosðy�yÞÞ2 for 9y�y9r p

2

0 for 9y�y94 p2

8<: ð33Þ

where y is the mean wave angle.

3.2. Stochastic response

The stochastic hydroelastic response of the VLFS is obtained byapplying the linear random vibration theory. Following theapproach adopted for the solution for a single frequency andwave angle, we first obtain the elements of the cross-spectralmatrix [SII(o)]q� q of the vector of incident potentials {fI}q�1 as

½SIIðoÞ�j,l ¼Z p

�pHIðoÞ 2e�ikðxlj cosyþylj sinyÞSðo,yÞdy ð34Þ

where xlj¼xl�xj and ylj¼yl�yj denote the difference of the x and y

coordinates of the locations corresponding to the lth and jthdegree of freedom, respectively. The function HI(o) is the transferfunction from the water surface elevation to the incident poten-tial, given by

HIðoÞ ¼g

ocoshðkðzþHÞÞ

coshkHð35Þ

Furthermore, we obtain the cross-spectral matrix of the forcevector as

½SFF ðoÞ�q�q ¼ ½HF ðoÞ�½SIIðoÞ�½HF ðoÞ�n ð36Þ

where []n

denotes the conjugate transpose operator and thecomplex transfer matrix [HF(o)]q� q is obtained by combinationof Eqs. (23b) and (27c) as

½HF ðoÞ�q�q ¼ i4por I½ �þ@½G�

@z

� ��1

q�q

: ð37Þ

Finally, the cross-spectral matrix of the response is obtained as

½SwwðoÞ�q�q ¼ ½HwðoÞ�½SFF ðoÞ�½HwðoÞ�n ð38Þ

The response transfer matrix [Hw(o)]q� q is given by

½HwðoÞ�q�q ¼ ½c�q�N½HBðoÞ�N�N ½c�TN�q ð39Þ

where [HB(o)]N�N is the harmonic transfer matrix, describing themodal response to a harmonic excitation, given by Eq. (26) as

½HB½o��N�N ¼ ð½c�T ð½Kf �þ½Ks�þ½Kw��o2½M��o2½Mw��io½Cw�Þ½c�Þ

�1N�N

ð40Þ

It should be noted that the inversion in Eq. (40) is trivial (i.e.the matrix to be inverted is diagonal), as the matrix [c] containsthe uncoupled modes of the system.

A discrete representation of the matrix [Sww(o)] is obtainedusing a finite number of frequencies. We can then compute the jthcomponent of the vector of the mth spectral moment of the

60

300

P5

P1

P2

P3

P4

x

y

case (a)

case (b)case (c) case (d)

Fig. 2. Mesh of the plate.

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–3730

response {lwm}q�1 in terms of the diagonal entries [Sww(o)]j,j of[Sww(o)], as follows:

flwmgj ¼

Z 10

om½SwwðoÞ�j,jdo ð41Þ

wherein the integration is performed numerically. The variance ofthe response can then be obtained by setting m¼0.

3.3. Response of stress resultants

Stress resultants are important in the practical design of VLFS.The stress resultants within a single Mindlin plate element of thefloating plate can be calculated by expanding the displacement inthe stress resultant–displacement relations using the obtainednodal displacements. The bending and twisting moments withinthe element e can be obtained as

Mxx

Myy

Mxy

264

375¼D

1 n 0

n 1 0

0 0 ð1�nÞ2

264

375½Bf �

ðeÞfwgðeÞ ð42Þ

where Mxx, Myy and Mxy are the bending moments and twistingmoment per unit length of the plate. The shear forces within theelement e can be obtained as

Qx

Qy

" #¼ k2Gh

1 0

0 1

� �½Bs�ðeÞfwgðeÞ ð43Þ

where Qx, Qy are the transverse shear forces per unit length of theplate. The elemental flexural strain–displacement matrix [Bf]

