composite plate with partial fluid khorsid

Composite Structures 104 (2013) 176–186

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Free vibration analysis of a laminated composite rectangularplate in contact with a bounded fluid

0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.04.005

⇑ Corresponding author. Tel.: +98 861 262 5050; fax: +98 861 417 3450.E-mail address: [email protected] (K. Khorshid).

Korosh Khorshid a,⇑, Sirwan Farhadi b

a Sound and Vibration Laboratory, Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-88349, Iranb Department of Mechanical Engineering, University of Kurdistan, Sanandaj 15175-66177, Iran

a r t i c l e i n f o

Article history:Available online 18 April 2013

Keywords:Free vibrationRectangular plateLaminated compositeSloshing fluidHydrostatic pressure

a b s t r a c t

In This study hydrostatic vibration analysis of a laminated composite rectangular plate partially contact-ing with a bounded fluid is investigated. Wet dynamic transverse displacements of the plate are approx-imated by a set of admissible trial functions which are required to satisfy the clamped and simplysupported geometric boundary conditions. Fluid velocity potential satisfying fluid boundary conditionsis derived and wet dynamic modal functions of the plate are expanded in terms of finite Fourier seriesfor compatibility requirement along the contacting surface between the plate and the fluid. Natural fre-quencies of the plate coupled with sloshing fluid modes are calculated using Rayleigh–Ritz method basedon minimizing the Rayleigh quotient. The proposed analytical method is validated with available data inthe literature. Using numerical data provided, effect of different parameters including boundary condi-tions, aspect ratio, thickness ratio, fiber orientation, material properties of the laminas and dimensionsof the tank on the plate natural frequencies are examined and discussed in detail.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays laminated composite plates are used widely in manyengineering applications including aerospace, civil, and marinestructures. Numerous studies have been performed to investigatefree and forced vibrations of thin isotropic plates in partial contactwith a fluid. Some of the most complete reviews on the subject arepresented by Khorshidi [1], Amabili [2], Jeong et al. [3–5], Kwak [6],Zhou and Cheung [7], Chang and Liu [8], Ergin and Ugurlu [9], Zhouand Liu [10], Ugurlu et al. [11], and Kerboua et al. [12].

For laminated composite plates, anisotropy of laminas intro-duces a coupling mechanism between the bending and the stretch-ing vibrating modes. During last decades, different laminatetheories have been developed to describe kinematics of compositeplates approximately. Many of these theories are extensions of theconventional single-layer theories of orthotropic plates. For exam-ple, the classical laminated plate theory (CLPT) [13] is derivedassuming that the normal to the middle surface before deforma-tion remains straight after deformation, which is an extension ofthe Kirchhoff–Love hypothesis.

Lina and Kin [14] used classical laminated plate theory to com-pute the natural frequencies of un-symmetrically laminated rect-angular plates. Bert and Mayberry [15] predicted the naturalfrequencies of un-symmetrically laminated plates with clamped

edges using Raleigh–Ritz energy method. Chow et al. [16] investi-gated the free vibrations of symmetrically laminated plates withRayleigh–Ritz method using admissible 2-D orthogonal polynomi-als. A review of the literature shows that, several methods havebeen investigated by researchers for the analysis of laminatedcomposite rectangular plates [17–19]. Rayleigh–Ritz method isone of the most popular methods to obtain approximate solutionsfor the natural frequencies of orthotropic and laminated compositerectangular plates [14–16,20]. A method based on superposition ofappropriate Levy type solutions for the laminated symmetric cross-ply rectangular plates was used by Gorman and Wei [21]. Whitneyand Pagano [22] used the first-order shear deformation platetheory (FSDT) for the free vibration analysis of composite plates.Reddy [23] derived a set of variationally consistent equilibriumequations for the kinematic models originally proposed by Levin-son and Murthy. Senthilnathan et al. [24] used Reddy theory topresent a simplified higher order theory, in which, a further reduc-tion of the functional degree of freedom is introduced by splittingup the transverse displacement into bending and shear contribu-tions. Kant and Swaminathan [25] presented an analytical solutionfor the free vibrations of laminated composite and sandwich platesbased on a higher-order refined theory. Carrera [26] presented acomprehensive review of such higher order theories in the analysisof multilayered plates and shells.

The governing equations and a detailed analysis of vibratingrectangular plates in contact with a fluid can be found in Khorshidi[1] and Jeong et al. [3–5] works. Such equations are not available

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K. Khorshid, S. Farhadi / Composite Structures 104 (2013) 176–186 177

for laminated composite plates in the literature. In order to fill thisapparent void, the present work is carried out to provide a theoryto calculate the natural frequencies of a laminated composite rect-angular plate partially contacting a bounded fluid in the bottomand vertical direction using Rayleigh–Ritz method. In the devel-oped model, the von Kàrmàn linear strain–displacement relation-ships are used in order to obtain kinetic and strain energies ofthe plate. The contributions given by the presence of the fluidand by the sloshing effects of the free surface are also includedin the model. In conclusion, results show that the fluid in contactwith the plate changes completely the linear dynamics. Therefore,fluid–structure interaction must be carefully considered. Thedeveloped numerical models are able to reproduce such resultswith good accuracy. Finally, the effects of boundary conditions, as-pect ratio, thickness ratio, fiber orientation, material properties ofthe laminas and dimensions of the tank on the plate natural fre-quencies are investigated.

2. Elastic strain and kinetic energies of a laminated plate

Consider a laminated composite rectangular plate with length a,width b, total thickness h, which is a part of the vertical side of abounded rigid tank filled with a fluid, as shown in Fig. 1. The tankhas width c1 and the fluid is of depth b1 and mass density qF and isconsidered to be incompressible, inviscid and irrotational. A Carte-sian coordinate system is used to describe governing equations.The coordinate system is placed so that the origin is located in thecorner of the studied plate on its middle surface, while axes x andy lie on plate’s edges and axis z is perpendicular to the middle plane.

2.1. Linear thin plate theory

Three independent displacements variables u, v and w in x, yand z directions, respectively, are used to describe deformationsof the plate; the geometric imperfection (or initial deformationdue to hydrostatic pressure, Appendix A) w0 in normal directionis also introduced. The displacements u1, u2, u3 of a generic pointof the plate at distance z from the z = 0 plane (see Fig. 1) are relatedto the middle surface displacements u, v, w by

u1 ¼ u� ðzþ h=2Þ @w@x

; u2 ¼ v � ðzþ h=2Þ @w@y

;

u3 ¼ wþw0: ð1a—cÞ

The von Kàrmàn nonlinear strain–displacement relationships areintroduced to describe the deformations of the plate. The straincomponents ex, ey and cxy at an arbitrary point of the plate are re-lated to the middle surface strains ex,0, ey,0 and cxy,0 and the torsionof the middle surface kx, ky and kxy by the following threerelationships

ex ¼ ex;0 þ ðzþ h=2Þkð0Þx ; ey ¼ ey;0 þ ðzþ h=2Þkð0Þy ;

cxy ¼ cxy;0 þ ðzþ h=2Þkð0Þxy ; ð2a—cÞ

Fig. 1. A laminated composite rectangular plate in contact with

where ex;0, ey;0, cxy;0, kð0Þx , kð0Þy and kð0Þxy are given in Appendix B.