(e) isgiven by

½Bf �ðeÞ3�24 ¼ ½Bf 1 Bf 2 . . . Bf 8� ð44aÞ

where

Bf i

� �¼

0 dNi

dx 0

0 0 dNi

dy

0 dNi

dydNi

dx

26664

37775 ð44bÞ

and the shear strain–displacement matrix [Bs](e) is given by

½Bs�ðeÞ2�24 ¼ ½Bs1 Bs2 . . . Bs8� ð45aÞ

where

Bsi½ � ¼

dNi

dx Ni 0dNi

dy 0 Ni

24

35 ð45bÞ

where Ni (i¼1,2,y,8) are the basis functions of the 8-nodeserendipity Mindlin plate element. Thus, the elemental transfermatrices for moments and shear forces can be expressed as

½HMxxðoÞ�ðeÞ

½HMyyðoÞ�ðeÞ

½HMxyðoÞ�ðeÞ

8>><>>:

9>>=>>;¼D

1 n 0

n 1 0

0 0 ð1�nÞ2

264

375½Bf �

ðeÞ ð46aÞ

½HQxðoÞ�ðeÞ

½HQyðoÞ�ðeÞ

( )¼ k2Gh

1 0

0 1

� �Bs½ �ðeÞ

ð46bÞ

By assembling the elemental transfer matrices, the globaltransfer matrices for moments and shear forces can then beobtained. Denoting the global transfer matrices of the momentsand shear forces by [HM(o)] and [HQ(o)], respectively, we cancompute the cross-spectral matrices of the stress resultants asfollows:

½SMMðoÞ� ¼ ½HMðoÞ�½SwwðoÞ�½HMðoÞ�n ð47aÞ

½SQQ ðoÞ� ¼ ½HQ ðoÞ�½SwwðoÞ�½HQ ðoÞ�n ð47bÞ

The spectral moments of the stress resultants can then beobtained in a manner analogous to Eq. (41).

3.4. Extreme value prediction

In the design of VLFS, the knowledge of the distribution of maximaof response quantities over a certain time period is required in orderto assess the serviceability and safety of the structure. The distribu-tion of the maxima of any response quantity can be derived using itsspectral moments, calculated by Eq. (41). Let Y(t) be the zero meanstationary Gaussian process that describes a response quantity at acertain node and let lm be its mth spectral moment. The distributionFYðTÞðyÞ of YðTÞ ¼ maxðYðtÞ,0rtrTÞ, where T is the period ofinterest, can be expressed as follows [17]:

FYðTÞðyÞ ¼ expð�ZþY ðyÞTÞ ð48Þ

where ZþY ðyÞ is the conditional rate of upcrossings of the level y giventhe event of no prior upcrossings. Also we should note that Eq. (48)neglects the probability of initial upcrossing of y. Assuming that theupcrossings of high levels are independent events, we can approx-imate ZþY ðyÞ with the unconditional upcrossing rate nþY ðyÞ given by

nþY ðyÞ ¼ nþY exp �

y2

2s2Y

!ð49Þ

where sY ¼ffiffiffiffiffil0

pis the standard deviation of Y and nþY is the mean

zero upcrossing rate given by

nþY ¼1

2p

ffiffiffiffiffil2

l0

sð50Þ

The derived approximation is the well-known Poisson modelwhich is shown to be asymptotically exact for large T. However,convergence to the Poisson model becomes slow for narrow-bandprocesses. Vanmarcke [18] gives a more accurate model that accountsfor the influence of the bandwidth. According to this model, theconditional upcrossing rate is approximated as follows:

ZþY ðyÞ � nþY ðyÞpðyÞ ð51Þ

where

pðyÞ ¼ 1�exp �ð1�a21Þ

0:6ffiffiffiffiffiffi2pp y

sY

� �� �1�exp

�y2

2s2Y

" # !�1

ð52Þ

wherein a1¼l1(l0l2)�1/2 is the bandwidth parameter which tends tounity for a narrow-band process.

In code-based design applications, partial safety factors are appliedto characteristic values of design quantities, such as displacementsand bending moments and designs are checked against serviceabilityand safety limit-state requirements. Characteristic values are usuallytaken as the expected maxima of the design quantities over a timeperiod that represents the duration of an extreme event. Accountingfor the fact that YðTÞ follows asymptotically a type I extreme valuedistribution, we can approximate the expected maximum using the

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–37 31

following expression [19]:

E YðTÞh i

� y0þgy0

s2Y ð53Þ

0

1

2

3

-1.5-1

0

11.50

0.4

0.8

1.2

1.6

S (�

,�)

� (rad/s)

� - � (radian)

Fig. 3. Plot of the applied directional wave spectrum for y¼ 03 .

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

S ww

P1P2P3P4P5SBM

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

S ww

P1P2P3P4P5SBM

�

�

Fig. 4. Input spectrum SBM and response spectra of the vertical displacements at 5 selected p

where g¼0.577y is Euler’s constant and y0 satisfies the followingequation:

y20

2s2Y

¼ ln nþY T� �

þ ln pðy0Þ� �

ð54Þ

Eq. (54) can be solved for y0 by applying an iterative procedure.

4. Numerical example

The VLFS considered by Sim and Choi [20] is used as anexample for this study. The length, width and height of the floatingplate are 300, 60 and 2 m, respectively. The following materialproperties of the plate are assumed: Poisson’s ratio n¼0.13,Young’s modulus E¼1.19�1010 N/m2, and the mass density ofthe plate rp¼256.25 kg/m3. The water density is r¼1025 kg/m3

and a water depth H¼58.5 m. The finite element mesh of theplate, consisting of 2000 8-node Mindlin elements, is shown inFig. 2. A total number N¼30 of modes is chosen for the presentstudy. The finite element mesh was chosen fine enough, from ourexperience with deterministic calculations, so that an accuratesolution is obtained for each of the considered wave frequenciesfor the discrete representation of the transfer matrices.

The chosen parameters for the spectrum of Eq. (32) areH1/3¼2 m, T1/3¼6.3 s. In the numerical examples, four cases ofmean wave angle y, namely (a) 01, (b) 301, (c) 601 and (d) 901, areconsidered. In Fig. 3, a plot of the utilized directional wave

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

S ww

P1P2P3P4P5SBM

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

S ww

P1P2P3P4P5SBM

�

�

oints for different mean wave angles. (a) y¼ 0o, (b) y¼ 30o, (c) y¼ 60o and (d) y¼ 90o.

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.2

0.4

0.6

0.8

1

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.2

0.4

0.6

0.8

1

x/L

y/L

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.2

0.4

0.6

0.8

1

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.2

0.4

0.6

0.8

1

x/L

y/L

�� w [m

]� w

[m]

� w [m

]� w

[m]

Fig. 5. Standard deviations of the vertical displacements for different mean wave angles. (a) y¼ 0o, (b) y¼ 30o, (c) y¼ 60o and (d) y¼ 90o.

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L

Fig. 6. Standard deviation of Mxx for different mean wave angles. (a) y¼ 0o, (b) y¼ 30o, (c) y¼ 60o and (d) y¼ 90o.

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–3732

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L

Fig. 7. Standard deviation of Myy for different mean wave angles. (a) y¼ 0o, (b) y¼ 30o, (c) y¼ 60o and (d) y¼ 90o.

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L

Fig. 8. Standard deviation of Mxy for different mean wave angles. (a) y¼ 0o, (b) y¼ 30o, (c) y¼ 60o and (d) y¼ 90o.

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

10

20

30

40

50

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

10

20

30

40

50

x/L

y/L

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

10

20

30

40

50

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

10

20

30

40

50

x/L

y/L

�� Qx

[kN

/m]

� Qx

[kN

/m]

� Qx

[kN

/m]

� Qx

[kN

/m]

Fig. 9. Standard deviation of Qx for different mean wave angles. (a) y¼ 0o, (b) y¼ 30o, (c) y¼ 60o and (d) y¼ 90o.