2.2. Linear first-order shear deformation theory

Five independent variables, three displacements u, v, w and tworotations /1 and /2, are used to describe the plate’s middle planedeformation. This theory may be regarded as the thick-plate versionof the von Kàrmàn theory since the nonlinear terms are the same.

The hypotheses are: (i) the transverse normal stress rz is negli-gible; in general, this is a good approximation of the actual behav-ior of moderately thick plates and (ii) the normal to the middlesurface of the plate before deformation remains straight, but notnecessarily normal, after deformation; this is a relaxed version ofthe Kirchhoff’s hypothesis.

The displacements u1, u2, u3 of a generic point at distance z fromthe z = 0 plane (see Fig. 1) are related to the middle surface dis-placements u, v, w, /1 and /2 by

u1 ¼ uþ ðzþ h=2Þ/1; u2 ¼ v þ ðzþ h=2Þ/2; u3 ¼ wþw0: ð3a—cÞ

where /1 and /2 are the rotations of the transverse normals aboutthe y and x axes, respectively, and w0 is the geometrical imperfec-tion in z direction. A linear field in z is assumed for the first-ordershear deformation theory. In Eq. (3c) it is assumed that thenormal displacement is constant through the thickness, whichmeans eZ = 0.

The strain–displacement equations for the first-order sheardeformation theory are given by

ex ¼ ex;0 þ ðzþ h=2Þkð0Þx ; ey ¼ ey;0 þ ðzþ h=2Þkð0Þy ;

cxy ¼ cxy;0 þ ðzþ h=2Þkð0Þxy ;

cxz ¼ cxz;0; cyz ¼ cyz;0

ð4a — eÞ

where ex;0, ey;0, cxy;0, cxz;0, cyz;0, kð0Þx , kð0Þy and kð0Þxy are given inAppendix B.

Eqs. (4d,e) show a uniform distribution of shear strainsthrough the shell thickness, which gives uniform shear stresses.The actual distribution of shear stresses is close to a parabolicdistribution through the thickness, taking zero value at the topand bottom surfaces. For this reason, for equilibrium consider-ations, it is necessary to introduce a shear correction factor withthe first-order shear deformation theory in order not to overes-timate the shear forces.

2.3. Linear third-order shear deformation theory

A nonlinear, third-order shear deformation theory of plates hasbeen introduced by Reddy. The displacements of a generic point ofthe plate are related to the middle plane displacements by

u1 ¼ uþ ðzþ h=2Þ/1 þ ðzþ h=2Þ2w1 þ ðzþ h=2Þ3c1 þ ðzþ h=2Þ4h1;

u2 ¼ v þ ðzþ h=2Þ/2 þ ðzþ h=2Þ2w2 þ ðzþ h=2Þ3c2 þ ðzþ h=2Þ4h2;

u3 ¼ wþw0:

ð5a — cÞ

liquid dimensions, coordinate and displacement systems.

178 K. Khorshid, S. Farhadi / Composite Structures 104 (2013) 176–186

where /1 and /2 are the rotations of the transverse normal at mid-dle plan about the y and x axes, respectively, and the other termscan be computed as functions of w, /1 and /2. After satisfying thezero shear strains at z = 0 and h, the following expressions areobtained

u1 ¼ uþ ðzþ h=2Þ/1 �4

3h2 ðzþ h=2Þ3 /1 þ@w@x

� �;

u2 ¼ v þ ðzþ h=2Þ/2 �4

3h2 ðzþ h=2Þ3 /2 þ@w@y

� �; u3 ¼ wþw0:

ð6a — cÞ

where the geometric imperfection w0 in the normal direction hasbeen introduced. Eqs. (6a,b) represent the parabolic distribution ofshear effects through the thickness and satisfy the zero shearboundary condition at both the top and bottom surfaces of theplate.

The strain–displacement equations, keeping terms up to z3, canbe written as

ex ¼ ex;0 þ ðzþ h=2Þ kð0Þx þ ðzþ h=2Þ2kð2Þx

� �ey ¼ ey;0 þ ðzþ h=2Þ kð0Þy þ ðzþ h=2Þ2kð2Þy

� �cxy ¼ cxy;0 þ ðzþ h=2Þ kð0Þxy þ ðzþ h=2Þ2kð2Þxy

� �cxz ¼ cxz;0 þ ðzþ h=2Þ ðzþ h=2Þkð1Þxz

� �cyz ¼ cyz;0 þ ðzþ h=2Þ ðzþ h=2Þkð1Þyz

� �ð7a — cÞ

where ex;0, ey;0, cxy;0, cxz;0, cyz;0, kð0Þx , kð0Þy , kð2Þx , kð2Þy , kð0Þxy , kð2Þxy , kð1Þxz and kð1Þyz

are defined in Appendix B.

2.4. Elastic strain and kinetic energies for laminated plates

The stress–strain relations for the kth orthotropic lamina of theplate, in the material principal coordinates, under the hypothesisr3 = 0, are given by

r1

r2

s23

s13

s12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ðkÞ

¼

c11 c12 0 0 0c21 c22 0 0 00 0 G23 0 00 0 0 G13 00 0 0 0 G12

26666664

37777775

ðkÞ e1

e2

c23

c13

c12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;; ð8Þ

where G12, G13 and G23 are the shear moduli in 1–2, 1–3 and 2–3directions, respectively, and the coefficients cij are given inAppendix C; s13 and s23 are the shear stresses and the superscript(k) refers to the kth layer within a laminate. Eq. (8) is obtained (i)under the transverse isotropy assumption with respect to planesparallel to the 2–3 plane, i.e. assuming fibers in the direction paral-lel to axis 1, so that E2 = E3, G12 = G13 and m12 = m13 and (ii) solvingthe constitutive equations for e3 as function of e1 and e2 and theneliminating it.

Eq. (8) can be transformed to the plate coordinates (x,y,z) by thefollowing vectorial equation:

rx

ry

syz

sxz

sxy

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ðkÞ

¼ ½Q �ðkÞ

ex;0

ey;0

cyz;0

cxz;0

cxy;0

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;þ ðzþ h=2Þ½Q �ðkÞ

kð0Þx

kð0Þy

00

kð0Þxy

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

þ ðzþ h=2Þ2½Q �ðkÞ

00

kð1Þyz

kð1Þxz

0

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;þ ðzþ h=2Þ3½Q �ðkÞ

kð2Þx

kð2Þy

00

kð2Þxy

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;; ð9Þ

where [Q](k) is the 5 � 5 matrix of the material properties of the kthlayer transformed in the plate principal coordinates and it is givenin Appendix C. For the first-order shear deformation theory, theterms in z2 and z3 do not appear in Eq. (9).