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

10

20

30

40

50

x/L

y/L 00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

10

20

30

40

50

x/L

y/L

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

10

20

30

40

50

x/L

y/L0

0.10.2

0.30.4

0.50.6

0.70.8

0.91

-0.10

0.10

10

20

30

40

50

x/L

y/L

Fig. 10. Standard deviation of Qy for different mean wave angles. (a) y¼ 0o, (b) y¼ 30o, (c) y¼ 60o and (d) y¼ 90o.

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–37 35

spectrum for y¼ 0o is shown. The four corner points (P1–P4, asshown in Fig. 2) and the center point (P5, as shown in Fig. 2) arechosen to illustrate the stochastic behavior of the floating plate.

4.1. Stochastic response

The response spectra of the vertical displacements at the fiveselected points (P1–P5, as shown in Fig. 2) of the floatingstructure are obtained as shown in Fig. 4. Fig. 5 shows thestandard deviation of the vertical displacement for the four meanwave angle cases considered. It can be seen that in general, theoverall response of the floating plate increases as the mean waveangle increases from 01 to 901. In the four cases studied, largerresponses are observed in the corner points than the centerpoints. Due to the symmetry of the directional spreading function,we obtain symmetric response spectra for mean wave angle case(a) with respect to the x-axis and case (d) with respect to they-axis, respectively, as the effects of oblique wave angles arebalanced, as shown in Fig. 4(a) and (d). The same results areobserved in the plot of the standard deviation of the response, asshown in Fig. 5(a) and (d). Owing to this cancellation effect, theresponse results for these two mean wave angle cases are similarto those obtained by deterministic hydroelastic analysis usingthese two mean wave angles as distinct wave angles (see resultsin the Ref. [21] by Gao et al.).

However, for other mean wave angle cases, the effects ofoblique wave angles cannot be balanced and hence result indifferent hydroelastic response for the same floating plate.Considering the mean wave angle case (b) for example, the

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

0.5

1

1.5

2

2.5

x/L

y/L

Extr

eme

resp

onse

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

2

4

6

8

x/L

y/L

Expe

cted

max

imum

Myy

Fig. 11. Expected maxima of response quantities predicted by the Vanmarcke a

response spectra as shown in Fig. 4(b) is not symmetric, while thelargest response is obtained at the corner point P4. This is due to thefact that in this case the directional spectrum includes a largernumber of waves coming from other directions which trigger thetwisting vibration modes of the plate. This effect can only be capturedif a directional spectrum is considered. For the example case withmean wave angle of 301, neglecting the probability of occurrence oflarger oblique wave angles would lead to significantly smallervariances. The same conclusion applies to the mean wave angle case(c), as shown in Figs. 4(c) and 5(c).

4.2. Response of stress resultants

In Figs. 6–10, the standard deviations of the stress resultantsMxx, Myy, Mxy, Qx and Qy are plotted.

Similar to the stochastic response of the displacements, sym-metric responses are obtained for the stress resultants Mxx, Myy, Mxy,Qx and Qy in the cases of mean wave angle y¼ 01 and 901 as shownin Figs. 6–10. The effect of directional wave is significant in the caseof mean wave angle y¼ 601 as shown in Figs. 6(c) and 10(c), wherelarger standard deviations are obtained as compared to those fory¼ 01. In deterministic analysis, however, smaller values of Mxx andQx are expected in the case of oblique waves because the dominatingcomponent (x-direction component) of the wave that results largervalue of Mxx and Qx is compensated by the y-direction component.