The elastic strain energy UP of the plate is given by

UP ¼12

XK

k¼1

Z a

0

Z b

0

Z hðkÞ

hðk�1ÞrðkÞx ex þ rðkÞy ey þ sðkÞxy cxy

�þK2

xsðkÞxz cxz þ K2

ysðkÞyz cyz

�dxdydz; ð10Þ

where K is the total number of layers in the laminated shell, a and bare the in-plane dimensions, (h(k�1),h(k)) are the z coordinates of thekth layer and Kx and Ky are the shear correction factors, which areequal to one (no correction) for the third-order shear deformationtheory. The shear correction factor used in the present calculations(for the first-order shear deformation theory) is K2

x ¼ K2y ¼

ffiffiffi3p

=2.The kinetic energy TP of the plate, including rotary inertia, is gi-

ven by

TP ¼12

XK

k¼1

qðkÞP

Z a

0

Z b

0

Z hðkÞ

hðk�1Þ_u2

1 þ _u22 þ _u2

3

� �dxdy dz; ð11Þ

where qðkÞP is the mass density of the kth layer of the plate and theoverdot denotes time derivative.

2.5. Boundary conditions and discretization

The boundary conditions for simply supported plates with mo-vable edges (SSM) are:

v ¼ w ¼ /2 ¼ Nx ¼ Mx ¼ 0; at x ¼ 0; a; ð12a—eÞ

u ¼ w ¼ /1 ¼ Ny ¼ My ¼ 0; at y ¼ 0; b; ð13a—eÞ

where Nx or y and Mx or y are the normal force and the bending mo-ment per unit length, respectively.

The boundary conditions for simply supported plates withimmovable edges (SSI) are:

u ¼ v ¼ w ¼ /2 ¼ Mx ¼ 0; at x ¼ 0; a; ð14a—eÞ

u ¼ v ¼ w ¼ /1 ¼ My ¼ 0; at y ¼ 0; b: ð15a—eÞ

The boundary conditions for clamped plates (CL) are:

u ¼ v ¼ w ¼ /1 ¼ /2 ¼ 0; at x ¼ 0; a; and at y ¼ 0; b: ð16a—eÞ

Three expansions of plate displacements are used to discretizethe system for the different boundary conditions. For simply sup-ported movable (SSM) edges, the displacements u, v and w androtations /1 and /2 are expanded by using the following expres-sions, which satisfy identically the geometric boundary conditions:

uðx; y; tÞ ¼XM

m¼1

XN

n¼1

um;nðtÞ cosðmpx=aÞ sinðnpy=bÞ;

vðx; y; tÞ ¼XM

m¼1

XN

n¼1

vm;nðtÞ sinðmpx=aÞ cosðnpy=bÞ;

wðx; y; tÞ ¼XbMm¼1

XbNn¼1

wm;nðtÞ sinðmpx=aÞ sinðnpy=bÞ;

/1ðx; y; tÞ ¼XbMm¼1

XbNn¼1

/1m;nðtÞ cosðmpx=aÞ sinðnpy=bÞ;


XbNn¼1

/2m;nðtÞ sinðmpx=aÞ cosðnpy=bÞ;

ð17a — eÞ


where m and n are the numbers of half-waves in x and y directions,respectively, and t is the time; um;nðtÞ, vm;nðtÞ, wm;nðtÞ, /1m;n

ðtÞ and/2m;n

ðtÞ are the generalized coordinates, which are unknown func-tions of t. M and N indicate the terms necessary in the expansionof the in-plane displacements and, in general, are larger than bMand bN , respectively, which indicate the terms in the expansion ofout-of-plane displacement and rotations.

For simply supported immovable (SSI) edges, the followingexpansions are used:

uðx; y; tÞ ¼XM

m¼1

XN

n¼1

u2m;nðtÞ sinð2mpx=aÞ sinðnpy=bÞ;

vðx; y; tÞ ¼XM

m¼1

XN

n¼1

vm;2nðtÞ sinðmpx=aÞ sinð2npy=bÞ;


XbNn¼1



XbNn¼1

/1m;nðtÞ cosðmpx=aÞ sinðnpy=bÞ;


XbNn¼1

/2m;nðtÞ sinðmpx=aÞ cosðnpy=bÞ:

ð18a — eÞ

Finally, for clamped edges (CL), the expansions take the follow-ing form:

uðx; y; tÞ ¼XM

m¼1

XN

n¼1

u2m;nðtÞ sinð2mpx=aÞ sinðnpy=bÞ;

vðx; y; tÞ ¼XM

m¼1

XN

n¼1

vm;2nðtÞ sinðmpx=aÞ sinð2npy=bÞ;


XbNn¼1



XbNn¼1

/12m;nðtÞ sinð2mpx=aÞ sinðnpy=bÞ;


XbNn¼1

/2m;2nðtÞ sinðmpx=aÞ sinð2npy=bÞ:

ð19a — eÞ

The boundary conditions on bending moments for clamped bound-ary condition of the plates can be approximated by assuming thatrotational springs of very high stiffness j are distributed alongthe plate edges, so an additional potential energy stored by the elas-tic rotational springs at the plate edges must be added. This poten-tial energy UR is given by

UR ¼12

Z b

0j

@w@x

� �x¼0

2

þ @w@x

� �x¼a

2( )

dy

þ 12

Z a

0j

@w@x

� �y¼0

" #2

þ @w@x

� �y¼b

" #28<:

9=;dy: ð20Þ

In order to simulate clamped edges in numerical calculations, a veryhigh value of the stiffness (j ?1) must be assumed. This approachis usually referred to as the artificial spring method, which can beregarded as a variant of the classical penalty method. The valuesof the spring stiffness simulating a clamped plate can be obtainedby studying the convergence of the natural frequencies of the line-arized solution by increasing the value of j. In fact, it was foundthat the natural frequencies of the system converge asymptotically

with those of a clamped plate when j becomes very large (in thisstudy the non-uniform stiffness is assumed j = 108).

The boundary conditions on the bending moments for SSI platescan be transformed into

Mx ¼XK

k¼1

Z hðkÞ

hðk�1ÞrðkÞx zdz ¼ 0; at x ¼ 0; a; ð21Þ

My ¼XK

k¼1

Z hðkÞ

hðk�1ÞrðkÞy zdz ¼ 0; at y ¼ 0; b; ð22Þ

where the stresses rx and ry, which are functions of z and of the dif-ferent material properties in each layer of the laminate, are relatedto the strains by Eq. (9). For the given expressions of kð0Þx , kð2Þx , kð0Þy

and kð2Þy , which are all zero at x = 0, a and y = 0, b for the givenexpansions, Eqs. (21) and (22) are identically satisfied for symmet-ric laminates. Additional terms must be added to expansions of thein-plane displacements u and v for asymmetric laminates, in orderto satisfy exactly Eqs. (21) and (22). In fact, bending and stretchingare coupled for asymmetric laminates.

The boundary conditions on the normal forces for SSM platescan be transformed into

Nx ¼XK

k¼1

Z hðkÞ

hðk�1ÞrðkÞx dz¼

XN

k¼1

Q ðkÞ11 ;QðkÞ12 ;Q

ðkÞ15

n o ex;0

ey;0

cxy;0

8><>:9>=>; hðkÞ � hðk�1Þ� �264

þkð0Þx

kð0Þy

kð0Þxy

8>><>>:9>>=>>;

hðkÞ2 � hðk�1Þ2

2

!þ

kð2Þx

kð2Þy

kð2Þxy

8>><>>:9>>=>>;


4

!3775¼ 0; at x¼ 0;a; ð23aÞ

Ny ¼XK

k¼1

Z hðkÞ

hðk�1ÞrðkÞy dz¼

XN

k¼1

Q ðkÞ21 ;QðkÞ22 ;Q

ðkÞ25

n o ex;0

ey;0

cxy;0

8><>:9>=>; hðkÞ � hðk�1Þ� �264

þkð0Þx

kð0Þh

kð0Þxy

8>><>>:9>>=>>;


2

!þ

kð2Þx

kð2Þy

kð2Þxy

8>><>>:9>>=>>;


4

!3775¼ 0; at y¼ 0;b: ð23bÞ

After eliminating terms null at the edges and linear terms (sincethey are satisfied by energy minimization), the following equationsare obtained

Nx ¼XN

k¼1

hðkÞ � hðk�1Þ� �

Q ðkÞ11@u@xþ 1

2@w@x

� �2

þ @w@x

@w0

@x

!"

þQ ðkÞ12@v@yþ1

2@w@y

� �2

þ @w@y

@w0

@y

!

þQ ðkÞ15@u@yþ @v@xþ @w@x

@w@yþ @w@x

@w0

@yþ @w0

@x@w@y

� �2#¼ 0; at x¼ 0;a; ð24aÞ

Ny ¼XN

k¼1

hðkÞ � hðk�1Þ� �

Q ðkÞ21@u@xþ 1

2@w@x

� �2

þ @w@x

@w0

@x

!"

þ Q ðkÞ22@v@yþ 1

2@w@y

� �2

þ @w@y

@w0

@y

!

þ Q ðkÞ25@u@yþ @v@xþ @w@x

@w@yþ @w@x

@w0

@yþ @w0

@x@w@y

� �2#

¼ 0; at y ¼ 0; b: ð24bÞ

where u and v are terms added to the expansion of u and v, given inEq. (17a,b), in order to satisfy exactly the boundary conditionsNx = 0 and Ny = 0. Because u and v are second-order terms in the pa-nel displacement, they have not been inserted in the second-orderterms that involve u and v. For symmetric laminates, Eqs. (24a)and (24b) lead to


uðtÞ ¼ �XbNn¼1

XbMm¼1

ðmp=aÞ 12

wm;nðtÞ sinðnpy=bÞXbNk¼1

XbMs¼1

smþ s

ws;kðtÞ sinðkpy=bÞ

8<:� sin½ðmþ sÞpx=a� þwm;nðtÞ sinðnpy=bÞ

�XeNj¼1

XeMi¼1

imþ i

�Ai;j sinðjpy=bÞ sin½ðmþ iÞpx=a�); ð25aÞ

vðtÞ ¼ �XbNn¼1

XbMm¼1

ðnp=bÞ 12

wm;nðtÞ sinðmpx=aÞXbNk¼1

XbMs¼1

knþ k

ws;kðtÞ

8<: sinðspx=aÞ

� sin½ðnþ kÞpy=b� þwm;nðtÞ sinðmpx=aÞ

�XeNj¼1

XeMi¼1

jnþ j

�Ai;j sinðipx=aÞ sin½ðnþ jÞpy=b�): ð25bÞ

3. Formulation of the fluid oscillations

Using the principle of superposition, the fluid velocity potentialUO, can be obtained as

UO ¼ UB þUS; ð26Þ

where UB describes the velocity potential of the fluid obtained byneglecting free surface waves and US is the velocity potential dueto fluid sloshing in the presence of the rigid plate. The fluid velocitypotential can be separated into spatial velocity potential and a har-monic time function.

UOðx; y; z; tÞ ¼ /Oðx; y; zÞ expðixtÞ: ð27Þ

The fluid velocity potential must satisfy the three dimensional La-place equation in the fluid domain.

r2/O ¼ r2/B þr2/S ¼ 0: ð28Þ

The boundary conditions on the bottom and the vertical walls of thetank are given by

@/B

@x

��x¼0¼ @/B

@x

��x¼a

¼ 0;@/B

@y

��y¼0¼ 0;

@/B

@z

��z¼c1

¼ 0;

@/S

@x

��x¼0¼ @/S

@x

��x¼a

¼ 0;@/S

@y

��y¼0¼ 0;

@/S

@z

��z¼0;c¼ 0:

ð29 — 36Þ

For the liquid upper surface with neglecting of the liquid sloshing,we obtain

/Bjy¼b1¼ 0; ð37Þ

For the liquid-contacting surface of the elastic plate,

@UB

@z

��z¼o

¼ @wðx; y; tÞ@t

; ð38Þ

where w(x,y, t) is the transverse deflection of the plate. Applying themethod of separation of variables based on the boundary conditionsof Eqs. (29-37), general solution of Eq. (28) is given as

UBðx; y; z; tÞ ¼X1l¼0

X1k¼0

Al;kðtÞ coslpxa

� �cos

ð2kþ 1Þpy2b1

� �fexpðrzÞ

þ expðrð2c1 � zÞÞgfor l; k ¼ 0;1;2; . . . ; 0 6 x 6 a; 0 6 y 6 b; 0 6 z 6 c1

ð39Þ

USðx; y; z; tÞ ¼XN1

i¼0

XM1

j¼0

Bi;jðtÞ cosipxa

� �coshðsyÞ cos

jpzc1

� �for i; j ¼ 0;1;2; . . . ; ð0 6 x 6 aÞ; ð0 6 y 6 bÞ; ð0 6 z 6 c1Þ;