The standard deviations of the moments Myy and Mxy aresmaller than those of the bending moment Mxx, as shown in Figs.6–8. This is due to the large aspect ratio of the floating platewhich indicates that the hydroelastic response is significant when

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

2

4

6

8

x/L

y/L

Expe

cted

max

imum

Mxx

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-0.10

0.10

2

4

6

8

x/L

y/L

Expe

cted

max

imum

Mxy

pproximation for a 2 h period for y¼ 03 . (a) w, (b) Mxx, (c) Myy and (d) Mxy.

0

2

4

6

8

10

12

Max

(ext

rem

es)

MxxMyyMxy

Mean wave angle � [degree]

0

50

100

150

200

250

0 15 30 45 60 75 90 0 15 30 45 60 75 90

Max

(ext

rem

es)

QxQy

Mean wave angle � [degree]

Fig. 12. Maximum value of extremes of stress resultants in terms of the mean wave angle.

I. Papaioannou et al. / Probabilistic Engineering Mechanics 33 (2013) 26–3736

the waves are coming along the strong axis of the plate, i.e. thex-axis. In the case of mean wave angle y¼ 901, the motion of theplate is dominated by rigid body motion, as can been seen inFigs. 5(d) and 7(d). However, it is still necessary to investigate theresults of twisting moments Mxy in the case of oblique meanwaves. As shown in Fig. 8(b) and (c), the magnitude of thestandard deviation of twisting moments Mxy are in the sameorder as those of bending moments Mxx results shown in Fig. 6.Moreover, the results of twisting moments Mxy are larger formean wave angle cases (b) and (c) than those for mean waveangle cases (a) and (d).

The blowups at the corners of the plate for the shear forces asshown in Figs. 9 and 10 are due to the strong variations nature ofshear forces near the free edges. As has been discussed by Rameshet al. [22], the shear forces do not vanish at the free edges,especially at the free corner. However, this effect would not affectthe overall observations of the spectra results for shear forces,because it only affects the small portion of the entire domain.Moreover, the exact results of shear forces at these affectedregions are known to be vanished.

4.3. Extreme value prediction

The expected maxima of the response quantities are obtainedbased on the Vanmarcke approximation by applying Eq. (53).The period T of interest is set to 2 h. Fig. 11 shows examples ofexpected maxima of response quantities in the case of mean waveangle y¼ 01.

The maximum value of the expected maximum of stressresultants is extracted for each case of mean wave angles.The results are plotted against the mean wave angle as shownin Fig. 12. It can be seen that bending moment Mxx and shear forceQx are the two dominating components of the stress resultants.Large values of these two components are obtained in the cases ofmean wave angle y¼ 601 and 751. This implies that the worstscenario of the floating structure might be in the cases of thesemean wave angles and this should be taken into account in thepractical design.

5. Concluding remarks

Based on the linear random vibration theory, a framework forstochastic hydroelastic analysis of very large floating structuressubjected to multidirectional irregular waves defined through adirectional wave spectrum has been developed. The approachinvolves a discrete evaluation of the relevant transfer matricesthrough a numerical resolution of the fluid–structure interactionproblem that combines the boundary element method for thefluid potential and the finite element method based on theMindlin plate theory for the plate response. Spectra of response

quantities are obtained as well as extreme responses, assuming aGaussian input.

The proposed method is applied to the stochastic analysis of anumerical example and the influence of the mean wave angle onthe standard deviation and extreme values of response quantitiesis demonstrated. It is found that the hydroelastic behavior of verylarge floating structures is greatly affected when considering adirectional wave spectrum, which provides a realistic descriptionof the sea state. The developed framework can be applied to thestudy of very large floating structures with non-rectangularshapes as well as to investigate the behavior of very large floatingstructures with flexible connector systems. Future extensionsmay include the response to non-stationary sea states as well asthe application of nonlinear random vibration analysis to accountfor the effect of second order wave forces.

Acknowledgments

This work was supported by the International Graduate School ofScience and Engineering of the Technische Universitat Munchen. Thissupport is gratefully acknowledged.

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