ð40Þ

where r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl=aÞ2 þ ðð2kþ 1Þ=ð2b1ÞÞ2

q, here l2

1 and k21 are arbitrary

nonnegative integers, s ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiði=aÞ2 þ ðj=c1Þ2

q, here l2

2 and k22 are arbi-

trary nonnegative integers, and Al,k(t) and Bi,j(t) are the unknowncoefficients. Applying the compatibility condition of Eq. (38), oneobtainsX1l¼0

X1k¼0

Al;kðtÞr½1� expf2crg� coslpxa

� �cos

2kþ 1ð Þpy2b1

� �¼ _wðx; y; tÞ; ð41Þ

The associated Fourier coefficients Al,k(t) can be determined in theusual manner from those of the right-hand side of the Eq. (41)

Al;kðtÞ ¼coff1ab1

R a

0

R b10

_wðx;y;tÞ cos lpxað Þ cos 2kþ1ð Þpy

2b1

� �� dy dx

rf1�expð2crÞg

coff1 ¼ fð1 if l and k ¼ 0Þ; ð2 if l or k ¼ 0Þ; ð4 if l; k – 0Þg: ð42Þ

The total kinetic energy of the fluid respect to the bulging modes ofthe plate and the fluid sloshing can be written

Tf ¼ TfB þ TfS ¼12qF

Z a

0

Z b1

0Uojz¼0

@Uo

@z

��z¼0

dydx ¼ 12qF

Z a

0

Z b1

0ðUBjz¼0 þUSjz¼0Þ �

@w@t

� �dydx; ð43Þ

so by the assumption of the ideal fluid and no surface waves, thekinetic energy of the fluid respect to the bulging modes of the platecan be written

TfB ¼12qF

Z a

0

Z b1

0UBjz¼0 �

@w@t

� �dydx; ð44Þ

The kinetic energy terms corresponding to the fluid sloshing in thetank is expressed as follows:

TfS ¼12qF

Z a

0

Z b1

0USjz¼0 �

@w@t

� �dydx; ð45Þ

The linearized sloshing conditions at the fluid free surface of thetank is

@UO

@y

��y¼b1

¼ x2

gUO

��y¼b1

; ð46Þ

where g is the gravity acceleration and x is the circular natural fre-quency of the liquid-coupled plate. Substituting Eq. (26) into Eq.(46) and using Eq. (37), one obtains

@UB

@y

��y¼b1

þ @US

@y

��y¼b1

¼ x2

gUS

��y¼b1

; ð47Þ

Multiplying both sides of Eq. (47) by qFUS then integrating themover the free surface of the fluid in the tank leads

U/Bþ U/S

¼ x2T/S; ð48Þ

where

U/B¼ qF

Z a

0

Z c1

0US

@UB

@y

� �y¼b1

dzdx; U/S

¼ qF

Z a

0

Z c1

0US

@US

@y

� �y¼b1

dzdx; ð49;50Þ

T/S¼ qF

g

Z a

0

Z c1

0U2

S

� �y¼b1

dzdx: ð51Þ

4. Rayleigh–Ritz approach

The Lagrangian function of the fluid-plate coupled system is

P ¼X

Starin Energymax �X

Kinetic Energymax: ð52Þ


With the application of Ritz minimization method, an eigenvalueequation can be derived from Eq. (52)

@P@q¼ 0; ð53Þ

where q is the vector of generalized coordinates and containsthe unknown time variable coefficients of the admissible trialfunctions presented by Eqs. (17–19) and (40) (i.e. q ¼fum;n;vm;n;wm;n;/1m;n

;/2m;n;Bi;jgT ). Following the Minimization Eq.

(53) the subsequent equation is obtained

ðKp þ KRÞCm;n �x2½ðMP þMfBÞCm;n þMfSBi;j� ¼ 0; ð54Þ

where Cm;n ¼ fum;n;vm;n;wm;n;/1m;n;/2m;n

gT and

Kp ¼@2Up

@qi@qj; KR ¼

@2UR

@qi@qj; Mp ¼

@2Tp

@qi@qj;

MfB ¼@2TfB

@qi@qj; MfS ¼

@2TfS

@qi@qj: ð55—59Þ

Eq. (54) cannot be solved until an expression for Bi,j is obtained.Thus, Eq. (48) are added to the Galerkin Eq. (54). This, increasesthe dimensions of the associated eigenvalue problem fromðN � NÞ to ððN þ eNÞ � ðN þ eNÞÞ, where N is the dimension of thecoordinates vector {wm,n}T and eN is the dimension of coordinatesvector {Bi,j}T. Consequently, the following Galerkin equation isobtained

Kp þ KR 0K/B

K/S

Cm;n

Bi;j

� �x2 MP þMfB MfS

0 M/S

Cm;n

Bi;j

� ¼ 0;

ð64Þ

where

K/B¼ @2U/B

@qi@qj; K/S

¼ @2U/S

@qi@qj; M/S

¼ @2T/S

@qi@qj: ð65—67Þ

5. Comparison study

In order to validate the present formulation, the natural fre-quencies obtained by the present method are compared with thoseof Ugurlu et al. [11], Kant and Swaminathan [25], Reddy [23],Senthilnathan et al. [24], Whitney and Pagano [22], and ANSYS(Release 11.0) as listed in Tables 2–4.

In Table 2, the non-dimensional fundamental natural frequencyof a simply supported square isotropic plate in partial contact withwater are reported for different fluid depth ratios including b1/b = 0,0.2, 0.4, 0.6, 0.8, and 1. In this table calculations have beenperformed for a square plate with dimensions and material proper-ties a = 10 m, b = 10 m, h = 0.15 m, q = 2400 kg/m3, E = 25 GPa andm = 0.15, and width of the tank is c1 = 100 m. It is worth mention-ing, Ugurlu et al. calculated their results for a rectangular plate incontact with an infinite fluid (c1 =1). It is observed from Table 2that the fundamental natural frequency of the plate increasesmonotonically, as the fluid depth ratio tends to zero. It is also ob-served that the mode sequence changes according to the fluiddepth. Table 2 shows that there is a good agreement betweenthe present results and those of Ugurlu et al. [11].

In Table 3, the fundamental non-dimensional natural frequen-cies �x ¼ ðxb2

=hÞffiffiffiffiffiffiffiffiffiffiffiq=E2

p� �of a simply supported square symmet-

ric laminated composite plate (0�/90�/90/0�) are presented fordifferent thickness ratios including h/a = 0.01, 0.02, 0.05, 0.1 and0.25. It should be noted that the results of Table 3 are calculatedfor graphite/epoxy material with a/b = 1. In this table, The funda-mental natural frequency of the simply supported laminated com-posite plate are computed by use of the Classical Composite Plate

(CLPT), the First order Shear Deformation (FSDT) and the Third or-der Shear Deformation (TSDT) plate theories. From this table, it canbe observed that the non-dimensional natural frequencies de-crease as the thickness ratios increase. In addition, the fundamen-tal natural frequencies computed by CLPT are not at the sameaccuracy of those provided by TSDT.

The comparison of the fundamental wet natural frequencies ofthe laminated composite plates obtained by using the Third orderShear Deformation (TSDT) plate theory and Finite element analysisare presented in Table 4 for different thickness ratios including h/a = 0.01, 0.1 and 0.2. In Table 4, the numerical results are obtainedfor a Four layer (0�/90�)s squared plate (a = 1 m, b = 1 m, c1 = 0.5 m)with different depth of the fluid (b1/b) which is varies from 0 to 1.The results of Table 4 are calculated for graphite/epoxy materialwith simply supported immoveable (SSI) and moveable (SSM)boundary conditions. The Finite Element Method (FEM) resultshave been performed by using the commercial software, Ansys(Release 11.0). For the Finite Element Method (FEM) analysis, thethree-dimensional model is composed of three-dimensional con-tained fluid elements (FLUID80) and elastic shell elements(SHELL99). This comparison confirmed reliability of the proposedmethod and FEM results.

6. Numerical results and discussion

In this section, numerical results are calculated according to thedeveloped analytical solution for the free vibrations of a laminatedcomposite rectangular plate in air or in contact with the boundedfluid by the rigid container walls. Calculations have been per-formed by using the commercial software, Mathematica (version7), and the results are presented in tabular and graphical formsfor different boundary conditions, plate parameters, and fluidparameters. In the present study, three different boundary condi-tions are investigated: simply supported immovable (SSI), simplysupported movable (SSM) and clamped (CL) edges. Presented re-sults have been obtained by using a model with 33 dofs.

Calculations have been performed for symmetric laminatedplates. Material properties of the plate are defined as listed in Table1. Also, fluid density is considered as qF1 = 1000 kg/m3.

6.1. Effect of interaction between plate and fluid on the wet modeshapes

The typical wet mode shapes of an orthotropic rectangular platewith 40% fluid depth ratio are illustrated in Fig. 2 for both move-able and immoveable simply supported boundary conditions. Theresults given in this figure correspond to the case where a = 2 m,b = 1 m, h = 0.02 m, material orthotropy orientation 0� andc1 = 0.5 m. These figures show that interaction between plate andfluid causes the wet mode shapes to distort from the dry modeshapes of the plate. Also, it is observed that the mode shapeschange according to the fluid depth. Especially, severe distortionsfrom dry modes can be observed in the higher vibration modes.In the higher wet modes, it is very difficult to categorize an equiv-alent dry mode due to the distortion of mode shapes.

6.2. Effect of material properties and boundary conditions on the wetnatural frequency

Fundamental wet natural frequencies of a One Layer (0�), a TwoLayer (0�/90�), a Four layer (0�/90�)s and a Six Layer (0�/90�/0�)slaminated composite plates are presented in Table 5. Differentboundary conditions (simply supported immoveable (SSI) andmoveable (SSM) and clamped (CL) boundary conditions) are con-sidered. Materials chosen for the plate are graphite/epoxy (AS/

Table 1Material properties of the laminated composite plate.

Material E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) m12 q (kg/m3)

Graphite/epoxy (AS/3501) 137.9 8.96 7.1 6.21 0.3 1450Graphite/epoxy (T300/976) 150 9 7.1 2.5 0.3 1600Boron/epoxy 206.9 20.69 6.9 4.14 0.3 1950Glass/epoxy 53.78 17.93 8.96 3.45 0.25 1900Graphite/epoxy 40 1 0.6 0.5 0.25 1000

Table 2Comparison study of non-dimensional natural frequencies (X ¼ xa2

ffiffiffiffiffiffiffiffiffiffiffiffiqh=D

p; x (Hz)) for a simply supported square isotropic plate in contact with fluid.

Mode sequence Method b1/b

0 0.2 0.4 0.6 0.8 1

(1,1) Present 3.1415 3.0127 2.0746 1.35634 1.0172 0.8565Ugurlu et al. [11] 3.169 3.064 2.196 1.496 1.173 1.036









(1,4) Present 26.6411 25.2968 18.7991 16.3239 15.2898 14.8670Ugurlu et al. [11] 27.52 25.66 18.31 15.96 15 14.62

Table 3Comparison study of non-dimensional fundamental natural frequencies �x ¼ ðxb2

=hÞffiffiffiffiffiffiffiffiffiffiffiq=E2

p� �of a simply supported square laminated composite plate (0�/90�/90�/0�).

Method h/a

0.01 0.02 0.05 0.1 0.25

Present (CLPT) 18.8898 18.8851 18.8526 – –Present (FSDT) 18.8383 18.6821 17.7001 15.2386 9.5147Present (TSDT) 18.8356 18.6718 17.6466 15.1073 9.3235Kant and Swaminathan [25] 18.8357 18.6720 17.6470 15.1048 9.2870Reddy [22] 18.8356 18.6718 17.6457 15.1073 9.3235Senthilnathan et al. [23] 18.8526 18.7381 17.9938 15.9405 10.2032Whitney and Pagano [24] 18.8362 18.6742 17.6596 15.1426 9.3949

Table 4Comparison study of fundamental natural frequencies of a square laminatedcomposite plate between analytical and FEM results.

b1b

ha ¼ 0:01 h

a ¼ 0:1 ha ¼ 0:2

Present Ansys Present Ansys Present Ansys

0 29.9778 29.89544 240.440 239.2477 343.370 339.81310.1 29.6885 29.50947 240.209 238.6906 343.207 341.71480.2 26.0253 25.90472 236.976 233.9471 340.906 335.3840.3 18.6746 18.51837 225.097 224.3595 332.108 331.58820.4 13.1129 12.96254 202.967 199.8331 313.689 313.53510.5 9.5182 9.443441 176.190 174.5336 288.387 282.95490.6 7.33229 7.230984 150.209 147.9097 257.746 256.77610.7 5.84451 5.766942 128.156 127.4243 228.903 226.80650.8 4.85699 4.796181 110.932 108.9018 203.924 199.85490.9 4.18535 4.113669 97.9970 96.75008 183.728 183.53351 3.71927 3.656121 88.4552 86.98845 168.052 165.5156


3501), graphite/epoxy (T300/976), boron/epoxy, glass/epoxy, andgraphite/epoxy as shown in Table 3. The results given in this tablecorrespond to the case where a = 2 m, b = 1 m, h = 0.2m, b1 = 0.4 m

and c1 = 0.5 m. Third order shear deformation theory (TSDT) is em-ployed to obtain the results.

From the reported results, it is observed that in the all cases, thelowest and the highest wet natural frequencies correspond tographite/epoxy and graphite/epoxy (AS/3501), respectively.

According to this table, the wet natural frequency of the plateswith SSI and SSM boundary conditions give exactly the same re-sults. Moreover, it is observed that the highest values of fundamen-tal wet natural frequency correspond to clamped boundaryconditions.

6.3. Effect of plate aspect ratio (a/b) on the wet natural frequency

Fundamental wet natural frequencies of a laminated rectangu-lar plate versus plate aspect ratio are illustrated in Fig. 3 for differ-ent boundary conditions. Presented results are obtained for 40%fluid depth ratio and dimensions b = 1 m, h = 0.2 m, b1 = 0.6 m,and c1 = 0.5 m. Inspection of curves given in Fig. 3, gives an ideaabout how the wet frequency varies when the aspect ratio.

Pla

te in

Air

Mode sequence (1,1) Mode sequence (1,2) Mode sequence (2,1) Mode sequence (2,2) Mode sequence (1,3)


plat

e pa

rtia

lly in

con

tact

wit

h w

ater

for

50

% f

luid

dep

th r

atio



Fig. 2. Comparison of typical mode shapes of a SSI and SSM orthotropic rectangular plate in air and partially in contact with water.


According to this figure, we see that the higher the aspect ratio, thelower the natural frequencies.

6.4. Effect of fluid depth on the wet natural frequency

Figs. 4 and 5 illustrate fundamental wet natural frequencies of afour layer (0/90/90/0) square laminated plate (a = 1 m, b = 1 m andc1 = 0.5 m) versus fluid depth for simply supported and clampedboundary conditions and different values of thickness ratio. Thewet natural frequencies of the laminated composite rectangularplate in contact with fluid are always less than the correspondingnatural frequencies of the plate in air. Due to this fact, when nor-malizing the natural frequency with respect to the free plate natu-ral frequencies, one can see that defined normalized naturalfrequencies of fluid–structure coupled system always lie betweenunity and zero. In Figs. 4 and 5, the results are shown for the platepartially in contact with water where the depth of the fluid (b1/b)

Table 5The effect of material properties of the laminated composite plate on the fundamental we

Material properties Boundary conditions One Layer (0�)

Graphite/epoxy (AS/3501) SSI and SSM 676.765CL 1078.96

Graphite/epoxy (T300/976) SSI and SSM 657.788CL 1036.48

Boron/epoxy SSI and SSM 645.273CL 985.122

Glass/epoxy SSI and SSM 469.445CL 864.319

Graphite/epoxy SSI and SSM 301.197CL 471.694

varies from 0 to 1. From these figures, it can be realized that thewet natural frequencies decrease as fluid depth increases.

6.5. Effect of plate thickness ratio (h/a) on the wet natural frequency

Fig. 6 presents fundamental wet natural frequencies of a fourlayer (0/90/90/0) square laminated plate versus plate thickness ra-tio for 100% fluid depth ratio, dimensions a = 1 m, b = 1 m andc1 = 0.2 m, and different boundary conditions.

It should be noted that the result in Figs. 4–6 is shown for thegraphite/epoxy material. From Figs. 4–6, it can be observed thatthe fundamental wet natural frequency of the plate increasesmonotonically, as the thickness ratio increases. This is not a sur-prising result since we know higher values of thickness ratio in-crease the stiffness of the structure more effectively than itsinertia and thus, higher vibration frequencies should be expected.In addition, Figs. 4–6 show that the natural frequencies reduce as

t natural frequency (Hz).

Two Layer (0�/90�) Four Layer (0�/90�)s Six Layer (0�/90�/0�)s

575.082 651.245 627.475998.435 1050.87 977.136531.631 546.820 597.640883.420 845.647 962.746549.668 579.767 604.479883.226 884.509 942.492407.220 429.754 438.089760.791 747.039 811.432274.875 262.975 277.715444.277 407.249 435.574

Fig. 3. Variation of fundamental natural frequency versus aspect ratios for asymmetric four layer (0/90/90/0) laminated composite plates with three combina-tions of boundary conditions using TSDT (h/a = 0.2, b1/b = 0.6, graphite/epoxy,c1 = 0.5 m).

Fig. 4. Variation of fundamental natural frequency versus depth of the fluid for asquare SSI and SSM symmetric (0/90/90/0) laminated composite plates using TSDT(graphite/epoxy, c1 = 0.5).

Fig. 5. Variation of fundamental natural frequency versus depth of the fluid forsquare a CL symmetric (0/90/90/0) laminated composite plates using TSDT(graphite/epoxy, c1 = 0.5).

Fig. 6. Variation of fundamental natural frequency versus thickness ratios for asquare symmetric (0/90/90/0) laminated composite plates with three combinationsof boundary conditions using TSDT (b1/b = 1, graphite/epoxy, c1 = 0.2).

Fig. 7. Variation of fundamental natural frequency versus width of the tank for asquare symmetric (0/90/90/0) laminated composite plates with three combinationsof boundary conditions using TSDT (h/a = 0.2, b1/b = 0.6, graphite/epoxy).

Fig. 8. Variation of fundamental natural frequency versus material orthotropyorientation for a square symmetric (0/90/90/0) laminated composite plates withthree combinations of boundary conditions using TSDT (h/a = 0.2, b1/b = 0.4,graphite/epoxy, c1 = 0.5).


the value of depth ratio increases and for all values of depth ratioand thickness ratio the frequencies corresponding to CL boundaryconditions possess higher values in comparison with SSI and SSMboundary conditions, which agree with the results of previoussubsections.

6.6. Effect of tank width on the wet natural frequency

Fig. 7 plots Fundamental wet natural frequency of a squaredplate (a = 1 m, b = 1 m and h = 0.2 m) versus fluid width. From this

figure, one can see that the fundamental frequency increases as thetank width increases and approaches to an asymptotic value. Thismeans that for high enough values of width ratio, one can use theassumption of infinite fluid depth.

6.7. Effect of the material orthotropy orientation on the wet naturalfrequency

In Fig. 8, the numerical results are obtained for a single layerlaminated composite squared plate (a = 1 m, b = 1 m, h = 0.2 m,c1 = 0.5 m and b1 = 0.4 m) with different material orthotropy orien-tations (h), where orientation angle (h) varies from 0� to 180�. From


this figure, on can see that the lowest and the highest wet naturalfrequencies correspond to orientation angles 0� and 90�,respectively.

7. Conclusion

In this study hydrostatic vibration analysis of a laminated com-posite rectangular plate partially in contact with a bounded fluid isinvestigated. For this purpose, Classical Composite Plate (CLPT), theFirst order Shear Deformation (FSDT) and the Third order ShearDeformation (TSDT) plate theories are employed. Using numericaldata provided, effect of different parameters including boundaryconditions, aspect ratio, thickness ratio, fiber orientation, materialproperties of the laminas and dimensions of the tank on the platenatural frequencies are examined and discussed in detail. Obtainedresults show that the accuracy of fundamental natural frequenciescomputed by CLPT decreases as the thickness ratio increases. Theseresults show that interaction between plate and fluid causes thewet mode shapes to distort from the dry mode shapes of the plate.Especially, severe distortions from dry modes can be observed inthe higher vibration modes. Also, it is observed that the modeshapes change according to the fluid depth and the wet natural fre-quencies decrease as fluid depth increases. The fundamental wetnatural frequency of the plate increases monotonically, as thethickness ratio increases. For all values of depth ratio and thicknessratio the frequencies corresponding to clamped boundary condi-tions possess higher values in comparison with movable andimmovable simply supported boundary conditions.

Numerical results reveal that the fundamental natural fre-quency increases as the tank width increases and approaches toan asymptotic value. Moreover, the higher the aspect ratio is, thelower the natural frequencies.

Acknowledgment

The authors gratefully acknowledge the funding by ArakUniversity, under Grant No. 90/9423.

Appendix A. The effect of the hydrostatic triangular pressure

To account the effect of the hydrostatic triangular pressure, thevirtual work corresponding to the hydrostatic triangular pressureis estimated by the following equations:

Uh ¼12qF g

Z a

0

Z b1

0wðb1 � yÞdydx; ðA:1Þ

The virtual work due to hydrostatic triangular pressure can be takeninto account for the forced and nonlinear vibrations. In order to takeinto account the effects of hydrostatic triangular pressure to the lin-ear free vibration, the plate configuration due to hydrostatic trian-gular pressure has been developed. The plate configuration for thethin plates due to hydrostatic triangular pressure can be approxi-mated as

w0 ¼X1m¼1

X1n¼1

Ahpðm;nÞ sinðmpx=aÞ sinðnpy=bÞ; ðA:2Þ

where Ahp is the unknown constant coefficient. Applyingbi-harmonic equation r4w0 = �qFg(b1 � y) coefficient Ahp can bedefined as

Ahpðm;nÞp4ððm=aÞ2 þ ðn=bÞ2Þ2 sinðmpx=aÞ sinðnpy=bÞ ¼ �qF gðb1 � yÞ: ðA:3Þ

From Eq. (A.3), the associated Fourier coefficient Ahp can beobtained:

Ahpðm;nÞ ¼ �1

ab1

R a0

R b10 qFgðb1 � yÞ sinðmpx=aÞ sinðnpy=bÞdydx

p4ððm=aÞ2 þ ðn=bÞ2Þ2:

ðA:4Þ

The system has been then studied in the case of a plate contactingwater on both sides. In this case, the contribution of the initialdeformation of the plate, given by the hydrostatic pressure of thefluid, can be eliminated if the water level in both tanks is identical.

Appendix B. Stress–displacement relationships

According to the thin plate theory, the middle surface strain–displacement relationships and the changes in the curvature andthe torsion are given by

ex;0 ¼@u@xþ @w@x

@w0

@x; ey;0 ¼

@v@yþ @w@y

@w0

@y;

cxy;0 ¼@u@yþ @v@xþ @w@x

@w0

@yþ @w0

@x@w@y

;

kð0Þx ¼ �@2w@x2 ; kð0Þy ¼ �

@2w@y2 ; kð0Þxy ¼ �2

@2w@x@y

;

ðB1 — 6Þ

According to the first-order shear deformation theory:

ex;0 ¼@u@xþ @w@x

@w0

@x; ey;0 ¼

@v@yþ @w@y

@w0

@y;


@w0

@yþ @w0

@x@w@y

; cxz;0 ¼ /1 þ@w@x

;

cyz;0 ¼ /2 þ@w@y

; kð0Þx ¼@/1

@x; kð0Þy ¼

@/2

@y; kð0Þxy ¼

@/1

@yþ @/2

@x:

ðB7 — 14Þ

According to the third-order shear deformation theory:

ex;0 ¼@u@xþ @w@x

@w0

@x; ey;0 ¼

@v@yþ @w@y

@w0

@y;


@w0

@yþ @w0

@x@w@y

; cxz;0 ¼ /1 þ@w@x

;

cyz;0 ¼ /2 þ@w@y

; kð0Þx ¼@/1

@x; kð0Þy ¼

@/2

@y; kð0Þxy ¼

@/1

@yþ @/2

@x;

kð2Þx ¼ �4

3h2

@/1

@xþ @

2w@x2

!; kð2Þy ¼ �

4

3h2

@/2

@yþ @

2w@y2

!;

kð2Þxy ¼ �4

3h2

@/1

@yþ @/2

@xþ 2

@2w@x@y

!; kð1Þxz ¼ �

4

h2 cxz;0; kð1Þyz ¼ �4

h2 cyz;0:

ðB15 — 27Þ

Appendix C. Stress–strain relationships for layer within alaminate

The coefficients in Eq. (8) for a lamina are given by

c11 ¼E1

1� m12m21; c12 ¼ c21 ¼

E2m12

1� m12m21;

c22 ¼E2

1� m12m21; mijEj ¼ mjiEi: ðC1—4Þ

Usually, the lamina material axes (1,2) do not coincide with theplate reference axes (x,y), while the three axis is coincident with z.Then, the strains and stresses on material axes can be related to thereference axes by using the following invertible expressions:

r1

r2

s23

s13

s12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼ T1

rx

ry

syz

sxz

sxy

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;;

e1

e2

c23

c13

c12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼ T2

ex

ey

cyz

cxz

cxy

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;; ðC5;6Þ


where

T1 ¼

cos2 h sin2 h 0 0 2 sin h cos h

sin2 h cos2 h 0 0 �2 sin h cos h

0 0 cos h � sin h 00 0 sin h cos h 0

� sin h cos h sin h cos h 0 0 cos2 h� sin2 h

26666664

37777775; ðC7Þ

T2 ¼

cos2 h sin2 h 0 0 sin h cos h

sin2 h cos2 h 0 0 � sin h cos h

0 0 cos h � sin h 00 0 sin h cos h 0

�2 sin h cos h 2 sin h cos h 0 0 cos2 h� sin2 h

26666664

37777775: ðC8Þ

It can be shown that

T�11

� �T¼ T2: ðC9Þ

Therefore, the matrix [Q](k) in Eq. (9) is given by

½Q �ðkÞ ¼ T�11 C T�1

1

� �T ðkÞ

; ðC10Þ

where C is the 5 � 5 matrix of coefficients cij in Eqs. (8) and (C1–4).As a consequence of the discontinuous variation of the stiffness ma-trix [Q](k) from layer to layer, the stresses may be discontinuouslayer to layer.

References

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